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Preface
widely used techniques to solve the eigenvalue problems of quantum-mechanical Hamiltonians. Its popularity is due to its simplicity and flexibility. The most crucial point in the variational approach is the choice of a variational trial function. One usually attempts to construct the trial function from some adequate basis functions which contain a number of nonlinear parameters. The direct method of diagonalizing the Hamiltonian matrix on such a basis set may not be feasible, except for simple systems, because of the large number of degrees of freedom involved in specifying the system. One thus faces a problem of selecting the most suitable basis set. It is by the stochastic variational method, that is, by a trial and error procedure with an admittance test that we give an answer to this problem. The stochastic variational method has been developed through the search for precise solutions of nuclear few-body problems. The variational method is
In this method
it enables
us
we
one
of the most
set up the basis element
to test many
parameters
to monitor the energy convergence. The aim of this book is to give
as
after the other because
one
fast
as
possible
and
moreover
unified and
reasonably simple problems with the use few-body recipe of the stochastic variational method and to present its application to various few-body problems which one encounters in atomic, molecular,
nuclear, subnuclear and solid
a
bound-state
for solutions of
state
physics.
quantum systems is in general extremely difficult and challenging, great advances have been made in recent years, especially for systems of a few particles and it has become possible to obtain accurate solutions for the eigenvalue prob-
Though
a
unified
approach
to the diverse
quantum-mechanical Hamiltonians. The main interest the few-body problems lies in, e.g., finding an accurate solution for
lem of various
in
the system
so as
to understand the
dynamics
of its
constituents,
test-
VI
Preface
ing the equation of motion and the conservation laws and symmetries, or looking for unknown interactions governing the system. Quantum mechanics plays a fundamental role in atomic and subatomic physics. It is via quantum mechanics that one can understand the binding mechanism of atoms, molecules and atomic nuclei, that is, the structure of the building blocks of matter. The interaction between the particles depends on the system: For example, the longrange Coulomb interaction dominates in atoms and molecules but the
very different mass ratio of the electrons and the atomic nucleus plays a key role as well. In contrast, the protons and neutrons in nuclei have
equal masses and the interaction between them is short-ranged. The variational foundation for the time-independent Schr5dinger equation provides a solid and arbitrarily improvable framework for the solution of diverse bound-state problems. As mentioned above, the most crucial point in the variational approach is the choice of the trial function. There are two widely applied strategies for this choice: One is to use the most appropriate functional form to describe the short-range as well as the long-range correlations among the particles. Such calculations, however, are fairly complex for systems of more than three particles, and the integration involved is performed by the Monte Carlo method. Another way is to approximate the solution as a combination of a number of simple basis states which facilitate the analytical calculation of matrix elements. We follow the latter course almost
in this book and show that the stochastic variational method selects
important basis set without any bias, keeps the dimension of and, most importantly, provides a very accurate solution. The book is conceptually divided into two parts. The first seven chapters present the basic concepts of the variational method and the formulation using Gaussian basis functions. The latter four chapters of the book cover applications of the formulation to various quantummechanical few-body bound-state problems. In Chap. 2 a general formulation is developed to express the physical operators which are needed to specify the Hamiltonian in terms of an arbitrary set of independent relative coordinates. The linear transformation of the relative coordinates induced by the permutation of identical particles is also established in this chapter. In Chap. 3 we review the basic principles of the variational method with particular emphasis on the case where the variational trial function is given as a linear combination of nonorthogonal basis functions. We introduce in Chap. 4 a key algothe most
the basis low
rithm used in this
book,
the stochastic variational
method,
and show
Preface
that its trial and
error
search
procedure
makes it
possible to
Vii
select the
important basis functions without any bias in the function space spanned by the basis functions. Some other methods to solve few-body most
briefly introduced in Chap. 5. Chapter 6 defines the type of variational trial functions used extensively in the book, the correlated Gaussians and the correlated Gaussian-type geminals. They are chosen because they enable us to evaluate matrix elements analytically and because they provide. us with precise solutions for most problems of real interest. A simple but powerful angular function is introduced to describe orbital motion with nonzero orbital angular momentum. To facilitate the systematic and unified evaluation of matrix elements,
problems
are
it is shown that the above Gaussian basis functions
function. In
7
are
all obtained
show that the
generatChap. deriving the matrix elements of the Gaussian basis functions for an N-body system of essentially any interaction. Explicit formulas axe given in this chapter for the simplest possible Gaussian basis functions, because they are already found to be very useful. The matrix elements for a, general case are detailed in the appendix. We show also in this chapter that the. method can be extended to evaluate the matrix elements of nonlocal potentials and the seniirelativistic kinetic energy as well. Chapters 8-11 present application of the stochastic variational method to various systems: small atoms and molecules (Chap. 8), baryon spectroscopy (Chap. 9), excitonic complexes and quantum dots in solid state physics (Chap. 10), and nuclear few-body problems (Chap. 11). We hope that this book will be found useful by students who want to understand and make use of the variational approach to quantummechanical few-body problems, while it may also be of interest to researchers who axe familiar with the subjects. It will be our pleasure if this book serves to bridge the gap between graduate lectures and the literature in scientific journals, as well as to give impetus to further development in the deeper understanding of quantum-mechanical fewbody systems. We assume that readers have taken courses on quantum mechanics and mathematical physics at an undergraduate level. No special knowledge is assumed of, e.g., atomic physics or nuclear physics. To help readers to understand the text, we have attempted to make the book self-contained, put as much emphasis as possible on clarity, and given several Complements of an explanatory nature. The Complements are intended to further develop or to reinforce the arguments and ideas presented in the text. We have collected the derivation
from
ing
a
generating plays a
function
vital role in
we
VIII
Preface
of formulas that may possibly be difficult for readers solutions at the end of the chapters.
as
exercises with
Depending on their interest, readers may adopt several reading strategies. A thorough-going reader is advised to read all of the text including the Complements. A reader who wants to understand only the basic formulation can omit the Complements. Anyone who is familiar with the variational method may skip Chaps. 2 and 3. Experts, or those readers who are only interested in the performance of the stochastic variational approach or the physical consequences of the results, may skip Chaps. 2-7. The Gaussian basis has long been used in many areas of physics. The correlated Gaussians were first introduced in quantum chemistry by S.F. Boys and K. Singer. The application of the Gaussian basis is one of the key elements of the success of the ab initio calculations in quantum chemistry. The stochastic variational method is actually very similar to the so-called "random tempering", that has been used to find the optimal parameters of the basis in quantum chemistry. In the random tempering pseudo-random parameters axe generated, and the best basis functions are selected by sorting out the states which improve the energy. There exists another method which is also similar to the stochastic variational method, called the stochastic diagonalization. This method, originally developed in solid state physics, attempts to find the lowest eigenvalue of huge eigenvalue problems by randomly testing the contributions of different states. The random selection of the parameters of a Gaussian basis, named the stochastic variational method, was first used by V.I. Kukulin and V.M. Krasnopol'sky in nuclear physics. We are grateful to Prof, V.I. Kukulin for his interest and encouragement during this work. We are indebted to our collaborators, K. Arai, Y. FujiNvara, L.Ya. Glozman, R.G. Lovas, J. Afitroy, C. Nakamoto, H. Nemura, Y. 0hbayasi, Z. Papp, W. Plessas, G.G. Ryzhikh, N. Tanaka, J. Usukura, and R.F. Wagenbrunn for useful discussions and for many of the calculations which press
our
hearty
are
included in this book. We wish to
thanks to Prof. K.T.
Hecht,
ex-
Prof. D.
Baye, Prof. reading of the Despite all these
M. Kamimura and Prof. R.G. Lovas for their careful
manuscript and for much advice and many comments.
efforts mistakes may still remain. Needless to say, these are our own responsibility. Suggestions and criticisms from our readers would be welcomed. We
are
grateful
for the
use
of the computer VPP500 of (RIKEN). One of the
the Institute of Physical and Chemical Research
Preface
for the
authors
(K.V.)
ment of
Niigata University and
is
grateful
would like to thank the
Research of the
hospitality
the Linac
funding agencies,
Iffinistry
of
(Hu gary),
of the
Ix
Physics Depart-
Laboratory
of RIKEN. We
Grants-in-Aid for Scientific
Education, Science and Culture (Japan),
Japan Society for the Promotion of Science (JSPS), the Science and Technology Agency (STA) of Japan, the U. S. Department of Energy, Nucleax Physics Division, and the Hungarian Academy of Sciences (HAS), for their support which has been vital for the completion of the work. Finally, without the support and patience of our families, this book would not have been possible. OTKA Grants
Niigata Argonne September
the
Y. Suzuki K.
1998
Varga
Table of Contents
I..
Introduction
2.
Quantum-mechanical few-body problems
........................................
...........
7
2.1
Hamiltonian ......................................
7
2.2
Relative coordinates
..............................
9
..................................
15
2.3
Symmetrization
2.4
Permutation of the Jacobi coordinates
Complements C2.1 An N-particle Hamiltonian in the heavy-particle center
..............
.........................................
Variational
3.2
The variance of local energy The virial theorem
3.3 4.
principles
......................
19
4.2
21
21
.......................
30
................................
33
.......................
39
................................
39
...............................
43
......................
44
.........................
47
..............................
47
....................................
50
Basis
optimization A practical example 4.2.1 Geometric progression 4.2.2 Random tempering 4.2.3
Random basis
4.2.4
Sorting
4.2.5
Trial and
error
search
.......................
Refining 4.2.7 Comparison of different optimizing strategies Optimization for excited states
4.2.6
4.3
................
.............................
Stochastic variational method 4.1
18
18
Introduction to variational methods 3.1
16
...........
coordinate set
C2.2 Canonical Jacobi coordinates 3.
1
...................................
.................
50 53
.
54
....
56
.
I
Complements
.........................................
61
XII
Table of Contents
C4.1 Minimization of energy 5.
versus
variance
..............
Other methods to solve
5.2
few-body problems Quantum Monte Carlo method: The imaginary-time evolution of a system Hyperspherical harmonics expansion method
5.3
Faddeev method
5.4
The generator coordinate method
5.1
.........
...........
6.
..................................
Variational trial functions 6.1
6.2
6.4
65
67 70
72
..........................
75
Correlated Gaussians
Gaussian-type gerninals Orbital functions with arbitrary angular Generating function The spin function
Complements
..............
.
82
87
................................
94
C6.2 Solid
spherical harmonics C6.3 Angular momentum recoupling C6.4 Separation of the center-of-mass =
as a
basis
.....
96 96 105
.....................
106
motion
.........................
1/2
......................
arbitrary spin arrangement
112
115
......
115
...........................
116
............................................
118
an
C6.7 Six electrons with S
Matrix elements for
=
0
spherical Gaussians, generating function
............
123
..........
123
..............................
125
7.1
Matrix elements of the
7.2
Correlated Gaussians
7.3 7.4
Correlated Gaussians in two-dimensional systems Correlated Gaussian-type gendnals
7.5
Nonlocal
7.6
Semirelativistic kinetic energy
Complements
potentials
.
129
.................
131
.
.
.
.
...............................
......................
.........................................
C7.1 Sherman-Morrison formula Exercises
.
..........................
from correlated Gaussians
C6.5 Three electrons with S
Exercises
momentum
.........................................
C6.6 Four electrons in
75
..............................
C6.1 Nodeless harmonicoscillator functions
7.
65
...................
and correlated
6.3
.........
61
134 137
143
........................
143
............................................
145
Table of Contents
8.
Small atoms and molecules
.........................
8.3
Coulombic systems Coulombic three-body systems Four or more particles
8.4
Small molecules
8.1
8.2
.......
149
.....................
150
...................................
.........................................
C8.1 The cusp condition for the Coulomb potential C8.2 The chemical bond: The H,+ ion C8.4 9.
Stability hydrogen-like Application of global vectors
169
.................
to muonic, molecules
174
................................
177
One-gluon exchange model Meson-exchange model
.....
178 178
............................
181
quark
...........
188
...................................
191
....................
196
method
a
magnetic
204
..........................
213
...
213
.............
216
...............................
223
.........................................
230
11.2 Few-nucleon
systems with central forces
potentials
C11.1 Correlations in few-nucleon systems C11.2 Convergence of partial-wave expansions Pauli effect in s-shell A
C11.4 The "C nucleus
Appendix
202
field. 204
few-body systems Introductory remark on nucleon-nucleon potentials
Quark
....
.........................................
11. Nuclear
C11.3
187
..............................
C10.1 Two-dimensional electron motion in
as a
...............
230
............
233
..........
239
.
242
hypernuclei system of three alpha-particles
...............................................
Matrix elements for
A.1
model
........................
Few-body problems in solid state physics 10.1 Excitonic complexes 10.2 Quantum dots 10.3 Quantum dots in magnetic field 10.4 Quantum dots in the generator coordinate
Complements
171
....
9.2
11.3 Realistic
167
....................
The trial function in the constituent
11. 1
165
167
9.1
Complements
154
........
molecules
Baryon spectroscoPy
9.3
10.
of
149
........................
............................
Complements
C8.3
XIII
general
Correlated Gaussians
A. 1. 1
Overlap
A.1.2
Kinetic energy
Gaussians
..................
.............................
of the basis functions
247 247
247
................
247
.............................
249
Table of Contents
XIV
A.1.3 A.1.4
......................
250
..................
256
Two-body interactions Density multipole operators
A.2
Correlated Gaussians with different coordinate sets
A-3
Correlated
matrix elements
Spin
A.5
Three-body problem and spin-orbit forces
Complements
.................
262 263
with
central,
tensor
..............................
265
.........................................
280
CA. 1 Matrix elements of central potentials CA.2 Matrix elements of density multipoles
CA.3
...............
280
..............
283
matrix elements of the correlated Gaussians
Overlap a three-particle system
for
Exercises
References Index
257
.............................
Gaussian-type geminals
AA
...
.........................
285
............................................
288
..............................................
...................................................
299
307
1. Introduction
There
are a
countless number of examples of quantum-mechanical few-
body systems:
constituent
quarks
in subnuclear
physics, few-nucleon
few-cluster systems in nuclear physics, small atoms and molecules physics or few-electron quantum dots in solid state physics, etc. The intricate feature of the few-body systems is that they develop or
in atomic
depending on the number of constituent particles. The mesons and baryons, the alpha-particle and the 'Li nucleus, or the He atom and the Be atom have very different physical properties. The most important causes of these differences are the correlated motion and the Pauli principle. This individuality requires specific methods for the solution of the few-body Schr6dinger equation. Approximate solutions which assume restricted model spaces, mean field, etc. fail to describe the behavior of the few-body systems. The goal of this book is to show how to find the energy and the wave function of any few-particle system in a simple, unified approach. The system will normally be in the minimum energy quantum state. As forewarned, however, to find this state, the ground state, is a complicated matter. The present stage of the development of computer technology, however, makes a very simple approach possible: Searching for the ground state by "gambling". Without any a priori information on the true ground state, completely random states are generated. Provided that the random states axe general enough, after a series of trials one finds the ground state in a good approximation. The reader individual characters
may find this
a
little
suspicious but there are indeed a number of fine error procedure which makes the whole idea
tricks in the trial and
really practicable. Before bombarding the
reader with
sophisticated details, let us example. Let us try to determine the energy of a Coulombic three-body system: a positron and two electrons, the positronium negative ion, Ps-. The example is simdemonstrate the random search with
Y. Suzuki and K. Varga: LNPm 54, pp. 1 - 6, 1998 © Springer-Verlag Berlin Heidelberg 1998
an
1. Introduction
2
ple enough so t1i at a graphic illustration of the wave function can be given, but its solution is far from being trivial. The ground state of this particular system can be calculated by different methods accurately. In the rest of this chapter we will give just the outline of the gambling method without paying much attention to the details. To look for the wave function of the ground state of the system, we generate random functions. The functions we assume some
continuous a.re
parameters
or
randomly
chosen. For
Tf (a, 0, x,
y)
=
=
are
random in the
depends
on some
sense
that
discrete
or
"quantum numbers" and these paxameters example, we assume a simple Gaussian form
e`2 -
3Y2
(1.1)
7
1r2 -7'31 denotes the distance between the two electrons and I (r2 +r3) /2 r 1 1 denotes the distance between the center-of-mass
where Y
functional form which
x
=
-
of the electrons and the
positron.
randomly chosen generated states are calculated and compared. The one among them giving the lowest value is selected to be a successful paxameter set. Figures 1.1(a)-(d) show examples of four random states and their energy expectation values which are obtained with such successful paxameters. Not surprisingly, the energies of the "configurations" appreciably depend on the shape of the functions, that is, on the random paxameters. The lowest energy, -0.11 in atomic units (a.u.) in Fig. 1.1(d), is higher than that of the exact ground state (-0.262). Here in the atomic units M, (the electron mass), e (the electron charge magnitude), and h axe chosen to be basic quantities of units, so the unit of length is Bohr radius h?/(Mee2) 5.29x1O-11 m, theunit of energyis me4/h? 27.2 ao 2.42 x 10-17 S. (Actually the and the unit of time is h3/(me4) eV, minimum energy calculated analytically with one term of Eq. (1.1) is -0.177, but it may not be known for a general case.) That means that the trial function is not general enough to describe the ground state. To improve the trial function, let us take a linear combination of two of the above functions of Eq. (1.1), where one of the two is fixed as the one already selected and another is newly selected after a number of random trials. Figures 1.2(a)-(d) show examples for random wave functions which are calculated by the two terms. The energy improved because the model space increased but we still miss a substantial amount of binding energy. Increasing the model space further A certain number of parameter sets a and and then the energy expectation values of the
=
axe
=
-
=
by adding one,
we
more
and
more
functions to the linear combination
reach the exact energy and the
wave
function
one
(Fig. 1.3)
by
with
1. Introduction
E--0.09
E--3.15
E=0.18
E--0.11
Examples of the energy expectation value and the (normalized) xyTf (a, 3, x, y) of Ps- for one random basis state. Figure 1(a) is in the upper left, 1(b) in the upper right, 1(c) in the lower left, and 1(d) in the lower right. In each section the x axis denotes the distance
Fig.
wave
1-1.
function
between the two electrons and the y axis denotes the distance between the center-of-mass of the two electrons and the positron. Atomic units are used.
1. Introduction
E-0.14 E=-0.18
0.1
0.0
B-0.12 E=-0.17
0.1 0.0
Fig.
1.2.
Examples
xyjcj-T1(aj, Oj
of the energy expectation value and the
wave
function,
y) +c2Tf(a2, 02 Xi Y) ji of Ps- for combinations of two random. basis states. Figure 2(a) is in the upper left, 2(b) in the upper right, 2(c) in the lower left, and 2(d) in the lower right. See also the caption of -,
Fig.
1. 1.
x,
1
1. Introduction
a
combination of about 150 functions of the form
(1.1).
The
5
conver-
gence of the energy versus the number of the functions in the linear combination is shown in Fig. 1.4. The energy gain is large in the first
few steps and then it
slowly approaches the
exact energy.
E=-0.262
X
Fig. and
1.3. The energy and the wave function of Ps- obtained selection of 150 basis states
by the trial
error
The alert reader may question the importance of the steps that are taken to increase the model space and can ask why not start say, a linear combination of 150 functions and then, by random of all the parameters, one may find the solution. The reason for trials
with,
increasing the number of functions in the combinations is to control "convergence" of the energy. By comparing the energy gain in the successive steps one can guess how far the exact ground-state energy the
is. There is
no
guarantee that the ad hoe 150 functions would be
1. Introduction
6
-0.250
-0.252 M 41
-0.254
-0-256
-0.258 Z
-0.260
-0.262 20
40
60
80
100
Dimension of the basis
Fig.
1.4. The energy convergence of Ps- as a function of dimension of basis are increased one by one with the trial and error selection. The
states that
solid, dashed and dotted
curves
correspond to
sufficient to reach the solution in
cases
three different random
paths.
where the exact energy is not
known. A
skeptic may say that one should, instead of the above gambling method, try a deterministic parameter search such as that furnished by the Newton or conjugate gradient method. While this may be true for small systems, the random trials the
axe more
successful in most of i
in the
I
a. by avoiding being trapped omnipresent local Moreover, one may have discrete paxameters or quantum numbers
cases
where the deterministic seaxch may not be suitable. One may also wonder if the solution we reach is the in
ground state energy. These the succeeding chapters.
and other
really (close to)
questions will be answered
Quantum-mechanical few-body problems 2.
first step toward the goal of giving a unified and reasonably simple recipe for solutions of few-body bound-state problems the basic
As
a
notations and concepts
The motion of the ton
operator
axe
introduced here.
few-body system
(Hamiltonian)
and described
governed by the Hamilby the eigenfunction of the
is
depends on the positions and other degrees of freedom of the particles. See, for example, [1, 21 for textbooks on quantum mechanics. One can define the positions of the particles in several different ways by using single-particle or relative coordinates. The single-particle coordinates are useful if the particles move ah-nost independently of each other. The relative coordinates are advantageous to emphasize correlations between the particles and Hamiltonian. This
to
wave
function
separate the center-of-mass motion. One
can
relate the different
coordinate systems to each other by linear transformations. The definitions and the properties of these transformations are elaborated in this
chapter.
indistinguishable particles. To comply with the Pauli principle the wave function has to be properly symmetrized. In the spatial paxt of the wave function the symmetrization induces permutations of the coordinates. It is shown that these permutations can be imposed by linear transformations in the relative coordinate One often deals with
space.
2.1 Hamiltonian isolated system of N particles with masses ml., and let 7,1,, ,,rN denote the position vectors of the particles. -, mAT All of the particles may be identical or different, or some group(s) of them may be identical. The Hamiltonian of the system, with the
Let
us
consider
an
*
center-of-mass kinetic energy Tcm
being subtracted,
Y. Suzuki and K. Varga: LNPm 54, pp. 7 - 20, 1998 © Springer-Verlag Berlin Heidelberg 1998
reads
as
Quantum-mechanical few-body problems
2.
8
N
N
2
H=E 2mi
-T,.
As the interaction
(2.1)
Vij.
+
unchanged by the displacement of the origin, the above Hamiltonian is translationaUy invariant and depends on the internal degrees of freedom only. One may encounter few-body, problems where the system is subjected to some external field or the particles move in a single-particle potential. The Hamiltonian then reads as IV
H
=
N
2
E2
remains
Vij
+
1: Ui + 1: i=1
i=1
N
(2.2)
Vij.
i>i=l
Even with the presence of the one-body potential Ui it may happen that one wants to remove the contribution from the center-of-mass motion. This will be considered later in
evaluating the matrix elements
of various operators. An extension of the nonrelativistic kinetic energy to the relativistic kinematics will be discussed in Sect. 7.6.
Three-body potentials can be treated in principle but axe suppressed for the sake of simplicity. The two-body potentials are assumed to be local (but nonlocal potentials will also be considered in Sect. 7.5) and can in general be expressed as
Vij
=
1: VP(ri
-
rj) 0?.
P
=
1: f VP (r) J(ri
-
rj
-
r) 0?. dr.
(2.3)
P
Here p is the short-hand notation to specify the component of the potential which is characterized by 0?. (e.g., central, spin-orbit, etc.
and
VP(r)
is the
corresponding form factor.
To
specify
the
particle
motion, several degrees of freedom are in general needed besides the spatial one. For example, the nucleon has spin and isospin degrees
freedom, and the quark has spin, flavor and color degrees of fre&dom. The specific potentials related to nuclear, subnuclear and other of
systems will be discussed in later sections.
2.2 Relative coordinates
9
2.2 RelatiVe coordinates The
separation of the center-of-mass motion
care
of most
is
important to describe intrinsic excitations of the system. The center-of-mass motion is taken mass
conveniently by introducing
coordinates;i
(XIi X2
relative and the center-of-
xN). The symbol- stands for a trans-
pose of a matrix. In particular, x is often used to stand for one-column matrix whose-ith element is xi, whereas ,'c a I row
an
x
N
IV
x
1
one-
matrix. Here XN is chosen to be the center-of-mass coordinate of
the system and the rest of the coordinates
relative coordinates.
independent single-paxtide coordinates
i
-=-
They
(rj,
-'
fXI,
axe
in
-
-
-,
X N-I
general
I
is
a
set of
related to the
TN) by a linear transformation:
N
Uijrj,
xi
ri
=
j=1
E (U-I)ijxj
(i
N).
(2.4)
j=1
We show two
examples
of relative coordinates
is the Jacobi coordinate set, which is defined
by
x.
See
Fig.
2. 1. One
the matrix
C
2
4 3
0
0
(a) Fig.
2.1.
Examples
(b)
of relative coordinates for the
four-particle systein. (a)
the Jacobi coordinate set, xi =rl -r2, X2:-- (MIrI +M2P2)/(MI +M2)'r3 7 X3 = (Tnlrl + M2r2 + M3 r3)/(Ml + M2 + M3) -P47 and (b) the heavyparticle center coordinate set, xi = rl r47 X2 = r2 r47 X3 = r3 r4-
-
-
2.
10
Quantum-mechanical few-body problems 1
-1
MI
M2
M12
M12
M1
M2
0
0
...
0
(2.5)
Ui M12
...
M12
N-I
...
N-I
M1
M2
Tal2---IV
Tnl2---.LV
MN ...
...
M12
...
N
given by
Its inverse matrix is
M2
M3
M12
M123
MIV ...
Tni
M3
M12
M123
M12
---
M
MN ...
(Uj)
M12
...
N
(2.6)
M 12
0
M123
-"M12
0
0
...
N-1
M12 ...N
where M12 i +mi and, especially, 7nl2---N is the total Tnl+Tn2+ the "heavy-particle center" coordinate is Another the of mass system. -
...
set defined
UC
by 1
0
0
0
1
0
-1 -1
...
(2.7)
-
0
0 M12
---
N
...
-1
...
MN
M2
Mj_
...
...
M12
...
given by
and its inverse matrix is
...
N
M12
...
M
N
(UC )-I
M12
...
M12
...
N
M1V-I
M2
MI ...
MN-I
M2
M, M12
M12
M12 ---N
M
N
M12
N
M2
M12
...
N
MN-I
M2
MI
M12
...
N
M12
...
N
M12
...
N
(2.8) The choice of the when the other
mass
particles.
heavy-particle
See
center coordinates may be natural
particle is heavier than the masses of the Complement 2.1. Note, however, that it is always
of the Nth
2.2 Relative coordinates
11
possible to transform from one coordinate set to another. The Jacobian corresponding to the coordinate transformation (2.4) is unity for both U matrices.
Corresponding to the transformation (2.4), the momentum pi is -ih ya conjugate to the expressed in terms of the momentum -xj axj =
coordinate xj: IV =
Pi
N
1: Ujilrp
Wi
=
1:(U-l)jipj
(i
=
1,
...,
N).
(2.9)
j=1
j=1 N
Note that -7rAT
=
Ej=1 p,
is the total momentum. The center-of-mass
kinetic energy is given by T,,n ir2,v/(2MI2 N). The kinetic energy with the center-of-mass kinetic energy subtracted is then operator =
...
expressed
as
P
(27ni 6ij
2mi
i=1
i=1
2m12
...
N)
Pi
Pj
j=1
M-1 N-I
Aij-xi
2 i=1
-
(2.10)
-7rj,
j=1
where N
1: UikUjk Mk
Aij
(i,j
=
11
....
N
-
1).
(2.11)
k=1
To evaluate the potential energy matrix elements for the tions that contain
no
dependence
on
the center-of-mass
wave
func-
coordinate,
it
is convenient to express the single-particle coordinate relative to the center-of-mass, ri xN, the interparticle-distance vector, ri -,rj, and -
difference, pi pp in terms of x and easily done by using Eqs. (2.4) and (2.9):
the momentum This is
-
-x,
respectively.
1V-1
ri
-
XN
=
1: (U-l)ikXk
-W X,
(2.12)
k=1 N-I
'ri
-
r3
1: k=1
((U-l)ik (U-I)jk) -
Xk
=
W(j) X,
(2.13)
Quantum-mechanical few-body problems
2.
12
N-1
2
(Pi
Pj
E (Uki
2
-
(ij )Ir-
Ukj)7rk
(2.14)
k=1
Note that the contribution from the term
momentum,
(Mi
is
when
legitimate
-
Mj)/(2Ml2 N)7rN, ...
we are
proportional to the total omitted in Eq. (2.14). This
is
in the center-of-mass
(,7rlV
system
=
0)
and
interested in the intrinsic motion of the system. The Hamiltonian of type (2.1) is now expressible in terms of independent relative coordinates alone. In what follows
and
x
7r are
relative coordinates unless otherwise
(i.e. (N
introduced to
the notation.
by the
W(i) EN-I k Xkk=1
=
they are (N-1) x 1
stated,
-
1)
1 one-column
x
I)-dimensional vectors) 0), Oj),
simplify
which is formed
-W X
-
represent only the
that
so
Eqs. (2.12)-(2.14), (N
one-column matrices. In matrices
meant to
Wx
E.g.,
is
a
and
(W)
are
Caitesian vector
multiplication of iv(i) and x, i.e., that Oj) and (W) satisfy the following
usual matrix Note
equalities M-1
-((ij)
3 W(ij) (k k
W( ij
=
(-ipw(ij)
k=1 N
W(ij)(1(ij)
W
k
i>i=l
(ij)
F(ij)
i>i=l
kI
N
where the relations UNTj
for k
=7 -
For
N
are
an
(N
-
1) x (N
-
1) symmetric
scalar
products
E xi-(Ax)i with this
i=1
=
1,
EN
Uki
=
0
A,
let the
quadratic
of the Cartesian vectors:
xv)
j=1
convention, that
1V -
matrix
Aijxi-xj. i=1
show,
ai(ri
(2.15)
N-1 N-1
i=1
It is easy to
1),
?ni/?nl2---N, (U-1-)iN
=
N-I =
-
used.
form, bAx, represent
..bAx
N
R, I
Jk, 2
2
i=1
ajw(')J))
x,
(2.16)
2.2 Relative coordinates N
1V
I:
a
(ri _,rj)2
ij
1:
=.i
a
ajjw
-(ij)
(ij)
(2.17)
X.
i>i=l
i>i=l
As
13
special case of the above relations, we have the following equalities
for the moment of inertia around the center-of-mass: N-1
N
N
Mi(ri
-
XN
MzMj
)2
M12
where the reduced is defined
mass
...
N(ri
Tj)
-
corresponding to
ILi
2
(2.18)
the Jacobi coordinate xi
by ?ni+17nl2 M12
...
z
...
(2.19)
1).
N
i+1
We also obtain 2
N
N
(ri
-
T'j )2
N
=
E(ri XN)2
(,ri
_
-
(2.20)
xiv)
i=1
The term are
the
E , (ri
-
XN)
masses
of all the
particles
same.
The total orbital mass
vanishes when the
system
(-xv
=
angular
0)
momentum
be
can
operator L in the center-of-
expressed
in terms of the
operators
relevant to the relative coordinates: N-1
N
hL
(ri
((,ri
-
X
Pj)
Tj)
(XN
X
2
(Pi
X
-
(Xi
7N)
X
7ri),
(2.21)
Pj))
i>i=l N
or i
-
X
'r
Rij)
-X
i>i=l
+
((ri
-
W('j)X
rj) XMi
-
2M12
X
Mj 7rN ...
N
(('j)-7r) =NU. 2
(2.22)
Quantum-mechanical few-body problems
2.
14
Eq. (2.22) use is made of Eq. (2.15) in the last step. Thus the equality of Eq. (2.22) holds in the case one considers the orbital angular 0 or treats a system of paxticles momentum in the system of 7rjV with equal masses. In atomic physics the trial function of Hylleraas type or correlated In
=
exponential type is often used with success. This function contains the exponential form expressed in terms of the interparticle-distance coordinates. T-nstead of the exponential function let us consider its Gaussian analogue IV
T
=
exp
E
2
aij (ri
-
rj )2
(2.23)
i>i=l
aij determine the falloff of the interparticle functions and may be considered variational parameters. Since not all of the N(N 1)/2 interparticle-distance vectors, ri rj, are Here
N(N
1)/2 parameters
-
-
-
(Irl T2) + (T2 -r3), it is convenient to T3 e.g., TI rewrite T in terms of the independent coordinates x. This is needed to independent,
calculate, e.g., the Eq. (2.23) as T
=
exp
aij
are
-1
(
(N
where the
=
-
-
2 -
related
1)
norm
-
of T.
Equation (2.17) enables
one
to rewrite
iAx), x
(N
-
(2.24)
1) symmetric matrix A
and the parameters
by N
N
-
(ij)
AkI
ajjWk
(ij) WI
=
I:
aij
(W
(ij) W(ij)
) kl'
i>i=l
i>i=l N-I N-1
aij
-
1: 1: UkjAkIU1j k=1
=-
-(CTAU)ij
(i
(2.25)
<
1=1
By substituting Akj of the first relation to the second relation one can check the validity of the second relation. The above two forms,
(2.23)
and
(2.24),
are
thus
equivalent.
We introduce both because in
applications the first form or in some others the second form is more advantageous. See Complements 6.4 and 7. 1. The norm (Tf I Tf) becomes finite if and only if A is positive-definite and is then given by some
((270N-1 /detA)
I 2
-with the
use
of
Eq. (6.32)
or
A necessary and sufficient condition of the a
symmetric
Exercise 6.2.
positive-definiteness
matrix A is that the matrix A is
expressible
as
A
of =
2.3
dDG,
where G is
an
(N
1)
-
x
(N
Symmetrization
1) orthogonal matrix containing is a diagonal matrix, Dij diJij,
-
1) (N 2) /2 parameterg and D including (N 1) positive parameters di. (N
-
15
-
=
-
closing this section, we remark that the matrix Uj of Eq. orthogonal in general. In some cases it is convenient to use a (2.5) coordinate system that is obtained from the single-particle coordinates by an orthogonal matrix. We show such an example in Complement Before
is not
2.2.
Symmetrization
2.3
function of the system must have a proper symmetry for the interchange of identical particles. This is achieved by operating on the The
wave
basis function with for bosons
or
the
suitable operator P, which is the symmetrizer antisymmetrizer for fermions. The operator P can in a
general be given by a combination of the permutations P with suitable 1
phases. The permutation transforms the Pr
=
P
=
(PI
2
'V
P2
PN
single-particle coordinates
as
ri
)
-+
of paxticle indices
rp,:
(2.26)
Tpr,
where the matrix Tp is
(Tp) ij E.g.,
=
(i, j
Jj pi
the matrices
Tp
are
1
0
0
0
1
0
T123
1,
=
...'
given as follows
1
(321
T123
(
The
T123
(2.27) in the
case
of three
0
1
0
1
0
0
0
0
1
0
particles:
213
123
T123'
N).
0
0
0
0
1
0
1
0
1
0
0
1
0
0
1
1
0
0
on
the
1
1
0
0
0
1
0
0
1
0
0
0
0
1
1
0
0
0
1
0
T123 132
T123
(3121
23 1
operation of P
wave
it is written in terms of the
function is thus not
(2.28)
a
single-particle coordinates.
problem
when
Quantum-mechanical few-body problems
2.
16
When the
ordinates,
wave
have to note
we
expressed in terms of the relative cothat the permutation P induces a lineaX
function is
transformation of the relative coordinates Px
=
follows:
as
(2.29)
Tpx,
by using Eqs. (2.4), (2.26) given by
where
(2.27)
and
the matrix Tp is
now
N
(TP)ij
1: Uik(U-1)Pki
(i,j
=
'I
...
I
N
-
1).
(2.30)
k=1
Though
we use
matrix is
which is
now
the
same
notation of
(N 1) x (N 1) -
-
Tp,
note that the size of the
because the center-of-mass
coordinate,
unchanged under the permutation, is eliminated. We Will show
Chap. 6 that the properties of Eqs. (2.26)-(2.30) can be used advantage for Gaussian functions. See Eqs. (6.28) and (6.29).
in
to
2.4 Permutation of the Jacobi coordinates
generate different sets of Jacobi coordinates by interchanging the paxticle labels by permutations. Figure 2.2 shows an example of One
can
three different sets,
x('), X(2)
,
and
X(3),
of the Jacobi coordinates for
three-body system. These sets are completely equivalent from the point of view of the dynamical description of the system, but they can be advantageously used in the calculation of the matrix elements. In the Jacobi coordinate set, defined by Eq. (2.5), the first Jacobi coordinate is equal to I'l -r2. By permutations one can create other Jacobi coordinate sets X(k) where the first Jacobi coordinate takes a
rj. The index k refers to the kth permutation and distinguish the permuted Jacobi coordinate set. As the two-
the form of ri serves
to
-
body potential depends on this coordinate, the set created by the permutation will prove to be useful. See Appendix A.5 for an example of using the permuted Jacobi coordinate set. Another peculiarity of the Jacobi coordinates is that if we are in the center-of-mass system (-xiv 0) then by using Eqs. (2.5) and (2.9) --xiv-11 that is, one of the single-paxticle momenta is equal to p1V the relative momenta. By permutations we can express any of one of the single-paxticle momenta in terms of a certain momentum of =
=
the relative momenta. This
simple expression can be exploited in sevevaluating matrix elements of operators
eral ways, because instead of
2.4 Permutation of the Jacobi coordinates
a-
is
*+-3
Mw-
2
3
Ak
a&-
71
IRW
Ah MW
1
2
3
2
2.2. Different sets of the Jacobi coordinates for the
Fig.
tem. From left to
right: x(l), X(2),
containing single-particle an
application
we can
do
so
See Sect. 7.6 for
versa.
three-particle sys-
x(3), respectively.
and
momentum
relative momentum and vice
17
with operators of more details and
to the calculation of matrix elements for the semirela-
tivistic kinetic energy. The permutation of the Jacobi coordinates is defined
by the corresponding cyclic permutation
of the column vectors of Uj and the
permutation exchange of the
(1, 2, the
...,
N)
=
For
masses.
1
(2
2
example, for
N)
3
the
the transformation matrix between
1
singlt-,particle coordinates
r
and the Jacobi coordinates
x(f)
takes
the form 0
1
-1
0
M2
M3
M23
M23
0 ...
0
U(k)
(2.31) M2
-1
MAF
M3 ...
M23
...
M
M23---.LV
N
M12
M2
MI
M23---N
MN
M3 ...
M12---N
M12
...
...
N
M12
---
N
The relative coordinates in the different Jacobi coordinate sets
expressed by
X(k)
=
each other
T(k) X
through
T(k)
=
can
be
the transformation:
UMU-1. i
(2-32)
symmetrization in Sect. 2.3 and the permutation of the Jacobi are formally very similar. The main difference is that while the symmetrization is assumed for identical (and therefore equal mass) particles, in the latter case we allow for particles with different masses. The
coordinates
Complements
18
Complements 2.1 An
N-particle
Hamiltonian in the
heavy-particle
center coordinate
set
that the
Suppose
mass
of
of the
one
particles,
say the Nth
paxticle,
is the heaviest among all the paxticles. In this case the choice of the heavy-paxticle center coordinate may be natural paxticularly when the
position of the Nth paxticle is close to the center-of-mass of the system, like in an atom, where the Nth particle is a nucleus. The Hamiltonian (2.1) can be reduced to the form which contains no explicit dependence the variables of the Nth
on
is
particle.
By using Eqs. (2.7), (2.10) and (2.11) the Idnetic energy operator expressed in the heavy-paxticle center coordinates as N
N-I N-I
2
1
,
P7
E 2mi
-
T.-
=
EE i=1
i=1
( 2mi k 2mjV) +
-
I-ii .7rj
j=1
N-1
1: i=1
N-1
70z
1 +
2jLjjV
-
TaN
E
(2-33)
Iri .7rj,
j>i=l
'where jLjN mimN/(Tni + Talv) is the reduced mass for the ith and Nth particles. The potential energy VjV (i 1, ..., N 1) is a function =
=
of ri -rN on
xi and
=
the ith
can
be considered the one-body potential Uj
On the other hand the
particle. lighter particles can
N-I
H=E i=1
Potential
energy
function of ri -,rj xi thus be reduced to the type of Eq. (2.2)
between the
HamiltoTdan
-
2AjN
=
a
-
xj. The
N-1
N-I
2
^7r'
is
acting Vij acting
+EUi+ 1:
(Vij+
MN
7ri-7rj).
(2-34)
j>i=i
=1
problem is thus reduced to that of an (N-1)-paxticle system with an external field Uj and an additional separable "two-body potential"
The
oo, jLjjv goes to mi and the 7ri .7rj term The effect of the finite nucleax mass in the helium atom
7ri -7rj. In the limit of m1V -
disappears. was
discussed in
-->
[3].
The above formulation
can
also be
applied
to
a
system of identi-
particles when some, or most, of them form an inert core. As an example let us take up the nuclear shell model, where the nucleons
cal
are
divided into two groups, the passive or inactive nucleons and a small number of active nucleons. The passive nucleons form
relatively
C2.2 Canonical Jacobi coordinates
19
by filling the lowest possible single-particle orbits and exert a potential field Ui on each of the active nucleons. The active nucleons may occupy several single-particle orbits belonging to several major shells outside the inert core. In this approximation it is desiran
inert
core
able to describe the motion of the active nucleons without inclusion
arising from the center-of-mass motion. It is clear can be excluded by taking the full Hamiltonian have the form (2.34) with the core as the heavy particle.
of any excitations
that such excitations to
2.2 Canonical Jacobi coordinates
As
was
2.2, the Jacobi coordinates
noted in Sect.
that the transformation matrix Ui of Eq.
is,
neither
following
UjUi-
nor
relations
UjA-'Uj
Uj-Uj
equal
(2.5)
have the property orthogonal, that
to the unit matrix.
Instead, the
fulfilled:
are
L-1,
=
is
x
is not
UiLUi
=
(2.35)
A,
with
Aij
Jij
Lij
(2.36)
Jij, Ai
Mi
N where pi (i 1, 1) is the reduced mass belonging to the ith Jacobi coordinate and is defined in Eq. (2.19), while 1LV is equal to the =
-
the transformation between
the singleUir and p UT-x, of paxticle and the Jacobi coordinate systems, x identities the lead above relations the to and following (2.9), Eqs. (2.4) total
mass
'MI-2
N.
...
By using
=
N
N
E Tnir?
=
z
N
E Nx
Alternatively,
N
pi2
E
-
Mi
one can
introduce
Ir2i
=
-(2.37)
Ai
"canonical" set of Jacobi
a
coor-
dinates: IV
Vijrj
with
V
=
A
2
UiL 2.
(2.38)
j=1
The square root matrices of the diagonal matrices L and A axe simply given by the square root of the elements. This system of Jacobi
coordinates
Vf"
=
belongs
f7v
and therefore
17
to
an
orthogonal
transformation:
(2.39)
20
Complements 1V
1V
EVji j
IV
E 2 Er2
and
(2.40)
=
i
i*
i=1
j=I
i=1
Let the momentum
Simfla,rly
to the
canonically conjugate to i be denoted qi. transformation of coordinates, we obtain the trans-
formation of momenta
follows:
as
N
71i
=
N
E ViiPi,
A
E Viini
=
j=I
(2.41)
j=I
The total Idnetic energy does not take to q and can be expressed as follows:
diagonal
a
form with respect
I?
E Mi
E E (VLfr) ij?7i -77j wLf7rl.
i=1
i=1
j=I
The two systems of the Jacobi coordinates ical Jacobi coordinates X
=
For
ujf '
a
=
=
M1
rT; 1'2
?T 1 M
M2M3
3
M123
12MI23
VMM1232:
1
2
particles with equal I
1
(2.43)
V
vf2-
I
I
v/6-
T6
matrix V reads
0
12
M12
M123
I
'1
73T
disadvantage
masses
mi
axe
M3
mass
2
V"6_
(2.45)
1
73
of the canonical Jacobi coordinates
is that if the
equal then the center-of-mass motion is not sepais not the center-of-mass coordinate).
not
easily ( N
(2.44)
V-Ml'23
-
v/3-
as
0
-
V/2-
rated
=
V(Uj)-Ix.
VMM1123
The
can
-VEM12
L2 M12
and for
Vr
Ujr and the canoneasily be related by x
three-paxticle system the transformation
V
(2.42)
=
3. Introduction to variational methods
The variational method is
popular approaches to tackle quantum-mechanical few-body problems. Though it gives only an approximate solution except for some special cases (the Ritz variational method, for example, gives only an upper bound of the energy), one can get a virtually exact solution with an appropriately chosen function space. The function space is defined by basis states and the wave function of the system is expanded in that, basis. In this chapter we briefly introduce the theorems requisite for obtaining a vaxiational one
of the most
solution.
3.1 Variational Let
principles
physical system whose Hamiltonian H is self-adjoint (Hermitian), bounded from below and time-independent. We are interested in finding the discrete eigenvalues of H and its corresponding (normalized) eigenstates: us
consider
H!P,, The
=
a
En(fi,
energies En
n
axe
=
(3.1)
1, 2,...
real and
are
ordered such that El :! E2<
....
We
that the ground state is non-degenerate. Although H is known, this does not mean that E,, and (fin axe known. In general, it is difficult to solve the eigenvalue equation (3. 1). When we do not know how to diagonalize H. exactly, the vaxiational assume
method becomes useful for any
Theorem 3.1 the state space the is such that E
=
RIHITI) R_ I TI)
type of Hamiltonian.
(Ritz Theorem). For an arbitrary function Tf of expectation value (Mean value) of H in the state Tf
> -
El,
Y. Suzuki and K. Varga: LNPm 54, pp. 21 - 37, 1998 © Springer-Verlag Berlin Heidelberg 1998
(3.2)
3. Introduction to variational methods
22
where the
equality
if and only if Tf
holds
is
an
eigenstate of H
with the
eigenvalue El. Proof.
elementary and can be found in of quantum mechanics. See for example [1, 2]. The ftmetion expanded in terms of the energy eigenstates
The
textbooks T1 may be
proof
of this theorem is
00
(3-3)
ai(fii.
In this
cluded and the
integration
over
(Tf jHn Itp) (TrI Tf )
the
with continuous
eigenvalues are sum must be extended appropriately to include them. Then we can show that for an integer n
expansion
eigenstates
0
Faci=2 A Eln)Jai 12 00, jai 12 Ei= n Y
inan
7
-
En I
-
-
(3-4)
I Clearly the right-hand side of Eq. (3.4) is non-negative for n and if for > if i It vanishes 2. > 0 because Ei 0 for E, only ai i > 2, that is, T, is an edgenstate of H with eigenvalue El. This proves I was used to prove the Ritz the Ritz theorem. Only the case of n theorem but other cases will be needed later to derive Temple's bound. =
=
-
=
approximate deexploited as ground-state energy El, namely the minimization principle of the mean value of H. Suppose that we choose a family of functions Tf (a) which are characterized by a finite number of parameters denoted a. In the case of Eq. (1.1), there axe two parameters a and P. We calculate the mean value E(a) of the Hamiltonian H in this trial function, and minimize E(a) with respect to a. The minimal value obtained in this way is an approximation to tile ground-state energy El of the system. Clearly, whether this variational method produces a satisfactory result or not substantially depends on the choice of the trial functions Tf (a). The Ritz theorem can be generalized to excited states as well: The Ritz theorem is the basis for
a
method of
it is
termination of the
Theorem 3.2
(Generalizecl
of the Hamiltonian H discrete eigenvalues.
value
Proof.
Let
E(TfIT)
=
us
(TfjHjTf)
sumed to be linear term
calculate
an
is
Theorem).
stationary
in the
increment JE of the
The
expectation neighborhood of its
mean
value us' g
+JT-1, where JTf is aschanged of higher order than a terms Neglecting
when Tf is
infinitely small. in JT1, we obtain
Ritz
to Tf
3.1 Variational
(TfITf)JE
=
=
The
(JTf IH
value E is
mean
that
J((TIIHITI)) -
-
23
principIcs
EJ((TIITI))
EITI)
+
(TfIH
stationary if JE
(3.5)
EIJTf).
-
=
0 for any infinitesimal
JT,
if
is,
(,NTtIH
EITI)
-
+
(TIIH
If JTf is chosen to be
the above
number,
-
-(H
EIST)
-
E)Tf,
(3.6)
0.
=
where
-
is
that the
equation implies
an
infinitesimal real
norm
of the function
(H E)Tf is zero, and thus the function (H E)Tf itself must be a null function, namely HTf = ET. Therefore the mean value E is -
-
stationary if and only if the state Tf from which E is calculated is an eigenstate of H, and the stationary value E is the corresponding
eigenvalue
of the Hamiltonian.
generalized Ritz theorem allows an approximate determination of the eigenvalues of the Hamiltonian. If the E(a) has several extrema, they are the approximate values of some of the energies E,,. In most cases for practical applications the. trial function is given as a linear combination of a finite number of independent functions This
!P(a): K
(3.7)
ciTf (ai).
Tf
independence of the functions will be discussed soon later. T' (a K) axe mutually orthogWe do not always assume that Tf (a I) onal because the use of nonorthogonal. functions is in fact quite useful. They can, however, always be made orthogonal if necessary, e.g. by a Gratn-Schmidt orthogonalization procedure. The variational method then reduces to the eigenvalue problem of the Hamiltonian inside the state space VK spanned by the set JTf((YI),...,Tf(aK)Ji that is, the space containing all linear combinations of T (a,), Tf(aK). The mean value E is given by The linear
,
...
i
...
Ctlic E
(3.8)
=
CtBC' where and
-
c
is
a
IC-dimensional column vector whose ith element is
ct is the Hermitian conjugate of c. The K
overlap
matrices W and B
are
defined
by
x
ci
K Hamiltonian and
3. Introduction to variational methods
24
Rij
=
(Tf(ai)JHJT'(cvj)),
Bij
(TI ((Yi) I Tf (aj)).
=
(3.9)
The linear parameter ci can be determined by the generalized Ritz theorem. The condition that E is stationary with respect to an ar-
bitrary, infinitesimal change problem
of ci leads to the
generalized eigenvalue
K
Wc
E&,
=
E(Rij
i.e.,
-
EBij)cj
=
0
(i
=
I,-, K).
(3.10)
j=J
The restriction of the thus
can
eigenvalue problem
of H to the
subspace VK
the solution.
simplify
We discuss the linear Tf (a K). The linear c
except for
c
0
=
can
function. In other
and
only
for
has
E
c
K
if c
independence of the functions Tf(al),..., independence of the functions means that no vector make
words,
0. This is
=
a
linear combination ,
EK i=1 ciTf(ai),
the combination becomes
equivalent
unique solution of c ci!F(ai) 0 is equivalent a
=
=
0.
a
a
null
null function if
that the
equation Bc 0 (To understand this, we show that
to
to Bc
saying =
0. Whe'n
=
EK ci!P(ai)
=
0,
0 for j obviously have Ef I ci (TI(aj) JTf (ai)) I,-, K, which but for 0 nothing j I,-, K. Conversely, when (Bc)j (&)i 0 for j I,-, K, by multiplying cj* (the complex conjugate of cj) and we
=
is
=
=
=
=
=
summing
over
j,
we
obtain ctL3c
=(.EK CiT,(Cv,) I EK I
K
which leads
us
solution
0 for the
Fj-1 ciTf (ai)
1
ciTf (ai))
=
0,
As the existence of
a unique 0.) equation Bc 0 is possible only when detB --A 0, the linear independence of the functions is assured by the condition detB :A 0. Because the overlap matrix B is at least positivectBc > 0 for any vector c, all eigenvalues semidefinite, that is (Tf JTf)
c
=
to
=
=
=
p of B become
Tf (a K) ar e real, positive when the functions Tf (a,), linearly independent. The positiveness of M is understood as follows: For the eigenvector c :A 0 corresponding to the eigenvalue JL, we have the relation pctc As the basis functions Tf are (Tf Ifl. linearly (ai) independent, Tf is not identically zero because otherwise we can make !rf EK ciTf (ai) vanish identically with c =A 0, which contradicts the i= I assumption of the linear independence of the basis functions. Thus (TfITf) is positive and of course ctc is positive, so the eigenvalue y has to be positive. We can also state that if there exists a vanisl-iing eigenvalue of B then the basis set JTf(a1),..-,Tf(aK) is linearly dependent. The solution of the eigenvalue problem, pc(A), gives us an ....
=
=
BcG l
orthonormal set:
=
3.1 Variational
principles
1: C(ii.') Tf (C'j),
01.1
25
(3-11)
2
i=l
where
c(l')
c(") tc(/")
is assumed to be normalized to
=
1. In many
practical problems it may happen that B has one or several very small eigenvalues. Then the eigenstate corresponding to the small eigenvalue has very large expansion coefficients cil") If this occurs then a - Flj,. small error in the matrix elements of R or B can lead to a larger error in the solution of Eq. (110). When the ill condition mentioned above does not occur, the generalized eigenvalue problem (3.10) can be solved safely. The eigenvalues q
(i
est
1,
=
...'
K)
eigenvalue
are
arranged be
El may
a
in
increasing order el :! good approximation to
62
:
the
The low-
...
ground-state
energy El if the state space V_T<- is chosen to include the physically most important configurations. Two functions T and TI' belonging to
the
and ej,
eigenvaluesq
(Tf I Tfl) where
CtBC'
-md c'
c
Rc
=
=
have the
overlap
(3.12)
1
satisfy
EiBc,
respectively,
the
and
following equation
'Hc
=,EjBc'.
(3.13)
I and likewise T', c has to be normalized to ctBc &t&' 1. Using the Hermiticity of R and B and the reality of the 0. Thus eigenvalues Ej in Eq. (3.13) leads one to (Ej cj)ctBc' the two eigenstates belonging to different eigenvalues Ei and ej are orthogonal. In the case of degeneracy, they can be made orthogonal by an appropriate procedure. The relationship between the eigenvaluesEi of the truncated problem and the eigenvalues E,, of the full Hamiltonian is elucidated by
To normalize
=
=
=
-
the Nfini-Max theorem. Theorem 3.3
(Mini-Max Theorem).
operator with discrete eigenvalues El :5 E2
eigenvalues of H restricted to independent set of K functions Tf (a,),
EK be the
the
...
E2:51E2i
El <'Eli
...
i
EK :
I
<
Let H be ....
a
Hermitian
Let El ! e2 <_
subspace Vrc of
Tf(aK).
a
-
<
linearly
Then
16K-
(3.14)
Proof. Let WK be the subspace spanned by the orthonormal eigenstates!Pl,(k I!PK of the operator H. We will first show that there ...
is at least
one
normalized function in Vrf with the property
3. Introduction to vaxiational methods
26
(TfJHJTf) To show this
p
WK;
(3.15)
consider the projected function PTf for any normalized VK, where P is the projection operator that projects
we
function Tf in onto
EK.
>
=,EK I!Pi) (!Pi 1. i= I
There
are
two
(i) there exists a function Tfo in Vr<- such (ii) PTf :A 0 for all functions T, in Vl<-. In
case
(i) Tro
is
a
(ii)
that
PTfo
=
0.
linear combination Of 43K+1,
the normalized Tfo has the case
possibilities:
T)lf+2,..., and hence (Trollll%) : : EK+, ! EK. In fimetions Tf, and Tf2 in Vl<- are
value
mean
any two different normalized
projected to different functions PTfl and PTf2 in _PVK because otherwise P(TV, Tf2) 0, which contradicts the assumption. Namely, any two different normalized functions in V.[<- axe projected by P to different functions in Wl,(. As both VK and )IVI<- have the same dhnension, it follows that there must exist a normalized function T, in V.Tf with the wPrf (a 4_ 0). Then !Tf can be expressed as a!PK + bo, property M =
-
=
where
0 is a PK+1,!PK+2,
(TfJHJTf)
=
normalized flinction
Next
2!
we
;
a
linear combination of
=
(T'IHIT')
largest eigenvalue >
of H restricted to
EK)
V_Tf,
Ell;C.
we
EK.
Vif which
by induction
It is easy to derive the
have
(3.16)
(K- I)-dimensional state space Vr.C-1
of all those functions in to 45K and
value,,.-
mean
_
define the
belonging
as
1. This function has the JaJ2 + JbJ2 Ja 12EK + lb 12(01H10) Erc + lb 12((01H10) ...
Because 16K is the IEK
given
are
can
following
orthogonal
show 'EK-1
to be the set
to the
eigenvector
! EK-1 and
so on.
theorem ft-om. the Nfini-Max the-
orem.
Theorem 3.4. Let E,
of
the lowest
! , E2
<
EK be
a
number K
Hermitian operator H. Let
eigenvalues of be a set Of linearly independent functions. a
Tl(a2)i..., Tf (aK)
Tf(al),
Then
K
1: Ei :! Tr(B`R),
(3-17)
i=1
where the matrices B and R
Proof. to
the
Let el :
162
subspace Vl<-
:5
---
:: -
are
defined by Eq. (3.9).
IEK be the
of H restricted
eigenvalues
of the K functions
Tf(Cfl),
...
i
Tf (Ci K) and let
3.1 Vi-triational
be the orthonormal.
eigenvalueS
6 11
E21
...
16K. Then
i
Oi
is
principles
27
eigenstates corresponding given by
to the
K
E UijT1(aj)
(3.18)
j=1
with U orein
the
satisfying
3.3
orthonormality condition UtBU
K,
K
J:Ej
=
1. From The-
obtain
we
K
EEi=E(OilHloi)=Tr(UtHU)=Tr((BU)-IHU)
:5
Tr(U-'L3-1WU)
=
Tr(B-1R).
(3-19)
right-hand side of Eq. (3-17) is determined only by the subspace spanned by the functions and not by the particular choice of the basis because any non-singular linear transformation of the basis does not change the trace (3.17). Namely, when a new basis It is noted that the
set Xl,...,XK is
K)
with
a
FC
given by a transformation Xi Ei= J Tij Tf ((--ij) (i T matrix non-singular (deff :A 0), we can show that =
Tr(B-'?i),
where
L3i'j
-
21
(XilXi)
and
R'ijtj
=
(XjjHjXj).
formalism where the
This leads to the
density can density-functional calculating the energies of excited states as well as the ground state. See [4, 5] for this interesting idea. The next theorem is fundamental to answer a question of how the eigenvalues obtained in a restricted subspace change as the basis be used
as a
basic vaxiable for
dimension is increased. Theorem 3.5. Let el ! - 162 mitian operator H restricted to the
of independent functions EI
K+1
be the
subspace VK of linear combinations
Tf (a,), If (a2),
...
i
Tf(01K).
of H restricted to the
eigenvalues of independent functions
combinations
Let El <
e2
:5
subspace VTC+l of
Tf (01)i llf (a2),
...
7
...
<
linear
Tf (aK), and
Then
!P(OK+I).
Ell
:! 'EK be Hie eigenvalues of a Her-
< El
:5
E12
:-5
C:2
:5
...
:5 '61K :-5
61K,(+i*
(3.20)
OK be the orthonormal eigenstates correspondProof. Let 011 02 16K- Any function !P in VK+j can be ing to -the eigenvalues 61,162 expressed as 1
...
i
K+1
Tf
CA,
(3.21)
28
1 Introdnetion to variational methods
where the normalized function inake, it.
orthogonal
01-<-+,
is constructed from
to any function in
Tf(aK+l)
Vfc: K
(Tf (a
OIC+I
to
2
1 (0i I Tf (Ct 1,;C+l 12
Tf (a 1,C+ 1))
K
(3.22)
x
The the
eigenvalue problem using simple form 61
0
0
62
-
0
-
0
the basis set
)
h, h2
takes
)
C,
)
C1
C2
C2
=E 0
0
-
h *1
h;2
...
EK
hK
CK
h I*c K
hrc+1
CTC+I
CK
CJf+
(3.23) where
hj
=
%JHJOrc+I) and hj* is the complex conjugate of 11j. The D(E) to determine the eigenvalues reads as
characteristic function
K
D(E)
(hK+1
-
K
E) II(Ei
-
E)
K
II(Ei
-
E)
hIc+i
i=1
(i
Ih-I2 EJ
_
E)
0.
E
(3.24)
I,-, K) axe nonzero. The (K + 1) obtained by finding the roots of the equation K
E-hK+I
E
-
j=I
Here it is assumed that all hi are
-
11 (Ej
Ihi 12
-
K
eigenvalues
K
=
Ihi 12 E
-
(3.25)
ei
The right-hand side of the above
equation becomes negative for E
< C,
and in
positive for E > CK. A graphic solution of Eq. (3.25) is displayed Fig. 3. 1. It is clear that the cross points of curves, namely the new
eigenvalues E , satisfy the relation (3.20). When there are n vanishing hi's, the corresponding ei's become the solutions of Eq. (3.23), i.e., E'i Ei. The remaining (K + I n) new ==
-
3.1 Variational
1N
29
principles
2
'=1E-Cj .................
.......I.. ............V..
.........
....
F-21
611'
.....
...
...........
..................
I 'EK
E-hK+1
Fig.
3.1.
Graphic
solution of
Eq. (3.25)
to obtain the
eigenvalues
of
Eq.
(3.23)
eigenvalues the
same
are
structure
also in this
through the eigenvalue. equation, which has Eq. (3.23). The relation (3.20) trivially holds
obtained as
case.
The significance of this theorem is
that, by including a further term in the basis, the K lowest eigenvalues; cannot become worse. Therefore, this theorem implies the Mini-Max theorem because, in the limiting case that the subspace approaches the full Hilbert space as the basis dimension increases, the K lowest-eigenvaluesEj converge to the exact eigenvalues; of the Hamiltonian. Suppose that one wants to see how the lowest eigenvalue changes when one vaxies only the (K+l)th function-!V(arc+j) while keeping the rest of all. the K functions unchanged. In this case one does not need to solve the generalized eigenvalue problem of type (3. 10) but only needs to find the smallest root of Eq. (3.25). -Of course the latter is much simpler and faster than diagonalizing the matrix. This advantage is used to sample more random trials in selecting a suitable basis function in the stochastic variational method, as explained in Sects. 4.2.5 and 4.2.6.
30
3. Introduction to variational methods
3.2 The variance of local energy It would be nice if there
method to
judge the accuracy of variationally. The expectation value E is the upper bound of the ground-state energy El. If we can calculate a lower boun,d, the. difference between tile, two bounds gives an estimate for how dose E may be to El. This makes sense if and only if the lower bound can be calculated as closely as possible to the ground-state energy. In order to discuss the lower bound, we define the vaxiance of the energy expectation value, o-', by the norm of the residue function were a
the solution obtained
(H
E)TfI(Tf JTf) '21
-
2
(T1J(H-E)2JTf) (TIITf)
-
0'
According
to Weinstein
(TfJH2JTf) R- I Tf
[6, 71
the
following
Theorem 3.6. There ii at least val
[E
theorem
exact
one
(3.26) can
eigenvalue
be
proved.
in the inter-
o-J.
o-, E +
-
E2
-
Proof. By using Eq. (3.3),
the variance
can
be
expressed
as
EMI (Ei E)2 JaiJ2 Fli= 00, JaiJ2 -
2 0'
_
Suppose that Ek
is the
(3.27)
eigenvalue which is
00
i
we
we
have
CO
J:(E Thus
closest to E. Then
2
_E
jai 12
have o-2 >
>
(Ek
(Ek -
-
E)2
E)',
jai 12.
(3.28)
which proves the theorem.
The Weinstein criterion guarantees that there is
an
eigenstate
whose energy is in the interval [E G-, E + o-] but does not indicate which one. In case E is sufficiently dose to the ground-state energy, -
the theorem E-
G-
<
gives
the lower bound
as
El.
(3.29)
Another lower bound called
Temple's
in terms of the variance of the energy
bound
[8]
is also
expressed
as
0'
E
<
-
e-E
-
El.
(3.30)
31
3.2 The variance of local energy
is the energy such that E < e < E2, where E2 conserved quantum energy of the first excited state which has the same numbers as the ground state. The best bound is clearly obtained by Here
-
setting
is
an
e
equal
arbitrary
to
E2. Temple's bound is easily derived
as
follows:
0.2
(E- --E )
E,
1 =
-
e
-
I
E
[
-
(,- + Ej)(E
-
Ej)
+
(TI I H2 I TI) (Tf I Tf)
EZ i=2 (Ej-,-)(Ej-Ej)jajj2 E Ei= 1 jai12
>
00
where
use
Eq. (3.4)
is made of
for E
(TfIH2lTf)l (Tflyf-) _EJ2 (with n 2). The two bounds, (3.29) and (3.30),
-
(3.31)
0,
(with
El
2]
Elf
n
=
1)
and for
=
indicate that the trial function
2
gives the smallest value of (7 is best among various trial functions giving the same expectation value. Temple's bound often gives better that
lower bounds than the Weinstein bound. See
[91
for the extension of
variational bounds. The variance of the
expectation
value
can
also be defined
through
the local energy HTf
Ej., (R) ,
!P7
(3-32)
I
where R stands for the
"configuration?'
of the
particles.
More precisely,
by ri and pi, where pi particle stands for the coordinates other than the position coordinate ri, then R stands for Ir I i P1 ilr2) P2 i 1. The variance defined by Eq. (3.26) is equal to the variance of the local energy if the coordinates of the ith
are
denoted
...
(TfjjEjo,(R)
2 0'
-
E121!Tf)
=
P(R)IEI.,:(R)
-
Ej2dR (3-33)
with
P(R) where as an
E
--
I Tf (R) 12 R I Tf)
(3-34)
energy of H in the state local energy: of the average
E, the
=
mean
(TIjEjo,(R)jT1) 010f)
-
Tf,
I P(R)Ejoc (R)dk
can
also be rewritten
(3.35)
3. Introduction to variational methods
32
Here
P(R),
uration
the
probability density of finding the system at the configpoint R, is non-negative and satisfies the. relation f P(R)dR
1.
H the trial function Tf is the exact
ground state of the Hamiltonian,
the local energy becomes R-independent and equals the ground-state energy El. A similax statement holds for excited states as well. If Tf is
the exact to
eigenstate
E,,. This
with
opens up
energy-&, the. local energy would be equal another possibility of the vaxiational method
instead of the minimization of the
mean
energy: The niinimization of
the variance of the local energy [101. The adjustable parameters of the trial function may be varied to minimize the variance. The calculation of the variance is in general much more difficult than that of the mean energy. An advantage of minimizing the variance of the local energy is, however, that the quantity to be minimized has a known lower bound,
Since in any eigenstate the vaxiance of the local energy is zero, this minimization principle applies to obtain excited states as well as the ground state.
namely
zero.
The
condition for o-' is obtained
stationarity
increment So-' in the
Theorem 3.2.
same.
way
By multiplying
calculating the increment following equation
to
(Tf Jqf-)jO.2 =,6((Tf JH2 Ifl) =
(,5qf- I H2
_
=
(JTf JH2
-
(Tf JH2
+
Here
-
E2
_
2EH + E2
mean
energy in
-
2 _
Eq. (3.26) by (Tflfl and change in JTf, we obtain the
linear
a
_
done for the
as was
an
both sides of
J(E2 (Tf IT,))
0.2 1 Tf) +
2EH + E
by calculating
(Tf I H2
-
_
E2
0.2j((T1JTf))
_
0.2 1 jqf-)
-
2E(Tf JTf) JE
0.2 1 TI)
0.21,5fl.
(3.36)
is made of
Eq. (3.5) in the last step. The stationarity condiequivalent requiring (H2 2EH + E2)(y 0.2TI. When the trial function is given as a sum of a finite number of functions Tf (a) as in Eq. (3.7), this condition reads as the following equation which use
tion is
can
to
=
-
be used to determine the linear parameters ci
(Q
-
2EW +
E2L3)C
=
0.2BC7
Qij
=
(!rf(a,) 1112 1 Tf (Cj)).
(3.37)
Since E defined
by Eq. (3.8) depends on c, the matrix in the round equation also depends on c. Therefore the parameter c must be determined self-consistently: One assumes an initial value for c, calculates the energy-R, and solves Eq. (3.37) to deterbracket of the above
mine
c
and 0-2. As the value
c
obtained in this way may in
general
3.3 The virkd theorem
not be
equal
to the initial
c
value,
one now uses
the solution
33
c as
the
initial value and repeats this cycle until both values of c become the same. The minimization of o-' becomes cumbersome even though the calculation of
Qij
is feasible.
The variational Monte Carlo
(3.35)
to determine the
mean
method of the
(VMC)
calculations
employ Eq.
energy. Tlie VMC method [111, or the of Two-body correlations into Multiple
Amalgamation Scattering (ATMS) [121 similar to it, usually chooses a trial function that attempts to incorporate both the sfi7ort distance behavior and the asymptotic behavior of the correct wave function, leading to complicated functional forms. Thus the analytic evaluation of the matrix elements is in general hopeless. Instead, the VMC requires only the evaluation of the wave function and its Laplacian to determine the local energy. If the interaction between the particles contains the spinorbit force, the first derivatives of tl-ie wave function are required as well. These derivatives are obtained numerically by appropriate difference formulas. The mean energy is then estimated by a Monte Carlo
integration. The sampling of a set of configuration points R is made by, e.g. the Metropolis algoritl-nn [131. The accuracy of the VMC or ATMS calculation is limited by the choice of the trial function as well as by the statistical error inherent in the Monte Carlo integration. To reduce the statistical error one has to sample a great number of points, which is computer time consuming. To optimize the variational parameters of the trial
wave
repeat the mean enof parameters. This is a hard job
function
one
ergy calculation using a different set considering that the VMC or ATMS has
particularly
in the
case one
has
The variational method is
which
can
be
adapted
has to
an
inherent statistical
error
number of parameters. very flexible approximation method
a
a
to diverse -problems. The variational method
physical intuitions lead us to an idea of particularly the qualitative form of the solution. It easily gives good values for the energy. The variational solution may, however, present unpredictable erroneous values for other physical observables. It is unfortunately valuable when
is
very difficult to evaluate their
error.
3.3 The virial theoren, summarize two theorems, the Hellmann-Feyntnan theorem and the virial theorem, which are valid for the exact solution.
In this section
They
can
we
be used to check the
quality
of the
wave
function.
3. Introduction to variational methods
34
Theorem 3.7
(Hellmann-Feynman Theorem).
Let
H(A)
be
a
Hermitian operator which depends on a real parameter A, and POO a normalized eigenstate of H(A) of eigenvalue E(A). Then the theorem states that
d
TA E(A)
=
(3-38)
( P(A)J DA
Proof. Differentiating,
with respect to
A,
the relation
(3.39)
E(A) we
have
a
d
dA
E(A)
=
( P(A)J
(9A
H(APKA)) 19
(9 +
OX The Iast two terms
(fi(A) IHN ON) on
the
+
R(A) W(A) 1
(3.40)
A
right-hand side. of Eq. (3.40)
vanish because
a
(9
PNIH(A)O(A)) E(A)
+
OPNIHNI
J-P(A))
IN
+
CA))
(!P(A) I OA jA-
P(A))]
d =
where
use
E(A) d,X ONON)
=
(3.41)
01
is made of the conditions that
H(A)!P(A)
When the evaluation of the matrix element of the
theorem
E(A)!P(A)
H(A)
and
does not
be used to
Oficulty, Hellmann-Feynman consistency of the vaxiational solution. If H(X) Ho +,XHI, then the matrix elements needed are just the terms of the energy expectation value. It is straightforward to generalize the 11611mannFeynman theorem to the case that the Hamiltonian contains a number of parameters AI A2 We note that the 11ellmann-Feymnan theorem can be stated in an integral form pose any check the
=
7
E(A2)
can
-
E(Al)
=
(!P(A2)JJff(A2) (!P(A2)J!P(A1))
3.3 The viri.-,U theorem
(3-42)
OA
"1
35
equality is trivial for the exact solution and has nothing to do with the I-Iellmann-Feym-nan theorem. The equality holds, however, only approximately for the variational solution and may be used to check the quality of the solution. The following theorem plays a key role in the derivation of the The first
virial theorem. Theorem 3.8. Let (P be
For any operator A
(T)I [H, A] I!P)
=
we
eigenstate of a Hermitian operator
an
obtain
(3.43)
0.
proof is easily obtained by noting
The
H!P
E. To test the accuracy of the vaxiational special operator of the form N
N
.
A=
=
ET, with
solution,
one
eigenvalue
often
uses a
(9 ri
ri-pi
h
H.
(3.44)
-
r. 'ri
i=1
because the commutator
[H, A]
is
easily calculated. We
operator A is a generator of the dilation operator, property, in a single-variable case,
e
aA,
d
exp
(ax dx ) f (x)
=
note that the
which has the
(3.45)
f (eax).
by expanding f (x) in power series and using for any integer value of n. valid nkXn, dX Assume that the Hamiltonian for an N-particle system takes the
This is
easily
(X_A_)kxn
shown
=
form N
H
Whether
or
Pi
E=I2mi
T+ W
=
-
T,,
.
+
W(ri, r2,
can
potential
W is assumed to contain
no
C7 ri-
Ori
P2(-2mi
-Pi'IrN),
'n Ar M12 --N ..
on
(3.46)
the virial theorem.
momentum
then show that
IT,
rN),
not the center-of-mass kinetic energy is subtracted from
the Hamiltonian is irrelevant to the discussion
The
.--,
operators. We
3. Introduction to variational methods
36
1W,
0
ri.
where M12
...
OW
1 c9ri N=
'=
(3.47)
-ri-
ari
EN
mi
is the total
EN
is the total momentum. ..,i= I Pi we obtain
of the system and 7rN
mass
Using Eqs. (3-44), (3.46)
(3.47)
and
N
aw
[H, Al
=
2T
-
with
WA
WA
(3.48) ri
i=1
Substituting
this into
Theorem 3.9 the Hamiltonian
2(!PjTjfl Therefore
77
a
-
(The
(3.46).
virial
Theorem).
following
Let!P be
theorem.
an
eigenstate of
of Eq. (3-44)we
obtain
0.
=
z7 defined
quantity
leads to the
For the operator A
((fijWAj(fi)
(!PIWAI'P) 2(!PjTj!P)
=
Eq. (3.43)
(3.49) by
'
1
(3-50)
vanishes for the exact solution and
be used to check the
can
quality
of the solution.
One has to know WA to make use of the virial theorem. It becomes particularly simple if the potential is a homogeneous function of degree s,
namely
W(Ar I Ar2
ArN)
i
=
i
=
because then
ating, to X
=
A*W(rjr2,
...
I
7 ....
ATNi AM AZN)
rN)7
(3.51)
obtain
with respect to
by the use of Euler's theorem or by differentiX, both sides of Eq. (3-51) and setting A equal
I
WA In this
we
W(AX1 AY1 AZI
=
SW-
special
(3.52) case
Eq. (3.49), together
yields the well-known
with
(!PjTj!P) + ((fijWjfl
=
Ej
relation 2
((!PITI! P) s
Because the
tial
(s
=
+2
E,
PIWI(P)
E
=
s+2
(s :A -2).
(3.53)
homogeneity condition is fulfilled for -the Coulomb potenthe quantity t7 is conveniently used to check the quality
-1),
3.3 The virial theorem
of the solution for
37
system of particles interacting via Coulomb
a
po-
tentials. For
a
general non-homogeneous potential IV
N
W(rwr2,
Ui(ri)
'rIV)
...
must be
E
+
i=t
Eq. (3.52)
Vij(ri
(3.54)
rj),
replaced by
Oui(r)
WA
-
i>i=l
N
IV
function of the form
r-
Or
)
OV,ij (r)
(r
+ r=ri
Or
)r=ri-rj (3-55)
spherically symmetric potential the operator r-ar- reduces simply r-4-. If the calculation of matrix elements of the. operator WA is not dr
For to
a
difficult,
the virial theorem
When the
potential
W
can
be used
depends
as
on a
parameter A,
(9
(!PIWAI(P) Here the
=
((filWA
+A
in the Coulombic we
case.
obtain
d
WI-P)
-
A
dA
(3.56)
E(A).
theorem is used to express the expecW in terms of the energy eigenvalue E(A). In a
Hellmann-Feynman
tation value of
A'&
molecular system X may be a set of parameters which stand for the positions of nuclei and if W consists of only the Coulomb potentials
-W simply reduces WA + A-bX plication
of this relation.
to -W. See
Complement
8.2 for
an
ap-
4. Stochastic variational method
approach to the variational solution of quantumproblems is to diagonalize the Hamiltonian in a state space spanned by some appropriate functions TV (a-,) Tf (a2) Tf (CeK). The applicability of this approach is, however, very limited, because the diagonalization may not be feasible if the dimension K of the state space is very large. This is typical in many-particle problems
The most direct
mechanical bound-state
7 ...I
I
such
as
the Hubbard model
or
the nuclear shell model. This method
approach", because one sets up a basis in a well-defined way, e.g., by using a complete set of states that contains no parameters, and then obtains the energy by a diagonalization. Another possibility is basis optimization, which is actually designed to avoid the problem of the huge basis dimension in the direct approach. In this case one specifically selects the basis states that are really essential to get the energy and the wave function of the syscan
be called
a
"direct
certain accuracy. It is obvious that this selection may be state-dependent: Some functions might be adequate to describe the
tem to
a
others would be
appropriate to approximate an excited state, particularly when the ground state and the excited state have different spatial extensions. The stochastic variational method uses this second route by selecting the most appropriate basis functions in a trial and error procedure. but
ground state,
4.1 Basis
some
more
optimization
It goes without saying that the quality of the variational approximation crucially depends on the choice of the basis functions. Our
primary aim here is-
choose basis functions 1.
They
can
be
IV-particle problems. For this that meet the following requirements:
to solve
easily generalized
for
Y. Suzuki and K. Varga: LNPm 54, pp. 39 - 63, 1998 © Springer-Verlag Berlin Heidelberg 1998
an
N-body system.
we
will
4. Stochastic variational method
40
2. Their matrix elements
3.
They
are
analytically calculable. easily adaptable to the permutational symmetry are
of the
system. 4.
They
are
flexible
enough
to
approximate
even
rapidly changing
functions.
important condition (3) is non-trivial because the permutational symmetry looks very complicated when expressed in terms of the relThe
ative coordinates to be used.
A
possible
choice for the basis functions
tions is the correlated Gaussian of
fillfilling the
above condi-
Eqs. (2.23)-(2.25):
-1 N-1
1
c
exp
Ax)
NE E Aij
exp
i=1
xi
-
xj
j=1
IV
expf -2
E
aij (ri
-
(4.1)
rj
i>i=l
equivalently, aij are nonlinear paxameters are spherically symmetric. To take into account non-spherical states, the basis function has to be multiplied by an appropriate orbital angulax function. In addition to the spatial degree of freedom, the particles may have other degrees of freedom, such as spin and flavor, and thus the function has also to be multiplied by suitable trial functions in these additional spaces. These functions may bring other parameters or sets of quantum numbers. These sets of quantum numbers (e.g., total and intermediate spins, orbital angular The matrix elements
Aij
or,
of the basis. These functions
momenta,
etc.)
be considered
can
often be referred to
as
trial function will be
as
channels in the
given
in
discrete parameters, and will following. (More details on the
Chap. 6.)
The actual form of the basis function is not very important at this stage. The above discussion serves to draw the reader's attention to the fact that the basis function
depends
on
many
and discrete parameters. The parameters define the
linear, nonlinear shape of the basis
function and determine how well the variational function space contains the true eigenfunction. To find the best possible solution, one has to
optimize the paxameters. To have a crude guess of how much optimization amounts to, let us consider an N-particle syswith the simplest basis function (4.1). This correlated Gaussian
work the tem
has ea,r
N(N
-
1)/2 parameters, and, by assuming
combination of K
functions,
-we
face
an
that
we
need
a
lin-
optimization problem of
4.1 Basis
K(N(N 1) /2) -
the number of
41
paxameters. Tb is number increases quadratically with
particles. By taking
N
=
4 and K
end up with 1200 parameters. The main problem of the minimization of
case,
optimization
=
200
as a
typical
we
nipresence of local minima. A local function reaches
a
minimum in
a
There
function is the
om-
point where the
finite interval of variables and the
number of such minima tends to increase
of the
a
minimum is the
exponentially
with the size
are plenty problem. optia function, and these optimizations can be divided into two categories: the deterministic and the stochastic optimizations. A deterministic optimization moves downwards on the slope of the function according to a certain well-defined strategy. There exist many elaborated algorithms (conjugate gradient, Powell (direction set), etc. [141) and they are deterministic in the sense that, starting from a given point, they always reach the same (local or global) minimum. The drawback of these techniques is that they are time consuming and tend to converge to whichever local minimum they first encounter. The solution in these cases may not be the global minimum but a local minimum. These methods are sensitive to the starting point and are
of different methods for the
mization of
unable to search further for
a
better solution after
a
local minimum is
reached.
Stochastic optimizations address the problem of finding the global large number of undesired local minima
minimum in the presence of a by making random steps [15,
161.
One can, for
example, start deterstarting points and then pick up the minimum of these. Numerous strategies have been developed in the last few years, e.g., simulated annealing [171, genetic algorithm [18] etc. The simplest (and actually not very economical) stochastic optimization is a random search where one picks up random points and tries to find the minimum. This may not sound very sophisticated, but the optimization with a random trial and error procedure seems to be the most efficient one. In some cases it would be nearly impossible to apply other strategies than simple random trials. To reduce the load of optimization, an alternative is to shift the burden of minimization from a large number of parameters to a smaller number of more sensitive, "tempering", parameters. Several such possibilities have been explored, e.g., geometric progression [19, 20], random tempering [21, 22, 231, and Chebyshev grid [241. The common property of these methods is that they use a "grW in the parameter space. The grid is given by some ad hoe rule and each point of ministic searches from several random
4. Stochastic variational method
42
grids defines a basis function. The grid can be defined by some simple functions which may depend on some additional- parameters to be optimized. The number of parameters contained in the functions which define the grid is chosen to be much smaller than that of the original function to be optimized. For example, in the case of a three-particle system, each of the basis functions has three parameters: All, A12 A21 and A22 or a12, a13 and C923- Some COnVenient the
=
choices for basis parameters
are
shown in Table 4.1.
Table 4. 1. Choice of parameters. In the random tempering p is an index to a prime number that is used to generate pseudo-random numbers.
denote
See
Eq. (4.8) for the
Geometric
All
=
notation <
i, i
>.
progression -2
(aiqk-i) (a2q2-1) I
A22
=
A12
=
(k
mi)
(k
M2)
(k
I,- mi.)
(k
1:
(k
1
(k
17
mi)
(k
17
M2)
-2
k
0
Random tempering a12
=
OL13
=
a23
=
exp(d, < k,p > +d2 < kp + I > exp(d3 < k,p + 2 > +d4 < k,p + 3 > exp(d5 < k,p +4 > +d6 < k,p + 5 >
,
...
...
M2)
M3)
Chebyshev gTid All
=
A22
=
A12
=
( a2tan ( altan
0
7r
2k-1
2
2 mi
7r
2k-I
2
2M2
4.2 A
distribute
certain number of basis functions to
a
part of the
wave
function and then
the basis elements in
4.2 A
regions
one
43
practical example
might
approximate the bulk
increase the
density of
where finer resolution is needed.
practical example
point of the previous section, let us consider an example for basis optimization. The example will at the same time provide insight into some other aspects of the methods as well. We consider the system of a positron and two electrons, Ps-, used in Chap. 1. The energy of this system has been very accurately calculated by various To illustrate the
approaches and
it has been found to be -0.262005 in atomic units
given in a.u. throughout this chapter unless otherwise mentioned.) The quality of the different optimization strategies used in this section can be judged by comparing their results with this value. Let the positron be labelled I and the electrons labelled 2 and 3. With the electron spins coupled to S 0, the orbital of the trial function that the antisymmetry requires part ought to be symmetric with respect to the interchange of the electron coordinates. The wave function is thus expanded as
(a.u.). (Energy
and
length
are
=
K
1
Tf
=ECk(1+P23) exp (-2-;
,Akx),
(4.2)
k=1
exchange operator P23 is introduced to assure the symr2 and X2 rl metry requirement. The coordinates are x, matrix 2 with three 2 and x Ak is a symmetric (rI + r2)/2 r3, nonlinear parameters All, A12 and A22. The wave function Tf has 3K parameters to be optimized. The linear parameter Ck is determined by solving the generalized eigenvalue problem (3.10). To reach the accuracy required in atomic physics, the minimum basis size of a three-body problem is at least K 100, as will be shown later. That means that even in this simple example we would have 300 parameters. This is already almost beyond the capability of most of the computer codes for optimization and this "full" optimization is out of the question for larger systems. The fact that the optimization of a large number of parameters is not feasible is just a small part of the problem. To have an efficient optimization one needs fast function evaluation as many times as required. There are two steps which are where the
=
-
=
-
-
4. Stochastic variational method
44
necessary to calculate the energy expectation value: The calculation of overlap and Hamiltonian matrix elements and the diagonalization.
the In
a
fall
required
optimization,
one
time increases
(a K3-process)
as
has to recalculate all matrix elements K2) and to
at each calculation of the energy. This is
load
for
(the
the Hamiltonian
rediagonalize
a
considerably
small system. heavy computational Instead, one can try a partial optimization. One can, for example, fix the parameters of all basis states but one. In this partial optimizaeven
a
only one row (column) of the Hamiltonian and the overlap -nn atrix (the required time is of order K). Let -us assume that the Hamiltonian has been diagonalized over the fixed basis states. Then, in the successive step, only one row (column) is changed. As has been shown in Theorem 3.5, after the N x N diagonalization there is no need for an extra diagonalization to solve the eigenvalue problem on the (N + I)-dimensional basis. Consequently, the computational time required by the partial optimization is only a small ftaction of that of the full optimization, and, moreover, only a small number of paxameters (three in the present example) has to be optimized. tion
has to be recalculated
4.2.1 Geometric
First let can
us
try
to
progression
use a
grid defined by
a
define three different sets of Jacobi
which two
particles
are
connected first:
Sects. 2.4 and 7.6 and in these systems to be
given
Fig. by x('), X(2)
and
us
X(3)
(23)1, (31)2,
and
(12)3.
See
denote the Jacobi coordinates The trial function is
IT
assumed
in the form K
3
(P)
qf
Ckl P=1 k=1
with the
2.2. Let
geometric progression. One coordinates, depending on
exp(--l (-P)A(P)x(P))0(11)00(x(P)), 2
(4.3)
1
angular function
0(11)00 (X)
=
[Y1 (XI) X Y1 (X2)100 I
-
(-1)1-M
(4.4)
-: 2= ,+1Y1Ta(X1)Y1-m(X2)-
Here p denotes the arrangement channel and Y1. (,r) is
a
solid spherical
harmonic
YIM M
=
7"YM (P),
(4.5)
4.2 A
where Yl,,, (i
)
polynomial
a
is
a
spherical harmonic.
practical example
The solid
45
spherical harmonic is (see Complement
of degree 1 in the Cartesian coordinate
A() is always chosen to be diagonal. Therefore, the k spherical part of Eq. (4.3) is considered a special case of Eq. (4.1). (See also Complement 8.4.) The trial function of type (4.3) is used in the so-called Coupled Rearrangement Channel Variational Method [20, 25, 26]. The basic idea of this approach is that, by taking into account 6.2).
The matrix
the different Jacobi sets, one can introduce various correlations. This method has been used with great success especially for Coulombic
three-body problems [201. Because of the symmetry of the trial
function,
(2) C
(3) =
k1
The 2
x
cki
we
A (2)
may A
-
requirement imposed
on
the orbital part
that
assume
(3)
(4-6)
"k
k
diagonal matrix A(P) 1, (k K) has two nonlinear k and they are taken as a geometric progression
2
=
...'
rameters
(Ak(')),,
(P)
ai(p) (qi ) In
principle
one can use
k-1
(i
=
1, 2).
pa-
(4.7)
different geometric progression parameters,
ai(P) and qi(p), corresponding to the different arrangement channels and the Jacobi
that
is, they would depend on p and i in order even depend on the angular momentum 1. For simplicity, in this example we use the same parameters, a and q, for all. arrangement channels and all sets of Jacobi coordinates. The main reason for using the geometric progression as parametrization is clear: The number of parameters of the basis is reduced to just 2 (a and q), so the optimization is simple. Another reason for the choice of this specific parametrization is that the overlap integral of the basis functions can be easily controlled by a choice of q and thus the danger of the linear dependence of the basis functions can be avoided. (See Sect. 3 for the danger arising from the linear to
coordinates,
get better convergence. They might
dependence of the basis functions.) The disadvantage of the use of these diagonal matrices is that, without having some polynomials like the scalar product of Eq. (4.4), the energy does not converge to its accurate value. As is shown in Table 4.2, one reproduces only the first two figures (E -0.2618530) =
of the
ground-state energy without the polynomial part, i.e., by taking 0 in the expansion. only I
=
4. Stochastic variational method
46
Table 4.2. The energy of Ps- in different energy is -0.262005. E
arrangement channels. "Exact"
(a.u.)
Channel
Partial
(23)1 (23)1
1
1 =0,2
-0-2392726
(31)2+(12)3 (31)2+(12)3 (31)2+(12)3
1
=
0
-0.2609626
1
=
-0-2619622
1
=
0,1 0, 1,2
(23)1+(31)2+(12)3 (23)1+(31)2+(12)3 (23)1+(31)2+(12)3
At this moment
=
1
=
1
=
1
=
a
wave
-0.2068096
0
-0-2619717
0
-0-2618530
0, 1 0, 1, 2
-0.2619804 -0.2619816
comment is in order. As the
spherical harmonies
angular functions, complete may think that the partial-wave expansion using only one of the Jacobi coordinate sets form
set for
a
one
would be sufficient to represent the wave function. The convergence as a function of I is, however, very slow if only one set of the Jacobi slow convergence in the partial-wave expansion will also be studied in detail in Complement 11.2.) E.g., the energy calculated with the (23)1 channel using the partial waves of
(The
coordinates is used.
up to I
2 is
=
just -0.2392726, which
of the three-channel calculation
(E
higher than the energy -0.2618530) using only I 0.
is much
=
=
Since the lowest threshold of Ps- is the Ps+e- channel at the
en-
ergy of -0.25, this result indicates that this single-channel calculation cannot bind the system. Moreover, the expansion with I has for prac-
tical purposes to be truncated to low values because the calculation of the matrix elements of high I waves generally becomes more time
consuming. Table 4.2 shows that, by combining the expansion Jacobi coordinate sets, the energy converges faster and few terms of the partial-wave expansion are needed. By
in the
the first
only using
10
grid
points for A(') k be
a
600
=
(=
we have found the optimal values of the parameters to 2.6. The variational energy obtained with this 0.06 and q =
2
x
10
x
10
x
with
computational load 180300
alized
(=
x
601/2)
one
repeated
a
calculation is -0.2619816. The
particular basis
set is the calculation of
matrix elements and the solution of
eigenvalue problem
has to be can
600
3)-dimensional
a
gener-
of dimension 600. For the optimization this
few dozen times.
By increasing the basis
get closer and closer to the exact energy.
size
one
4.2 A
4.2.2 Random
practical example
47
tempering
Another
popular tempering method is random tempering [21, 22, 231approach involves the generation of (pseudo-) random numbers with the basis parameters defined by the following prescription: This
I
ak
1: dj < k, j >
exp
(k
=
1,
...,
K),
(4.8)
j=1
where <
k, j
> is a
pseudo-random numer, the fractional part of (k(k + 1)/2)V/P---(j) P(j) being a prime number in the sequence 2, 3, 5, 7,..., that is, P(I) 2, P(2) 3,... By using this tempering number I of one a formula, optimizes parameters, dj, instead of the original 3K parameters. A possible application of the formula for a three-particle case is given in Table 4.1. The origiTi of this formula is the following. One may assume that the ground-state wave function is an integral transform of some known with
=
function with
some
weight
=
function. The known function in
our case
is the correlated Gaussian and its nonlinear parameters are the integration variables. The simplest way to carry out the integral trans-
formation is to
the Monte Carlo method. The
quadrature points integration can be generated by the above formula. As this formula provides "good lattice points" for the integration, these nonlinear parameters can be thought to be adequate to represent the wave function in a variational approach. While the random tempering works in many examples in a superb way, one has to note that it has a serious problem: It often leads to (almost) lineaxly dependent bases. In our actual example, the best energy of Ps- with random tempering is -0.261872, and further hnprovement of this value was difficult due to the linear dependence. The parameters of the random tempering used are, in the notation of Table 4.1, ?nj 7, Tn3 5, di 4, d2 -8, d3 Tn2 4, d4
required
use
in this
=
-7, d5
=
-1, d6
The parameters
are
=
=
1, and the basis dimension nearly optimized.
-11,
di
=
p
=
=
is K
=
245.
4.2.3 Random basis
The above calculations have
suggested
that not all of the
grid points equally important grids give nearly the same energy. The explanation is simple: The basis functions are nonorthogonal to each other, none of them is indispensable; they are dense, that are
but the different
can
4. Stochastic variational method
48
will compensate any of thein can be omitted because some others for the loss. This property of the basis functions and the success of
is,
tempering suggest the idea of a completely random distribution of the parameters. To illustrate this possibility, let us generate K 100 sets of basis parameters in the expansion (4.2). The eleraents of Ak are randomly chosen from a "physical" interval: the random
=
1
1 <
0 <
10,
0 <
10,
< 20.
0 <
-\/FY23
fa13
V/a12 The
<
(4.9)
advantage of using the parametrization by aij instead of Ak 1/.\,ra-jj corresponds to the "distance' between the particles.
is that
e-l"2, the expectation value relevant to the distance is, e.g., (r) V 4/(7a), or A,/-(r-27) VF3/(2a).) The paxameters inside this interval describe the (e+e-e-) system,
(Note that,
for
a
Gaussian function
2
=
=
where the average distance between the positron and the electrons is expected to be less than 10 a.u. and the distance between the electrons is
expected to
be less than 20
a.u.
This limitation
serves
practical pur-
in pose, and it is based on the physical intuition that the paxticles bound system cannot move very fax away from each other.
The difference between the random
a
tempering and the random
distribution of the parameters is that in the former there axe several parameters which generate the parameters of the basis functions and these
optimized, whereas in functions are randomly chosen.
generating parameters
paxameters of the basis
are
the latter the
energies of different random bases are shown in Fig. 4.1. By increasing the size of the basis, the energy goes very close to the exact The
value. See Table 4.3. The energy is better than that obtained with the basis set of geometric progression type, but the comparison is diffi-
geometric progression, the matrix of nonlinear paxameters is diagonal and the partial-wave expansion is introduced to enable the trial function to span the full model space. This paxtialwave expansion increases the basis size significantly. When we use the full A matrix and randomly distribute aij, this partial-wave expansion is not necessary (see Complement 11.2). One may try to choose aij in a geometric progression, but our experience is that both the geometric progression and the random tempering of aij tend to lead to linear dependence. To avoid the linear dependence in the basis (which does not occur cult. In the
case
of the
frequently with the fully random basis), the overlap of the random basis fimctions should be smaller than a prescribed limit. Any random points which do not satisfy this condition are to be omitted and
4.2 A
practical example
49
-0.250
I -0.252-
-0.254-
Ci -0.256-
-0.258
-0.260
-0.269 20
40
60
80
100
Dimension of the basis
Fig.
4.1.
Energies of Ps- for different basis sets. The parameters of the are completely randomly chosen and no preselection is made.
basis states
Table 4.3. The energy of Ps- (in a.u.) for K basis states that are selected randomly. "Exact" energy is -0.262005. The energies Ei are obtained by
starting from different random points. The energy of "Best 100" is the one calculated by sorting the best 100 basis states from 400 basis states, where 50 random trials are probed at each step of the basis selection. Neal is the number of matrix elements evaluated during the optimization. K
=
100
K
=
200
K
=
400
Best 100
El
-0.2617619
-0.2619798
-0.2620032
-0.2619995
E2
-0.2617281
-0.2619793
-0.2620026
-0.2619978
E3
-0.2617918
-0.2619669
-0.2620016
-0.2619956
E4
-0.2618826
-0.2619675
-0.2620008
-0.2619953
E5
-0.2616285
-0.2619824
-0.2620024
-0.2619987
Nevai
5050
20100
80200
80200
4. Stochastic variational method
50
replaced by a new trial. Note that this cure cannot be applied to the procedures of the random tempering and of the geometric progression. In these procedures there is no such prescription. One can just omit some grid points depending on the actual parameters. The energies listed in Table 4.3 are surprisingly good, but the size of the basis seems larger than absolutely necessary. For the solution 400 is excessive. of a three-body problem a basis dimension of K We would like to decrease the basis size by selecting a minimal, indispensable set of basis functions. This will be described in the following =
subsections.
4.2.4
Sorting
One may wonder: Are all the basis states equally important? What happens if we omit a few states from these bases? To answer this ques-
tion,
we
reordered the basis states
by the following random selection
process: 1. A number
n
of basis states
giving the lowest energy restricted basis.
pool,
basis states, were tested in
state.
Again
2. Then
n
were
was
picked
up
randomly and
the
one
selected to be the first element of
a
again randomly chosen from the remaining a
2-dimensional calculation with the first
the state that
produced
the lowest energy
was se-
lected. 3. This
reordering
4.2.5 THal and
is continued until the last basis state.
error
search
randomly chosen nonlinear parameters give good results provided the basis dimension is big enough. For an N-particle problem, howthis ever, the basis dimension might become prohibitively large to use The
4.2 A
practical example
51
simple procedure. One can obviously improve the convergence by selecting the most important states and by not admitting all random trials in the basis set. This increase of basis dimension by searching the best among many random trials is a key of the stochastic variational method (SVM). The original procedure of the SVM, proposed in [271, has recently been developed further [281 and successfully applied to multicluster descriptions of light exotic nuclei [29, 30]. Learning from these applications to nuclear few-body problems, we have generalized and refined the method further to encompass diverse quantumfew-body systems emerging in nuclear and atomic physics
mechanical
[31, 32, 33]. The sorting method has shown that from
large number of random basis states one can select a smaller set without a significant loss of accuracy. Moreover, the accuracy can be improved by increasing the basis size. This experience motivates one to apply the following trial and error procedure, combined with an admittance test, in order to set up the most important basis set: Let Ak be the parameter set defining the kth basis function, and let us assume that the sets A,,- Ak-1 have already been selected, and the (k l)-dimensional eigenvalue problem has already been solved. The next step is the following: a
I
-
Competitive selection A number
s 1.
of different sets of
n
(Ak',
...'
Ank)
are
generated
ran-
domly. By solving the n eigenvalue problems of k-dimension, the corresponding energies (Ek,..., Ek) are determined. The parameter set Ak' that produces the lowest energy from
s2.
s3.
among the set
(El,...,Ek) k
is selected to be the kth parameter
set.
s4. Increase k to k + 1.
The essential different
motivating this strategy is the need to sample parameter sets as fast as possible. The advantage of this proreason
cedure is that it is not necessary to recompute the whole Hamiltonian matrix nor is it necessary to perform a new diagonalization at each time when
a new
parameter Ak is generated. See Theorem 3.5.
The convergence of the energy with the basis selected in this way is shown in Fig. 4.2. The convergence is much faster than in the previous case
in
number
Fig. 4.1, and n
it
of trials. The
dimensional basis sets
can
be made
even
faster
energies found by using K are
by increasing
=
shown in Tables 4.4 and
10- and K
=
the 100-
4.5, respectively.
4. Stochastic variational method
52
-0.250
-0.252
-
-0.254
Ca -0.256
-0.258
-0.260
-0.262 100
80
60
40
2-0
Dimension of the basis
Fig.
4.2.
trial and
Energies of Psprocedure
for different basis sets that
are
selected
by
the
error
Table 4.4.
Energy
of Ps-
(in a.u.) by
different
optimization strategies.
10. The energies "Exact" energy is -0.262005. The basis size is set to K Ei are obtained by staxting from five different random points. The number n denotes the random candidates probed in the trial and error procedure =
of SVM. Ne,,,,l is the number of matrix elements evaluated during the optimization. Tn the case of the full optimization by the Powell algorithm 3900
diagona,lizations were also required. The time in units of seconds is on a Digital Alpha 2100 (250MHz) workstation. The refining cycle is repeated 10 times.
Method
Powell
SVM(n=l)
SVM(n=10)
Refining(n=10)
El
-0.261251
-0.244364
-0.251083
-0.261180
E2
-0.261398
-0.234532
-0.250729
-0.261165
E3
-0.261283
-0.236207
-0.252659
-0.261040
E4
-0.261259
-0.243323
-0.252956
-0.261175
E5
-0.261193
-0.226697
-0.251260
-0.261039
N,-:.v.i
214500
55
550
88055
Time
60
0.8
1.0
11
4.2 A
Table 4.5.
Energy
of Ps-
The basis size is set to K
(in a.u.) by
=
100. In the
practical example
53
different optimization strategies. of the fiffi optimization by the
case
Powell
algorithm 7200 diagonalizations; were required. repeated only once. See also the caption of Table 4.4.
The
refining cycle
Method
Powell
SVM(n
El
-0.26200016
-0.26176191
-0.26199427
-0.26200231
E2
-0.26199947
-0.26172812
-0.26199382
-0.26200351
E3 E4
-0.26200164
-0.26179182
-0.26199733
-0.26200312
-0.26200135
-0.26188261
-0.26199778
-0.26200301
E5
-0.26200193
-0.26162811
-0.26199836
-0.26200271
Neval
35466150
11615
86150
805050
Time
7200
27
43
195
These tables also contain deterministic
procedure.
=
1)
SVM(n
10)
=
=
10)
compaxison with the performance of
a
One
can
conclude that
energy convergence with this simple selection the energy converges to the exact value.
4.2.6
Refining (n
is
a
reach
easily procedure. Moreover, one can
Refining
It is obvious that the basis size cannot be increased forever. Moreover, when the Kth basis state is
the previous states are kept fixed. This means that we try to find the optimal state with respect to previously selected basis states, but actually some of the basis states selected earlier
might
states took
not be
so
important
their role. So
succeeding procedure where the previous the following steps: over
selected,
states
are
one
anymore because the
may include
probed again
as
refining by
a
described
Refinement cycle
(A!,..., Ain)
random paxameter sets are r2. The parameters of the ith basis state ri.
generated. are replaced by
the
new
energies ElK ...Ek axe calculated. r3. If the best of the new energies is better than the original one, then replace the old parameters with the new ones, otherwise keep the candidates and the
r4.
original ones. Cycle this procedure through the
Note
that, again,
orem
3.5.
there is
no
need for
basis states from i
=
I to K.
diagonalization because
of The-
54
4. Stochastic variational method
Table 4.6.
the energy of Ps- in the refining steps. The 100 and 5 random states are probed. Atomic units are
Rnprovement of
basis size is K
=
used.
Refinement
cycle
E
(a.u.)
(Starting value)
-0.261992767
Ist
-0.262002742
2nd
-0.262003677
3rd
-0.262004063
4th
-0.262004178
5th
-0.262004255
6th
-0.262004314
7th
-0.262004339
8th
-0.262004372
10th
-0.262004398
"Exact"
-0.262005
4.2.7
Comparison
of different
optimizing strategies
4.2 A
for
a
longer time,
might get
we
practical example
somewhat better
55
energies, but the
available computer time is limited in any case. Moreover, the other methods reach at least the same energy in a fraction of this time as we
have
seen.
Tables 4.4 and 4.5 show that the full
to converge to different
indicates that the
points depending
procedure
on
optimization seems starting point, which
the
leads to different local minima. This is
by explicit inspections of the parameters. In the third column of Tables 4.4 and 4.5 the energies obtained by using different sets of completely random parameters are given. The
confirmed
next column shows the results of a trial and one
error
selection and the last
10 energy after refinement cycles. We have carried out cycles (each basis state has been cyclically probed 10 times) in
gives the
refining 10-dimensional, and
the
one
refinement
cycle
in the 100-dhnensional
case.
In summary, one may conclude that the stochastic selection of basis parameters leads to energy convergence independently of different random paths and, moreover, to very accurate results in a fraction
computational load than the full optimization. And while the latter is certainly superior in principle (one may need fewer basis states to reach a given accuracy), the former suits more practical applications. Moreover, in more complicated problems of time and
by much
less
where, in addition to the nonlinear parameters, one has to find the most adequate channels (quantum numbers, etc.) to describe the system, the random selection
seems
to be the
only
viable method. Since
the energy convergence attainable is of course not precise in a mathematical sense, one should check the quality of convergence of the wave
function
as
well.
competitive selection the steps is always admitted In the
in
the
relatively
to be
a
best candidate found
basis state
even
though
its
to be very small.
contribution to the energy may sometimes happen newly selected basis has very strong overlap with (linear combinations of) the previous basis states. To avoid this This indicates that the
an
alternative
approach called
a
utility selection [281
select the basis states. The steps go
as
can
be used to
follows:
Utility selection A. Generate
s2. Determine the
value
Ak. energyEl(k) by solving the k-dimensional eigen-
randomly
problem.
a
parameter
set
4. Stochastic variational method
56
s3. Admit the parameter set
gained by including ,El
(k)
< Ej
(k
-
it is
1)
-
Ak to be the kth element if the energy larger than a preset value Je, namely
JE.
Otherwise return to the step (sl) for the next attempt. If no parameter set was found to pass the utility test out of n consecutive
attempts, reduce JE to, say half of its original value, and to
return
(sl).
s4. Increase k to k + 1.
When competitive selection is used, convergence is inferred by the energy curve's flattening out, while when utility test is used, convergence is
signalled by
an
insistent failure to -find further elements that pass
the test. Instead of the random trial strategy described in (sl)-(s2)-(s3)-(s4) and (rl)-(r2)-(r3)-(r4), one may think of more efficient and sophisticated
approaches, like simulated annealing or genetic algorithms. approaches may give faster convergence, although the random selection strategy may have a slight advantage: It is easily implementable in a parallel way because the random trials are absolutely independent of each other. One should note that the basis elements selected by the above admittance tests are in general never optimal, not even locally. A better candidate which will gain more energy could be found in the neighborhood of the basis element admitted. The incorporation of a fine tuning, that is, an additional search for even better parameters I the vicinity of the element that passed the admittance test would clearly The latter
accelerate the energy convergence. When the evaluation of the matrix elements does not require heavy computational loads, this fine tuning is recommended to reach faster convergence. Or even a determini tic selection such as Powell's method [34, 351 might be used to find out
the
locally
4.3
best basis element.
Optimization
for excited states
The Mini-Max theorem the Hamiltonian in not
only for
Sect.
3.1)
shows that
K-dhnensional basis
ground
one
by diagonalizing
gets
an
upper bound
state but also for the excited states
happen predict the energies of the excited
may to
the
a
(see
that the basis set found for the
ground
state is
states with the
as
well. It
fairly good
same
conserved
4.3
quantum numbers
Optimization
for excited states
(angular momentum, parity etc.)
57
the
ground By optimizing only the ground state, however, the energies of the excited states will not necessarily converge (see Fig. 4.3). The energies certainly decrease because we increase the basis size, but except for as
state.
the energy of the second excited state, the upper bounds are not very accurate. (See Chap. 8 for the detail of the basis function used in the
present
section.)
0.4
171
Cd
0.3
CY)
0.2 W
0.1
0.0 0
200
400
600
dimension of the basis Convergence of the energies of the first five 'S states of the Helium atom when only the ground-state wave function is optimized. The energy difference Ej Ei'act is shown in the figure. See Table 4.7 for the exact energies.
Fig.
4.3.
-
As the ith
eigenvalue Ej
obtained
by diagonalization
is the upper
bound of the energy of the ith excited state, one may try to optimize this upper bound to get an accurate estimate for the energy of the excited state. In practice we repeat the same procedure as before, but now the basis selection is governed by the requirement that the ith
eigenvalue, get the ith
ground-state energy El, should be improved. To eigenvalue we need at least an i-dimensional basis to start not the
4. Stochastic variational method
58
with, but practical numerical considerations suggest that it is better to start with a basis in which all the lower eigenvalues (k i 1) axe already "stabld. This means that we need a first guess for the lower eigenvalues, otherwise it may happen that when improving the first excited state, for example, we pick up such components that lower the ground state. As the energies of the excited states of the Helium atom are known to a high accuracy, we will test our strategy in this case. First we 100-dimensional optimize the ground state of the He atom on a K basis. The energy of the ground state is very accurate and even the energy of the first excited state is acceptable. Starting from this K 100-dimensional basis, we increase the basis size one by one, picking up basis states which improve the energy of the first excited state. This procedure quickly improves the energy of the first excited state, while the energy of the ground state also improves a little bit (the important thing to note is that it does not get worse) because the basis size is increased. After reaching the basis size of K 200 we switch to the =
-
=
=
=
second excited state and
so on. The convergence of the result is shown th at the energy quickly converges to the exact value after its turn of optimization started.
in
Fig.
4.4. One
can see
The above
procedure gives us a basis where all the excited states are accurate up to a certain digit. We can of course create bases which give an accurate energy for an individual state, while the energy of the other states might be poor. In that case we simply carry out the stochastic search for a given state staxting from a first guess basis (like the K 100-dimensional basis in the previous case). Tbis optimization may include refinement cycles as well, if necessary. The results we obtained in this way are compared to the "exact" (i.e., the best =
calculation in the
literature)
values
[36, 37]
in Table 4.7. One
for the excited states
for the
can
t1aus
energies ground Examples of caculation of the energies of excited states win be given for the tdA molecule in Complement 8.4, the baryon spectroscopy in Chap. 9 and the four-nucleon system in Sect. 11.2. Before closing this section, we remark that the SVM assumes that the numerical accuracy of the energy calculation in each step ishigh. If the energy is calculated by, say the Monte Carlo method andhas an inherent statistical uncertainty, then the admittance test of the SVM will never work. Even when the energy is calculated by a deterministic get
as
accurate
method,
such
as a
as
state.
diagonalization, the accuracy of the numerical calgood enough to guaxantee the required precision
culations has to be
4.3
Optimization for excited
states
59
0.005
0.004
0.003 Ca >1 CD
0.002
0.001
k
0.000
400
200
0
dimension of the basis Fig.
4.4.
Convergence of the energies of the first five 'S states of the Helium one hundred basis states are selected to optimize the ground
atom. The first
state, the
next
one
hundred
(from
optimize the first excited state, and is shown in the
figure.
101 till 200) basis states are selected to so on. The energy difference Ej Eie"'t -
See Table 4.7 for the exact
energies.
Energies in atomic units of the ground state and the first four 'S excited states of the Helium atom. In column A the basis is optimized successively for all the states as described in the text (K 500). In column B the basis is optimized separately for each state, leading to five different bases (K 600) tailored for the respective states. The "exact" values are Table 4.7.
=
=
taken from
[36, 371.
State
A
B
"Exact"
EiE2 E3 E4 E5
-2.9037243758
-2.9037243769
-2-90372437698
-2.1459737740
-2.1459740452
-2.14597404605
-2-0612718887
-2.0612719880
-2.06127198974
-2-0335865085
-2.0335866779
-2.03358671702
-2.0211312479
-2.0211768312
-2.02117685157
60
4. Stochastic variational method
of the energy. An analytic evaluation of the matrix elements is thus an essential prerequisite in the SVM.
Recently an importance sampling algorithm called the stochastic diagonalization method [381 has been presented to compute the smallest eigenvalue and the corresponding eigenvector of extremely laxge matrices. It is interesting to note that though the algorithm used there has been developed independently of the SVM, it includes procedures analogous to those of SVM. The stochastic diagonalization was applied to matrices of order up to 1035 X 1035. All the above
procedures are based on the minirni a ion of the value. This is, however, not the only possibility expectation energy to find accurate energies and wave functions. As described in Sect. 3.2, the minimization of the variance of the local energy is another
possibility
in
case
will be studied in
the variance
a
can
simple example
be calculated
of the
analytically. This following Complement.
C4.1 Minimization of energy
versus
61
variance
Complements 4.1 Minimization
of energy
versus
variance
by treating a solvable m in the potential of a particle problem, an -Vo exp(-r/a) (Vo > 0). By expressing the wave function Tf V(r) as X/ /Tir, the Schr6dinger equation for this problem takes the form Here
we
compare the two ways of minimization
of mass
S-wave motion of
=
h?
d2X
-Voe'Xp I
2m dr 2
To
(-?')X=EX.
simplifT the notation, 8Tna2 VO
=
(4.10)
a
I
h2
we
and
introduce
V--8Tna
i/
2E,
(4.11)
ji2
exp(-r/2a) reduces Eq. (4.10) change of variable z differential equation of the Bessel functions
The
=
d2X Z-2 The
1 +
dX
--
z
+
dz
1,2)X (1 -Z2 _
=
to the
(4.12)
0.
boundary conditions for bound states, X(r)
=
0 at
r
=
0 and oo,
lead to the solution
X(r)
=
cJ,
( exp (_
r
with
C-2
2a
r
=f'j ( exp(-- ))2
dr,
2a
tj 0
(4.13) where the value of
J-(O
=
is determined
by
the condition
(4.14)
01
and the smallest
(4.11).
v
v
value
In order to have
gives the ground-state energy El through Eq. bound state, the potential strength VO must
a
larger than about 2.405. ground-state energy by the variational method, let us assume a simple basis function for X, rexp(-ar/a), where a is 2for a single basis a variational parameter. There is no minimum of oobtained by variance and the the function. Table 4.8 compares energy using the minimization principle for E or o-2. The parameter values 5.0, a of a are determined by the SVM. For a deeper potential of combination of two basis functions already reproduces the energy rea2.6, however, at least sonably well. For a very shallow potential of
be such that
is
To estimate the
=
=
62
Complements
Table 4.8.
Comparison of variational solutions obtained by E-minimizattion. 2-minimization for an exponential well. Units of E and o-2 are e/(8ma2) and (h2/(8ma2) )2 respectively. The mean values, (T) and and
u
,
(T 2
1/2
,
are
in unit of
with the exact
one
K
5.0
2.6
2
5
a.
is also
The
overlap integral given.
of the variational solution
E
0-2
(r)
(T2) '21
Overlap
E-min.
-3-5840
0.3236
1.4362
1.6169
0.99997
0-2_Min.
-3.5824
0.05437
1.4301
1.6044
0.99983
Exact
-3-5848
0
1.4376
1.6213
1
E-min.
-0.016446
1.275x 10-4
9.4174
12.236
0.99999
U2_min.
-0-016415
3.49Ix 10-6
9.1025
11.619
0.99940
Exact
-0.016447
0
9.4331
12.285
1
C4.1 Minimization of energy
versus
variance
63
(T2-minimization E -minimization
0
............................................... ......... ...............................................................................
i 0
10
20
wave
function-I
30
40
r1a
Fig. exact
4.5. The local energy curve for the shallow potential: wave function is also shown in an arbitrary unit.
2.6. The
5. Other methods to solve
few-body
problems
chapter we briefly show the essential points of other approaches to few-body bound-state problems. There are many different approaches and it is far beyond the scope of tbis book to discuss all of them. Included here are either only those methods which have some sort of connection with our approach or their results are frequently compared with that of SVM, and therefore a short explanation might
In this
be useful.
Quantum Monte Carlo method: The imaginary-time evo lution of a system
5.1
quantum-mechanical system is governed by the time-dependent Schr6dinger equation. When the time t is replaced by an imaginary time, -iha (a > 0), the Schr6dinger equation transforms into a diffusion-like equation The evolution of
Xf (a) aa
a
(5-1)
-HTf (a),
which has the formal solution Tf (a)
=
e-Ha Tf,
(5.2)
th-ne-independent. Here T is an initial wave function Tf (a 0). The imaginary-time evolution of the estimate the ground-state energy. The to used be function can wave
provided
that the Hamiltonian H is =
basis of this idea is Tf lim a-W
that wave
(a)-
a,
(qf- (a) I qf- (a) ) -i
is, the
wave
function at
jail
fimction a
--+
(5.3)
_
oo.
TI(a) approaches To prove
Y. Suzuki and K. Varga: LNPm 54, pp. 65 - 73, 1998 © Springer-Verlag Berlin Heidelberg 1998
this,
the exact
we
only
ground-state
need to
use
the
5. Other methods to solve
66
expansion (3.3) for TI
few-body problems
Eq. (5-2)
that a, :7 0, namely the initial wave function has a non-vanishi -n g overlap with the groundstate wave function. In what follows we assume that this condition is in
and
assume
fulfilled.
If
we
consider
imaginary
T/"(a)
time a, the
bound to the
a
variational trial function value
mean
ground-state
energy
E(a) El.
depending
on
the
of H in Tf (a)
gives an upper In fact it is easy to show t'h at
(Tf (a) I H I Tf (a)) R(a) I Tf (a))
E(a)
-2a(Ei-EI) aj 12 E+E-E ie I I i=2 1+EI i=2 e-2a(Ei-EI)JLi!J2 ai a
E' i=2 (Ei EI+
E1)e-2a(Ei-E1.) I
-
ai e -2a(Ei-El )I 1+1:' i=2 a,
ai
12
2` 12
>
(5.4)
El.
Clearly E(a) goes to E, when a goes to infinity. Another quantity called the asymptotic energy estimator is defined by 1
E(a, -r)
2T
This has the Jim
In
=
(Tf(a) I Tf (a)) (Tf(a + T) I Tf (a + -F))
(5-5)
*
following properties:
E(a, -F)
=
E(a),
(5-6)
-r--+O
+
E(a, -F)
=
E, + -In 2,r
-2a(Ei-El) e EOO I ai 12 i=2 af
(5.7)
'
e -2(a+-r) (Ei -El) I.Es! 12 J:" i=2
I+
a,
Equation (5.6) can be derived by noting that the left-hand side is equal to ( '1/2) -da.Lln(Tf (a) ITI(a)) and that the derivative can be easily calculated by Eq. (5. 1). It is clear that, for T > 0, E(a, 7) ! El and E(a, -r) converges to EI_ as a goes to infinity. In fact, the quantum Monte Carlo (QMC) method or the Green's function Monte Carlo cle to
out the
project point here is not
a
In
general,
method
ground
one
a
[40-44]
uses
state from the initial
construction of
mates the solution but
e-H'.
(GFMC)
a
good
e-Ha
wave
as a
vehi-
function. The
state space which
approxi-
calculation of the imaginary-tiTne evolution of
cannot
compute e-Ha
and H, do not commute, but
by dividing
=
e-(-ffo+H')'
the time
a
into
because rn
Ho
any smaH
Hyperspherical
5.2
steps Aa
=
a/n,
we can
(RIe-H0A'e-HIA'Ik) (RIe
-H,
IRI
=
expansion method
harmonics
approximate G(R, k)
in
=-
(RIe-"0'Ik) by
factorized form. Then the full propagator
a
ff _f G(R, Rn-1) G(Rn-1, Rn-2)
x
-
-
G(RI,k)dPfn-IdRn-2
*
..
be evaluated
use
several time steps Aa and
extrapolate
to
'
-
(5-8)
dRI
the Monte Carlo method. In
can
by
67
practice,
da
=
one
must
0 in order to
eliminate time step errors arising from the non-commuting nature of the kinetic energy and potential energy operators. the propagator (5.8) to calculate both the numerator and denominator of Eq. (5.4)
When
we use
statistical
Eq. (5.4), subject to the
E(a) a're
in
Then the nice feature
of Monte Caflo
integration. ground-state energy may be lost in method. One of the most serious problems in the QMC is the so-called minus-sign problem [43, 451. The propagator G(R, k) is not always positive. This makes it difficult to use importance sampling techniques based on the classical stochastic process such as a Maxkov process or a Molecular Dynamics technique in computing the propagator by the Monte Carlo method. of
having the QMC
an
upper bound to the
Hyperspherical
5.2 We
errors
briefly
introduce
an
harmonics
approach
expansion
which is based
on
method
the
the trial function in terms of hyperspherical harmonic
expansion of
(HH)
functions.
spherical harmonics are useful to expand the angular funca single-particle motion, the basic idea of the HH method is to generalize this simplicity to a system of particles by introducing a global length p called the hyperradius and a set of angles Q. Let us suppose that all of the particles oscillate harmonically with angular frequency w Just
as
the
tion of
N
N
2
Pi
HHO
rniw2Ir2
+
Using
the Jacobi coordinate set,
one can
into the intrinsic and center-of-mass N-I
1 +
2yi
_2
separate the Hamiltonian
parts
N-1
7r2
HHO
(5.9)
2
2mi
/-Ziw2X2i +
7r2N 2rn12
...
2
N+ 2 'rn12---NWXNi (5.10)
5. Other methods to solve
68
few-body problems
where Ai is defined by Eq. (2.19). Among the 3(N 1) degrees of freedom for the intrinsic motion, -
the
p is defined
hyperradius
by
N-I 2
[lix?
P
(5.11)
The
hyperradius contains only intrinsic coordinates, but it can also expressed as E i_=, Mi(Ti XN )21A, which is proportional to the moment of inertia of the system (see Eq. (2.18)). The choice of IL can be axbitrary, The hyperradius is apparently symmetric with respect to the interchange of particle labels. The potential energy of the intrinsic 2P2. Just as the single-particle kinetic part is simply expressed as -Ilzw 2 radial and angular parts, the intrinsic is into separated energy operator be
-
kinetic energy operator is reduced N-1
h2
7r2
I: 21.Lii
2jL
i=1
02
[5;;72
3N
-
as
follows
4 0
+
L
-ap
P
2
(5.12)
p2
where L is called the
grand angular momentum, which is expressible in terms of 3N 4 angles S2. there is no unique way to choose the angles, one possiAlthough bility is to choose 2(N 1) angles &j and 0j, the polar coordinate of N k 1, 1) and the other N 2 angles -yk (0 < -yk E, xi (i 2 2,3,..., N 1) called the hyperangles as follows -
-
=
-
-
...,
-
-VFILIV-1 XN-I V Ak Xk ,//-Ll x,
=
=
=
V/jL P COS7N-1,
- ,FA P Sin7N-1
Vrj-Lpsin-yjv-j
...
...
Then L is related to the usual
(k
Sin'Yk+1COS7k
angular
momentum
LN-D
a2
2
Lk
3(k
-
2
a7k
-
2)cot-yk
Lk-I +
+ COS
'Yk
sin -YA;
...,
N
-
+
(k
4cot2-yk
2,
2), (5.13)
2
=
2,
siny3sint2.
follows
L2
=
la-lk
N
2),
1k
=
"Xk h
X
-irk
as
5.2 2
L1
The
Hyperspherical
harmonics
expansion method
=121.
69
(5-14) of L 2, called the HH
eigenfunctions L 2y/C (S?)
=
K(K + 3N
-
functions,
5) Yjc (Q)
(5.15)
by the quantum number )C which comprises K (K 5 eigenvalues of the angular operators. 0, 1, ...) Expressing the intrinsic wave function as a sum, over k, of products are
chaxacterized and 3N
-
of radial functions
RK(p)
and the HH functions
Y)c(Q),
3N-4
Ep-
-V
(5-16)
RIC (P) Yk
2
/C
N-body Schr8dinger equation, after eliminating the center-of-mass motion, is reduced to the matrix equation for the radial functions the
d2
'C(C + 1) P2
(_W [ dp2 21L
E)
(p)
N
+
1: (YK 11:
Vii I yic) RIC, (P)
=
0,
(5-17)
with
L=K+
3(N 2
-
(5.18)
2).
The volume element is
given by
N-i
P3N 4dpdQj
fj dxi
MIL2,
AN-I
N-1 x
fj df2kSin3k-4,YkCoS2,YkdYk,
(5-19)
k=2
df?k Sin?ykd'OkdOk is the usual surface element. Because C is positive for N > 3-particle systems, it turns out that the "centrifugal baxrier" is always present in N > 3-particle systems even for zero total
where
orbital
=
angular
For the HH ation of the over one
momentum states.
expansion method to be practical, of course, the evalu-
potential
has to have
energy matrix elements must be feasible. Moreguide for the truncation of the HH basis, even
a
5. Other methods to solve
70
few-body problems
the
expansion converges with a few K values, because the deof the HH functions is high. It is also necessaxy to construct generacy the HH functions of proper symmetry for a system of identical par-
though
ticles. This is
a
difficult
HH
expansion method
(N
<
5).
A
sophisticated
a
to
with FT. In
it limits the
application of the systems of rather small numbers of particles
version of the HH
correlation factor F
(5.16)
problem, and
expansion method [461 employs
N
=
[36,
113'>i=l hi
ri
-
rj) [471, replacing
TV in
481 the correlation factor is chosen to
Eq. satisf ,
special boundary condition like the cusp condition for the Coulombic system. (See Complement 8.1 for the cusp condition.) The use of the correlation factor accelerates the convergence and leads to precise a
solutions. The HH expansion method was introduced in 1935 by Zernike and Brinkman and reintroduced 25 years later by Delves and Smith. See
for
example [49, 50]
for details and
[511
for recent
developments.
5.3 Faddeev method
T
=
'01
+
V)2
+
(5.20)
?P31
where for the bound state each component is related to Tf
*j
GoViTf,
=
where the free
or
(E -110)0i
three-body propagator
=
is
by
(5.21)
ViTf, given by
I
Go
-
E Here
Ho
-
is the
netic energy
-
Ho' three-body kinetic
being
(5.22)
energy with the center-of-mass -kisubtracted. The potential Vi is expressed in terms,
5.3 Faddeev method
V12 It V23 V2 V31 V3 notation, that is, V1 is easy to see that Eqs. (5.20)-(5.22) is equivalent to the Schrbdi-nger HO + VI + V2 + V3 equation (E H) Tl'= 0 with H We show two forms of the Faddeev equations which are widely of "odd
man
out"
71
: --
:::--
i
-
i
=
-
used for numerical solutions. One is the differential form which results
by rewriting the
(E
(E (E
-
Ho
-
-
Ho
-
-
Ho
-
Another is the
Oi
=
second form in
Eq. (5.21):
VI),01
=
V1 (02
+
03)
V2)02
=
V2 (03
+
01)
V3)7P3
=
V3 (01
+
02)
integral
(5.23)
form of the above
(j 7 ij
GiVi(Oj + Ok)
k
equation
=7 i, j :7 k),
(5.24)
with
Gi The
(5.25)
=
E
integral
-
Ho
form
-
Vi'
can
G,
02 03
be
simply expressed 0
0
0
G2
0
0
0
G3
in
matrix form
a
0
V,
Vi
V2 V3
0
V,2
V3
0
'01 2
03
(5.26) satisfy the boundary condition that Oi large distances. This condition is, a particular value of energy E, which is notbing but the energy eigenvalue of the three-body system. The Faddeev method is thus a direct method to solve the eigenvalue problem. It does not rely on the variational principle. There are three possible choices for the Jacobi coordinates X(1), X(2)7 X(3). See The bound-state solution has to all of the components however, met only for
Fig.
vanish at
2.2. It is natural to express the Faddeev
of x(l), and likewise
02 by X(2) and 03 by X(3)
component 01 in .
The most difficult part
in numerical works of the Faddeev method then
comes
uation of matrix elements between the functions
of the different Jacobi coordinate sets. Readers for
some
details for nuclear and Coulombic
problems. In Chap.
11
we
will show that
related Gaussians with the
use
a
are
from the eval-
expressed
in terms
referred to
three-body
[55, 561
bound-state
optimization of the corSVM gives solutions that are
careful
of the
terms
5. Other methods to solve
72
few-body problems
in excellent agreement with the results of the Faddeev method for
nuclear
three-body problems.
A real power of the Faddeev method is
probably not only in boundstate problems but in applications to scattering and continuum problems. See [541 for the recent developments on this subject. It is possible to extend the Faddeev method to four-particle system. The Faddeev equations are then called Faddeev-Yakubovsky equations which consist of seven components, four for I + 3-partition of four particles and three for 2 + 2-partition. Numerical solutions become increasingly difficult.
5.4 The
generator coordinate method
There is
a
which is
a
called the generator coordinate method (GCM), natural extension of Eq. (3.7) to the case of continuous
theory
superpositions. The GCM is a powerful method and has a variety of applications such as the restoration of symmetries or good quantum numbers (e.g., angular momentum and particle number) In many-body wave functions, the microscopic description of reaction dynamics between two composite particles and collective motions of many-body systems. The GCM
assumes
that the trial function is
a
continuous superpo-
sition of the basis function Tf (a), which is often called the
generating
function: Tf
=
f f (a)
Tf (a) da.
(5.27)
f (a) is called a weight function. A real or complex parameter a specifies the generating function. It is called the generator coordinate to distinguish it from a physical coordinate. The symbol The function
a
may
tegral
represent in
more
Eq. (5.27)
than
is then
one a
parameter:
the
equation known
principle as
=
multidimensional
the parameters. The choice of the important in the GCM. The variational
a
generating
a2l
...
).
integration
is used to determine
f H(a, a')f(a)da' El B(a, a')f (d)da', with
,
functions
the Hill-Wheeler equation
=
(al
f (a)
Tf(a)
The inover
all
is most
and leads to
[571 (5.28)
5.4 The generator coordinate method
(a, a) Here
=
li(a, a)
(Tf (a) I H I Tf (a)), and
13(a, a')
B (a,
are
the Hamiltonian kernel and the
a)
=
(Tf (a) I Tf (a)).
73
(5.29)
linear integral operators and called
overlap (or norm) kernel, respectively.
the GCM looks very similar to the diagonalization of the Hamiltonian in the basis states TV (a). This is true if the parameter a
Formally
is discretized and if
only a finite number
of functions Tf (al) ,
...
i
Tf(aK)
are used. Then the HiU-Wheeler equation reduces to the generalized eigenvalue problem (3.10). For further details see [57-601. Examples
of
using discretized generator
10.4.
coordinates
are
given
in Sects. 8.4 and
6. Variational trial functions
Due
care
of the correlation between the
particles
is
important
for
a
precise variational solution. The variational trial functions used in this text, correlated Gaussians and correlated Gaussian-type geminals, are formulated
by using a generating function g. Orbital functions with arbitrary angular momenta or Cartesian polynomials around arbitrary centers are constructed from g by using a simple, well-defined prescription. It is also shown that the generating function can be related to the product of single-particle Gaussian wave-packets through an integral transformation. This 'uncorrelated' form is useful to extend to a many-body system of identical particles. The spin function for an N-fermion system is briefly discussed.
6.1 Correlated Gaussians and correlated
Gaussian-type geminals approach is the choice of the trial function. To solve an N-particle problem, it is of prime importance to describe the correlation between the particles properly. The correlation between the particles can be described by functions of appropriate relative coordinates. The correlation is then conveniently represented by a correlation factor, F rl3',,i=, fij (ri rj). There are two widely applied strategies: (1) One is to use this form of F directly by selecting the most appropriate functional form to describe the short-range as well as long-range correlations. Such calculations are, however, fairly involved for systems of more than three particles and the integrations involved require the Monte Carlo technique. (2) An alternative way to incorporate the correlation is to approximate fij by a number, possibly a large number, of simple terms which facilitate the analytical calculation of the matrix elements. We follow the second course by using an expansion over a correlated Gaussian "basis". Most crucial for the variational
=
Y. Suzuki and K. Varga: LNPm 54, pp. 75 - 122, 1998 © Springer-Verlag Berlin Heidelberg 1998
-
6. Variational trial functions
76
important that two basic requirements axe satisfied in the second course. First, in the limit of large dimensions the basis functions should become complete, so that the results obtained by means of some systematic increase of the number of basis functions could converge to the exact eigenvalue. Second, for the approach to be practicable, computational effectiveness is required for the basis functions. The It is
matrix elements have to be evaluated with ease; otherwise calculations witli combinations of a great number of basis functions would be
extremely difficult. In our opinion only the correlated Gaussian basis meets these requirements. The usefulness of Gaussians was already suggested in 1960 independently by Boys [611 and Singer [621, and since then exploited by many authors [63, 22, 64, 65, 661. Though a mathematically sound proof may not be available for the completeness of the Gaussian functions, a heuristic argument for it is possible as follows: As is discussed in Complement 6.1, any square-integrable, well-behaved function with angular momentum 1m can be approximated, to any desired accuracy, by a linear combination of nodeless harmonic-oscillator functions eter
a
(Gaussians)
(Eq. (6.37)). By generalizing
N-particle
continuous size paramthis to the N-particle system, the
basis function then contains
N
rj)21
F=
(r113" ,,j=jexpf--!a-2
This
simple correlated Gaussian
U
calculations. For
a
specific
for
cases,
see
-
?,
discussion
is
on
of
=
a
a
product
eXpj
-.1 2
of these Gaussians:
EN )2j. j>i=l ozij(ri -,rj
actually widely used in variational the completeness of Gaussians for
example [67, 351.
Of course, we have to keep in mind that the Gaussian is not economical in describing the asymptotic behavior of the wave function at
laxge
distances
(see Fig. 6-2). Moreover,
it does not
predict
a
correct
specific quantities such as the cusp ratio. See Comasymptotics and the cusp ratio could be well plement described with exponential functions. For example, welinow that the Rylleraas-type functions give very accurate results for Coulombic fewbody system (see, e.g. [68, 691). The correlated exponential functions are not, however, amenable to analytic evaluation of matrix elements for a system of more than three paxticles. This makes it difficult to use the exponential functions as a vahational trial function for a general N-particle system. value for
some
8.1. Both the
We extend the above argument further to define the correlated we consider separately two cases for two types
Gaussians. For this
of Hamiltonians. First in be
expressed
hn
terms of
case a
of
set of
Eq. (2.1) the Hamiltonian can independent relative coordinates
6.1 Correlated Gaussians and correlated
,7
x
(xi,
=
...
,
xN-1).
As
was
shown in
Gaussian-type geminals
Eqs. (2.23)-(2.25),
77
it is then
con-
venient to express F in terms of x, instead of N(IV- 1) /2
interparticlefunction, a so-called
distance vectors, ri rj. An N-particle basis correlated Gaussian, then looks like -
I:
Type where A is
TI
(N
an
exp
=
-
1)
(_2 :TbAx)
x
(N
0 (x),
(6-1)
1) positive-definite, symmetric
-
matrix
of nonlinear parameters, specific to each basis element. As mentioned in Sect. 2.2, the matrix A with these properties can in general be written
6DG with the use of an orthogonal matrix G and a diagonal
as
matrix D with all
positive diagonal elements. The function O(x) is a generalization spherical harmonics Y of Eq. (6.56) to the many-particle case. More details will be given below. of the solid
The second each
is
is suited to the Hamiltonian
case
governed by both
(2.2).
The motion of
the
single-particle potential and the two-body Eq. (6.1) to include an independent motion of the ith paxticle around some point R, Here PI, is not a dynamical coordinate but just a parameter vector. In this particle
interaction. It is thus useful to extend
case we
but
do not need to
can use
the
use
the relative and center-of-mass coordinates
single-particle coordinates. Therefore we are led to
the
following type of correlated Gaussians which are often called correlated Gaussian-type geminals N
expf
XeXpf Here
r
-
aij (,ri
-
2
-
2
Y
R stands for
i(ri a
N
E j>i=, aj(,r, _,rj)2 A defined N
3=i+l
can
through Aij as
Type
11
:
Tf
=
x
Ri
)210(,r
be written
=
exp
-
T3)2
set of vectors
aij, the correlated
expressed
_
Aji
=
firl
R). RI,
-
compactly
-aij
(i
<
as
-
R),
Ar
-
2
(r
-
---,,rN
-
Ar with
j), Aii
Gaussian-type geminals
1 -2
O(r
_
=
can
R)B(r
-
RNJ_ a
matrix
E''-=, aji
in
R)
As
general
+
be
I (6.2)
78
6. Variational trial functions
6.1 Correlated Gaussians and correlated
Gaussian-type geminals
79
optimization of nonlinear parameters allows us to obtain high-quality solutions with expansions containing not too many terms. The Gaussian of type I is a special case of type IL To unify the description for both cases, we change the notation in what follows throughout this chapter according to the following conventions: As it is cumbersome to use x or r depending on the type of problems, we in the
of the presence of an external field. To have N as the number of xi coordinates in both cases, we consider (N + 1) particles for the basis of type I and N particles for the basis of type II,
will
use x even
respectively.
case
Table 6.1 summarizes
our
conventions for the notations.
Table 6.1. Convention of the coordinates jo
`:
(XII X21
---I
XN)
in the
cor-
related Gaussians and the correlated are
abbreviations of
Number of
Meaning
Gaussian-type geminals. SP and CM 'single-particle' and 'center-of-mass', respectively.
particles
of
x
T ansformation matrix
from SP coordinates
SP coordinates relative -
ri
Gaussian-type geminals (Type II)
N+I
N
Relative coordinates U: Ur X =
SP coordinates U=1:
X=r
IV
I
XN+1
Ei==1 mixi
7nl2---N
(U-1X)i
Xi
-
XCM
XCM
Relative distance vector -
Correlated
(Type I)
r
CM coordinate: xcm
to CM: ri
Correlated Gaussians
(U-I-X)i
-
(U-'x)j
Xi
-
Xj
rj
Before
discussing the
function
O(x)
in
Eq. (6.1)
we
distinguish two
types of variational calculations: One is a type of calculation called a vaTiation before projection. Here the trial function used is not neces-
sarily an eigenstate of some of the conserved quantities belonging to the symmetries of the Hamiltonian such as paxity and angular momentum. After the variational calculation, the conserved quantities are proje-ted out from the trial function. For example, we know that an eigenstate of a rotationally invariant Hamiltonian should have a good angular momentum. Yet one often uses a variational trial function that does not have the proper rotational symmetry but, after the
6. Vaxiational trial functions
80
variational calculation, the symmetry is restored by angular momentum projection. The restoration of good angular momentum is ensured
superposition of the rotated variational solutions and the weight function involved in the superposition can be determined, e.g., by the
by
a
generator coordinate method of Sect. 5.4. Another type of calculation
after projection, where the trial function
is what is called variation
constructed
so as
is
quantum numbers of the conthe variation after projection is superior to
not to mix different
served quantities. Clearly the variation before projection because the variation in the former
case
only in the state space which has the same symmetry as solution, while in the latter the variational solution tends to reach a minimum in the space including the basis states with different quantum numbers. Even a calculation of variation before projection type, if thoroughly done, would reach a solution that has the proper symmetry in the limit that the state space is complete. See Sect. 10.4. Of course calculations of the variation after projection type become more challenging because it is in general not easy to calculate matrix elements with trial functions that are eigenstates of the conserved observables. Our objective is, however, to use trial functions with the
is carried out
the exact
proper symmetry. The function O(x) in
Eq. (6.1) describes non-spherical motion. To angular momentum L and its projection M, a direct generalization of solid spherical harmonics Y (4.5) (see Complement 6.2) to a many-particle case is a vector-coupled product of the solid spherical harmonics of the relative coordinates: describe the orbital motion with the orbital
[[[Yi, (XI)
OLM*
X
)(
...
Y12 (X2)IL12
X
Y1' *V)
I
X
Y13 (X3)IL123
LM
N
C.
r-=JM1.IM27 where c,, is
a
...
(6.3)
Ylimi (xi),
7MNI
product, (IIM112M2 IL12 M1+M2) (L12 M1+M2 13M3 IL123
M1 +Tn2 +M3)
-
-
-
(L12 N-IMI+?n2+-+TnN-1INmNJLM),
of the
...
Clebsch-Gordan coefficients needed to couple the orbital angular mospecified quantum numbers. Here each relative motion
inenta tG0 the
has
a
angular momentum. See, e.g., [2, 70, 71, 721 for the deangular momentum algebra. Angular momentum recoupling
definite
tails of
coefficients used in t'his book will be defined in
Complement
6.3.
6.1 Correlated Gaussians and correlated
Gaussian-type geminals
81
Since the
angular momentum of the relative motion is not a conquantity, it may be important for a realistic description to include several sets of angular momenta (117 12 IN; L12, L123 ) This is the case especially in nuclear few-body problems [32, 251 because the non-central components of the nuclear potential necessitate higher partial waves. For example the tensor force couples S waves to D waves. It is also noted that a faster convergence is in general obtained by allowing the use of different sets of relative coordinates together with suitable sets of angular momenta [30, 20, 251, because a parserved
7 ....
i
...
-
ticula,r type of correlations can be best described in the coordinate set conforming to the type of correlation. From the fact that OLM(X) can
be
expressed by
different
paxtial-wave decompositions
in different
relative coordinate systems, one can conclude that the use of partial waves may not be so important after all. Besides, the various possible
partial wave contributions
increase the basis dimension.
calculation of matrix elements for this choice Of OLm (x) becomes too
Moreover, sooner or
This choice is therefore
complicated. especially as the number
the
later
obviously particles increases and/or different sets of relative coordinates are employed. This difficulty can be avoided by adopting a different generalization of Y for OLM(x) [33, 311: nient
OLM(X)
inconve-
of
V2KYLM(V) N V 2K+LyLob)
Only
the total orbital
number in most
cases
pression. The real
with
v
Uixi
which is
angular momentum,
(at
vector ii
a
=
fix.
(6.4)
good quantum
least =
approximately), appears in this ex(ul,..., uAr) defines a global vector, v, a
linear combination of the relative
coordinates, and the wave function angle b. The vector u may be
of the system is expanded in terms of its considered a variational parameter and
one
may
try
to minimize the
energy functional with respect to it. The energy minimization then amounts to finding the most suitable angle or a linear combination
of
angles.
The
of the parameter u can be more advantavariational calculation than the discrete nature of the set of
continuity
geous in a the angular momenta
(111 12
IN; L12, L123 -) because the change of the energy functional can be continuously seen in the former case. The factor of v2K+L plays an important role in improving the short-range 7 ...,
i
.
behavior of the basis function. A remarkable
advantage
of this form
6. Variational trial fimetions
82
of
is that the calculation of matrix elements becomes much
OLm(x)
simpler
than in the former
momenta is
completely
case
coupling of N angular appendix for details.
because the
avoided. See the
Eq. (6.2) the construction of the trial function with good orbital angular momentum may not be an For the function
0(,r
-
R)
in
immediate concern, because the vector R is intended to determine a specific shape of the system. The function 0(r R) is chosen in the -
spherical
basis
as
N
O(x or
-
R)
_nz.12kiy1 imi (X,
IX,
in the Cartesian basis N
0 (x
-
i=1
(6-5)
IZZ
as
3
fl 11 (xip
R)
_
-
(6-6)
Rip)nip,
P=I
where the index p = 1, 2, 3 stands for x, y, z components of the vectors xi and R, and nip is a non-negative integer for the p-direction of the
simple transformation for the polynomial parts between the spherical basis and the Cartesian basis, the above two representations are actually equivalent. See Complement 6.2. In the following we assume that 0 is given by Eqs. (6.3) or (6.4) for the correlated Gaussian of type I and by Eq. (6.5) or Eq. (6-6) for the correlated Gaussian-type geminal of type II. The Gaussians of type I have definite parity of either ith
or
particle.
Because there is
a
(_l)L, depending on the choice Of OLM(X),
type II
parity
are
not
while the Gaussians of
eigenstates of the parity operator. To project out good function, one has to take a combination of two
from the trial
Gaussians with the centers at R and -R.
6.2 Orbital functions with
arbitrary angular
momentum The construction of
a
trial function with
good angular
momentum
is very important for obtaining solutions of a rotationally invariant Hamiltonian. In the previous section we introduced the forms of
OLM(X) Since
to describe the orbital fimction with
Eq. (6.3)
good angular momentlan. Eq. (6.4) takes
appears to be better established but
6.2 Orbital functions with
arbitrary angular
momentum
83
particularly simple form and makes the calculation of matrix elements very simple, it is useful to understand the relationship between the two angular functions. a
Any functions of type
Theorem 6. 1.
N
v2KyLM (V) can
be
with
expressed
2ki
2k2
-
-
all
*
*
X
*
X
Y12 (X2)] L12
YIN
of terms
Y13 (X3)] L123
X
(XN)l LM7
non-negative and satisfy 2k,
+
11 + 2k2 +12 +
-+2kv+IN =2K+L.
Proof. We V
are
linear combination
a
[[[Yl (XI) X
where ki and li
uixi
of
in terms
x2k'
X2
XI
v
=
prove the theorem
by induction.
For
U1XI + U2X2, the assertion is true because
equality [73, 747 75] (see Exercise
6.1 for
v2KYLM(V)
D
a
a
we
two-variable vector know the
following
simple proof):
KL
kj.Ijk212
U2k3-+llU2k2+12 2 1
2kl.+11+2k2+12=2K+L
XX
where
DKL
kjIjk212
DKL kjIjk212
-
-
is
-
[Y11 (X1)
same
Y12 (X2
LM7
Bkj.,,Bk2l2 (2K + L)! Q1112; L). BKL (2k, + I,)! (2k2 +12)1
(6.7)
(6.8)
by
47r(2k + 1)! 2kk!(2k + 21 + 1)!!'
and C is the coefficient needed to
the
X
given by
Here Bkj is defined
BkI
2klX2k2 2 1
(6.9) couple two spherical haxmonics with
argument [70]:
1YI 00
X
and reads
as
YI'001 LM
=
C(Il; L) YLm (i),
(6.10)
6. Variational trial fimetions
84
1 +::1: (21+1)(211+1) L (101'OILO). 4 4v(2L 47(2 v(2 + 1)
F2
Qll'; L)
The coefficient C vanishes unless I + I' + L is For
general
a
of the vector
case
VN-1 +UNXN With VN-1
vector-coupled products and
x
U1X1 +
=
V2KyLM(V)
formula to show that
2kN
v *
*
even.
-,LulvxN , we put v +UN-lXN-1 and use the above ulxl +
=
*
(6.11)
can
be
*
-
expressed 2K12
of the terms of
v
...
N-1
1
in terms of the
YL12
...
(VN-1)
Y1N (XIV) with 2K12 m-1 + L12 N-I 2K + L 2kN 1Ndecomposing VN-1 to VN-2 + UN-IXN-I, one can apply =
N
N-I
...
By
further
the
same
2KI2 N-1 to vN-1 YL12 Then it is clear that we ...
argument
repeatedly.
-
-
...
...
N-1
can
(VN-1)
th is process show that the statement is and
use
true.
Theorem 6.2. A
two solid
vector-coupled function of
spherical
harmonies
[Y11 (Xl) be
can
X
(X2)]
Y12
expressed
(-1)11+12
with
=
(_I)L
LM
in terms
of a
linear combination
of terms
X2ki.x2k2V2qyLM(V), 2 1 where
2k, + 2k2 the- form Of V =
Proof.
For
a
2q
+
=
(6.12) 11
+
UIXI + U2X2
L
given
L and the vector
of each term has with appropriate coefficients ul and U2
12
value, 11
-
v
-
+
12
-
L is
even
and non-negative, and
thus 11 + 12 may be set equal to 2k + L with a non-negative integer k. We use induction with respect to k. First we show that the statement is true for k 0 (11 + 12 L). As Eq. (6.7) shows, for the K = 0 =
case
there
are
=
with
(see also Eq. (6. 108)), and (L + 1) terms of [YI(xi) x By taIdng (L + L), each multiplied by UI1 UL-1. 2 L with + + X2 (i aix, 1) ai =7 0, we
only terms
k,
=
k2
=
0
V
=
UIXI + U2X2 consists of
YL-I (X2)] LM (I
=
0,
YLM(V)
With
...,
1) different vectors, vi obtain
=
L
YLm(vi)
=
-21+
71 + 1)! 4-Ax(2L 1. 21 + I)! 1 1+ 1) (2L
E(aj)' 1=0
X
One
can
[YI(XI)
view the above
equations for [Yl(Xl)
X
X
-
YL-1(X2)]LM-
equation as
a
(6.13)
system of simultaneous linear
YL-I(X2)]LM (multiplied by
the square root
6.2 Orbital functions with
arbitrary angular
factor).
The determinant of
nothing
but Vandermonde's determinant and it becomes
(L + 1)
(L + 1)
x
85
momentum
(ail)
coefficient matrix
nonzero
is
for
different
ai's. Therefore the solution of the linear equation exists, and hence it is possible to express [YI (XI) X YL-1 (X2)] Lm as a combination of terms of YLm(vi). Thus our assertion has been proved. Next we assume that the statement holds for k < K 1, and will show that it also holds for k K. To prove this, we note that a general [YII (XI) X Y12 (X2) I LM With 11 + 12 2k + L 2K + L which satisfies the triangular relation Ill -121 :! L < 11 +12 takes the form [YK+l (xi) x YK+L-1 (X2)] LM (1 0, L). What we have to show is that the term -
=
=
=
=
...,
[YK+1(Xl)
YK+L-I(X2)]LM is expressible as a combination of terms of the form of Eq. (6.12). Now, looking at Eq. (6.7) in its full generality, the expansion of V2KyLM(V) contains all of these (L + 1) functions, each
X
multiplied by
u
K+I u
K+L-1
2kt+li
and
further terms. These have
some
the form const. x u 1
2+12X 2kjX2k2 lyll (XI) 1 2
at least
nonzero
one
U2 of k, and k2 is
+ L with k
k2)
=
K
-
ki
k2
-
< K
Y12(X2)ILM
in these terms
appropriate
v
are
k,
-
1.
-
Equation (6.7)
vectors.
where -
Thus, by the assumption that 1, all factors [YI, (xi) x expressible in the form of Eq. (6.12) -
statement of the theorem holds for k < K
with
Y12 (X2)]LM7 equal to 2 (K
X
and 11 + 12 is
can now
be rewritten
as
L
E
V2KyLM(V)
u
K+L-1
K+1
U2
1
CKLI [YK+l
(X1) X YK+L-1 (X2)ILM
1=0
+
(6.14)
...'
where CKLI is a suitable constant factor and the symbol indicates the terms that axe already expressed in the form of Eq. (6.12) as stated above. in
exactly
By taking (L
the
+
different vectors vi in V2KyLM(V) before, Eq. (6.14) can be viewed as
1)
same manner as
system of lineax equations for [YK+I(XI) X YK+L-1(X2))LM, which is also solvable. The solution yields [YK+I(XI) X YK+L-I(X2)lLm as a
combinations of terms of is clear from the
uniquely
proof,
Eq. (6.12).
note that the
completes the proof. As vector v in Eq. (6.12) is not
This
determined.
It is easy to see that the following result.
repeated application
Theorern 6.3. A vector-coup led product
of Theorem 6.2 leads to
of solid spherical haT7non-
ics
[[[Yll-(Xl)
X
Yl,2(X2)IL12
X
Y13 (X3)IL123
X
X
YIN (XAr)lrm
6. Variational trial functions
86
(-1)11+12, (_j)L123 (_l)L12 (_I)Ll2 N-I+IN can be expressed in
With
=
...
(_I)L12+13,
=
...,
(_ 1) L
and
of a linear combination of
terms
terms N
I
(Xi Xj) kjj V2qyLM(V) -
with 2
of each Uj's.
li i= Ei> j=1 kij + 2q X:N is given by EN 1 UiXi With
kii
j=1
term
+2
=
v
Since the factors roles in lish the
-
1
xj2
=
(xi xj)
and
-
are
L, where the
pprop,riate.
a
vector
co
scalar, they play
v
j cient8
no
active
the rotational motion. Theorems 6.1 and 6.3 estab-
desci ibing equivalence
between the
angular
Eqs. (6.3) and angular momenta a-Te the parity oL the basis
functions of
(6.4)
under the condition that the intermediate
restricted
stated in Theorem 6.3 and that
as
L
given by (_ 1) through the angular momentum L. The basis function whose parity is given by (_l)L is called tohave a natural parityThe construction of a general angular function with unna ural parity must be based on the vector-coupled form of Eq. (6.3). Unfortunately there is no simple function analogous to Eq. (6.4) for the unnatural parity case. One way to construct the angulax function with unnatural parity is for L > 1 function is
OLM (X)
V2K [YL-1(V)
.
X
W]LM
with v
=
iix
and
Sij (xi
w
x
(6.16)
xj),
skew-symmetric matrix which satisfies Sij 0-, YL-I(v) must be replaced -Sji. For the special case of LI with YI(v). A slightly simpler angular function would be possible by introducing another vector v' as follows:
where S is
an
N
x
N
=
OLM(X)
=
V2K [YL(V)
X
VILM)
Vr
=
I?X-
(6-17)
Eq. (6.7) that in the case of K 0 both ki and k-2 are limited to zero and only the stretched coupling, namely 11 + 12 L, is allowed. See Exercise 6.1. With an increasing K value the possible values of partial waves 11 and 12 increase including the case of nonWe note in
=
=
stretched
coupling.
This
applies
to the
case
of many variables
as
well.
6.3
To increase K is thus
one
way to include
Generating ftmction
higher partial
waves
87
in the
calculation. The matrix A of the correlated Gaussian is often assumed to be
diagonal in order
In this
case
to reduce the number of nonlinear
parameters.
has to increase K when the contribution of
one
high
important. However, in the case where expected partial A is not diagonal, additional and important partial-wave contributions come from the cross terms of the exponential part of the correlated 0. E.g., the term exp(-Aijxi xj), when Gaussian even with K expanded into power series, contains many terms of the form (X,.X,)n, which can describe high partial waves associated with the coordinates relation xi and xj. This is easily understood by noting the following to be
is
waves
-
=
for
arbitrary
vectors
(a .,r)n
a
and b:
Bkja
2k
r2k
2k+l=n
E
Yj,,,(a)*Yj,,,(r) M=-1
Bkja 2kr 2k(_1)1,,
F21-+l[Yl(a)xYl(r)loo, (6.18)
2k+l=n
Eq. (6.45) and the addition theorem (6.54) for spherical harmonics. This implies that even the basis function with 0 is expected to be useful if a general matrix A is used in the K variational calculation. The calculation for the dtl-L molecule given in Complement 8.4 will clarify the point discussed in this paragraph.
which results from
=
6.3
Generating function
The calculation of the matrix elements becomes
generating
function for the correlated Gaussian.
simpler if one uses a In fact, the following
function g, which contains an auxiliary "vector" A (8 1, ..., SV), is found to generate the correlated Gaussians of both type I and type 11 =
conveniently:
g(s; A, x)
=
exp
To relate the use
the
following
(-2 ;Mx 9x).
(6.19)
+
generating
function g to the correlated Gaussian
formula
B kja2k+l r 2ky1M (,r)
=
fYjm (ol) (a
.
r
)2k+lda
we
6. Variational trial functions
88
f Y,.(
=
,
(
92k+l
Aa-r
dal
e
a)
OA2k+l
(6.20)
A=O
which is
easily proved by using Eq. (6.18). Then the vector-coupled product (6.3) can be generated from the factor egx of the function g as follows: By choosing si aiti with a unit vector ti we obtain =
N
-I bAx)
exp
YI,., (xi)
2
dii- Y ,,., (Fi)
B01i
Oai
g(a It; A, x)
(6.21) By a symbol alt we mean a "vector" such that given by aiti, where & (a, aiv) with ai =
I
(ti,
...,
tN)
is
I
a
The
x
N
-
- -
one-row
7
each component is a real number and
matrix of Cartesian vectors ti.
in the above
key point equation is Eq. (6.20) which relates the solid spherical harinonics to e ". Since Eq. (6.20) yields a more general term r2kyl., (,r) than just the solid spherical harmonics, it may also be useful to use a simpler relation, which generates just the solid spherical haxmonics. The relation (6.58) serves for this purpose. By choosing tj (1 T,j 2, i(I +,ri 2), -27-j) (j 1, N), we obtain =
=
...,
IV
exp
2
TbAx)
Y1,m, (xi)
N
a Ii-Mi
01i
I
H=1 Climi Oaili O-Fili-mi x
g(alt; A ,X)
1
t =(,-, 2,i(l+, j j j
U=11
---
2),-2-rj)
(6.22)
IN)
where
47r(I M) 1 (21 + 1) (1 + m)! -
Cim
=
(-2)111
-
(6.23)
Note that the vector ti satisfies ti-tj -2(-Fi-'F,j)2, particularly ti 2 0. This formula requires only differentiation in contrast to Eq. (6.21), =
6.3
where both differentiation and
integration
Generating
function
needed. To
are
89
couple
the
spherical harmonics to the desired function in Eq. (6.3), one has multiply Eq. (6.21) or (6.22) by q, and sum over x. To construct the correlated Gaussian-type geminals from g we note
solid to
2RBR
exp
=
exp
(6.5)
f
-2
1
iAx
-
2
BR; A + B, x)
---
(x
to
serves
+
-
R)B(x
-
Eq. (6.6).
f
-2 (x
Jc- Ax
-
2
-
R)
+
(x
-
R)j-
generate the function O(x
It is easy to show that
or
exp
9R) g(s
1 -
0('-R)
The factor
-
R)B(x
-
R)
-
(6.24)
R)
of
Eq.
I
IV X
IIIX,_Ri12ki y1irni (X,
-
Ri)
N
02ki+li
fj BkjIj f dtiYjj.j (ti) Oaj 2kj+1j
i=1
x
x
exp
(
-
-1 kBR
-
2
-J-tR)
g(alt + BR; A + B, x)
(6.25)
1
aj.=O,...,CXJV=O ItAr 1=1 it, 1=1-, ...,
or
fn (A, B, R, x) I
I
expf -2 j Ax 2(x -
-
R)B(x
-
R)j
xip i=1
N
3
anip
fj fj atipnip
i=1
x
-
Rip )nip
P=j
exp( 2kBR iR) -
-
P=I
g(t + BR; A + B,
x))
1
(6.26)
6. Variational trial fimetions
90
where
n
stands for the set of
In,,, n12,n13,...,nNj,nN2,nN31-
Tn the
Cartesian representation the parameter a plays no active role but the x,y, and z components of each vector ti, (tillti2lti.), serve to construct the function
O(x
-
R).
To construct the
vector-coupled product OLM(X) of Eq. (6.3), one has to sum over mi's with appropriate Clebsch-Gordan coefficients in Eqs. (6.21) or (6.22). Apparently this is a very tedious task particularly when the number of particles is large. The choice Of OLM(X) of Eq. (6.4) leads us to the following very simple equation which relates the correlated Gaussian to g. By choosAe with a unit vector e, we obtain for t2 tN ing t1 =
V
=
iiX
fKLM (u, A, x)
' =
BKL
f
=_
exp
YL M (' )
(_2I FcAx (
v
2KYLM(v)
d2K+L
g(Aeu; L WA-2_K+
A, x)
dL
(6.27) We
from
Eqs. (6.21), (6.22), and (6.25)-(6.27) that the are explicitly constructed from the generating function g. Depending on the choice of the vector s, g leads to different forms of the correlated Gaussians, when followed by suitable operations acting on s. These are surn-ma J ed in Table 6.2. The construction of fKLM (u, A, x) is simplest among others and it has a wide range of applications as will be shown in later chapters. The correlated Gaussian of Eq. (6.27) contains only the relative coordinates and the center-of-mass motion is dropped from the outset. Thus there is no problem arising from the coupling between the intrinsic motion and the center-of-mass motion. If one uses the singleparticle coordinatesr instead of x in Eq. (6.27), the coupling between them occurs in general and has to be taken caxe of appropriately in can see
correlated Gaussians
order to calculate the energy of the intrinsic motion. A suitable choice of A and u will, however, lead to the result that the center-of-mass motion in
separated from the intrinsic motion. This will be discussed Complement 6.4. Chap. 2 we discussed the linear transformation of the coordi-
can
more
In nates
x
be
detail in
induced
by tJae permutation
P. It is
important
to Imow the
6.3
Generating
function
91
Relationship of the two types of the correlated Gaussians to generating function g of Eq. (6.19). The symbol alt indicates a set of vectors f aiti amtN I. Bkj and Cl,,, are defined in Eqs. (6.9) and (6.23), respectively. Table 6.2.
the
Correlated Gaussians
-I.;v2
exp
Ax) rIN
i=1
(I rIN
1
P01-i f
-
i= 1.
2
i= I
i=1
x
Ylimi(Xi) ali-Mi
ali
I
rIN
jg(alt; A, x))
diiYiini(Z)
-!.,'cAx) IIN
exp
Y1 irni (Xi)
climi aaili ajli-mi
g(alt; A, x)
I
t =(I-, 2,i(l+, j j j
2),-2-rj)
i=O"ri=O (i=l,.. N) .,
ld Ax) IV12Ky f dgYLm( ,) (
exp 1 =
Correlated
Uixi)
i=1
,
x)
).X=O,e=je-j=3-
Gaussian-type geminals
-UMx
-
(In,iv exp
.1 2
2
x
EN
=
d2K+L j,-X2K+L g(Aeu; A,
F3 K-L
exp
(V
J(V)
L IV
2
I
-
,
Bkjjj 2
(x
-
R)B (x
f dii-Yi
RBR
-
I fT7
R)
=,
Ixi _RZ12ki Y1 imi(Xi_ 14)
a2ki+li iM
JiR) g(a It
-
+
BR; A + B, x)
expf .1.7cAx I(x--RR)B(x R) I IIN JJ'=j(Xip =ffrff, lip-=, t-,77ri- jexp(-1:YZBR-!R) g(t+BR; A+B, x))
-j=O,jtjj=j
3
-
-
-
-
, i=-
2
2
3
a7"P
i=
2
p
P
x
ti=o
p
-
Rip )ni,
6. Vahational trial functions
92
effect of P
the correlated Gaussians. Since the correlated Gaus-
on
generated from the generating function, it suffices to examproperty of g due to P. By using the relation Tpx (see Eqs. (2.26) and (2.29)), we obtain a very simple result
sian is
ine the transformation
Px
=
Pg(s; A, x)
=
g(Tp_.9; Tp-ATp, x).
(6.28)
One
only needs to change the matrix A and the vector s appropriately. An important fact is that the generating function preserves its functional form under P. This is also true for a more general linear transformation T of the coordinates x, e.g., a transformation from one set of coordinates to another. Combining this fact with Eq. (6.27), we
obtain
a
very useful
property of the correlated Gaussian fKLM-
Namely for the transformation of Tx
TfKLM(u, A, x)
7--
=
Tx,
we
have
fKLM(TU TAT, x).
(6.29)
Thanks to this nice property one only needs to redefine the parameters A and u of the basis function to construct the transformed wave function. This
property plays
important role
an
in
evaluating
the matrix
elements. The
generating function (6.19) plays a key role in generating the and, moreover, facilitates the evaluation of the
correlated Gaussians matrix elements of
physical operators.
the formulation based
It is therefore desirable that
the
generating function In a many-body system as well. In extending the results to laxger systems of identical paxticles we need to cope with the symmetry adaptation one can use
of the
the
wave
function. Tn such
generating
function
were
a case
on
it would sometimes be useful if
expressed
in
an
"uncorrelated7 form of
the coordinates x, because the symmetry adaptation can then be shnplified by using the technique of Slater determinants or permanents. In fact to the
we can show that the generating function can be related product of the Gaussian wave-packets centered around -4 =
(R,,..., RN) through an integral transformation. Using the definition Eq. (6.51) of Complement 6.1, we can express the product of the
of
Gaussian
wave-packets
as
N
det-V
_'i
ORj (Xi)
)
4
1 exp
:iFx + 2
k_Vx
1 -
2
kFR) (6-30)
with
an
N
x
N
diagonal
matrix
6.3
-YJ
0
0
72
Generating ftmetion
93
0
...
(6.31)
0
'YN
A direct calculation using
1:Mx +
exp
2
2
x)
dx
-9A-18
exp
detA
following equation which relates product of the Gaussians of Eq. (6.30): proves the
2
g of
Eq. (6.19)
(6.32) to the
g(s; A, x)
(detr)3
4
expf 'g(r, Arisl -
-
(47r-) N (det(rT
2
N
g(_V(.V-A)-1s;A(F-A)-'FR)
X
'Y'
ORi (xi)
dR.
(6.33) Note that
A(r
From this
we see
A)-' r
r(r
A)
-1
r
F is
symmetric matrix. The function g in the right-hand side of the above equation depends on the integration variable R and serves as just a weight which is needed to convert the product of the Gaussian wave-packets to the generating function. Equation (6.32) is proved in Exercise 6.2. We have seen that the correlated Gaussians are all generated from the same generating function as summa ized in Table 6.2. Furthermore, the latter can be obtained by the integral transformation of Eq. (6.33) involving the product of single-particle Gaussian wave-packets. -
=
-
that the calculations
-
can
a
be reduced to the matrix
N-particle wave functions involving the product of the wave-packets. The width parameters -yi of the wave packets can be chosen arbitrarily. To choose uniform width parameter for all of them is most convenient if xi indicates the single-particle coordinate of the identical Particles, because then the permutational symmetry of the wave function is simply reduced to that of the "generator coordinates" R. Even when x denotes the set of relative coordinates, this elements of
Gaussian
6. Variational trial fimcdons
94
nice property
be used
by including the center-of-mass coordinate as [311. integral representation of the generating function has been successfully employed in accurate solutions of few-body problems [311 as well as in microscopic descriptions of nuclear systems can
shown in
The
in multicluster models
6.4 The
X.L M
[29, 30].
spin fanction
=
1
cosO -1 .1 (0),
2
Alternatively,
22
.1; 2
-IM) 2
.1 + sin# .1 22 (1),
may set up the
we
.1; 2
.1 2
M).
spin function with
(6.34) a
continuous
parameter such as that of Eq. (6.34) by an elementary method instead of using the successive coupling. Its merit is that the construction of the
spin function is simple, that the evaluation of spin matrix elements
is easy, and
V in
moreover
Eq. (6.34),
that
one can use
continuous parameters such
The construction is done
as
follows. The
Young diagram
spin function with spin S for the N-fermion system
[(N12)
+S
as
in the variational calculation.
(N12)
-
S].
The maximum
weight function
is
for the
[n+ n-I
with M
=
=
S,
XSS, must have n+ spin-up functions and n- spin-down functions. The number of terms distributing n+ spin-up functions among the N
particles terras:
is
NS
=
(,+). Therefore xss is expressed n+
as a sum over
these
6.4 The
spin fimcdon
95
NS
XSS (A)
=
E Ai ii)
(6.35)
-
i=1
characterizing the spin function, satisfy the condition, are not independent 0, where S+ is the spin-raising operator. Acting with S+Xss(A) S+ on XSS (A) yields n+ + 1 spin-up functions and n- 1 spin-down N functions in each term and thus leads to (n++,) independent terms.
ANs)
The coefficients
of. each other but must
=
-
Since the coefficient of each term must N
I
V (n+ 1) (nl+) +
1
=
vanish,
one
(2S + 1)N! + S + 1)! (IN (IN 2 2
has
-
S)!
(6-36)
independent parameters to specify the vector A completely. Here the last minus one of Eq. (6.36) comes from the normalization of the spin function. XSM is easily obtained from Xss(/\) with the use of the spin-lowering operator and thus denoted as XSM(A). The overlap of two spin functions is independent of M and sh-nply given by iV. The independent pa(XSS(A)IXSS(/X')) (Xsm(/\)IXSM(/\')) varied be /X can continuously in the variarameters needed to specify =
=
tional calculation. The ner.
isospin
exactly the same manthe spin and isospin parts in Eq. (6.35), leading to a
function is also constructed in
permutation P on reordering of the terms
The action of the
simply produces
a
linear transformation of the vector /X to another vector denoted
A(P).
96
Complements
Complements 6. 1 Nodeless harmonic-oscillator functions
as a
basis
In variational calculations for bound states it is very important to have a set of basis functions which can approximate square-integrable
functions to any desired accuracy. For a single-variable function one may use the well-known complete, orthonormal eigenfunctions of, e.g., the harmonic-oscillator
(HO)
Hamiltonian. Or
one may try to use functions which may not be complete mathematically able to cope with many problems flexibly and, from a practical
nonorthogonal but
are
point of view, accurately. The amine the
possibility
functions. We will
by studying
purpose of this
of nodeless HO functions the
see
performance
Complement
is to
ex-
set of such "basis"
as a
of the nodeless HO functions
how well
they approximate a given function f (r). The nodeless HO functions with angular momentum Im have
continuous
parameter
-Palm (T)
=
Nal
( V3)
14 I/r
exp
2
T
with
a
solid spherical harmonic
the normalization constant
(2 1+2 a"
a
a:
2)y7n(,VF
(6.37)
I/r
1
(see the following Complement). where
Nj
is
given by
2
2
Nal
Here
(6-38)
=
(21 + 1)!! is introduced to scale the
length. The Gaussians employed in Complement 8.1 to solve the ground state of the hydrogen atom are 0. The overlap of two nodeless HO functions is special cases with I simply given by v
=
(-Vaiml-Va-'im)
aa1
Na I IVa'I
2
2
(6.39)
_
=
.
+.,1)2
(a+a)2 2
2
We attempt to approximate a normalized function momentum ITn in terms of combinations of
fl,,, (r)
with
angular
K
Am (T) The ance
error
cirailm (T) of the
approximation
(6.40) is estimated
by calculating
the vaxl-
C6.1 Nodeless harmonic-oscillator functions
0-
2(f)
=
fI
cillil,,-,
,,
97
a
set of
Or)Idr.
parameters
(P.41)
jal, a2,..., aKI by
the trial and
may set them up
by SVM. Once they are selected, minimizing 0-2 to determine the ci's priate way
basis
2
fl (r)
One may choose or
as a
error
leads to
some
appro-
procedure a
linear
of the
equation
K
E(Failml-Vajim)cj =' (Faiiralfim)
(i
=
1,
...,
(6.42)
K),
j=1
andthena 2 is As
a
I
given by
test function
K
-Y:ij=iCi*Cj(-Vai1mIrajIm)-
fim(r)
we
first take up the HO
wave
functions
angular frequency w hy/m, where n is the number *,,Im (r) of radial nodes. We employ a phase convention such that the radial HO wave functions are all positive for r greater than the outermost nodal is needed to point. The overlap of the functions Fal (r) and calculate o-2. It can be obtained by using a generating function of the with the
=
..
HO functions
Ik
4
A(k,,r)
exp
2
+
Nf2-vk-r
2
1 VT
-
2
21 (6.43)
0,,jm(r)*P,,jm(k), n1m
polynomial of the complex Bargmann space variable k, is the Bargmann transform [76, 771 of the HO wave functions in a spherical basis and its explicit form is given in terms of the solid spherical harmonics of (4.5) [781
where
Pnjm(k),
a
:(2
P,,lm(k) where
Bn1 n -
1) n + 1)! (2n
(6-44)
(k k) ny1m (k), -
Bnj, defined in Eq. (6.9), is the coefficient needed to Legendre polynomials Pl(x):
express xm
in terms of the
xm
21+1
E
=
4r
(6.45)
BnIP1(
2n+l=m
Expanding
(6.44),
we
the
overlap (F ,lm (r) I A(k, r))
obtain
in terms of the
polynomials
Complements
98
(-PaimlOnlm) Obviously,
(2n + 21 + 1)!! (2n)!! (21 + 1)11.1
the
1
(1+ )n (
2n)+!!
-
approximation
in
-
a
Eq. (6.40)
2
2Va
a
1 +
(6.46)
a
becomes less trivial
the number of radial nodes of
as
1 f increases. For Onlm (r) wit"h n 2, the combination of merely a few (::5: 3) terms leads to very good 5 it is hard to make u 2 less than approximations, whereas for n 10-10 with a double precision calculation. This is so because all ai's tend to take a value of unity, reflecting the fact that such a HO function can be expressed by combinations of higher order derivatives of Falm(r) with respect to a at a 1, and the optimization procedure tends to construct such derivatives out of -Tal,,, of almost equal parameters. One thus has to deal with an almost singular overlap matrix ((Faiiml-Vajim)) to obtain the solution of ci's. In stead of approximating each of the 0,,,. (r) with different sets of =
or
=
=
1),
(j:IV n=O 0r2(0nIM))/(N-.L
minimized the average value, (U2 )N with a single set of ai's. The values of the
ai's,
we
follow
a
=
geometric progression and
cessive terms
were
determined
of ten nodeless HO functions The next test
example
k
ai's
were
assumed 11-o
its first term and the ratio of
by Powell's method [141.
yielded (a 2)8
=
0.6
x
suc-
A combination
10-10 for I
is the shifted Gaussian defined
=
0.
by
I
( 3)4 (_IV(,r2+S2))jI(I/'Sr)y V
(r)
=
Fj
exp
-
where the function
ii(x)
(6.47)
7 IM
2
7F
is the modified
spherical
Bessel function of
the first kind
ij(X)
=
F '7-x'I,+.! (X) 2
X2k+1
1: (2k)!! (2k + 21 + 1)!!'
(6.48)
k=O
and where the normalization constant Fj is 4e
2
'us
with
F,j
2
(6.49)
2
To get the normalization constant, 00
fo
x
e-ax21, (bx) 1, (ex) dx
which is valid for Rea
>
Gaussian
wave-packet
I =
2a
0, Rev
shifted Gaussian because it is
we
>
have used the formula 2
exp(b +c2),,'(bc)'
-
4a
1. The function
closely
centered around
2a
(6.47)
related to the s
(6.50) is named
single-particle
C6.1 Nodeless harmonic-oscillator functions 3
,ps' (r)
99
(6.51)
11
-
2
T
through
basis
1 exp,
-
as a
the relation
4V'7-r Os
(6.52)
1M
Fj 1ra
Here
use
eVr_5
is made of the
=
evrscos'o
=
equation
E(21 + 1)ij(vrs)Pj(cosO)
(6.53)
1
with the rem
angleO between r and spherical harmonics
s, and the well-known addition theo-
for the
I
4w-
Pi (CosO)
1:
=
21+1
yjm(i )Yjm( )*
ra=-l
=
4-x-(-I)' /2_71-+I
lyl(p)
X
(6.54)
YIM100.
8 The radial paxt of the shifted Gaussian is hence peaked at r shifted the be to therefore would It approximate challenging -
/_2, Iv.
Note that large s in terms of the combinations of A e-211 A of of in terms as Eq. (6.43) expressible (NF2'91 T). 0,v(r) The overlap between the functions F,,la(r) and 7p,jn(r) is
Gaussian with
1
2
is
(r,im 1,0SW
v/-2-
N,, 1 Fs
1(1 + a)'+' 2
P(_ 1+a
ex
V I/
(6.55)
Assuming again that the ai's follow a geometric progression, 0-2 was minimized by Powell's method for a given value of ;. For the case 2 0 it is possible to make o- < 10`0 with ten nodeless HO of I =
functions for the shifted Gaussians of up to :! 10. All ai then tend to take values close to each other for large,;, and this requires a very
equation (6.42). Increasing the number 2 of nodeless HO functions gives us even smaller o- values and widens precise solution of the
the range of maximum approximated well.
linear
(;
value in which the shifted Gaussian
can
be
examples strongly suggest that the nodeless HO functions can approximate square-integrable functions to any desired accuracy, though the number of nodeless HO functions needed depends The above
Complements
100
on
the
shape of the
test function. A remarkable
shows that the linear function
approximated to high
r
itself, defined in
accuracy in terms of a
in
example given
[66]
finite range, can be combination of Gaussians a
e-ar2 ) We stress two remarkable points of the nodeless HO functions. One a combination of a few nodeless HO functions can approximate
is that
the shifted Gaussian is easy to make
2 a
even
with
less than
nodeless HO functions. In
an
a
large
value of
10-1-0 with expansion
a
;.
For
example, it only 15 HO functions,
combination of
in terms of
would need many more terms to obtain the same accuracy. This indicates the flexibility of a nonorthogonal Gaussian "basis" compared one
to the
orthogonal basis such as the HO functions. Another point is that there are many, possibly an infinite number, of sets f a,,..., aKI which approximate a given function equally well, even though the number K is fixed. This is what we have experienced in the above examples. We display graphic illustrations of Gaussian expansions for different functions with I 0. The most appropriate nonlinear 0, m paxameters ai are determined by optimization, while the linear ones ci are given by the solution of the least square equation (6.42). To point out the dependence on the number of Gaussians, we use different numbers of terms in the expansion (K 5, 10 and 20). In the first example we approximate the HO function 05oo(r) by =
=
=
Gaussians. This function is smooth but oscillates and
asymptotically falls
off like
a
(it
has five
nodes)
Gaussian function. As shown in
10 and K 20 Gaussians give a perfect fit to the 6.1, K function, so these curves are practically indistinguishable. =
In
=
the second
to fit
Fig.
exact
an exponential function e-', the wave function of the ground state of the hydrogen atom. The asymptotics of this function is quite different from that of a Gaussian. To approximate the asymptotic part of this function one needs many terms of Gaussians as is illustrated in Fig. 6.2. By increasing the number of Gaussians one has better and better agreement in the asymptotics. After a certain distance, the Gaussians fall off much more rapidly than the exponential function. In many practical applications, especially for bound states, however, one can always use enough Gaussians to reach the required accuracy. We note that the Gaussian fit gives a poor value (zero) for the derivative at the origin (the exact value is -1) as will be discussed in Complement 8.1. In the next example we try to approximate the absolute value fimction f (r) 12.5-rl. The Gaussian expansion, again, does a pretty
f (r)
=
=
case
we
attempt
C6.1 Nodeless harmonic-osefflator functions
as a
basis
101
10
5
0
-5
-10 0
4
2
6
8
r
0, m 5, 1 0) and its Fig. 6.1. The harmonic-oscillator function (n approximations by Gaussian expansions. The solid curve is the (UM3.ormalized) harmonic-oscillator function f (r) V 500, and the dotted, dashed and 5-, 10- and 20-term Gaussian expansions. long-dashed curves are the K The 10- and 20-term. expansions are practically indistinguishable from the -
=
exact
curve.
100
IT
-C
20
10-40
--60
10
40
10
0
10
30
20
40
50
r
Fig. 6.2. The exponential function and its approximations by'Gaussian exp(-r), expansions. The solid curve is the exponential function f (r) and the dotted, dashed and long-dashed curves are the K 5-, 10- and 20-term Gaussian expansions. =
=
102
Complements
good job (Fig. 6.3), would expect
of further Gaussians
by the inclusion
and
an even
one
better fit.
3
2
V
1
0
0
1
2
I
I
.
3
4
5
r
Fig. 6.3. The absolute value function and its approximations by Gaussian expansions. The solid line is the absolute value function f (r) 12.5 -,rl, and the dotted, dashed and long-dashed curves are the K 5-, 10- and 20-term Gaussian expansions. =
=
The approximation for the step function, f (r) 1 if 7- < 2.5, 2 if r > 2.5, is less impressive (Fig. 6.4), but it goes without f (r) that it is not trivial to fit that function. One can improve the saying =
=
fit
by increasing
the number of
Gaussians,
be pretty slow. The last example shows that
adequate
in
some cases.
Let
us
with Gaussians. This function is it simulates the behavior of
a
try
but the convergence
might
Gaussian expansion may be into
approximate f (,r)
practically
a wave
zero near
function of
a
the
12 =
r
2
e-T
origin, and
system -with very
core. The Lenard-Jones or other hard-core potentials such a function. Figure 6.5 shows that the Gaussians produce may generally give a good fit to this function. By scrutinizing the inner
strong repulsive
part of the approximation
reality
(see Fig. 6.6)
the Gaussians do not
produce
one can
exact
see,
however, that
zero near
the
in
origin, but
C6.1 Nodeless harmonic-oscillator functions
as a
basis
103
3
2
C, -
0 0
Fig.
6.4. The
1
2
3
4
5
step function and its approximations by Gaussian expansions.
The solid line is the step function f (r) = I if r < 2.5, f (r) and the dotted, dashed and long-dashed curves are the K 20-term. Gaussian
2 if
r > 2.5, 5-, 10- and
expansions.
the
approximate function oscillates around the exact one even for 20 Gaussians. This oscillation may be very unpleasent: To tame the hard core potential, we need a wave function which is effectively zero near the origin. The oscillating function leads to numerical problems, which makes it rather difficult to obtain the solution of interaction has
a
very
It is
problems
where the
strong hard-core part.
to know the
important completeness of the basis functions in L (square integrable functions), H1 and H2 (first and second Sobolev) spaces because the bound state in quantum mechanics is traditionally 2
formulated in L 2, and the mathematical solution of the
equation is formulated in H' and H2
spaces.
(The
Schr8dinger
space HI is
a
set of
functions whose derivatives of up to the mth order are all square integrable. Note that the kinetic energy operator requires the second order derivative of
a
basis
variational method
ple
in
[791,
and the
function.) are
The convergence properties of the Ritz demonstrated in H1 and H2 spaces, for exam-
completeness
in L 2 is not sufficient to
guarantee
the convergence of the Ritz method. The proof of the completeness in the space of L 2 and in the spaces of H' and H2 is presented in
[801.
This work proves that any function
can
be
approximated
to any
Complements
104
200
150
100
50
0
-50
-j
-100 0
1
2
3
4
5
r
Fig.
expansions. dashed
The solid
curves are
10 and 20 term
curve
the K
=,r
and its
is the ftmction and the
=
expansions
12e_,r2
approximations by Gaussian dotted, dashed and long5-, 10- and 20-term Gaussian expansions. The
f (r)
6.5. The fimction
are
practically indistinguishable from
the exact
curve.
0.002
.
.
.
.
.
i
.
.
I
I
0.001
0.000
-0.001
-0.002 0.0
0.1
0.2
0.4
0.3
0.5
0.6
r
in Fig. 6.5, but the inner paxt is 5- and 10-term Gaussian expan i n scope and not drawn here.
Fig.
6.6. The
curves
same as
with the K
=
magnified. axe
The
out of the
C6.2 Solid
prescribed
accuracy with
spherical
linear combination of
a
that the number of Gaussians in the the parameters of the Gaussians
harmonics
105
Gaussians, provided
expansion is sufficiently large and
are
appropriately
chosen.
6.2 Solid
spherical harmonics The solid spherical harmonic Yl,(r) r1YI,(i6) of Eq. (4.5) is a 0. 0, of the Laplace equation, V2f (,r) solution, regulax at r The irregulax solution is given by r-1-'YI (f). The solid spherical harmonic is a homogeneous polynomial of degree I in the Caxtesian =
=
==
..
coordinate:
21+1 +
rlyl.(,P)
(1 + Tn)! (I
47r
-
m)!
p! q!
p
-
q)!
pq
(_X+i )p (X _iy)q Y
X
2
where p and q p + q <
1,
p
-
=
(6.56)
7
positive integers which satisfy the conditions The complex conjugation leads to Yl,,(,P)*
are
q
ZI-p-q
2
m.
(-I)MY1-09It is easy to show that with 7-21i(l + 7-2) -2-r) we obtain
(t-T)i
=
(X
+
iY
2 -
-F
(X
_
iY)
I
M)! where C1. is it
(
given
in
vector t
(I
-
2TZ)I
-F'-'YI,,, (r),
C1
=
(6.57)
Eq. (6-23).
the condition 0 < 1
-
m
<
al
aal This
_
special complex
For a given 1 the power of -F must 21, which guarantees the range of m should. Equation (6.57) leads to the equality
satisfy as
a
eat-r
=
CIMYIM(r)-
(6-58)
a=O
simple relation was used
in Sect. 6.3 to
generate the angular part
of variational trial ftmctions. 2
By expressing the Laplacian as V2 the solutions of the
I =
a
T2- Yr-
a (r 2 -a-,
we see
that
equation
[V2_(n-1)(n+1+1) I f(r)=0 r2
(6-59)
Complements
106
expressed
be
can
When
n
1
=
in
or n
polar coordinates as r'Y,,,,(r) or -1 1, the equation reduces to
=
-
the
Laplace
equation. The inverse relation of
Eq. (6.56)
where 1 takes the values n,
given
it is
as
follows:
n
-
2,
...,
0
or
(6.60)
npq 1. A hint to calculate BIM
in Exercise 6.3.
jjIM1j2TtI2j3Ta3)
IT
expressed
BPq Yl,,, (i6),
XPY qZn-p-q =,rn
is
is
(6.61)
j
=
U1 +j2) +j3
=
il
U2 +j3)
+
possible
J12 +j3
to choose
ii
a
+
(6.62)
J231
basis in which
j212, j2, J, j2, 2 j2, 3 1 j2,
are
diagg-
onal
(6.63)
ljlj2 (J12) h; jM) 7
or a
basis in which
2 j2, 1 j2, 2 j2, 3 123, J2i Jz
jil j2h (J23); JM) i
expressed
in terms of
ljlj2 (J12), h; JIVI)
diagonal
(6.64)
-
The transformation from the it is
axe
=
a
one
basis to the other is unitary and
U-coefficient
as
E U(jlj2 Jh; J12 J23) jil, j2h (J23); JM) J23
C6.3
Angular
momentum
recoupling
107
(6-65) The U-coefficient is often called
by
the
overlap
U(jlj2 Jh; J12 J23) and is in fact
a
unitary Racah coefficient.
It is
given
of the two basis states
(il j2h (J23); JM ljlj2 V12) h; JM) (6-66)
:::::
independent
1
i
i
of the
magnetic quantum number
M.
By using Clebsch-Gordan coefficients, the unitary Racah coefficient can be expressed as
(jITn1j2?n2jjI2MI2
U(jlj2 Jh; J12 J23) MIM2M3MI2M23 X
(JI.2Ml2j3M31JM)(j2M2j3M3lJ23M23)UIMIJ23M231jm)(6-67)
Note that in the above
equation M
is fixed to
certain value in the
a
range of -J < M < J, so that actually only two m values are independent in the summa i n. As the Racah coefficient is real and
unitary, the (U-1 Ut =
inverse transformation of
Eq. (6.65)
is
simply given by
CT)
=
U(j1j2 Jh; J12 43) ljlj2 (42) h; JM)
jil j2h (J23); JM)
1
7
J12
(6-68) If the basis of the it is
Eq. (6.63)
is to be transformed to
and
angular momenta j, coupled with i 2 to
transformation coefficient is
ljlj2 (42) h; JM) 1
=
=
j3
axe
the total
first
angular given by
coupled
a
to
basis in which
J13 and then J, then the
momentum
(-l)jl+j2-J12 ji2il (J12)
(-l)jl+j2-J12 1: U(j2jI Jh; J12 J13) jj2
i
i
h; JM)
j1h (J13); JM)
J13
=
E(-J)jl+J-J12-JI3 U (j2jl Jh; J12 J13) ljlj3 (J1-3)
7
j2; JM)
J13
(6-69) unitary Racah coefficients are transformation coefficients becomplete sets of states, so that they obey orthonormality and
The tween
Complements
108
completeness relations. The orthonormality condition (jl,j2j3(J231); JMjjI j2h (J23); JM) Jj ,3 j,,,,, reads, with the use of Eq. (6.68), as 7
E U(jlj2 Jh; J12 J23) U(jlj2 Jh; J12 J231)
:_
(6-70)
JJ23 J23'-
J12.
The
relation may be
completeness
E (jlj3 (J13)
7
expressed
as
j2; JM ljlj2 (J12) h; JM) i
J12
(jlj2 V12) h; JIVII-ljlij2j3(j23); JM)
X
7
-
i
i
in terms of the Racah coefficients to
equation can be transcribed following relation
This the
(6.71)
Ulh V13) j2; JMIj1 j2h (J23); JM)
E (-I)jl +J-J12 -JI3 U(j2jI jj3; J12J13) U(jlj2Jj3; J12J23) J12
(-1)32+j3-J23 U(jlj3Jj2; J13J23)There
are
symmetry relations with respect
among the six seen
curly
(6.72) to the
interchange
momentum labels. These relations
angular by introducing the
so-called
6j symbol [70, 72, 751
can
be best
written
by
brackets
U(jlj2Jj3;j]-2J23)=(-I)jl +j2 +j+j3 Af(2JI2+1)(2J23+1) X
1
il h
j2 J
because the symmetry relation of the one can show that
U(jlj2Jj3; J12J23)
=
(-l)j2+J-JI2-J23
=
is rather
+ 1)(2J, J1 223+1)
(2j2
+
simple. E.g.,
U(Jj3jlj2; J12J23)
U(jlJj2j3; J2342)
=
rL2
U(Jl2jlj3 J23; j2 J)
(6.73)
6j symbol
U(j2jlj3J; J12J23)
U(j3Jj2jl; J12J23)
X
J12 J23
1) (2J + 1)
C6.3
If
one
U(j2 J12 J23 J; j1h)
of the
(6.74)
-
efficient has the value +1. If J12 =
=
109
recoupfing
2
labels, ji, j2, j3,
U(jlj2jlj2; J120)
momentuin
(2JI2,+ J,2+ 1)(2J23 + 1) + 1) (2j3 + 1) (2j, +
(_l)jl+j3-J12-J23 X
Angular
or or
J, is zero, the unitary Racah J23 is zero, then we have
co-
U(jljlj2j2; OJ12)
(-l)ll+j2-J12
p2
2J12 +1
+ 1) (2j2 2j,+ (2j,
The above discussion is extended to the
case
+
(6.75)
1)
involving
four
com-
muting angular momentum operators, jlj2,j3,j4. A basis with a good total angular momentum J and its z component J, is constructed as, for example,
jj1j2(JI2)J3j4(J34);JM)
or
ljlj3(J13)ij2j4(J24);JM)- (6-76)
The transformation ftom. the basis in which J12 and
quantum numbers numbers is
to the basis in which
J13 and J24
are
J34 are good good quantum
again real and unitary:
ljlj2 (J12) j3j4 (J34); JM) i
J13 J24
31
32
h J13
j4 J24
J12 J34
ljlj3 (43) j2j4 (J24); JM) 1
-
J
(6-77) The transformation coefficient is
recoupling coefficient between four angular momenta called a 9j symbol in unitary form. It is independent of M and given by (jIj3(JI3)J2j4(j24); JMjjIj2(j12)J3j4(J34); JM)i a
that is
il h J13
j2 j4 J24
J12 J3 4 J
M I M 2 M3 M4
IVI 2 M3 4 MI 3 M2 4
(jlMlj2M2ljl2MI2)(j3M3j4M4lJ34M34)(Jl2Ml2J34M341JM) (il7nlj3Tn3lJl3Ml3)(j2M2j4M4lJ24M24)(Jl3Ml3J24M241JM)(6-78)
Complements
110
again that M is fixed to a certain value in the range of -J < M < J, so that only three m values axe independent iTi the above summation. The inverse transformation is given by
Note
ljlj3 (J13) j2j4(J24); JM) i
A
j2
h
j4
J12 J34
J24
J
J13
J12J34
ljlj2 V12) j3j4 (44); JM) i
(6-79) The
orthonormality and completeness relations for the unitary 9j
coefficients
can
il h J13
J13 J24
be derived
as
before:
J12 J34
j2 j4 J24
il h J13
j2 j4 J2 4
A
J2
h J13
j4 J24
J12 J3 4
J
(-l)j3+j4-J34 J12J34
L
j3-j4-J23+J24
There
are
many
il j3 J13
JJ12JI2"JJ34JS4"*
J
J
J
A
j3
j4 J14
j2 J23
L
il j4 J14
j2 h J2 3
J1-2 J3 4 J
J13 J2 4
j
(6.80)
J
symmetry properties of the unitaxy 9j coefficient.
These will have their
[70, 72, 751
J121 J341
simplest form in curly bracket
terms of the familiaz
9j symbol
written in
j2 j4 J24
J12 A4
-\/(2JI2+1)(2J34+1)(2Jl3+1)(2J24+1)
J
X
1
il h J13
j2 j4 J24
J12 J3 4
Using the symmetry properties of the 9j symbol, il h
j2 j4
J12 A4
j2
j4
J13 J24
J13
J24
J
J12
J34
J
J3
(6-81)
J we
have for
example
C6.3
Angular
momentuin
j + 1) (2J34 + 1) (243 + 1) (2J12 (231 + 1) (2h + 1) (2J + 1) (2jj -
-
(243 + 1) (2J24 + 1) (2J34 + 1) =V, 3
(2j3 (233(2. 3.3 + 1) (2j4 + 1) (2J + 1)
(-l)jl+j2+j3+j4+JI2+J34+JI3+J24+J
Ill
recoupling
j2 j4 J24
J12 J3 4 J
j3 J13
J13 il j3
J24
i
j2 j4
J12 J34
33
34
il J13
j2 J24
31
J34 J12 J
(6.82) For
One may consider other basis states than those given in Eq. (6.76). example, the basis in which J12 and J34 are diagonal can be trans-
formed to
pled
as
basis in which the
a
angular
momentum is
successively cou-
follows:
ljlj2 (42) j3j4 (J34); JM) 7
E U(J12j3 Jj4; J123 J34) jj1j2 (J12)
7
j3 (J123) j4; JM) 7
-
J1,23
(6-83) The
9j coefficient
rewrite the state in
coupling,
as
be
expressed in terms of products of unitary Eqs. (6.69) and (6.68) enables one to Eq. (6.83), which is obtained by the successive can
Racah coefficients. The
use
of
follows:
jj1j2 (J12) h (J123); j4; JM) i
(-J)jl+J123-JI2-JI3 U(j2jlJl23j3; J12J13) J13 X
jj1j3 (43), j2 (J123), j4; JM
(-j)j1+J123-J12-J13 U(j2jlJl23j3; J12J13) J13 J24 X
U(J13j2 Jj4; J123 J24) jj1j3 (43), j2j4 (J24); JM) (6.84)
Complements
112
Substituting this result leads to
useful
a
il h J13
j2 j4 J24
into
Eq. (6.83) and comparing with Eq. (6.77)
identity J12 J34 J
E(_j)jI+JI23-JI2-JI3 U(Jl2j3Jj4; J123J34) J123
U(j2jljl23j3; J12JI3) U(Jl3j2Jj4; J123J24)-
X
(6.85)
application of the above equation, we note that if one of angular momenta is zero the 9j coefficient is expressible i-n terms of Racah coefficients. E.g., we have As
an
nine
ilh J13
0
31
j4 j4
A4
U(jlj3Jj4; J13J34)i
J 0
h J13 0
h h
6.4
j4 J24 j2 j4 J24
:
6
J
2 243 + 1
2j, (2j,
J
+
1) (2j3
+
1)
U(J13jI Jj4; h J24)
i
32
(-l)j4+J-J24-J34 U(j2j4Jj3; J24J34)- (6-86)
J34 J
Separation of the center-of-mass
motion from correlated Gaussians
The correlated Gaussian
(6.27) in the global vector representation is, among the several possibilities of the correlated Gaussians, most easily constructed from the generating function. Thanks to this simit has
wide range of applications. The relative coordinate (xi, ___j xiv-1) is used to represent the correlated Gaussian, so that the center-of-mass motion is dropped from the beginning. The
plicity ;
a
=
aim of this
Gaussian is
Complement is to expressed in terms
fKLM (u, A, r)
=
exp,
show
that, even when the correlated of the single-particle coordinates r as
(_2'fAr) V2Ky M(V) L
C6.4 Separation of the center-of-mass motion
113
N V
(6-87)
wri,
possible to separate the intrinsic motion, expressed in terms of the relative coordinates, from the center-of-mass motion by requiring some special conditions on the parameters A and u. Here A is an N x N symmetric matrix and u is an N x I one-column matrix. Therefore, in it is
the correlated Gaussian basis the
always absolutely necessary used
use
of the relative coordinates is not
but the
single-particle coordinates can be following. correlated Gaussian of Eq. (6.87) can be
well. We shall show the conditions in the
as
By using Eq. (2.4), the expressed by the relative coordinates x and the center-of-mass dinate xN, which are again denoted as x:
fKLM (Ui Ai 7)
=
fKLM (U A! X) i
coor-
(6.88)
7
with
A'
=
U---IAU-'
The matrix elements
U,
and
A!Nj
=
=
A!jN (i
U--IU. 1, give
=
...'
and center-of-mass coordinates and
(6.89) N
-
1)
rise to
pendence of the function. The element A!Nlv
is
a
connect the relative a
center-of-mass de-
parameter describing
the center-of-mass motion which has Gaussian form. Thus the separation between the intrinsic motion and the center-of-mass motion is made
possible by requiring that
XNi where
c
=
is
0
an
(i
=
I,-, N
1)
and
XNjV
=
c,
arbitrary positive constant. As (U-I)iN=l (i
the above conditions N
-
are
rewritten
as
=
11.... N),
follows:
N
I:I:Ajk(U-I)ki=O
and
j=1 k=1 N
N
E 1: Aik
=
(6-90)
C-
j=1 k=1
The center-of-mass contamination in the
YLm(v)
can
be removed
by requiring that u1V
angular function V2K =
0, that is,
IV
Euj i=1
=
0.
(6.91)
Complements
114
and
(6.90)
The conditions
(6.91)
ensure
that the correlated Gaus-
is free from any contaminations due to the center-ofmotion. The intrinsic motion is described by the correlated
(6.87)
sian mass
Gaussian of type (6.27) and the center-of-mass motion is given by exp(-cx'N /2). The number of parameters contained in the function
N(N + 1)/2 + N, whereas since there axe (N + 1) condi(N 1)(N + 2)/2 free parameters. This number is of course equal to the one which the correlated Gaussian (6.27) has: (N 1)!V/2 + (N 1) (N 1) (N + 2)/2. It is instructive to see how Eq. (6.90) leads to the separation of the (6.87)
tions
is
we
have
-
=
-
-
-
center-of-mass motion. For this purpose part of Eq. (6.87) can be rewritten as
note that the
exponential
1
I
(- 2 FAr)
exp
we
exp
a
-2
j(,r, _,rj)2
2
(6.92) where aij
and,3i
are
related to the matrix A
by
N
aij
-Aij (i 7 j),
=
gi
=
1: Aki-
(6-93)
k=1
The
coupling
between the intrinsic motion and the center-of-mass
tion arises from the second term in the exponent. As ri is
mo-
expressed
as
IV-1
7'i
E (U-l)ijXj + XN7
(6.94)
j=1 we
have N-I
N
X2N
Ep,r? i
i=1
N
(
Pi(U-l)ij
+2 j=1
xj
*
xN +
i=1
(6-95) Here the symbol
...
denotes the terms that
coordinates and have
quadratic in the relative xN. Substituting 3i of Eq. are
dependence (6.93) and using the condition (6.90), we obtain that 0. EN EN j= 1_ Aki(U-l)ij k= I EN j= 1)3i ((,T-l)ij =
no
on
=
EiN=1,3i
=
c
and
C6.5 Three electrons with S
6.5 Three electrons with S Let
M
construct the
us
=
1/2
11)
1/2
115
1/2
=
function with S
spin
=
=
The basic terms for
1/2.
are
I ITI),
=
12)
=
13)
1111),
=
(6.96)
I ITT),
assumed to be in increasing
order, e.g. in the state 11) the first and second electrons are in a spin-up state while the third electron is in a spin-down state. The requirement that S+X(A) must vanish leads to the condition Of /XI + -X2 + A3 0. By taking account of the normalization the spin function can be parametrized by a single variable &(-,xl2 < 6 < 7r/2) as follows:
particle indices
where the
are
=
1
(V/jcos,0 VilsinO) I TIT) ( F, coso 4 sin i ) I IT). 2
X.L
i 22
sinO
(A)
+
-
The,O value is chosen such that,&
=
with the intermediate Spin S12
0 whereas V
S12
(6.97)
I
+
=
0
corresponds =
to the
7r/2
spin function
to the
one
with
I-
=
By acting with S-
on
Eq. (6.97),
we
obtain
2
X.1
_.L
2
2
(A)
=
+
These
baryon
:3sin,& 1111) (61 41sin,&) (Vj cos,0 4 sin,&) 1111). -
_
-
+
spin functions
wave
c Os'&
can
functions. See
6.6 Four electrons in
be used for the
Chap.
spin part of three-quaxk
9.
arbitrary spin arrangement Complement 6.5 gives us the following of the four-electron system. For function general spin an
Arithmetic similar to that of result for the S
=
0:
Xoo (A)
(2
(6.98)
in,&
+
COO + 2
sinO)
(2Icos,&
I -
inO)
cosO + 2
ril-isind)
Complements
116
I
+
(2
COO
-
FWIi.6)I lilt) F3 sinO I JITT),
(6-99)
+
where the paxameter 6 satisfies -T/2 < V < -Ft/2 and is chosen such that,O 0 corresponds to the spin function with S12 07 S123 1/2; =
=
=
1i S123 corresponds to the one with S12 and two of electrons molecule two consisting positronium 1/2. positrons, the most important spin function is such that the spins of 0. the identical paxticles are coupled to zero, which corresponds toO
whereas V
=
-r,/2
=
=
In the
See Sect. 8.3. For S
=
1:
ezzsindsin o
X11 (A)
2 +
3
F1!is 4's 4S
sin,& cos o
( ilc -(61
+
osv
-
cos'O +
-
in,&
sin o)
in,& cos p
-
r!2is r!is
in ? sin
p)
in 6 sin
W)
1
inO cos W +
7
(6-100) where the parameters V and W satisfy --F,/2 < 0 < -F,/2 and W < 7r/2. The three independent spin functions with definite
-Ti-/2
<
S12 and 0 S123 values correspond to the V and (p values as follows: for S12 and S123 O for I and 6 and S12 S123 0; 1/2, 1/2, T-/2 I and for and 6 and S12 S123 0; 3/2, z/2 Ti-/2. W o =
=
=
=
=
=
6.7 Six electrons with S
=
=
0
spin functions may be constructed from those of a smaller group of electrons through angular momentum couplings. As an example, let The
spin function of the six-electron system with S 0, Xoo (A), from two groups of three electrons. Clearly X00 (A) is given as us
a
construct the
=
combination of two terms with xoo (A)
As two
tion
=
122 00)
cosO .1 .1;
S123(= S456)
+ sin V
=
1/2
or
3/2:
1!!; 00).
already mentioned, the spin function with S123 0 or 1, and similaxly independent states with SI-2 with S456 has two independent states with S45 1/2 was
=
=
(6.101)
22
=
1/2
has
the fimc=
0
or
1.
C6.7 Six electrons with S
Therefore the first term of the
parametrized I I
00)
as
=
right-hand
side of
=
0
Eq. (6.101)
117
can
be
follows:
COS
sin 61
611 (S12
COS
62
=
1 (S12
0)"17 (S45 2 =
00 0) '1; 2
=
0) 2 (S45
sinVI sin V2 COS
V31 (S12
sin 01 sin #2 sin
631 (S12
=
=
=
00 1) '1; 2
1) 2 (S45 1) "2 (S45
=
=
I
00 0) *'; 2
1) '1; 2
00)
(6-102)
hand, the spin function with S123 3/2 or S456 3/2 has just one independent state and needs no angle parameters. Each term of the above equation and the second term of the right-hand side of Eq. (6.101) can be easily written down in the form (6.35), e.g., On the other
(S12
=
=
=
0) 211 (S45 I -
2v,f2-
00 0) '1; 2
=
(I
-1 ITIJIT)
+
+
(6.103)
+
The four parameters of the five
6, #1-, 02,,03 determine the relative weight independent spin states for the six-electron system with
S=0It is of
course
possible
to start with other
groups of four electrons and two electrons trons.
or
groupings such
as
two
three groups of two elec-
Exercises
118
Exercises Eq. (6.7).
6.1. Prove
Solution. Let
IT
Of
= -
f
course
us
consider the
V)2K+Ldb
YLm (&) (a
I is
equal
E
=
With
U
=
2 1 + -C2-
BKLa 2K+L V2KyLM(V) from Eq.
to
calculate I in another way,
(a.V)2K+L
quantity
(6-104) (6.18).
To
the relation
we use
(2K+L)! p!q!
(a-xj)P(a-X2)q.
(6.105)
p+q=2K+L
[YI, (a)
x
x
Y1, (xl)loo [Yi, (a) 11 12
0
A
A
0
x
Y1,,(a)],\ x1YI1 (XI-)
2A+I
(211+1)(212 X
IYX(a)
X
Y12 (X2)] 00
0
11 12
[[YI, (a)
X
X
Y12 (X2)]AIOO
+,)a11+12C(l, 12; A)
1Y1J-(XI)
X
Y12(X2)1,\100-
(6.106)
coupled form [Y,,_ (a) X Y12(a)],x, is reduced to a 11+12C(II 12; A) Y by using Eqs. (6.10) and (6.11). When using Eq. (6.106) in Eq. (6.104) and integrating over b., we see that A is restricted to L, because otherwise the integral vanishes due to the orthogonality of the spherical harmonics. We thus obtain The
f YLM01)[YA(a)
X
1YII(XI)
X
Y12(X2)],X]00d&
119
Exercises
[YI XI)
JAL
X
Y12 (X2)1 LM
(6-107)
-
Combining these results leads to Eq. (6-7). 0. Then p + q is Consider Eq. (6.105) for the special case of K equal to L. Since 11 and 12 have to satisfy the condition L < 11 + 12 ":: L. Equation (6.7) then simplifies p + q, 11 + 12 must also be equal to =
to the well-known formula
YLM(XI
X2)
+
=
JX1
+ X2
I'F'YLM( XI +X 2)
L
EDOL
=
010 L-I
1XI (Xl)
YL-1 (X2)] LM
X
1=0 L
1+41')
4-) 47r(2L + 1)! [Yl(Xl) 7) 1)! (2L 21 + 1)! (2 + 1) (21
E
=
1=0
X
Yr,-I(X2)]LM-
-
(6-108) 6.2. Prove
Eq. (6-32).
positive-definite, symmetric matrix A can be diagonalized by a suitable orthogonal matrix T, namely tAT becomes a diagonal di being positive. With the matrix D with its diagonal element Dii use of a transformation x --> y (x Ty), the integral.T of Eq. (6.32) Solution. A
=
=
is evaluated
I
where
as
follows:
exp
(deff)
Though
3
Dy
2
is the Jacoblan
detT is 1 in
Y) (detT)
+ Ts
general,
3
(6-109)
dy
(functional determinant) det(ax/Oy). we
may
assume
dy. preserve the volume element, dx the Since D is diagonal, integration
it to be +1 in order to
=
rately
j
in each
component of y, which reads 1
exp
Therefore
2 we
can now
djyj2 + (ts)j -yj dy
be
performed
sepa-
as
(27r)
3 2
e 2di
(6.110)
di
obtain
N
(27r) d-
3 2
exp( 2di
(6-111)
Exercises
120
N
n,'=, di
It is easy to note that
and
E Jtsfi/dj j=
to detD
equal
expressed
=
det(TAT)
=
detA,
E ,(D-1)jj(!fs)j ( fs)j
as
-
=
j=
-
TsD-lts
9T(TA 'T)ts
9A-1s. Using these results
-
in
Eq. proof. It is useful to generalize Eq. (6.32) for the case where all the vectors and si are d-dimensional. Following the above derivation we get
(6.111) xi
be
can
Z
-Z-
is
=
-
-
leads to the
d
1.,r-,Ax +
exp
where the scalar
(a,, a2,
--.,
x) dx
2
ad)
detA
I
)
exp
2
M-1s),
(6.112)
(inner) product (a b) for d-dimensional vectors d (bl, b21 bd) is to be understood as (a b) -
and
-
=
=
....
i:d Some useful formulas related to this
integral axe collected below. By differentiating both sides of the above equation with respect to the mth component of the vector si,
(xi)m exp
(si)m,
(-2I: Mx 9x) +
we
obtain
dx
d
((A-'s)j)m
(
detA
)
exp
(2 9A-1s)
Further differentiation with respect to
(xi),a (xj),, exp
(
I
TcAx +
-
2
(sj),,
leads
(6.113) us
to
9x) dx
f(A-l)ijJm,, ((A-1S)iW(A-18Wnj +
d 2
exp(19A-1s).
X
detA
Setting
m
=
n
and
f (xi xj) ( -
exp
(6.114)
2
summing
over m
I
7cAx +
-
.
2
leads to
9x) dx
d(A-')jj ((A-1s)i (A-1s)j) +
-
d
X
detA
exp
2
s).
(6.115)
Exercises
121
bl,,,,, with ln 1, 0, -1) a,,,b,, (&,) (M /-4-x/3aYj"' a,,, I rIL) (Im -L: etc. for three-dimensional vectors a and b that is, a, (a,; +iay), \/2 --!-(a,, iay). Here n 0 or I or 2 and -n < [t < r,. ao a, a-, v/2Tn the
case
of d
=
3, let
define
us
product, [a
tensor
a
x
=
=
=
=
-
,
--
=
The scalar
(a b) -
=
=
-
product
-vf3-[a
is
special
a
of the tensor
case
I
bloo
x
that
product,
whereas
=
r,
=
is,
I and
give a vector (outer) product and a second rank tensor. See also Eq. (11.2). The use of Eq. (6.114) leads to the following result r,
=
2
1
[xi
x
xjl,,,,exp
9x) dx
:IAx +
-
2
v/3- &,OJ ,O (A-1)ij
[(A-1s)i (A-1s)j]
+
x
3
2exp 2 gA`s).
X
detA
6.3. Derive
B,,,Pq
a
Eq. (6.60).
npq explicit formula for Bj'.
Solution. The
give
in
hint for its derivation. First
x=
(6-116)
we
r V:r(Yj--j(P)-Yjj(P)), V
is
bit
a
lengthy,
and here
we
recall that
F -r(Yj-j(P)+Yjj(P)), V V
y=i
3
3
VL37'y 7F
Z=
(6.117)
10
Thus the power
XpyqZn-p-q
can
be
expressed
q
P
(7 )n r
XPY qzn-p-q
=
rn
p+q
2
2n-
2
iq
3
E t1=0
X
(yi, (i6))
The power of the k
"+"
(YI -1 (,P))
(k,m)
(Y1jr (j )) =ED L a
(p) E (q) (-1? A
p+q-g-v
spherical harmonics
as
I/
V=O
(y10 (,,))
can
be
n
-p-q.
expressed
(6-118) as
YL km (f)
(6.119)
L
(k,m) where D L
can
be derived
calculated ftom the equation
by using Eq. -(6.10) successively
and it is
Exercises
122
(k,m)
DL
V/47r
=
E LIL2
...
Lk-I
k
xll(C(Li-ll;Li)(Li-l(i-1),ralmlLiim)),
(6.120)
i=1
where
LO
0 and
=
determined
as
Li
=
L. In the case ofTn Lk 1, Li is uniquely 0 there are several possibilities for i, but for rn =
=
=
product of the spherical the following:
the value of Li. Thus the
(6.118)
can
be reduced to
(Y11 (i6))
A+-
(Y,
-1
(f))
p+q-A-v
(Y10(f))
harmonies I
Eq.
n-p-q
(p+q-,p-t/,-I) D(A+"")Dp+q-IL-il E D L(n-p-q,O) 11+V
L
X
Y[t+v li+v(P) Yp+q-A-v
-p-q+IL+v
(f) YLO (i )
product of the three spherical harmonics spherical harmonic by using Eq. (6.10):
The
YIL+z,
IL+v
(i6) Yp+q-tt-71
-p-q+IL+zl
is
(6.121)
-
coupled
to
a
single
(i ) YLO (i6)
E C(IL+v p+q-1-t-7,1; L) C(L'L; 1) L11
X(A+v1-t+vp+q-tz-v -p-q+A+YIL'-p-q+2A+2z;) x
(L' -p-q+21j,+2v L 0 11 -p-q+2A+2v) Y, (6.122)
The above formulas
B npq I.
provide
us
with the
ingredients
needed to derive
7. Matrix elements for
spherical
Gaussians
chapter we show that the generating function introduced in the previous chapter can be used to advantage to derive the Hamiltonian matrix elements of an N-particle system for both the correlated Gaussians and the correlated Gaussian-type geminals. Matrix elements of various physical operators between the generating functions will be tabulated in a compact form. Examples of matrix elements for spherical or primitive Gaussians are given in this chapter and more general matrix cases will be treated in the appendix. We can thus calculate for interaction a many-body system. We elements of essentially any also show that the matrix elements involving the correlated Gaussians become particularly simple in two-dimensional problems. An extenIn this
sion to the matrix element for nonlocal potentials is also presented. We discuss the kinetic energy operator for the relativistic kinematics and calculate its matrix element for the correlated Gaussians as an
example.
7.1 Matrix elements of the
generating
As the correlated Gaussian basis function
can
function
be constructed from
function g (6.19), it is convenient to derive the matrix element between the basis functions from that between the functions This is true even though we use more than one set of coordinates
the
generating
g.
because the transformation of the coordinates
simply leads to
a
change
of the parameters A and s, as shown in Eq. (6.28). It is convenient to use g not only because thereby all types of correlated Gaussians can be constructed in a simplified manner but also because the matrix elements between functions g can be evaluated with ease. Table 7.1 lists formulas for the matrix elements of some basic operators 0
Y. Suzuki and K. Varga: LNPm 54, pp. 123 - 148, 1998 © Springer-Verlag Berlin Heidelberg 1998
7. Matrix elements for
124
spherical
elements, M (g(s; X, x) 101g(s; A, x)), o-Foperators generating functions g of Eq. (6.19). Here we take all vectors d-
Table 7.1. Matrix 0 between
dimensional.
(a
b)
x
and
x is
=
a
N
E,=1 wi xi.
short-hand notation for
tensor
a
product [a
x
b]2,,
A vector
product
defined for three-dimensional
are
XB-1s AB--s. P is A + X, v a and b. B s + s' and y Tpx. permutation operator and the matrix Tp is defined by Px
vectors a
Gaussians
=
=
=
-
=
0
eXP(2!' B-1V)
detB
fvB-1-vMo
X
IdTr(B-'Q) +i B-1QB-1vjMO
aiQx (Q:
symmetric matrix)
a
( X [fV-X
X)
(iv-B-1-v x B-lv) Mo
CX121L
[fvB-lv x (B-1Vj2ttMO -ih&Mo
X X
(7rj
=
-ih
axj
h2
-iA7r
(A:
d 2
(2,X)
MO
a
-
VAy
JMO
symmetric matrix)
L.
(rzL
IdTr(AB-XA)
-i =
Ei(Xi
X
E,, (B-')ij(s'i
x
sj),Mo
7ri)) 2
L2
(fVX
(d X
J(iv-x
J(i7vx P
-
-
-
I)WB-1-s
-
( Ej(B-1)jjZz Sj) I.M0 X
r)
-ih
(fvB-1-v &) Mo
r)
Mi.
=-
(27riv-B-1w)-2
x
exp
r)
(iv-x
x
r)
x
.4
-ihf (r
X
f
1
2iv-B-lw
&) +
(r
AB-lw ivB--Iw
-
(r
I MO ibB-lv) IM,
bB -lV)2 x
(g(s ; A, x) Ig(Fp-s; TpATp, x))
125
7.2 Correlated Gaussians
A4
=
(7.1)
(g(s; W, x) 10 Ig(s; A, x) ,
positive-definite, symmetric matrices. The fxj,..., XNJ, can be integration over a number N of vectors, x carried out by using Eqs. (6.112)-(6.116). where A and X
are
N
x
N
=
leaving this section, we note that the formulation with the generating function, though quite useful and powerful, is still severely restricted in its application to a many-body system by the number of paxticles because of the following properties of the method: Before
1. The number of nonlinear
of
one
basis function is
in the matrix A
parameters contained
N(N + 1) /2. By increasing
particles the optimization hibitively time consuming. of
the number
of these parameters becomes pro-
2. To calculate the matrix elements between the basis
functions,
one
needs to calculate the determinant and the inverse of the N x N matrix B A+ A. The number of operations needed to calculate =
the determinant and the inverse increases
as
N3. The symmetrizaof this calculation N!
the
repetition postulate would require overlap and the kinetic energy and (N(N 1)12)N! times for the potential energy matrix elements, respectively. As the system gets more complex, the number of basis functions needed to represent the wave function increases considerably and
tion
times for the
3.
more
As
was
-
matrix elements have to be calculated.
shown in
Chap. 4,
a
number of random trials
are
probed in the
SVM to select a basis element. To calculate the energy for each trial, the determinant and the inverse of the corresponding matrix B have
computational load mentioned in the second item would become very heavy even for systems containing a relatively small number of particles. Fortunately, when one repeats the calculation of detB and B-1 by changing only one element or a few elements of B, as is often the case in the SVM selection procedure, to be calculated many
one can use
times,
so
the
the Sherman-Morrison formula
involved in the calculations. This will be
[141
to reduce the load
explained
in
Complement
7.1.
7.2 Correlated Gaussians We
assume
form of
that the Hamiltonian of
Eq. (2.1).
an
N-particle system takes
The set of coordinates :Z
=
(xl,..., xN-I)
the
stands
126
7. Matrix elements for
spherical Gaussians
for the relative coordinates chosen to describe the basis function. As a
example of correlated Gaussians we basis function is given by Eq. (6.27):
concrete
that the
fKLM (u, A, x)
exp
(-2I bAx
assume
in this section
IEX12K YLM(ijX).
(fKILIMI (U 7 X7 X-)IOIfKLM(U; A7 X)) I
BKLBK"L'ff X
de--d
f
YL m (e^-) YL, lw,, (, f)
(g(A'eu; X, x) 10 Ig(Aeu; A, x))
*(
d2K+L+2K'+L"
dA2K+LdAj2KI'+L-' (7.2)
X=O,,N,=O' e=jej=1
e"=jeIj=I
As the matrix element
(g(Aeu'; X, x) 10 Ig(Aeu; A, x))
between the
generating functions is given as a function of A, A', e, and e' for various operators by using Table 7.1, one only needs to perform the operations prescribed in the above equation. As will be shown in the appendix, the operations can be done very systematically for many cases of practical interest for arbitrary L. The correlated Gaussiajas with L 0 what we call spherical Gaussians take the particularly simple form in the special 0. As case, K they give a fairly good solution for a variety of few-body problems, we list the explicit for-m of the matrix elements for this case in the following. Because the parameter u of Eq. (6.27) is redundant in this approximation, we denote v'r4-7rfooo (u, A, x) simply as =
=
FA(x)
=
exp,
(-2 Jc'Ax).
(7-3)
127
7.2 Correlated Gaussians
Since this function is
equal
to
g(s
0; A, x),
=
special
a
generating function (6.19), the needed matrix elements given in Table 7.1. The overlap of the spherical Gaussians is given by
(FA' IFA )
(
=
)
A)
in fact
(7.4)
.
The matrix element of the kinetic energy is obtained
(2.10)
are
3 2
(2T)N- I det (A +
of the
case
through Eqs.
(2.11) by
and
2
P
(FAI
-
Tm IFA)
2mi
3OTr (A(A + A) -'A!A) (FA, I FA).
(7-5)
2
The matrix element of the
two-body potential
(FAIIV(ri-rj)IFA)=(FAIIFA)(
is
given by
j)3j V(r)
Ci
2
.1
e-2 C,,r2 dr
(7.6)
27
where I
-
=
w(ij) (A + W) -lw('j)
(7.7)
Cii
w('j) defined by Eq. (2.13). The matrix element of the total potential energy is easily obtained by summing Eq. (7.6) over i, j. As a by-product of Eq. (7.6), we show that the matrix element of Iri rj I' can be easily obtained because then the integral in Eq. (7.6) can be reduced to is
given through the column
vector
-
a
r
-
e
LCT2
2
dr
47r
f
r
a+2
LCr2dr
e- 2
0
a+3
2-x
(2)
x
e -xdx
2
----
C
0 "+3
27r
(2 ) C
2
(a+
where F is the Gamma function. Thus
1
(7-8)
2
we
obtain
7. Matrix elements for
128
(FA'117'i
rjl'IFA)
-
spherical Gaussians
(FA, IFA)
=
2
2
T
Cij
X
where cjj is given by Eq. (7.7). For a spherically symmetric
potential
the
(7.9)
2
integral
W
I(k, c) can
is
a
>
-1
e-
!
CT2
(7.10)
dr
analytically for
a
certain class of
potentials.
If
V(r)
in the form
V(r) then
k
V(r)r
be calculated
given
n(n
fo
=
re-"
=
2-br
(7.11)
expression for I(k, c) becomes possible for -k) by using the following formula closed
an
integer
00
1
ee-ar2-brdr
0
n+I 1
-
T
7F(- I)n
ni
n
Ek=O (n-k')! k! fk gn-k
2 ,fa-
n!
for
for
bn+l
a
> 0
a=O
b>0
(7-12) with
V
fo
go
=
exp
b
(4a) erfc ( 2V/a-
and for k > I
[k121
E V (k
A
k! -
i=O
b
2i)!
k-2i
)
2
=(-I) kH'k-I
A
Here
the
Hn(x)
is the Hermite
complementary
erf (x)
(7.12).
=
(2/V9T f
In
(7.13)
case
error
polynomial and erfc(x) function, where erf(x) is the
e-t2dt.
where
an
1
-
error
erf(x)
fis
function:
See Exercise 7.1 for the derivation of
analytic
evaluation of the
integral
is
Eq. impossi-
ble, one has to rely on a numerical integration. Examples of application of FA(x) to various quantum-mechanical few-body problems and 11.3.
are
presented
in
[811.
See also
Complements
11.1
7.3 Correlated Gaussians in two-dimensional systems
129
7.3 Correlated Gaussians in two-dimensional
systems In this section ments in the
we
case
review the trial
wave
of two-dimensional
function and its matrix ele-
(2D) problems. A real application
of the results in this section will be made to the in
problems In 2D
on
few-body
quantum dots in Chap. 10.
problems
the motion of the
The vector
xy-plane.
study
r
particles
is constrained to the
has therefore two components
r
=
(x, y),
and
for relative coordinates and other vectors introduced
all
expressions 0. The scalar previous sections remain valid by setting z will include variables and over only the spatial integrations products in the
xy
=
components. The main difference between the 2D and the 3D momentum. In 3D the basis function
bital angular eigenfunction
2
of the operators L
,
case
was
is the
taken
L ,:
rlyi-00In 2D L
2 =
as
or-
the
(7.14) L
2
so
Z ,
that
one
of these operators becomes redundant
the fact that the system may be invariant under rotations only. The corresponding orbital angular functions
reflecting
around the z-axis take the form
r'e"mw
=
(X
W,
(M
>_
0),
(7-15)
is, the states are characterized by the quantum number "m". This function, apart from normalization, can be derived from its 3D coun0 (or 6 m and z r/2 in polar coordinate terpaA by setting I that
=
=
=
system). The matrix elements derived for the 3D
case
used because they already include an integration
cannot be
over
directly
the z-component.
One can, however, derive the matrix elements in a very similax way by using generating functions and exploiting the simplifications present in the 2D
function.
the
The first step toward this is to construct a generating Similarly to the 3D case, we assume a trial function, by using
global
case.
vector
fm(u7 A7 x) where the
x
x
representation,
=
(V1
+
in the form:
iV2)' exp
I -Je' A x
(_2
,
(7.16)
is the vector of relative coordinates in 2D and v, and V2
and y components of the vector
are
7. Matrix elements for
130
spherical Gaussians
N-1 V
(7.17)
Uixi.
Note that the
complex conjugate of fm (u, A, x) is an eigenfunction of L, with eigenvalue -mTo derive matrix elements, one can use a particularly simple generating function for f,,, (u, A, x): dr"
f (u, A, x)
--
..
dtra
g(s, A, x)
(7.18) t=O
with
g(s, A, x)
=
exp
1-2
- CAX +
where the 2D vector sj (j vector (tuj, ituj). Note that
=
*1
+
I,-, N
iV2)
-
1)
(7.19) is defined
by
a
*
complex 2t-eui Uill
they satisfy si sj 0, si sj'3 generating function remain valid as listed in Table 7.1 by substituting d 2 and assuming 2D vectors. By using the matrix elements of the generating function all the necessary matrix elements can be easily calculated by carrying out the differentiation prescribed by Eq. (7.18). The matrix elements for some basic operators -
=
-
=
The matrix elements of the
=
are
collected in Table 7.2.
Table 7.2. Matrix elements A4 ...... B is equal to A+ X. A is defined in same as
be
defined in
changed
0 If,,) for two-dimensional
Eq. (2. 11). Q, R,
Eqs. (A.2), (A.9), and (A.18), integral I is defined in Eq. (7.10).
to 2. The
0
Mram' M,,,J IV
P2
Ei=l 2mi IiA7r
-
T,,.,n
..
.,
with
Mm
(27r)N-1 detB
1(2p)m
+ mQ)MmJmm, -lh2(R 2 P
=
L;,
V(J ri
MMmJnm' -
ri 1)
(27r)'V-'-c(7nI)2I5mM, J:M detB
case.
p, p, -y, -y, c are the but the factor 3 in R must
n=O
(2#)m-'(-y-y')' (m-n)!(nl)2 1(2n+1,
c)
7.4 Correlated
7.4 Correlated We
the form of
(Irl,
...
Gaussian-type geminals N-particle system now takes single-particle coordinates j
that the Hamiltonian of the
assume
Eq. (2.2). We
the
use
=
in this section. The transformation matrix U defined in
rN)
Eq. (2.4)
131
Gaussian-type geminals
is
now
redundant
or can
be set
equal
to the unit matrix.
The basis function is chosen to be the correlated
Gaussian-type geminal of Eq. (6.26). The function is characterized by three parameters. The matrix A describes the correlation among the particles, the diagonal matrix B determines the spread of the single-particle wave(RI, RN) give the centers of the packets. This packets, and k type of basis function is employed in molecular physics and nuclear cluster models. The permutation P of particle indices transforms the coordinates as defined by Eqs. (2.26) and (2.27). This transformation changes the basis function to =
...,
(TpATp, TpBTp, TpR, r),
Pf,,(A, B, R, r) where
use
stands for
p-I Thus the
is made of the relation
fnq,,, nq,-2 PI
P2
1
2
(7.20)
'n
,
nq13 ...
...
, ....
N
permutation acting
=
nqN, ,
PIV
Tp-i (Tp)-' for nq,3) nq,2, 1
) (q, on
2
and
Tp-n
N)
...
q2
Tp-
=
qN
...
the basis function
just leads
to the
simple transformation of the parameters, A, B7 R, and n and still keeps the form of the basis function invariant. It is not nice, however, that the value of n changes under the permutation. The integral needed to evaluate the matrix element with the correlated Gaussian-type geminal is 3N dimensional and of multi-center type. Such integrals are extremely hard to calculate and to handle. See, e.g., [82, 831 for some developments made for those multi-center integrals which appear in molecular physics. To avoid the complication, a few simplifications are introduced in practice. For example, only one pair of particles is allowed to be correlated in each basis function. This is equivalent to assuming that the only non-vanishing elements of the matrix A are, Aii = Ajj correlation between the ith and
expf-aij(ri _rj)2 /21.
Another
=
aij and
Aij
=
Aji
=
-aij, if the
is taken into account
jth particles simplification
is to
use
by only primitive
0. Primitive Gaussians Gaussians, that is, the basis function with n yield solutions of good quality for few-electron systems. In this section we calculate the matrix element in the primitive Gaussian basis =
7. Matrix elements for
132
spherical Gaussians I
fo (A, B, R, r)
=
=
allowing the
most
exp
-2
exp
2
I
FAr
-
2(r
-
R)B(T
-hBR) g(BR; A
general type
R)
-
B, r),
+
(7.21)
of correlations. The formulation pre-
sented here will be found to be very simple and lead to compact results for various matrix elements. The matrix elements with higher angular
momenta, when needed,
be obtained from the matrix elements
can
generating function by straightforward but tedious differentiaalgorithm for a systematic evaluation of the differentiation is presented in Appendix A.3. The overlap matrix element is readily available from Table 7.1. It in the
tion. An
reads
as
MG
(fo (A!, B', k, r) Ifo (A, B, R, r))
=
3 2
exp
detC
2'DC-1v 2RBR 2kB -
(7.22)
-
where
C=A+B+X+B',
v
=
BR+ B'k.
Note that formulas derived in this section metric matrix B
or
are
(7.23)
valid for
a
B' provided that A + B and A! + B'
general are
sym-
positive-
definite. The matrix element of the Idnetic energy is also available from Table 7.1: N
(fo (A! B'; k 7
h2 =
2
JKI ((A
I? P'
2mi
+
I fo (A, B, R,
B)C-'(X + B)A)
-
VAyjMG
(7.24)
with y
=
(X + B)C-'BR
where A is
a
-
(A + B)C-'B'k,
diagonal matrix with Aii
=
I/mi.
(7.25)
The matrix element of
the center-of-mass Idnetic energy is calculated very easily as -well. As Eq. (2-10) shows, it is calculated by the above formula with a shnple
7.4 Correlated
modification: The matrix A is all i and
now
to be taken
we
make
Aij
:--
1/?n12
...
for
N
one-
and
two-body poten-
following relation
of the
use
(fo (W, B/I k, r) I V (I iv- r =
as
j.
To calculate the matrix element of both tials
133
Gaussian-type geminals
B, R, r))
S 1) 1 fo (A,
-
f V(x) (fo (A!, B1, k, r) IJ(,Cvr
-
S
-
x) I fo (A, B, R, r)) dx, (7.26)
where iv-
(WIi
:--
...
i
WN)
is
an
auxiliaxy parameter
and S is
a
3-
dimensional vector. Substituting the matrix element of the J-function leads to
Eq. (7.26)
in Table 7.1 into
(fo(W, Bf, k,,r)IV(Ifvr
27r)
=MG
-
2
Sl)lfo(A, B, R, r))
expf
V(X)
C
C _
2
(X + S
_
VC-IV)2ldX
17
0,0
FTCVS C
1CS e_2
2f
-I
V(x) x e-!i
CX2
(ecx
-
e-
csx)dx,
(I 0
(7.27) where
c-1 =,W-1w were
and
s
=
jW-1v
-
(7.28)
S1
introduced.
two-body matrix elements are calculated in a unified way with an appropriate choice of w. The matrix element of the one-body potential, V(Iri SI), can be obtained by N). If V(x) takes the form of Eq. (7.11), choosing wj Jij (j then the above integral is obtained analytically for n (n > -1) As the most important application of this formulation, we give below the matrix element for the Coulomb potential: We show that both
one-body
and
-
=
=
-
1
(fo (X, BI k' r) 1 I
Iri
-
S1
1 fo (A, B, R,
( F?)' c
erf
MG S
where c-' is
now
defined
by
(C-1)jj
(7.29) and
s
is
given by I (C-lv)i
-
S1
-
7. Matrix elements for
134
The
the
matrix element
two-body
by
same manner
the
the matrix element of as
Wk
=
-
Coulomb
the substitution of c-1 s
J(C-'v)j
=
tor e-P'
can
also be obtained in
of
exactly
Eqs. (7.27) (7.28). needs to choose w one only SI) rj the of the matrix element 1, E.g., rj 1, can be calculated from Eq. (7.29) With use
V(Iri
Jik Jjk (k potential, 1/ Iri
:--
spherical Gaussians
-
and
To derive
-
-
(C-1)jj + (C-l)jj (C-1)jj (C-1)jj (C-lv)jl. When V(x) takes the Gaussian form
-
=
-
-
and fac-
2, the matrix elements for both the central and the spin-orbit
potentials
can
be
expressed
particularly simple
in
forms:
(fo(X,Bf,k,r)je-P(' '_-r)2 Ifo(A,B,R,r)) -9
c
c+2p
)
2
CP
expf
-
c
(fo (A!, Bf, k, r) I e-P(fVr) 2(for
+
x
2p
(iV-C-1V)2j.
(7.30)
p) I fo (A, B, R, r)) Ii
=
-ih( W-lv
X
&)-MG (
C
c
+
CP
2
2p) eXpj
c
+
2p
(CVC-1V)2j. (7-31)
It is clear that the above formulation
N-body
matrix elements
as
applies
to the calculation of even
well.
7.5 Nonlocal
potentials
In this section
show
simple example of the extension of the calculational technique to nonlocal potentials. The application of nonlocal potentials will be presented for positronic atoms in Sect. 8.3 and for the nucleus 12C iTj Complement 11.4. The potential is assumed to be of separable form we
a
N
E ki
V
(Ti
-
rj)) (W. (ri
-
Tj)
(7.32)
a
with
a
Gaussian form factor
O a Or)
-
=
This form is to other
e
-21
ar
2.
(7-33)
quite useful in many practical cases, but generalization is straightforward by using the same trick as for the
cases
7.5 Nonlocal
that is
potential functions, function
J(ri
-
rj
-
135
potentials
by substituting the function
o,,
by
a
delta
r).
The first step of the calculation of the matrix elements of the above potential is to substitute the relative coordinates x by a new set of
Jacobi coordinates Y fY1; --- YN-11. The Jacobi coordinates y are the from obtained original set of coordinates x in such a way that yj =
=
,ri
The matrix of the
(see Sect. 2.4).
-rj
corresponding transformation
y=T (k)X,
(7.34) 1, is defined in Sect. 2.4. The kth (k) T is defined in such a way that the first
det(Ox/Oy)
whose Jacobian is in
permutation
deriving equal
Jacobi coordinate is
to ri
=
-
rj.
The next step is to show the matrix elements for the generating function (6.19). The matrix element of one term of the potential reads as
(g(s'; X, x) I
o,, (ri
-
rj)) (W,, (ri
-
rj) Ig (s; A, x))
(g(ts; tXT, y) IW,,,(yj) (W,,(yjjg(ts; tAT, y)),
(7.35)
integration has to be carried out first over yj in both sides and then over the remaining coordinates, yj's. We use the same notation y to denote these remaining (N 2) variables, as well. To carry out the integration implied in Eq. (7.35), we introduce the following where the
-
notations: -
B:
an
row -
b:
(N
-
2)
x
(N
-
2)
matrix obtained
and the first column of the matrix
an
(N
-
by suppressing tAT.
2)-dimensional vector I (TAT) 12,
the first
(TAT) I N-1
-
bi: the first diagonal element of the matrix (TAT), i.e., (TAT),,.
-
t:
-
tj: the first vector
a
set of
(N
-
2)
vectors
j(t8)2,
---I
(tS)N-11-
(Ts) 1.
corresponding to the bra side are introduced in exactly the same way and distinguished with a prime symbol from the quantities of the ket side. By separating the yl-dependence explicitly from the generating function g as The notations
g(Ts; TAT, y) g(t; B, y) exp and
by using
the formula
(-2 bjy2 (6.32)
1
by. yj
one
obtains
_
+ tI. yj
(7-36)
7. Matrix elements for
136
Gaussians
spherical
(w,, (yJ lg(i s; i AT, y)) i 2
(2T
(2d
exp
d
t2
g
(t
-
d
bti; D,
y),
(7.37)
where 1
D
d=a+bl,
B
=
-
bb.
(7.38)
d
As this function is again a generating function, the integration over the remaining variables y is just the overlap of the generating functions, and from Table 7.1
one
(9(8'; A!, X) kPa (Ti
obtains the final result
-
Ti)) ((Pa (Ti
ddI det (D -+D 1)
exp
rj) Ig(s; A, x))
3 2
(27r)N
X
-
as
-
)
I 1 Iti2 + 2d' t/ 2+ iTo (D + D) -1vO 2
(2d
(7.39)
1
with vo =t-
Ibt, + t'
I -
d
&
b't'l.
The above result becomes Gaussian with L
(FAd Wa (ri
-
=
(7.40)
paxticularly simple for
0 used in Sect. 7.2 and reads
TA) (Wa (?'i
-
ddIdet(D + DI)
Ii) IFA)
)
(7.41)
To calculate the matrix element for nonlocal
factor,
(J(y,
-
we
as
32
(27r)N
form
the correlated
note from
Eq. (7.36)
potentials
of an
arbitraxy
that
r) lg(i s; i AT, y))
g(t
-
br; B, y) exp
which leads to the
general
(g(s'; A!, x) 1,5(,ri
-
rj
(-2 blr2
+tl-r
result
-
r)) (J(ri
-
rj
-
r) Ig(s; A, x))
(7-42)
7.6 Semirelativistic kinetic energy 3 2
(2r)N-2 det (B +
x
B)
(_2Ibir2+tl-r
exp
137
-
Ib' rt2 + t/ rf+Iij (B + B') -1v .
1
2
1
2
(7.43) with v
=t- br
+t'- b'r".
By introducing the
(7-44)
short-hand notations
ti .Z
W
C
-
b(B + B)
-1
(t + iV)
=
I
ti b,
-
6(B + B')-b
-
(B + B)-'(t + t)
-6(B + B")-lb'
=
1
41(B + B)-lb Eq. (7.43)
can
be
expressed
(g(s'; X, x) I J(ri
-
rj
-
V,
-
(7.45)
61 (B + B') -Ib'
in terms of g in
compact form:
r')) (J(ri
r) Ig(s; A, x))
-
rj
-
3
N
2 (2 det(B + i ')
X
2
exp
-
2
(t + t) (B + B)
-1
(t + t)
g(W; C' Z).
(7.46)
7.6 Semirelativistic kinetic energy This section shows the calculation of the matrix element of the semirelativistic Idnetic energy. This will be needed in the
nuclear systems among others. See Chap. 9. We assumed the nonrelativistic Idnematics in
application
Chap.
2. The
to sub-
single-
Idnetic energy T is then related to the momentum p by p 2/(2m). In the relativistic Idnematics it is replaced by T
particle T
=
V4c__2p2+,rn2c4
=
mc2. The speed of light c is set to unity and the term, -mc2, is dropped unless otherwise stated as it just shifts the -
138
7. Matrix elements for
energy
expectation value. The kinetic
spherical Gaussians energy
we
consider takes the
form IV
T.r
I"
=
rIzP12 +Ira?
(7.47)
z
The evaluation of the matrix element of the operator (7.47) is never trivial, because the separation of the center-of-mass motion requires
special care.
The contribution of the center-of-mass motion is removed
for the nonrelativistic
case
in Sect. 2.1
by subtracting
the center-of-
kinetic energy from the Hamiltonian. This prescription led us to the expression (2.10) for the intrinsic kinetic energy, which, as it mass
should, does not contain any dependence on the total momentum -r, IV. However, this method no longer applies to the present case and must be replaced by a more general procedure. We assert that the sought procedure is to evaluate the matrix element in the center-of-mass system, that is, in the system of IV :=
,7r,v
E pi
(7.48)
0.
i=1
Note new
that, when applied to the nonrelativistic kinetic energy, this, procedure reduces to the previous prescription of subtracting
the center-of-mass kinetic energy from the total kinetic energy as expected. This is easily seen as follows: By substituting Eq. (2.9) into N
(1/2) ENJ j= E3=,Aij-7ri--rj with Aij being defined by Eq. (2.11). (The center-of-mass kinetic energy was subtracted from the beginning in Eq. (2.10), so that the suffices i and j go up to N 1 and Aij was defined for i, j :! , N 1. In Tnr
=
i:N p2/ (2mi), j= I
i
we
obtain Tn,
=
-
-
the
case
where the center-of-mass kinetic energy is included, it is not see that i and j take I to IV and Eq. (2.11) is still valid for
difficult to
requirement (7.48) then restricts the sum over practice instead of up to N. Thus the matrix j element of the operator T,,, evaluated in the center-of-mass system is equal to that of the intrinsic kinetic energy of Eq. (2.10). Now we will show a method of calculating the -matrix element of T,
(i, j
=
i and
11
....
N).)
up to N
The
-
I in
in the center-of-mass
system in
two
steps. The first step is
to express
the operator in terms of the operators defined in the relative coordinates. As Eq. (2.9) shows, the single-particle momentum operator pi can use
be
expressed
in terms of the N momenta 7rj (j x defined by the tran
the Jacobi coordinate set
N).
=
r-ma
i
-n
If
we
matrix
7.6 Semirelativistic kinetic energy
139
(2.5), the momentum PN becomes just --7rN-1 thanks to the condition (7.48). This peculiarity was already noted in Sect. 2.4. Note that none of the other momenta takes such than
one
-7r's. Let
denote this
us
a simple form; they contain more particular Jacobi coordinate set x, the
and the momentum 7rIv_1 defined in this co(N) Ov) I and -7r N_ respectively. Obviously we can
corresponding matrix Qj, as
X(N),
1)
other sets of Jacobi coordinates
ordinate set
(N
define
-
each of which
J
x(k) (k
1,
=
1
acting with the cyclic permutation (1, 21 k-times
on
2.4 and
Fig.
tion
be obtained to the
original Corresponding
2.2.
froM
permutation (1, 2, of the
U(1) U(3) J 1 7
to each set
Eq. (2.4) defines TT(". i
to
IV)
=
...'
masses
U(2) i
from
-
a
N)
and
so on.
same
Likewise,
1), by
2
...
N
3
...
1
x(M. See
also Sect.
linear transforma-
It is easy to
and at the
mi,...' miv..
,
X(k)
(2 see
the N column vectors of
by rearranging
permutation
...,
pattern of the Jacobi set
analogous
N
to the relative coordinate set obtained
corresponds
the
...'
U(1) i
can
N)
U(i
time
U(2)
that
according by cyclic
can
the
be obtained
As the transformation between
and relative coordinates is
the
single-particle always given in the form of Eq. (2.4) for any Jacobi set X(k), the corresponding transformation of the single-particle and relative momenta is also given as in the form of Eq. (2.9): N
E (U(
Pi
i
))
7r,(k)
i
=
3
ji
1,
...'
N).
(7.49)
i=1
we
Using the special form of the matrix U(k) and the condition (7.48), following useful relation
obtain the
Pk
=
(k) -7r.-i
(k
=
1,
...,
N).
(7.50)
That is, the kth single-particle momentum is equal to the negative of 1-th relative momentum defined in the kth Jacobi coordinate the N -
set.
Therefore,
energy
(7.47)
with the
can
be
help of Eq. (7.50) the semirelativistic kinetic effectively replaced in the center-of-mass system
by N
T.r
)2 +M2 _(7rN-02 J2 Mi' MI (')
+
Note that this operator has
be paid is that
one
(7-51)
no cross terms such as -7ri--7rj. The price to has to transform the coordinate systems conforming
7. Matrix elements for
140
W
to 7rN-,. There is
work with
can
spherical Gaussians terms when
difficulty in evaluating the cross
no
one
definite set of the relative coordinates. That is the
a
followed that route for the nonrelativistic kinetic energy reason why in Sect. 2.2. In the case of the sen-lirelativistic kinetic energy the cross we
term has to be avoided
will be shown below.
as
The second step is to show how to evaluate the -matrix element of the operator (7.51). One has to calculate the matrix elements term by show
method
term. For
brevity
Gaussian
FA(x)
set of the
Jacobi coordinates X(N)
we
of
Eq. (7.3).
Let
N-
only
=
simplest correlated
for the
x
denotes the last
jX1'...' x1v-11.
To calculate the
us assume
(7r(i) J2 +,rn?,
matrix element of nate
a
we
that
have to transform the coordi-
Z
system conforming to the ith Jacobi
set. This
can
be achieved
by
using the relation X
where last we
=
V(i)X(i)
V(')
row
is
(7.52)
1
an
(N
-
1)
and coWmn of
x
(N
-
1)
matrix obtained
U(N) (U('))-l. By using
by omitting
the
this transformation
have
(FA, (x) I
(7r(')
N-I
)2
+
Ta2i IF ,(X))
(detV(') )3 (FA,(i)(x('))IV(,7r(') N- 1)2+Ta,21FA (,)(X(i))), where
(detV('))'
is
Jacobian
a
corresponding
to the
(7-53)
change of
inte-
variables and
gration
A(')
=
A!(')
00AV('),
=
00WO).
The kinetic energy operator and the relative coordinates of the lated Gaussians are now expressed in the same coordinate set.
To cope with the square root,
we
(7.54) corre-
go from the coordinate space to
the momentum space:
(FAf(,)(X(i))l
/(7r(i)
N-I
)2+Ta3jIFAW (X(i)))
ff (FA,(i)jk')(k'j (-7r'(') 1)2+mj2jk)(kjFA(i))dkdk, IV-
where k as
(k" I k)
(7.55)
(27t-)-2R-'V-1-)6i 0 is normalized kiv-11 and (xlk) Iki of the correlated Gaussian The Fourier transform J(k -k). =
. ....
7.6 Sen-Arelativistic kinetic energy
and the matrix element of the square root operator a very simple form:
( kIFA)
(xlk)* exp
=
=
(detA)
-,)2 fixj(v'
(k'I
+
-
2
(-2 FcAx)
V/ (w
=
(7.56)
Vh 2 k,2v-
=
+
into
j
I
+
;j J(k
Eq. (7.55),
-
k').
(7.57)
one can
reduce the
M2 FA (x))
(detV(') )3 (detA(')detX('))-32 x
=
dx
,
mj2 I k)
_1)2
be written in
FA- (k),
By substituting these expressions integration over k as follows:
(FA, (x)
can
(FA,(j)
(k) I
-i
Vr(7r,(i 1)2)2
+
M3 mj2i IFAM
(detV('))3(detA(')detA!('))--:32i x
(FA,(j)
(k) IJ(klv-l
-i
-
(k))
f vh2q2
q) IFA(j)
-,
+
Ta2i
(k))dq.
(7.58)
To calculate the matrix element of J(kN-I-q), werewriteit
q)
with
make
141
an
use
I x (N
auxiliary
-
1)
one-row
matrix Cv
of the formula in Table 7. L Then
(FA, (x) I
V (7 (Vi r
1
)2
+
M
if 2
27r
e-2Iciq2
(0,
as
J(iv-k-
0, 1) and
obtain
i2 I FA (x) )
(detV(') )3 (detA(') detA! x
we
=
(FA, (j)
vh2q2
+
-1
(k) I FA(j)
-1
(k))
Mi2 dq 3
(FA, I FA) f (ci, mi)
(2-x)
(det(A
IV-
+
A)
f (Ci, 7ni),
(7.59)
with
Ci
(f,7(i)A(A+A!)-'A'V('))
where the function
f
is defined
by
N-1
N-1,
(7.60)
142
7. Matrix elements for
spherical Gaussians
(2v) X je-2-Ixq2 2
Ax M)
X
47r
( ) 27-1
and
use
a2
j,)O
h2 -+M2 dq I
(FAIIFA) =
-=
\//-h2q2 + M2 q2 dq,
(7-61)
0
is made of the fact that the
with respect to the
2
e-!! xq
change
overlap has the following property
of the Jacobi coordinate set:
(FAI(x)IFA(x))
(detV(') )3 (FA,(i) (x(')) IFA(i) (x('))) i
(detV(i))3 (detA(')detA'())-i (FA,(j)-i- (k)IFA(i)-i (k)). 2
(7.62) The
overlap (FA, I FA)
is
explicitly given by Eq. (7.4).
143
Complements 7. 1 Sherman-MorTison
fonnula
To show that the Sherman-Morrison formula
vantage let
in
us use
selecting
a
[14]
can
be used to ad-
basis element from among many random trials, (4. 1) or Eq. (7.3) as the basis
the correlated Gaussians of Eq.
functions. The matrix elements
Aij or, equivalently, aij are nonlinear and of the they are related to each other through Eq. basis, parameters (2.25). As is shown in Sect. 7.2, the calculation of the matrix elements, (FA, I FA)
and
(FA, I HI FA), requires
and inverse of the matrix B Let
that
=
the evaluation of the determinant
A + X.
optimize the symmetric matrix A of nonlinear paxameters by changing only one element, say Ak1 Alk, or aij (j > i), but by keeping all other parameters unchanged. Instead of changing all of the nonlinear parameters randomly at once, we change one particular element randomly and then proceed to other elements step by step. The latter type of optimization of A is certainly us assume
we
attempt
to
=
a
very restricted way, but in this
the computer time
case
required for tremendously reduced, as will useful in selecting a successful
the evaluation of the matrix elements is be
seen
below. And it is
actually
very
candidate from among a number of random trials. We see from Eq. (2.25) that changing only one element aij to aij + A produces a change in A as follows: A
--+
A+
where the
(2.13). k,
Alk
--+
-
is just
OPOP
a
(and 1, k) Alk + A) can e(')
as
e(')
-
1)
x
(N
-
1) matrix,
Eq.
whereas
AN
,
=
follows:
A
where
(N
is defined in
-
e(k)
A+ 1 +
an
Oi)
hand, changing directly only Alk of A as AN AN + X (and be achieved by introducing (N I)-dimensiOnal
element
unit vectors
is
column vector
number. On the other
I
A
(7.63)
I)-dimensional
Note that
Oi) 0A the
(N
Aw(0w(ii),
+
e(l) eo(k)
(7-64)
Jkl
is defined
ui and vi stand for
by
(N
-
given by Eqs. (7-63) and
(e('))i
1). By letting I)-dhnensional vectors, both of the changes (7-64) are in general expressible as =
Ji, (i
N
-
P
A
--->
A+
Aiuigi,
(7.65)
Complements
144
where p is either I or 2. As B is equal to A + leads to the
following
X,
the
change B,
of A
as
given by Eq. (7.65)
modification of
P
B
---+
B+
(7.66)
Aiuigi.
Therefore the calculation of the matrix element for the above
change
of A results in the calculation of the inverse and determinant of the
special form of matrix, B + a, Aiuigi. When the modification is given by just one term of the form AuO, the Sherm an-Morrison formula can
be used to obtain
Aub)-'
(B
+
det
(B + Aub)
B-1
-
-AB-1u,6B-1,
1 + AbB-lu
(7.67)
and
(I + A,5B-1u)detB.
(7-68)
See Exercise 7.2 for the derivation of the Sherman-Morrison formula.
advantage of these formulas is appaxent: By knowing B-' and detB one can easily calculate the right-hand side of the equations, and the A dependence is given in a very simple form. To change A, therefore The
there is
no
need for the evaluation of inverses and determinants of the
modified matrix B
(which would require (N
get the desired results by
_
1)3 operations),
but
we
simple multiplication and division. When the modification of B is given by Eq. (7.66) in fiL11 generality, then the inverse and determinant can be calculated by using the Woodbury formula [141, which is the block-matrix version of the Sherman-Morrison formula. If one wants to change a few of the aij's or one-column (and one-row) elements of A at the same time, the summa ion in Eq. (7.66) has to be further extended appropriately. a
Exercises
145
Exercises
Eq. (7.12).
7. 1. Derive
Solution.The the
integral
case
as
of
a
=
0 is
1 00rnCar2-brdr
dn 1
(_I)n x
case
of
a
>
0,
we
may
get
dn
(-I)n
=
0
By putting
In the
simple.
follows:
=
dbn 2
bl(2,Va-),
1000e-ar2-brdr I-exp ( ) erfc( \,Fa
dbn
V
b
4a
a
the above
2
(7-69)
-
equation becomes
00
fo
rne-ar2-brdr
=
V,-(_I)n
(
1
2
- fa- )
n+1
n
nt
k=O
(n
-
k)! k!
A (X)
gn-k (X) ,
(7-70)
where
fk (X)
=
A (X)
=
(_X2) (eXp (X2))
eXp
(k)
eXP(X2) (erfC(X))(k).
(7.71)
fk (x) defined Eq. (7.13). Remembering that the
It is easy to show that
above is
in
Hermite
not
(n)
(-I)n eX2 (exp(_X2)) difficult to check that gk(x)
.(X)
=
7.2. Derive
and
can
(erfc(x))
be
equal to the one given polynomial is given by (1)
expressed
_X21 V/,-X-,it is -
=
as
-2e
in
Eq. (7.13).
Eqs. (7.67) and (7.68).
Solution. Let X represent u,&. Then the inverse as follows:
(B + AX)-1
may be
calculated
(B + AX)
-1 =
(I + I\B-'-X) -'B-1 CO
=
j:(-A)n(B-IX)nB-1.
(7-72)
n=O
special form of X, XB-'X reduces to cX, where the constant factor c. is given by OB-'u. Therefore repeated use of this relation leads to the follwoing result: Because of the
Exercises
146
(B -I.X)n B-1
B-'(XB-'XB-'X
=
d'-'B-'XB-1
=
By substituting
this result into
(B + AX)-' which is
nothing
=
......
(n
Eq. (7.72),
>
we
B-'XB-17
-
(7.74)
Eq. (7-67).
but
given by Bij
P(A)
obtain
I+Ac
+ Auivj, let
us
(B + AX)
=
-
=
ao +
aj.X +
-
-
+
-
(i, j)
with its
suppose that the determinant is a The function P(A) is a polynomial
P(A) det(B+AX). degree (N 1) and can be expanded
function of A:
of at most
(7.73)
1).
A
B-1
To calculate the determ In ant of the matrix
element
B-'X)B-1
as
follows:
aN-O:I F-1,
(7-75)
where the coefficient ak is calculated by k!p(k) (0). The rule of dif0 ferentiating determinants leads us to the conclusion that ak =
for k ao
=
2 because of the
>
P(O)
=
special
form of the matrix X. We have
detB. The coefficient a, is obtained
as
B1,
B12
B, N-1
V1
V2
VN-1
Biv-, I
BN-1 2
BN-1N-1
follows:
N-1
Ui
a,
N-1 N-1
IV-1 N-1
E Y uivj,6ij
E T ujvjdetB(B-')jj
i=1
i=1
j=1
j=1
(,DB-1u) detB.
(7.76)
Aij is the (i, j) cofactor of the matrix B and use is made of the detB (B-1)jj. Thus P(A) is eq ial to (I+ADB-'u) detB, relation Aij Here
=
which is what
we
7.3. Calculate the
want to derive.
one-body density matrix for the generating function
g: Pi W,
r)
(g(sf; X, x) IJ(,ri
-
xN
-,r'f)) (,5(,ri
-
xN
-
r) lg(,s; A, x)).
(7.77)
147
Exercises
(2.12) shows,
Solution. As Eq. W The
ri
-
can
xN
be
expressed
as
EN-I 1=1
argument made for the nonlocal potential in Sect. 7.5 in exactly the same way. The can, therefore, be applied to this case only necessary change is to replace w('j) with 0). The density matrix When the wave function pi (r, r) takes the same form as Eq. (7.46). has the proper symmetry for a system of identical particles, the onebody density matrix does not depend on the suffix i.
w, xj.
Reproduce the nonrelativistic kinetic energy formula (7-5) by using the formulation presented in Sect. 7.6. 7.4.
Solution.
By using Eq. (7.50), the matrix element of the nonrelativistic
kinetic energy operator in the center-of-mass system becomes IV
N
2
(FA'(-X)JE
IFA(X))
2Tnj
=
(FA, (x) 11:
(i)
(7rN-l)
IF",(X))
2m,
3h2
(7.78)
(FAI IFA)
2mici
3h2 / (2mx) for the nonrelativistic kinetic energy is f (x, m) obtained by replacing Vlfh-2q2 + M2 in Eq. (7.61) with h2q 2/ (2m). To perform the summation over i in Eq. (7.78), we need to know the special matrix element M9 N-1: k
where
=
N
VM
(U(N) )kI (TT(i)
-
us
vectors of
Uj(N)
TnI , M2 ,
....
(1, 2,
N)
can
0) i
recall that
...,
mv
can
Uj and
EE
(7-79)
i
kN-i
Let
-I
IN-I
be obtained same
time
with the
i times. Rom this construction it is easy to
be obtained from
constructed from
N)
Uj('
(N)
in
Uj'
but
the column vectors. Then
the N column
by rearranging the masses the operation cyclic permutation
at the
according to
by rearranging
exactly
the
see
that
same manner as
by rearranging the row obtain (see Eq. (2.6))
U(') i U
M is
vectors instead of
we Mi
for
1: k i
for
I
M12---N
UW
(7-80) I N-1
-(I
'i
-
) M12---N
=
i.
Exercises
148
Using
this result in
0 for k
Eq. (7.79) and noting the relation
N enables
VW siimma
IV
i
n
3h2 -
2mici
over
3eT 2
which
was
(
Eq. (7.78)
(A(A
+
X)
can
easily
j(kv)
U
as
(7.81)
(UJ)ki
be done to obtain
-'A!A)
being
(7.82)
defined
kinetic energy from the total kinetic energy. This exercise serves as an indirect evidence for the
formulation
-
by Eq. (2.11). This agrees with Eq. obtained by explicitly subtracting the center-of-mass
with the matrix A
(7.5),
i in
I
to obtain the desired matrix element
(U'(V))ki
-
kIV-1
The
us
E'V I=
given
in Sect. 7.6.
validity of the
8. Small atoms and molecules
examples for the application of the method to atomic and molecular systems. The interaction between the charged particles is the Coulomb force. The long-range character of this force makes the solution of the few-body problems difficult, especially in the case of scattering. We restrict our attention to bound states, where This
chapter
contains
many different methods have been elaborated in the
These calculations
provide an
excellent
past decades.
possibility to test the
of our method. Relativistic effects in atoms
perimental precision of today, which calls
are
efficiency
withi-n the reach of ex-
for very accurate theoretical
calculations.
8.1 Coulombic
systems
The systems of charged by the masses (MI jn2 ,
particles
can
be characterized and classified
mN) and the charges (qj,q2,...,qN) of distinguish systems formed by either equal (unit) or unequal charges, and depending on the masses of the particles one can classify the systems as adiabatic and nonadiabatic ones. ITI the unit charge systems of more than two particles the constituents form a molecule and the binding energy depends only on the mass ratio(s) of the particles. Atoms are good examples for systems with unequal charges. The distinction between the adiabatic and nonadiabatic cases is dictated by the possibility of a simplified treatment in the former case. In the adiabatic case the masses of a group of particles are considerably heavier than those of the rest. The classical examples are the molecular ion (see Complement 8.2). In these H2 molecule and the H+ 2 cases the electrons move faster, while the protonic frame may rotate and vibrate by moving considerably slower. This physical picture is expressed in mathematical form as the Born-Oppenheimer approximation, where the electronic motion is first calculated by assiimin the constituents. One
....
can
Y. Suzuki and K. Varga: LNPm 54, pp. 149 - 176, 1998 © Springer-Verlag Berlin Heidelberg 1998
8. Small atoms and molecules
150
form
CkA
JeXP
2
i AkX) jVk 12K+L YL
M
(f,-I") X Slus
k
(8.1)
UkiXi
Vk
The operator A is introduced to impose proper symmetries on identical particles (antisymmetry for fermions and symmetry for bosons) of the system. In most calculations the index K is assumed to be zero or at most zero or one. In the case of the PS2 molecule of Sect. 8.3, K is set
equal
to
zero.
The details of the calculations
can
be found in
some of the system presented in this chapter. The vari[33, 84, 851 ational parameters included in each basis function are the elements of
for
the -matrix
Ak and the coefficients
Uki, which define the
vk. Since the Hamiltonian used in this
chapter
global
vector
commutes with the
spin operator, the variational trial function can be chosen to 'have a definite spin value S. The value of S influences the symmetry of the orbital part of identical particles. Therefore, possible S values 'have to be tested in general to obtain the ground state. For treating the adiabatic system of small molecules in Sect. 8.4, we use combinations of the generating function g itself as the variational trial function.
8.2 Coulombic
three-body systems
8.2 Coulombic
13-16
figures, that the solution of
lem is
one
151
three-body systems
the Coulombic
three-body prob-
of the most useful benchmark tests to compare different
methods. The accuracy pursued in Coulombic cases is not of purely academic interest, but highly motivated by the high precision of the For
experiments.
example,
the fine-structure
splitting of the
state of the Li atom has been measured with
(parts
per
million).
include relativistic
including
an
the
18 22p2pj
accuracy of 20 ppm numbers one has to
experimental and quantum electrodynamics (QED) corrections, To
explain
terms of the second- and third-order in the fine-structure
require very high accuracy. In addition to the accurate reproduction of the energies, another motivation is that in the variational calculations, even though the energy is good, other physical observables might be less accurately determined. The increased accuracy of the energy, as we will demonstrate, eventually will lead to constant, which
very accurate values of other observables
as
well.
three-body general (m+M+m-)-type stability C A B system with unit charges has been thoroughly explored [861, and the like requirement for stability can be phrased as an empirical rule: In masses" ac[861. charges have to be borne by equal or nearly equal cordance with this prediction, systems such as H- (pe-e-), H2+ (ppe-), Ps-(e+e-e-), HD+(pde-), HT+(pte-), or tdg- are all bound, while (ppe-) or (pe+e-) are most probably not. Here p, p, d, and t are proton, antiproton, deuteron (2 H) and triton (3H) respectively. In the latter case, the particles with opposite charges form an atom, which does not bind the third particle. Some systems, e.g., the muonic molecules such as (ttft-) or (tdft-), remain bound even for L =-;,k 0 orbital angular domain of
The
a
"
,
momenta.
by
A second group of the Coulombic three-body problems is formed systems where not all the particles carry unit charges. The rep-
resentatives of this category are the helium atom (ae-e-) and the helium-like ions, where a stands for the 'He nucleus. These systems often form bound states with L atom
(ape-)
angular
has been observed
momentum states
We have
challenged
(L
our
=
=,4
0
as
[871 30
-
well. The
antiprotonic helium
and studied in very
bigh orbital
40) [88, 891.
method to calculate the
energies of
some
of these systems. In the calculations to be presented the number of basis functions superposed is mostly limited to be modest because our
primary
purpose is to demonstrate the overall
performance of the
cor-
related Gaussians and not to compete with well-established methods sharpened for these systems. We will increase the basis dimension to
8. Small atoms and molecules
152
reach
high
accuracy
only
in
cases
where such calculations
sidered to be important. The results of the calculations
are con-
axe
listed in
Table 8.1. Table 8.2 presents the parameters used in the calculations. The results are compared with those of other (mainly Hylleraas-type or
correlated
hyperspherical haxmonics basis)
calculations. Our results
reasonable agreement with other calculations. In most cases a basis size of K 200 was used in the SVM. The precision of the axe
in
=
results
be
improved by increasing the basis size as can be seen on the selected examples of Table 8.1. In these cases we reach almost the same precision as the other methods. The calculation extends to nonzero orbital angulax momentum states as well, including an e,,c 31 state of the antiprotonic helium atom or the a,mple for the L 3pe bound of the H- ion. These nonzero orbital angustates slightly can
=
lar momentum states the
as
well
as
L
=
0 states have been
investigated
by using global representation. The recovery of the results of other calculations (which are based on several different represenvector
tations of the orbital part of the
usefulness of the
wave
function)
convinces
us
of the
global vector representation. See Complement 8.4 for tdl-t molecule with the global vector
calculation of the
compaxative representation. a
To illustrate the convergence of the energy and the expectation values of average separation distances, the results at different basis sizes
are
tabulated in Table 8.3 for Ps-.
increasing the itself. One
Actually
accuracy is the conventional
can
the limit of further
precision of the computer
notice that at the basis size of K
100 the energy the first four figures of =
is accurate up to six decimal digits, but only the separation distances can be precisely determined.
the basis size, the virial coefficient falls below
high
accuracy of the calculation that all the
calculation
are
By increasing
10-'0, showing by the digits
of the reference
recovered.
Quite a few very accurate methods have been developed to solve the three-body Coulomb problem. It is very difficult to go beyond their precision. This is especially true for the methods which have been elaborated for a given system only, incorporating as much physical intuition as possible into the trial function or into the solution. In contrast with these methods, we use the same trial function, which is of Gaussian nature and therefore it is not tailored to Coulomb problems at all. Still, as the examples prove, one can get a sufficiently good solution in a unified and automatic way knowledge about the systems to cope with.
in
without
a
priori built-
The real power of the
8.2 Coulombic
three-body systems
153
Energies of different Coulombic three-body systems in atomic basis dhnension. See Table 8.2 for the constants which are the K is units. used in the calculations denoted superscripts a, b and c. Table 8.1.
System
State
K
SVM
Other method
K
Ref.
Ps-
Ise
600
-0.262005070226
-0.2620050702328
1488
COI-i-
600
-0.527710163
-0.527751016523
850
200
-0.1252865
-0.1252865
90
200
-112.97300a
-112.9730179a
200
-110.26210a
-110.2621165
a
ttA
Ise 3pe Ise 1PO
200
-105.98292
b
UIL
1D'
-105.982930b
2250
ttA
IF'
200
-101.43131'
-101.43'
200
MIL MIL td[t
IS'
200
-111.36444a
-111.364511474a
[691 [90] [911 [23] [231 [921 [931 [231 [20] [20] [941 [951 [681 [68] [681 [68] [68] [681 [681 [681 [681 [891 [96]
H-
UP
b
b
500 500
1400
IP,
200
-108.17923
1D'
200
-103.40849a
-103-408481a
1566
'He
Ise
600
-2.9037243769
-2.903724376984
700
He
Ise 3se
200
-2.9037242
-2.903724372437
100
He He He
He 'He
-108.179385
2662
200
-2.1752291
-2.175293782367
700
IP0 3po
200
-2.1238423
-2.123843086498
700
200
-2.1331635
-2.133164190779
700
1-D' 3D'
200
-2.0556201
-2.055620732852
700
700
-2.055338993068
-2.055338993337
700
IF' 3F'
200
-2.03125504
-2.031255144382
700
He
200
-2.03125506
-2.031255168403
700
He
IGe
200
-2.02000069058
-2.020000710898
700
He
3Ge
200
-2.02000069062
-2.020000710925
700
VHe+
L=31
300
-3.50760
-3.50763486
1728
'Li+
Ise
300
-7.279913
-7.279913
He
Table 8.2. The constants used in the calculations. The of the electron
mass.
energy in eV.
Set
a
masses are
m,,=7294.2618241, mp=1836.1515. R.
Set b
Set
c
Mt
5496.918
5496.92158
5496.918
Md
3670.481
3670.483014
3670.481
MI,
206.7686
206-768262
206.769
2R,,
27.2113961
27.2113961
27.2116
is the
in units
Rydberg
8. Small atoms and molecules
154
Energy and different separation distances for the (e+e-e-) three-body system as a function of the basis dimension K. The virial ratio 71 is defined by q 11 + (V)/(2(T))I. See Eq. (3.50). Atornic Table 8.3. Coulomble
=
units
are
-E
(,r2+_) (7-2
1 2
used. SVM
SVM
SVM
Hylleraas
(K=100)
(K=200)
(K=600)
[691
0.26200465
0.2620050648 0.262005070226 0.2620050702328
5.489
5.48962
5.489633252
5.489633252
8.548
8.54856
8.548580655
8.548580655
6.958
6.95832
6.95837
6.95837
9.65284
9.65291
9.65291
1
9.652
0.46
77
approach will
x
10-4
be
0.34
more
number of particles is most
as
The
or more
expedition
nium molecule
to
10-6
0.54
x
10-10
x
10-10
where the
than three and the method still works al-
while the other methods need tedious efforts.
particles
larger Coulombic systems
(PS2).
0.23
highlighted in the following sections, more
easily as before,
8.3 Four
x
starts with the
positro-
This exotic molecule consists of two electrons
and two
positrons. The possibility that the PS2 molecule or in general electron-positron system consisting of p positrons and q electrons form a bound system was originally suggested by Wheeler [97], and this question has been extensively studied since then. The existence of the positronium. negative ion Ps- (p 1, q 2) has experimentally been observed [981. The binding energy Of PS2 was first calculated by Hylleraas and Ore [991. To date, it has not been observed yet due to the dffficult experimental circum tances, and this fact has intensified the theoretical interest in solving- this Coulombic four-body problem [65, 100, 69, 101-1041. Actually the positron-electron annibil i n limits the
=
=
the lifetime of Ps2 to few nanoseconds. I-n
obtaining the
solution for the
PS2 molecule, it is useful to note PS2 is invariant with respect to the charge permutation, that is, the exchange of positive and negative charges. that the Hamiltonian for
The trial function should therefore either remain
unchanged or change
sign under the charge permutation operation. The ground PS2 turns out to be even under the charge permutation. its
state of
8.3 Four
or more
particles
155
The convergence of the energy Of PS2 against the increase of the basis dimension is shown in Table 8.4. The fact that the best vari-
ational calculation
[103]
in the correlated Gaussian basis is
already
at the basis size of 400 illustrates the power of the random
surpassed trials.
Table 8.4. The total energies (in a.u.) of the ground state and the bound excited-state of the PS2 molecule in atomic units. K is the basis dimension.
PS2
Method
(K (K SVM (K SVM (K SVM (K SVM (K CG [1031
SVM SVM
=
=
=
=
=
=
100) 200) 400) 800) 1200) 1600)
(L
0)
=
PS2
(L
=
1)
-0.516000069
-0-334376975
-0.516003119
-0.334405047
-0.516003666
-0.334407561
-0.516003778
-0.334408177
-0.5160037869
-0.334408234
-0-516003789058
-0-3344082658
-0-5160024 -0.51601+-0.00001
QMC [1041
possible existence of bound excited-states of the PS2 molecule [84, 85] by taldng all possible combinations of states with L 0, 1, 2 spins. By 0, 11 2,3 orbital angular momenta and S a bound excited state we mean such a state that cannot decay to any dissociation channels. The results of the calculation were negative in I (with negative parity) and all but one case. In the case of L of a second bound-state the existence S 0, the calculation predicts We have examined the
=
=
=
=
of the PS2 molecule. This unique bound-state has been found to be odd under the charge permutation operation. The convergence of the excited-state energy is shown in Table 8.4. Figure 8.1 summa izes the energy spectra of the bound states made up of two positrons and two electrons
together with the
relevant thresholds.
One may ask the question of why the second bound-state cannot decay to two Ps atoms in spite of the fact that it is located above the threshold of Ps (1S) +Ps(IS). Since the total spin of the state is to zero, it
that
they
dissociate into two Ps
can
have
equal spins
(ground state)
and the relative orbital
atoms
coupled provided
angular momentum
1. (Recall that the Hamiltonian preserves spin, between them is L orbital angular momentum and parity.) However, this is apparently =
impossible
equal spins are bosons and their L. Consequently, the PS2 molecule
because two Ps atoms of
relative motion
can
only have
even
156
8. Small atoms and molecules
0
Ps(2P)+e++e-0.1
-0.2
-
Ps(lS)+e++e-'
Ps-+e+
Cd -0.3
IP0
-
Ps(IS)+Ps(2P)
>1 0)
-0.4
-
W
-0.5
-Ps(IS)+PS(IS)
-0.6
Se
PS2
-0.7
Fig. 8.1. The energy spectrum are given in atomic units.
with L
=
I and
of electron and positron systems.
Energies
negative paxity cannot decay into the ground states of
two Ps
atoms, that is, the lowest threshold of Ps(IS)+Ps(IS). Since the energy of this L 1 state is calculated to be E -0-3344 a.u. (see =
Table
=
8.4), (-0.3125 au.) of Ps(IS) + Ps(2P), this state is stable against autodissociation into this which is lower than the next threshold
channel. The
binding energy of this state is 0.5961 e-V from this second threshold, about 40% more tightly bound than that of the ground state Of PS2 whose binding energy is 0.4355 eV from the lowest threshold. are
The expectation values of vaxious listed in Table 8.5. In the
quantities for the PS2 molecule
PS2 molecule we deal with antiparticles, so the electronpositron pair can annihilate. The second bound-state may decay either by annihilation or by an electric dipole transition to the ground state [851. The most dominant annihilation is accompanied by the emission of two photons with energy of about 0.5 MeV each. To have an estimate for the decay width due to the annihilation we have substituted
8.3 Four
or more
particles
157
Properties of the ground and excited states of the PS2 molecule. positrons; are labelled I and 3 and the electrons are 2 and 4. Because (r14) of charge permutation symmetry, some equalities hold, e.g. (r12) Table 8.5. The
=
(r32)
=
(r,34).
Atomic units
PS2
(L
are
used.
0)
=
PS2
(L
=
1)
(r12) (r13)
4.4871530
7.56881891
6.0332070
8.8575844
(r212
29.112633
80.173836
46.374735
96.085514
2
r13 3
( r12) 3 (r 13 )
253.04611
1041.3251
443.85244
1226.7955
(412)
2807.2718
15612.112
4 13
5202.0371
17939-574
(r 21) (r.3' -2) r12 -2) r13
0.36839693
0.24082648
0.22079007
0.147244820
0.30310361
0.16081514
0.073444303
0.032230158
(1'12'TI3) ('r12 'T14) (5(rl2)) (5(rl3))
23.187368
48.042757
(r ) 12
5.9252651
32.131079
0.0221151
0.0112091
0.0006259
0.00014591
(V2) 1
-0.258001894
-0.16720401
(V1 V2)
0.1307732538
0.091656853
(VI'V3) 11 + (V) /(2(T))
-0.0035446132
-0-016109693
*
the
0.3
10-9
x
probability density
(J(r12))
,
of
into the formula
rannihi
=
47
x
10-6
electron at the
position of
a
positron,
[1021
(MC622 ) 2hc(TIjJ(rj-T2)jTf) 62
=
an
0.36
4ir-(hc)
4
hcaOI(5(rI2))i
(8.2)
equal to J0 (Tf 15(rl r2) ITf) with ao Roughly speaking, the Metime is inversely probeing portional to the probability. The Metime due to the annihilation is estimated to be 0.44 ns. This is twice that of the ground state (0.22
where
(,S(r12)), given
in a.u., is
-
the Bohr radius.
ns). dipole transition from the excited state emits one photon with energy of 4.94 eV. The decay width Idjp&,, for this transition The electric
8. Small atoms and molecules
158
is calculated
through the reduced transition probability B(EI) dipole operator D.:
electric
16v
Tdipole
':--
(E) 3B(EI;
I-
0+),
-->-
he
9
for tile
(8-3)
where I
B(EI; 1-
--+
0+)
=
)7 I (Oind I DI, ,
_
M) 12
(8.4)
1
with 4
qi 1ri
Djz
X4
-
I Ylp (ri
-
X4)
IIIere X4 is the center-of-mass of the
(4.94 eV)
tation energy
B(El)
dipole
P-92 molecule and E is the exci-
of the second bound-state. We calculated the
value and obtained
electric
(8-5)
-
B(EI)
=
0.87e2a 2. The lifetime due 0
transition has been found to be 2.1
ns.
The
to the
branching
dipole transition is thus about 17 % of the total decay rate. Therefore, both branches contribute to the decay of the excited state of the PS2 molecule. Its lifetime is finally estimated to be about of the electric
The excitation energy of 4.94 eV found for PS2 is different by 0.16 eV from the corresponding excitation energy (5.10 eV) of a Ps 0.37
ns.
atom. This difference
to be
large enough to detect its existence, e.g. in the photon absorption spectrum of the positronium gas. Before discussing the spatial distribution of the PS2 molecule, let us recall that the average distance (r+-) between the positron and the electron is 3 a.u. in the ground state of the Ps atom, while it is 10
a.u.
first excited state. The root-mean-square radius
in its
(ri
-
X'j
)2),
culated to be 3.61
surprising a
system of
of the
a.u.
5.66 a.u., 1.5 times
if
seems
larger
one assumes a
ground
The
rms
PS2 molecule
is cal-
radius of the second bound-state is
than that of the
ground
state. This is not
that the second bound-state is
Ps atom in its
(spatially extended)
state of the
(rms),
ground
state and
a
excited state. To check the
essentially
Ps atom in its first
validity
of this
as-
sumption, we have restricted the model space to include only this type of configurations. This can be achieved by a special choice of the uki parameters of that is, the
Eq. (8-1). The energy converged to -0.323 a.u., Ps(1S) +Ps(2P) system with zero relative orbital angular
momentum forms state of the
a
bound state with energy close to that of L
PS2 molecule, therefore this configuration
is
lik-ely
=
to
1
be
8.3 Four
the dominant
configuration in this molecule.
uration, the Ps- + e+
or
Ps+ + e- with L
=
or more
There is
particles
159
second
config-
a
I relative orbital
motion,
intuitively may look important because two oppositely charged particles attract each other, but it is barely bound (E -0.315 a.u).
which
=
1 state The average distances in Table 8.5 show that in the L the two atoms are well separated. In fact we can estimate the root=
mean-square distance d between the two atoms
by
2
d2
I'l +7'2
7'3 +7'4
2
2
(2(r12 ) 2
4
(,r213)
+
-
) (8-6)
2(rl2*rl4)
The symmetry properties of the PS2 wave function are used to obtain 6.93 a.u. the second equality. Using the values of Table 8.5 yields d =
for the L
=
I excited state and d
state. One cannot
give
a
direct
Arij
The correlation function defined
gives
(TfIJ(ri
=
more
-
-
(r,2
of the
(,rj)2
ground ground or
is
0
laxge.
by
(8.7) on a
system than just various average
quantity can be calculated for the correlated Gaussians
by using Eq. (A.30) =
=
=
r) ITf)
detailed information
distances. This
Tf with L
rj
for the L
a.u.
geometrical picture
excited state because the variance
C(r)
4.82
=
or
(A.136).
0, C(r) becomes
monopole density.
a
ground-state wave function of only r, which is called the
For the
function
For the excited-state
wave
function with L
=
1,
monopole and quadrupole densities. and the electron-positron electron-electron the 8.2 displays Figure correlation functions r 2C(r) for the ground-state of the PS2 molecule. The peak position of the electron-electron correlation function is shifted to a larger distance than that of the electron-positron correla-
C(r)
consists of the two terms of
tion function. The latter has much broader distribution and reaches
farther in distances
compared
to the
corresponding function
of
a
Ps
atom.
Figure
8.3
displays
the electron-electron and
electron-positron
cor-
relation functions for the second bowid-state of the PS2 molecule. As I state consists above, the correlation function for the L of the monopole and quadrupole densities and their shapes depend on the magnetic quantum number M of the wave function. Of course the M-dependence of the shapes is not independent of each other
mentioned
=
8. Small atoms and molecules
160
0.020
0.015
Cd 0.010
0.005
0.000 0
Fig.
4
8.2. The correlation functions
molecule. The solid dashed
the
2
curve
r
6
8
r
(a.u.)
2C(r)
for the
12
ground
14
state of the
PS2
denotes the electron-electron correlation and the the electron-positron correlation. For the sake of comparison, curve
electron-positron correlation function for
dotted
10
a
Ps atom is drawn
by
the
curve.
but is related
by the Clebsch-Gordan coefficient. See Eq. (A.136). quadrupole density is contributed from only the P wave of the electron-positron relative motion, while the monopole density is contributed by both S and P waves. Figure 8.3 plots the correlation funcThe
tions for both
(a)
M
=
0 and
(b)
M
=
1. As the correlation function
axiaUy symmetric around the z axis and has a reflection symmetry with respect to the xy plane, the correlation function sliced on the xz plane is drawn as a function of x (x > 0), z (z > 0). The electronelectron correlation function has a peak at the point corresponding to the average distance of 7.57 a.u. The electron-positron correlation function has two peaks reflecting the fact that the basic structure of the second bound-state is a weakly coupled system of a Ps atom in the L 0 state and another Ps atom in the L 1 spatially extended state. The peak located at a larger distance from the origin is due to the P-wave component of the PS2 molecule. The hydrogen and positronium molecules can be considered as members of the same family as both are quantum-mechanical fermio-nic is
=
=
four-body systems of two positively and two negatively charged identical particles. But they are at the opposite ends of the (M+M+m-m-)-
8.3 Four
x
x
(a.u.)
(a.u.)
x
z
(a.u.)
z
(a.u.)
0 x
or more
(a.u.)
161
particles
z
(a.u.) z
(a.u.)
(a.u.)
Fig. 8.3. The correlation fimctions rC(r) in atomic units, raultiplied by one thousand, for the second bound-state of the PS2 molecule. The z comI for 0 for (a) and M ponent of the orbital angular momentum is M (b). Drawn on the xz plane are the corresponding contour maps. =
=
8. Small atoms and molecules
162
type Coulombic systems called biexcitons biexciton and two
(or
biexciton
holes,
molecule),
is observed in
a
a
or
bound
variety
excitonic molecules. The
complex of
two electrons
of semiconductors
[105, 106].
See also Sect. 10.1. The biexciton molecule is characterized mass
ratio
o-
=
m/M. Apart
by the connection, however, their
from this
properties are radically different, e.g., H2 is an adiabatic but PS2 is a highly nonadiabatic system. Moreover, while in the case of the H2 molecule many bound excited-states have been observed experimentally and later studied theoretically, in the case of the PS2 molecule only the ground state and the unique excited state discussed above have so far been predicted theoretically. See also Complement 8.3 for the stability of the biexciton molecule. Figure 8.4 displays the dependence of the binding energy of the biexciton molecule on the mass ratio a m/M. The changes of the binding energies in the ground state (L 0) and the excited statues with L 1 and negative parity is si-milar. Both the ground and excited states become less bound by changing the mass ratio from H2 to PS21 though the binding of the excited state decreases to a somewhat lesser =
=
=
extent. The energy of the transition from the excited state to the
ground state is also shown in this figure. This transition may take place in an external field, for example. By increasing the mass M of the positively charged paxticles tbowa,rd infinity, one arrives at the energy of the C I H,, 2p-x state of the H2 molecule. This state is formed by an excited H-atom and a groundstate HI-atom. Consequently, a statement similar to the case of the PS2 molecule is valid for the biexcitonic molecule: The second bound-state of the biexciton molecule is of
a
ground-state
dominantly formed by
exciton and
an
L
=
an
interacting
pair
1 excited-state exciton.
The rule that the Pauli
principle forbids odd partial waves between identical bosons also applies to the biexciton with L I and negative The second bound-state of the biexciton molecule cannot decay parity. to two ground-state excitons. A somewhat similar situation exists in the 3p, state of the H- ion as well, where its second bound state cannot decay due to the parity conservation. By changing the mass ratio in that (M+m-m-) system, however, this kind of state disappears for oI and the Ps- ion is known to have only one bound state. =
=
Tables 8.6-8.8 show tem
our
results for various other Coulombic sys-
.
The tivated
investigation of the stability of positronic atoms has been nioby the use of positrons as a tool for spectroscopy (positron
8.3 Four
or more
163
particles
0.15
0.10 CU
0.05
0.00 0.0
0.4
0.2
0.6
0.8
to
M/M
Fig.
8.4. The
mass
ratio
o-
binding
=
m/M.
state, and the solid
energy of the biexciton molecule as a function of the curve is the binding energy of the ground
The dotted
curve
is that of the first excited state with L
Table 8.6.
curve
Energies of different Coulombic four-body systems
I and
=
is the energy difference, multiplied negative parity third, between the first excited state and the ground state.
The dashed
by one
in atomic
units. K is the basis dimension.
System
State
K
SVM
Other method
K
Ref.
PS2
Ise IP0
800
-0-516003778
-0.516002
400
[102] [107] [1071 [1071 [1031
800
-0.334408112
600
-7.478058
-7.47806032
1589
Li
Ise 1PO
1000
-7.410151
-7.410156521
1715
Li
'D'
1000
-7-335520
-7.335523540
1673
'HPS
Ise
1200
-0.7891964
-0.7891794
PS2 Li
8. Small atoms and molecules
164
Table 8.7.
Energies of different Coulombic five-body systems
in atomic
units. K is the basis dimension. The Li- energy with K = oo is the extrapolated one [1091, where E = -7.500577 is given by multiconfiguration
Hartree-Fock calculations with K
=
2997.
System
State
K
SVM
Other method
K
Ref.
Be
Ise Ise
500
-14.6673
-14.667355
1200
600
-7-50012
-7-50076
00
[1081 [1091
Li-
Ise Ise
(27r+, 3,-1-) Li +
e+
Table 8.8.
Energies
200
-0-5493
1000
-7.53218
of different Coulombic
six-body systems
in atomic
units. K is the basis dimension.
System
State
K
SVM
(37,-+,37r-)
IS' Ise Ise
300
-0.820
600
-7.73855
1000
-14.692
Li + Ps
Be+e+
annihilation
spectroscopy) is whether
in condensed matter
physics.
An
intrigu-
or a chemically stable system containing a ing question positron or a positronium could be formed in the various targets. This
not
be answered
only by a sophisticated calculation or experiment because the mechanism responsible for binding the positron to the neutral atom is the polarization potential present in the atom+e+ system. The boundness of the hydrogen positride (positronium hydride) HPs was predicted theoretically by Ore [99] in 1951 and it has recently been created and observed in collisions between positrons and methane [1101. The properties of HPs is discussed in [851. The use of the SVM proved for the first time that the positronic lithium (Li+e-r [1111 and the positronic beryllium (Be+e+) [1121 are stable. We see from the tables that the positron separation energy of the positronic lithbun is 0.054 a.u. Below the Li+e+ channel the Li++Ps channel is question
can
open and the
energy of the
positronic lithium is only 0.0022 a.u. against the dissociation into the Li++Ps channel. A calculation has to be accurate at least to 10-3 a.u. to answer the stability of the
binding
positronic litbium. Likewise, the positron separation energy of the
Be+e+ system
typically
is
only
about 0.01
0.025
a.u.
a.u.
one
Due to the
tiny binding energies of
has to. be able to reach
high
accuracy
8.4 Small molecules
165
6-particle systems. A naive picture of these systems is that the positron orbiting around the neutral atom slightly polarizes the negative electron cloud, and the positron is bound by the resulting in these
attraction.
(as a fall N-body solution) for the investigation of the stability of much larger systems (e.g. Sodium plus positron) To extend the method
question. One can, however, try to use a "frozen core approximation?'. In this approximation the positively charged core is considered to be passive (its polarization is neglected) and the problem is is out of
model space where the single-paxticle orbitals are orthogonal to the core orbitals. One has to solve the modified Schr6dinger
solved in
a
equation of the form
(H + AP)Tf
=
with
ETI-
P
(8-8)
Oi) (Oi iGoccupied
produces wave functions that are orthogonal to the core orbitals provided the positive constant A is large enough. The projection operator P is an example for the nonlocal potentials discussed in Sect. 7.5. See also Complement 11.4. One can validate this approximation by comparing it to the "exact" fuU N-body calculation for Li+e+. This approximation turns out to be very accurate, reproducing the first six digits of the result of the full calculation [1121. Assuming that the accuracy holds for larger systems, one seems to find the stability of positronic sodium (Na+e+) [1121. which
8.4 Small molecules As it case
was
mentioned at the
demands
a
special
beginning
of this
assumed to take combinations of the form 1
g(s; A, x) where
Aij
and
functions. The x.
exp
=
Note that
s
(_2
=
is
a
Mx +
f8li S21
"generator x
chapter, the molecular
treatment. The variational trial function is
-7
(6.19)
9x),
SN-11
coordinates"
(8.9) are s
parameters of the basis
are
chosen
conforming
to
set of relative coordinates. Our aim here is
to calculate the energy of the
system which
can
be
directly
com-
pared to experiment without recourse to the adiabatic treatment like the Born-Oppenheimer approximation. Each basis function includes !V(IV 1) /2 + 3 (IV 1) parameters to be optimized. These parameters -
-
8. Small atoms and molecules
166
describe various correlations. The matrix A describes the electronic
correlations and motions, while the generator coordinate s makes the function flexible and allows us to represent several "peaks" of
wave
the
distribution
density
the holes
when, for example
in the
well separated and the electrons them. around 0 the function By choosing s are
=
hydrogenic limit, "atomic orbits"
axe on
(8.9)
at the
'has its
maxhnum
origin and this limit is suitable around a1, when m/M the paxticles with nearly equal masses are moving equally fast. At the hydrogenic limit, when the motion of the heavy particles are very slow compared to the light ones, the density distribution has several peaks =
=
axound the attractive centers, and to represent these configurations need to shift the maximum of the trial functions out of the origin
we
by choosings appropriately. The usefulness of the generator coordinates in the basis function (8.6) can be understood by the following example. Let us try to calculate the energy of the IH+ this basis with and without 2 by using (that is by setting them to zero) the generator coordinates. The latter
form
corresponds
to the correlated Gaussians for L
of that system is -0.6026
basis of K
=
a.u..
300 Gaussians
0. The energy Without the generator coordinates a
give
-0.5999
a.u.
=
for this molecule. The
inclusion of the generator coordinate immediately h-nproves the con10 basis states vergence and one can get -0.6024 a.u. by using K =
only! Table 8.9 shows ions
consisting
examples
of calculations for the molecules and the
of protons and electrons..
Table 8.9. Ground state
energies of small molecules
in atomic units. K is
the basis dimension.
System
K
"OH+ 2 0OH2 "OH+ 3
SVM
Other method
K
Ref.
50
-0.602634429
-0.602634214
160
100
-1-17445
-1-174475714
1200
100
-1.34351
-1.343835624
600
[1131 [1131 [1141
167
Complements 8.1 The cusp condition for the Coulomb potential It is desirable that the trial function satisfies the proper asymptotic behavior or the special boundary condition as demanded by a given
special boundary condition, the cusp condition [115] known for the Coulomb potential, by using the hydrogen atom as an example. The local energy for the hydrogen atom is given by Hamiltonian. We discuss
h2 1
Ej"'c
a
192 Tf
-
-r-2
2m Tf
where hl is the
angular
h2
2 (9Tf + r
-5r- )
momentum
1
e2
2
(8-10)
1 IIf
+
.M r2 Tf
r
operator. As
was
discussed in
Sect. 3.2, the local energy for the exact wave function turns out to be a constant. The Coulomb potential in the local energy gives a singular 0. For the local energy to be a constant, this singular behavior at r =
behavior must be
for
compensated
by
the kinetic energy term. For
an S wave, where the wave function Tf has no angular dependence, 12Tf 0 and the constancy of El.,r requires that the second term in =
the bracket in
Eq. (8.10) cancel
the
singular
behavior of the Coulomb
potential: I
2
aTf)
Tf 9r
-me h2
r=o
ao
where ao is the Bohr radius. It is easy to see that an exponenexp(-r/a) for the radial part of the ground-state wave
tial form of
function leads to sen
a
constant local
density
if and
only
if
to be ao. The constant of the local energy is then
-0/(2ma2) drogen
=
_Me4/(2h2)
=
-El
as
expected,
atom ionization energy without the
a
is cho-
equal
where E, is the
to
hy-
proton recoil effect, that
is, the well-known energy of 13.6 eV.
angular dependence, it has to take care I/r singularities. Then Tf may be expressed I/r as a product of radial and angular parts: Tf r'R(r)Y(S?), where s 0. is a positive constant and R(r) is assumed to be nonzero at r Substituting Tf into Eq. (8.10), we obtain For
a
general
case
when Tf has 2
and
of both the
=
=
h?
Eloc 2m
+
1
a2 R
( R -5r-2
h2 1 2 __1 Y 2Tar2 Y
2(s + 1)
8(8+1)
1 M
+
+ r
R 9r
r2
2 -
_.
r
(8-12)
Complements
168
As R does not vanish at
r
0, 1IR gives
=
origin. The condition that the local following result: 1
I
( OR) 9r R
We know that the second
12y equation
no
singularity
at the
energy is constant leads to the
=
(s + 1)ao'
=o
rise to
(8.13)
S(S + 1)y
is satisfied if and
only
if
s
is
a
positive integer 1, and then the first equation determines the correct behavior of R near the origin as R(r) oc exp(-r/(l + I)ao). Equations known to be the cusp conditions. Let us attempt to solve the S-wave hydrogen atom variationally with Gaussian basis functions, exp[-(a/2)(r/a0)2J' where a is a vari-
(8.11)
and
(8.13)
are
ational parameter. We functions that lead to
use a
Gaussians
as
an
example of such basis
rather accurate solution but
are
poor in
satisfying the cusp condition. When a single basis function is used, the optimal value of a is 16/(9z), giving the minimum energy of -8/(37t-)EI -0.849EI. A combination of a few terms approximates the ground-state energy quite well. The parameter values of a are determined by the SVM. Table 8.10 shows sample results of such calculations obtained with the code given in [81]. The calculation with five Gaussians already reproduces the energy up to three digits. The wave function obtained with ten Gaussians can reproduce both the energy and the mean values of r and I/r fairly accurately. The over=
lap of the wave function with the exact wave function is very close to unity. We may conclude that the Gaussian basis can predict physical quantities to high accuracy. Of course, the solution does not satisfy the proper asymptotic behavior at large distances and, moreover, always gives zero for the cusp value of Eq. (8.11). The local energy displayed in Fig. 8.5 for the variational wave function indicates that with increasing K it tends to show smaller and smaller deviations from the exact wave function except for the singular points mentioned above. The local energy at large r deviates from the correct value because the Gaussian basis has the wrong asymptotic behavior. It is possible to generalize the above arguments for the cusp value in a system of particles interacting via Coulomb potentials. Evaluating the cusp value for a pair of particles with charges qj and qj, we obtain
(If 16(ri
-
2'j) alriarj I Of)
(T/IJ(ri
-
rj)ITI')
I-tijqiqj h2
(8-14)
where tzij is the reduced mass of the two particles. The left-hand side of Eq. (8.14) is expressed with the matrix elements involving
C8.2 The chemical bond: The H+ ion 2
J(r)
=
J(r)/(47rr2)
used to test the
The cusp values for a pair of paxticles quality of the variational solution at the .
are
169
often
particles'
coalescence.
Table 8.10. Variational solution for the
hydrogen
of Gaussian basis functions. The last
shows the exact values. Ei is the is the Bohr radius.
hydrogen K
atom ionization energy and ao
E
E,
((_L_)-2) ao
atom with
a
number K
row
ao
((_E_)2) ao
ao
Overlap
1
-0.8488264
1.131774
0.8488284
1.499996
2.650706
0.9568351
3
-0.9939585
1.903352
0.9939409
1.491519
2.922759
0.9987560
5
-0.9996191
1.986219
0.9995692
1.499147
2.991284
0.9999446
10
-0.9999998
1.999700
0.9999958
1.500004
3.000014
0.9999999
-1
2
1
1.5
3
1
K= 1
0-, IN.
-4-
K=3
K=10
1 %
0
5
10
15
20
rlao Fig. 8.5. hydrogen
The local energy
curve
plotted for the variational solution
of the
atom. K denotes the number of Gaussian basis functions.
8.2 The chemical bond. The
H+ 2
ion
Quantum mechanics enables us to understand the chemical bond, which is responsible for the formation of molecules from isolated atoms. The chemical bonding phenomenon involves the delocalization of electrons in an atom to gain attraction from the other nuclei when
Complements
170
the atoms
close to each other. We take up the simplest possiH2+ ion, to understand what an important role the
come
ble molecule, the
Hellmann-Feynman
and virial theorems
[1]
of the chemical bond. See
play for clarifying the origin
for detail.
ffilly quantuin-mechanical description of a molecule is a complex problem. This problem is usually simplified by using the BornOppenheimer approximation, where the electronic motion is separated from the nuclear motion, considering the fact that the electron mass is much smaller than that of the nuclei. One starts with determining the motion of the electrons for a fixed configuration R of the nuclei and The
ground state, of energy U(R), of the electronic system. Then one assumes that,when R varies, the electronic system always remains in the ground state corresponding to R, that is the electrons follow adiabatically the motion of the nuclei. The chemical bond is then determined by studying the nuclear motion in a potential energy V(R) which comprises the Coulomb repulsion between the nuclei and obtains the
U(R)Ht 2
Let R be the distance vector between the two protons of the ion and
v
be the
vector of the electron with
position
respect
to the
center-of-mass of the protons. The electron motion is determined the Hamiltonian
2[t
Note that R is
equal
is
the
m
to
e2
e2
p2
H,e( R)
IT
just
a
-
by
stage. The reduced mass jL where M is the mass of the proton and
parameter
2Mm/(2M + m),
(8-15)
IT + RI 2
Al 2
at this
of the electron. The electronic energy U(R) is the lowest of the Hamiltonian (8.15). It is clear that U(R) becomes
mass
eigenvalue
only. The Schr8dinger equation for the Hamiltonian completely separated in elliptical coordinates with respect (8.15) to the foci, R/2 and -R/2. We do not need its exact solution in the following discussion, but note that it is well approximated by the variational calculation using a trial function of the form a
function of R is
Tf
-01"
where at
s.
(Z' R) +'OL' (Z' R),
01, (Z, s) is the charge Z is
The
(8.16)
_
2
2
ls a
hydrogenic orbital
of radius
variational parameter and its
ao/Z
centered
optimal
value
is deternUned to miniraize the energy for each R. The optimal value 1 for R ---* co. At 0 to Z 2 for R of Z decreases froin Z --+
cc
the system will switch
=
=
=
R
over
to
a
configuration
of the
hydrogen
C8.3
atom and the
Stability
proton. Between these
of
hydrogen-like
molecules
extremes, Z is
two
function of R. The energy U(R) is -4E, (2M/(2M + and approaches -E, (MI (M + m)) for R oo.
a
m))
171
decreasing for R
=
0
By using the virial theorem (3-49) and Eq. (3.56) with A R, Ry7 R, we can show that the expectation values, (T) and (W), of the kinetic energy and potential energy of H,..(R) satisfy the relation d
2(T)
+
(W)
+R
dR
U(R)
=
(8.17)
0. '9
Here we have used the fact that WA+R. aR W R d dR to
=
-W and
Rr-aR -U(R)
U(R), enables us equation, together with (T) + (W) U(R). the protons, between of the potential express (T) and (W in terms This
V(R)
=
U(R)
+
=
(e 2IR),
as
follows:
d
d
(T)
=
-U(R)
-
R
dR
U(R)
=
U(R)
=
-V(R)
-
R dR
=
2U(R)
+R
dR
For the chemical bond to must have
(T)
,
clude from
occur
2V(R)
in the
+R
dR
V(R)
-(8.18)
-
R
H+2
system, the potential V(R) --+ oc) -Er at some point
V(R V(Ro:) < -E-r have to be met. Since oc (see Eq. (3.53)), we can con-2E, at R at equilibrium (R Eq. (8.18) that, RO), the electronic
a
deeper
minimum
that is, V(Ro) Er and (W)
Ro,
e2
d
d
(W)
V(R),
=
than
-
0 and
-- -
-
=
kinetic energy is increased and the electronic potential energy is decreased. The lowering of the electronic potential energy is large enough to cancel the
repulsion
for the chemical bond.
and
V(Ro)
-
V(R
-+
between the protons, and that is
According to
oo)
=
an
exact
responsible 2.00ao calculation, Ro -
-2.79 eV.
challenging to perform a nonadiabatic calculation in which no separation of the electron and nuclear motion is made. The validity of the Born-Oppenheimer approximation can be tested in such a calculation. Furthermore, the development of the nonadiabatic treatment for a smaU molecule is of importance in its own right because the adiabaticity may be questionable when the electron is replaced with heavier particles like the muon or the pion. The excellent results obIt is
tained in Sects. 8.2-8.4 indicate that realistic nonadiabatic, calculations are
in fact
possible
in the correlated Gaussian basis.
172
8.3
Complements molecules
Stability of hydrogen-like
The existence of bound states of systems composed of particles with unit charge attracts considerable attention. We discuss this problem here
by applying
of the
some
principles discussed
in
Chap. 3. always bound
The system of the hydrogen-like atom, (M+m-), is and its binding energy is equal to jL/2 in units of e
1, where p stability of a
Mm/(M+m)
=
is the reduced
-mass.
=
I and h
=
What about the
hydrogen-like molecule (M+M+m-m-)? This system is characterized by the mass ratio om/M. Two well-known examples include the hydrogen molecule (a < 1) and the positronium molecule PS2 (o1). Another example is the biexciton molecule [105, 1061. See =
=
Sect. 8.3. The value of 0 <
Emits,
< I.
c-
o-
of the biexcitons
A molecule is bound
can
vary between the two
provided that
the threshold
of any dissociation channel is higher than the lowest energy of the system. The lowest dissociation channel is (M+m-) + (M+m-) for this system, and its threshold energy is Eth the hydrogen-like molecule has been studied
(see Fig. 8.4). However,
-ft. The
stability of
numerically
in Sect. 8.3
=
theoretical argument [1011 makes it possible to prove that the system is bound for arbitrary values of o-, that is, the ground-state energy E of the system is lower than Eth. The proof relies
on
stability
the
a
scaling property of the Coulombic Hamiltonian and the point of the proof given in [101] will be
Of PS2. The basic
shown below. The Hamiltonian of the system -ff
=
-ffS
+
1
_as
=
4[t +
(P21 +P22+P32 +P42)
(
2 e
1
1 +
the
mass
nothing
I -
r34
7'12
1
4-M
is
4m
1 -
r13
) ( 2+ Pi
r14
-
T'24
r23
),
a
-+
=
A H(PS2), M,
(8.20)
2
P44:) P22_P2_ 3
but the Hamiltonian of
mX of
I
I -
a
(8.21)
-
system
(X+X+X-X-)
particle X being equal to 2p. By applying transformation ri Ari (,X m,/(21L)), we obtain I-Is
as
(8-19)
-
HS
is written
HA,
I
HA
(M+M+m-m-)
a
with
scaling
=
(8.22)
C8.3
where the
m,.
is the electron
Stability and
mass
molecule. This
positronium HS, ES,
state energy of
by Es
=
(2A/Tne)E(PS2).
from the the
can
of
hydrogen-like molecules
H(PS2)
is the Hamiltonian for
indicates that the
equation
be obtained from that Of
Note that this relation
groundPS2, E(PS2),
also be obtained
can
Helh-nann-Feynman theorem and the virial theorem. In fact Eqs. (3.38) and (3.53) enables one to obtain (mx 2/-t)
of
use
173
=
d
a
ES
=
dTax
1
( fisjTax-Hsjfis)
=
-- !PSjTSjPS) Tax
I
(8.23)
ES,
-
Tax
which leads to the solution that is the
(Ps
ground-state
wave
E,,/mx
independent of mX. Here function of HS and TS is the kinetic is
energy of HS.
According to the Ritz theorem following relation
the
ground-state
energy
El of H
satisfies the
El :! ((fiS IHS + HA PS)
(8.24)
-
The term
HA is odd under the interchange P of the masses M and -HAP. Then by using that (PS is invariant under i.e., PHA
m,
=
(0SjpHApj(pS) _((pSjHAp2j0S) ((PsIHA10s) The side of 0. Eq. (8.24) is thus equal right-hand -(0sjHAj(Ps) to ((fisjHsj0s) ES, and we obtain P
we
have
El
-
=
Eth :! ES
-
(-A)
=
2A =
M,
(E(PS2)
-
(-
=
M.
2
))
(8.25)
-
The energy of -m,/2 is the threshold for P,92 to decay into two Ps atoms. If the stability Of PS2 is established, that is, E(PS2) +
(m,/2)
0, then
<
we can
the
hydrogen-like
[991
calculated the energy
showed its
immediately from Eq. (8.25) that always stable. In fact Hylleraas and Ore Of PS2 by using a simple trial function and
conclude
molecule is
stability.
It is of interest to examine the
stability of a hydrogen-antihydrogen
like molecular system, (M+m+M-m-) [1041. The lowest threshold of this system is now Eth -(M/4) (m/4) corresponding to the dis-
sociation
(M+M-)
+
(m+m-)
and gets
deeper
and
deeper
as
M in-
creases, which is in contrast to the case of the hydrogen-like molecule. It is un I ikely that the hydrogen-antihydrogen system gets lower energy
Complements
174
than the threshold. It is thus
expected
that there is
a
critical
mass ra
M/m beyond which the system becomes unstable. According to the calculations hn [104, 1161, the stability limit is M/m < 2.1.
tio,
Application of global vectors to muonic molecules The aim of this Complement is to show the utility of the global vector v of Eq. (6.4) for describing the angular part of the variational trial function. As a test example we take up a Coulombic, three-body system, the tdIL molecule, which has attracted much attention in relation to muon catalyzed fusion [1171. The excited P state especially plays a key role in the fusion since it lies close to the threshold for the decay 8.4
to the
ttt
atom and the deuteron.
The basis function
we use
is
fKLM of Eq. (6.27).
matrix elements for this function
Chap.
7 and the
appendix.
can
be calculated with the method of
Without loss of
parameters ul and U2 which define U +U22 1. Each basis function for =
The Hamiltonian
v
generality
can
be normalized to
given
a
the variational
satisfy
set of KLM values thus
contains at most four nonlinear parameters, three of which
come
from
positive-definite to assure the 6DG, general be expressed as A
the matrix A. The matrix A has to be finite
where G is one
Of
norm
a
fKLM,
2x2
and
can
in
=
orthogonal matrix
(
cosO
-sinO
sin,& cos
?
specified by just
parameter V and D is a diagonal matrix including positive d, and d2 values.
two
diagonal
elements of
The accuracy of the variational solution depends on how the parameters ul, V, di, and d2 are given [331. The most naive choice would
be to take G
ing only
a
as a
single
unit matrix
(6
=
0)
,
which is
equivalent
set of relative coordinates x, and then to
to
us-
try
to
im-
by including successively higher partial appropriate choice of K and ul values. Many examples show [20, 29, 30, 311, however, that this type of single-channel calculation does not work well, especially in the case where the adiabatic approximation is questionable as in the present example. Gx correspond to other The matrix G may be chosen to let y particular coordinate sets such as the so-called rearrangement cha n nel [201. (For Gx to correspond to the rearrangement channel, the length scales of x and y in general have to be modified appropriately.) Three possible rearrangement channels expressed in the Jacobi reach convergence
plied by
waves
an
=
particles d, t, and tL are labelled particle 1, 2, and 3, the three patterns in Fig. 2.2 correspond to (tlL)d, (/-td)t, (dt)IL arrangements, respectively. If x is understood coordinate set
are
drawn in
Fig.
2.2. If the
C8.4
Application
of
global
vectors to muonic molecules
175
to stand for the coordinate set of the
(tl-t)d arrangement,
ordinate set
arrangement is obtained
to the other
the
co-
corresponding by choosing such an appropriate,& value that is uniquely determined by the change of the coordinates. With this choice of ?Y, we can write ,'N DGx i Ax dj y', + d2y22 Dy The following three simple types of bases were chosen: =
=
=
*
(i)
K
0 and A
=
dDG,
=
trices
connecting only explained above.
K
(ii) (iii)
=
K
0
or
1 and the choice of A is the
0 and A
=
where G is restricted to the
=
special
ma-
the three sets of rearrangement channels
dDG,
where G is
same as
now an
in
case
as
(i).
arbitrary orthogonal
matrix.
As
mentioned in Sect. 6.2, the angular part with K 0 describes only the stretched configuration, 11 + 12 L, and therefore was
=
-
the basis of type (i) allows rather limited angular correlations between the particles. In fact, the possible (Ili 12) values are given by
(1, L
-
1), (1
=
0,
...,
L).
(ii)
Basis
is
an
clude the non-stretched
extension of basis
(i) to in1, possible (11,12)
coupling. With K 0,..., 1), (1 L) are also allowed. Though K is set to in basis zero (iii), the angular correlation can be taken into account through the cross term of the exponential part of the basis function. In bases (i) and (ii) the factors exp(- FbAx/2) are always "Correlationfree" in a particular channel, that is, it contains no cross term in the exponential part. In these bases it is through the inclusion of all rear-
(I
+
1, L
-
I +
rangement channels that type ment
for
(iii), (that
=
=
=
one
the contrary, is, coordinate
on
through the general
takes
one
care
of the correlation. In
set) explicitly;
case
(i)
contained in basis
form of A.
expected, the
re-
to other bases. The correlation
poor compared is too restricted to obtain
are
(i)
basis of
just any one arrangethe correlations are allowed
Table 8.11 shows the results of calculations. As sults with
a
needs to consider
a
realistic
description
of the system. The angular correlation, which is taken into account in basis (ii), certainly improves the energy over case (i). The basis functions
first six for L
(iii) give
figures 0
even
better
of the most
energies. In fact they reproduce the precise variational calculations [20, 23, 921
2 states. Even with K
0, the use of the full A matrix incorporate important correlations between the particles. A Gaussian basis of type (ii) is employed in the Coupled=
enables
--
one
to
=
the
Rearrangement-Channel Gaussian (CRCG) basis variational calculation of [201, where the angular part is, however, represented by the
176
Complements
coupling of type (6.3). The fact that the D state energy with the type (iii) calculation of dimension 200 becomes slightly lower than that of [20] with 1566 basis functions confirms the importance of successive
optimization of the nonlinear parameters. Further increase of the dimension can improve the accuracy rather easily in the angulax function of Eq. (6.4) as seen in the case of S and P states. The basis function used in [23, 921 is correlated exponential (CE) which a
caxeful
takes the form of
exp(-air,
-
T21
-
JT2
-
T31
-
7IT3 -'r1j)
for the orbital motion with
Multiorbital
nonzero plied by some polynomials determined The are momentum. by the , parameters a, -y angular calculations These 4.2.2. in Sect. random tempering explained using the exponential function give very precise results.
Table 8.11. The total
energies of the tdIL molecule. Parameter set a of by superscript b, where set
Table 8.2 is used except for the case indicated b is used. Atomic units are used. Dimension
E
200
-111.29346
200
-111.36398
200
-111.36444
600
-111.3645077
CRCG[20] CE[231
1442
-111.364507
1400
-111.364511474
(i) SVM (ii) SVM (iii) CE[231
200
-108.13122
200
-108-17914
200
-108-17940
1800
-108.1795424
800
-108-179361
CRCG[20]b CE[921b
2662
-108.179385
1900
-108.1793881
(iii)b CRCG[201b CE[921b
800
-99-660367
2662
-99.660548
1900
-99.6605507
SVM
(i) (ii) SVM (iii) CRCG[201
200
-103.37067
SVM
200
-103.40824
200
-103.40849
1566
-103.408481
L
Method
S
SVM SVM SVM
P
SVM
SW
P*
D
(i) (ii) (iii)
(ifl)
b
SVM
9.
Baryon spectroscopy
This
chapter is devoted to applications of the SVM to the constituent quark model of baryons. There are several different models of baxyons and this is a very hot topic nowadays. The discussion of the different models and their relative merits are far beyond the scope of this book, and our only aim here is to show the applicability of the SVM to solving tbree-body dynamics in constituent quark potential models. The constituent quark model assumes that the baryons comprise three quarks with dynamical quark masses and the quarks interact via phenomelogical potentials. These potentials contain a confining term and other terms which determine the fine structure of the spectra. are no free quarks observed in nature, the confining inter-
As there
action hinders the "ionization7 of the the
baryons in the constituent usually taken as a power
models. This interaction is
quark potential potential (V(r) rP). The second part of the interaction is traditionally chosen as the one-gluon exchange potential (OGEP) which is a color analogue of the Breit interaction of quantum electrodynamics (QED). There is an alternative possibility, the meson exchange interaction, which is motivated by the fact that, despite its quite successful reproduction of the ground-state spectra of the mesons and baryons, the OGEP may not be adequate for describing the excited states as mesonic effects or quark-antiquark excitations become important. This chapter includes examples for both choices to illustrate law
-
how the SVM
can
be used for these systems with different interac(nonrelavistic and semirelativistic) forms of
tions and with different
the kinetic energy. These simple quaxk models have obvious limitations
(e.g., large relativistic corrections can be anticipated for light quarks), but their success
indicates that relativistic and field theoretical effects
incorporated
in the paxameters. In
so
Y. Suzuki and K. Varga: LNPm 54, pp. 177 - 186, 1998 © Springer-Verlag Berlin Heidelberg 1998
far
as
are
somehow
sophisticated approaches
9.
178
Baryon spectroscopy
quantum chromodynamics (QCD) do not provide results for these systems, the application of constituent quark models is justified. based
on
9.1 The trial function in the constituent
quark
model carries space-, spin-, flavor-, and color degrees of freedoTn. The flavor is an extension of the isosphn to include up(u),
quark (q)
The
down(d), strange (s), charm(c), bottom(b),
and
top(t) quarks.
baxyons are colorless: The three quarks in a baryon form a color-singlet state, that is, they are totally antisymmetric with respect to the interchange of the color coordinates. For example, the eight baryons of spin 1/2, N (p, n), A, Z (Z+, V), Z-), and EF (EE,", belonging to a family of octet baryons, consist of three quarks of u, d and s flavor varieties. The product of the spin-, flavor-, and spacepart of their wave functions has to be symmetric to comply with the generalized Pauli principle. The spin paxt is represented by the spin functions X(S,,)Sm, (see Complement 6.5): The
TIT),
xmi a
X(O).1.1 22
2
r6 ( -V
11 X(1) 'TT
The flavor
IR
=
I (
S exp
where X and metrizer.
9.2
same
wave
-
I III)
functions
wave
baryons with the The trial
21111)
chaxge
function for
-1 -
are
2
are
-v/2
(I TIT)
-
I
ITT)), (9-1)
I ITT)
shown in Table 9.1, where the
arranged baxyons is a
are
in each column.
combination of the terms
6Ax) X(S12)SM (T12)TMT
the
(9.2)
spin and flavor functions, and S is
a
sym-
One-gluon exchange model
one-gluon exchange potential (OGEP) model has proved to be quite successful in describing the spectra of the ground states and
The
various
properties of
the
baryons [118, 119, 1201.
In this model the
9.2
Table 9. 1. Flavor to
wave
model
One-gluon exchange
ftmctions of baryons
179
arranged columnwise according
charge +2
Baryon N A
UUU
+1
0
uud
ddu
uud
ddu
-1
ddd
71,= (ud du) d + du) 71, 72 (u
A
-
s
2 '
UUS
dds
s
ssd
SSU
S?
SSS
7L (ud
Ac
-
2 ,
El-
72= (ud
UUC
+
du) c du) c
ddc
72 (ud
Ab
1
uub
Eb
vr2-
-
du)b
(ud + du) b
ddb
quarks exchange massless but color-charged gluons and the qq potenas the color analogue of the Breit interaction [3, 1211. Consequently, the simplest version of the OGEP includes a Coulomblike I/r piece and a color-magnetic ("hyperfine") piece:
tial is taken
7r -
i<j
6mimj
(AI7. AC j)(ffi-0j)j(rj-rj)'
where the Gell-Mann matrix
generator for the ith uct,
1:8
=1
introduce masses
dence,
(9-3)
S
quark
AF (a) AC (a). (In
A9 (a) (a
and
(A'9 Ajc) -
a more
tensor and other
spin-orbit,
=
z
is the color
SU(3)
scalar
prod-
stands for
a
elaborated framework order
higher
essentially
terms.)
one can
The
quark
very weak flavor depenflavor-blind. In a nonpertur-
mi in the denominator introduce
but the interaction is
8)
1,
a
bative treatment the contact interaction leads to
a
Because
singularity.
of this and because of the finite spatial extent of the constituent
quarks
the delta function should be smeared out, and it is to be substituted by a Gaussian or a Yukawa form factor.
Among the several different parametrizations of the OGEP we have potential [1221 as an example:
chosen the AM
3(A9 8
Vij H
-
A(7) 2rr.'
x
--+Ar-A+ r
3mpaj
exp
Tij ij 1.
7F 2
a
3
Pij
-
-
O'j
,
(9.4)
180
9.
Baryon spectroscopy
where the
length r is given in units of inverse energy (I fin 1/197.327 MeV-'), and the smearing parameter pij is given iTi the forioa Pij
=
-B
(
A
_
"
2mimj
mj + Taj
)
(9-5)
The
315 MeV, quark masses used in the calculation are Tn,, Tnd 577 MeV, m, 1836 MeV and ?nb 5227 MeV. The parameters m , were determined by a fit on a large sample of meson states in every ffavor sector [1221: They are n 0.1653 0.5069, rs' 1.8609, A GeV2 A 0.8321 GeV, B 0.2204 and A 1.6553 GeVB-'. The matrix element of the color part can be easily obtained because the baryons are in a color singlet state. By noting the relationship between the color-exchange operator P53 and (AF Ajc), =
=
=
=
=
=
=
=
=
=
=
7
-
PC== zy
2
C (A9-Af)+-, 3 '
(9.6)
J
the matrix element of
(A9.A'7) 3
between the
S
simply
to
states reduces
-8/3.
The SVM results
[1221
color-singlet
[81, 1231
are
compared
with the calculations of
in Table 9.2. The
agreement between the two results is good. data [1241 are included for guidance. One has to
The
experimental compaxe the energies relative to the nucleon ground state and one has to bear in mind that this potential gives good results for the mesons and no free parameter has been introduced to fit the baryons. Table 9.2. The
tial
[1221.
masses
of baryons
The numbers in
(in MeV)
parentheses
axe
as
the
predicted by the AU poten-
masses
relative to the nucleon
mass.
Baryon
Ref.
N
[1221
SVM
Experiment [1241
998
995
939
A, Z. Ab Zb
1154(156) 1231(233) 1343(345) 1674(676) 2296(1298) 2466(1468) 5642(4644) 5849(4851)
1306(311) 1149(154) 1228(233) 1339(344-) 1674(679) 2290(1295) 2467(1472) 5635(4640) 5849(4854)
1232(293) 1116.(177) 1192(253) 1318(379) 1672(733) 2285(1346) 2455(1516) 564150 (470250)
-Fb
5808(4810)
5803(4808)
A
A E
0
9.3
9.3
Meson-exchange
model
181
model
The OGEP model is successful in and
Meson-exchange
describing ground-state energies
properties of baryons, but for the excited
states
a
number of
delicate problems remain unsolved. No model has been able to
explain, orderings in light- and strange-baryon spectra in a simple three-quark description of baryons. The source of the problem can be easily understood by the following argument: The ground state of the nucleon N(939) has positive parity followed by the positive paxity 1/2+ Roper resonance N(1440) and the negative-parity 1/2- and 3/2- states N(1535) and N(1520), respectively. The parity of the states follows as (+, +, -, -). In the case of the A, the order of states follows a (+, -1 +1 -) pattern. The order of parity of the spectra in a simple harmonic-oscillator quark model is (+, -, +, In the case of the linear confinement the level spacing changes, but the order for
example,
the correct level
of states remains the
The
same.
wave
function of N and A differs
only in flavor, but the OGEP is flavor independent (though it slightly depends on flavor through the mass difference of light and strange quarks in Eq. (9.4)). Therefore it predicts the same level order for both N and A in contradiction to the experiments. To resolve this problem an explicitly flavor-dependent mesonexchange interaction has been introduced [1251. The essential difference between the two approaches is that while the OGEP of type (ai o-j) V(rij) acts on the color and spin space, the Ei<j (AF AjC) 3 -
-
%
meson-exchange potential Ei<j AFA- 37(aj-o-j)V(rjj) acts on the flavor and the spin degrees of freedom. The meson-exchange potential combined with a semirelativistic form of the kinetic energy is the second example of the application of the SVM for baryon spectroscopy. In this example no gluon-exchange mechanism is explicitly introduced to describe the qq interaction but only a meson-exchange potential is employed to generate the baryon '1
spectra. The
three-quark Hamiltonian is a sum of the semirelativistic Idmeson-exchange potential V and the confinement potential VO + Cr: netic energy, the
3
3
H=E /--2+ ?
Mi2 P71i +M?
where the
+
(Vii
meson-exchange potential
+
V"
+
Clri
V reads
as
-
ri
1)
(9.7)
9.
182
Baryon spectroscopy 7
3
Vij (r)
V7r (r)
=
ij
1: AF (a),XjF (a) + VI
a=4
+ VI7 (r) AF i (8) ij
The form factor of the
g2
f
1
47-,
2
AjF (8) + 3 Vj'j7' (r)
Oj
-
aj
meson-exchange potential
V
ij
AF i (a) AjF (a)
(r)
3
a=1
12mimj
2
I
6 -Ay
e-A-yr
A 'I
A2'Y
r
r
I
paxametrized
is
as
(9-9)
1
coupling constant gy is taken to be g8 for the pseudoscalar octet mesons (7r, K, n) or go for the pseudoscalar singlet meson (n,), respectively and the cut-off mass Ay is assumed to be given by
where the
Ay
=
Ao
(9.10)
+ riLy
q, 77. For the constituent quark masses we take the 500 MeV. Parameters typical vdues, Mu, Md 340 MeV and m, of the Hamiltonian are listed in Table 9.3.
for each -/
=
K,
-Ft,
=
=
=
Table 9.3. Parameters of the semirelativistic constituent
quark model with
meson-exchange potential Fixed parameters
Quark
masses
Meson
[MeVI
masses
[MeVI 928
Mui Md
M,
Ar
AK
An
A'a
340
500
139
494
547
958
41r
0.67
Free parameters
(golg8)2 1.34
Ao
[fin-]
K
Vf0 [Me,V
fln-21 2.33
-416
0.81
2.87
C
The -matrix elements of the semirelativistic kinetic energy in Eq. (9.7) can be calculated with the method presented In Sect. 7.6. The matrix elements of
culated
by-using
AF(a)AjF(a) 71
in the flavor space
Table 9.1 and the
explicit
can
be
easily
calm-
form of the GeH-Ma=
Meson-exchange model
9.3
183
matrices. It is useful to know the action of the
AF(a)A-3 (a)
ator
the flavor
on
wave
ffavor-changing operfunctions. As is summarized in
Z
9.4, for example the ffavor function uidj, when acted AF (a) M'(a), transforms to 2djuj uidj. I
Table
E
3
upon
by
-
a=
Z
3
Table 9.4. Effect of
Operator 3
flavor-changing operators
on
quark pairs
UU
dd
SS
ud
us
ds
Ar (a) X3 (a)
UU
dd
0
2du-ud
0
0
1
AF (a) AF j (a)
0
0
0
0
2su
2 sd
4
-1 UU
-1 dd
4UU
lud
-!us
-ids
3
3
Z
/\F(8),XjF(8)
3
3
3
3
_
predictions of this model axe shown for all light- and strange-baryons with mass up to M < 1850 MeV; the nucleon mass is normalized to its mass of 939 MeV, which determines the value of the confinement potential parameter V0. All masses corresponding to three- and four-star resonances in the most recent compilation of the Particle Data Group [124] are included. The result is good, reproducing the spectra of all low-lying light and strange baryons [123, 126]. In particular, the level ordering of the lowest positive- and negative-parity states in the nucleon spectrum is reproduced correctly, with the 1/2+ Roper resonance N(1440) falling well below the negative parity 1/2- and 3/2- states N(1535) and N(1520), respectively. Likewise, in the A and Z spectra the positive-parity 1/2+ excited baryons, A(1600) and E(1660), fall below the negative parity 1/2-In
3/2-
9.1 the SVM
Fig.
states,
A(1670)-A(1690),
and the
1/2-
state
E(1750),
respec-
tively. In the A spectrum, at the same time, the negative-parity 1/2-3/2- states A(1405)-A(1520) remain the lowest excitations above the A
ground
state.
remark is in order about the necessity of employing a semirelativistic kinetic energy operator in the three-quark Hamiltonian (9.7). Certainly, this is only an intermediate step toward a fully At this
place,
a
covariant treatment but it
already
allows
us
to include the kinemat-
ical relativistic effects. In any nonrelativistic approach these effects get compensated by the potential parameters, but there one faces a
disturbing
consequence of
constituent
quark
and
c
v1c > 1,
is the
where
velocity
of
v
is the
light.
mean
velocity of the
184
9.
Baryon spectroscopy
18001700-
El
E----1
L-J
=
16001500M
[MeV]
14001300-
12001100-
N
1000-
A
900-
1
1+
1-
3+
3
2
2
Y
Y
5+ 5 Y Y
1-
3+ 3i -2
Fig. 9.1. Comparison of low-lying baryon spectra predicted by the mesonexchange potential model with experiment. The solid lines denote the calculated spectra, and the shaded boxes show the error bars.
experimental
masses
with
their
(Continued
on
the next
page.)
One may think that the idea of the meson-exchange interaction acting between quarks is unfamiliar though it has produced the good results. To
judge
its merits the present calculations have to be
tended into several directions
ex-
(prediction physical observables, e.g. factors, decay widths, etc.). The complexity of the NN interaction is basically due to the compositeness of the nucleon. As N, A or Z belong to the same octet family, it is natural to attempt a unified description of the NN, AN or EN interaction from the underlying qq potentials. This goal is at present too difficult to achieve directly on the basis of QCD. In so of
form
far
as
QCD as yet presents no results, the effective theory formua microscopic quark-cluster model [1271 seems to be justified.
lated in
9.3
(Continued
from
Fig
Meson-exchange
model
185
9. 1.)
E] 1800
1700 1600
-
-
F-7
17.71
t
H
r--.7.
-
r__,_T
p G=
r_.j
0
p"
-
-
1500M
[MeVj
1400-
13001200-
1100-
A
1000900
1
1
_2
1
2
_2
_2
1+
1-
3+
1
_2
1800
1700 1600
1500 M
[MeVJ
1400
1300 1200 1100
1000
900 3-
1
11
3+
_2
_2
1
_2
186
The
9.
Baryon spectroscopy
intermediate-range
attraction of the
cannot be accounted for
by
tribution between the colorless the
baryon-baryon interactions produces no con-
the OGEP because it
It is not clear yet whether introduced above leads to a realistic de-
baryons.
meson-exchange potential scription of the NN interaction or not. It is probable that the OGEP combined with the meson-exchange potential is a useful, effective vehicle to obtain a reasonable description of both baxyon spectra and baryon-baxyon interactions at the same time. See, for example, [128] for the attempt along this line.
in solid state
10.
Few-body problems physics
This
chapter is a collection of examples of the application of the SVM to few-body problems in the field of solid state physics. The examples include excitonic complexes, biexcitons, and quantum dots with and without magnetic field. The biexciton and two
(or excitonic molecules), the bound state of two holes
electrons, provides
an
interesting few-body problem,
since it
may be thought as a positronium. or a hydrogen molecule with variable electron and hole masses realized in semiconductors. The biexcitons
also often confined and
are
can
be modelled
as
two-dimensional
(2D)
systems. In recent yeaxs there has been much excitement in the possible applications of ultrasmall. systems with a length scale of 10-100
A,("nanostructures"). The technological motivation (in electronics and optoelectronics)
is that the smaller components
to manufacture very small
contain
very few electrons
only a
are
faster. It is
possible
nanostructures, often called "dots", which
(N
<
10).
The
quantum-mechanical
effects in these small systems axe very important and their properties are strongly N-dependent. The energy spectra of few-particle quantum
dots call for theoretical interpretation. In confined in
are
a
box of sizes
L, L.,
a
nanostructure the electrons
and L,. If
L,
-
Ly
>>
L, the
quantum dot is quasi 2D. The confinement is usually modelled by a harmonic-oscillator potential. This system is somewhat similar to an atom in
nature, where the "confining" potential is the Coulomb
traction of the electrons to
quantum dots
are
Throughout this chapter
we use an
tron. When the dielectric constant of
length =
a
effective
mass
me*
for
material is denoted
an
by
elec-
r., the
and the energy will be measured in "atomic units" with Bohr h2n/ (M*e2 ) and haxtree (two times the Rydberg energy) e-
radius a*
2R*
at-
nucleus in the atom, and therefore the often referred to as "artificial atoms". a
=
e2/(Ka*), respectively.
Y. Suzuki and K. Varga: LNPm 54, pp. 187 - 211, 1998 © Springer-Verlag Berlin Heidelberg 1998
188
10.
Few-body problems
10.1 Excitonic
in solid state
physics
complexes
The excitonic
complexes have considerable importance in the development of semiconductor physics and spectroscopy. The bound state of a positively charged hole with an effective mass m and an electron with an effective mass m,* is called an exciton in semiconductor physics. The mass ratio ocan between 0 (hydrogenic limit cchange m*/m*h and oI (positronium limit) depending on the material and other factors. Like in the case of hydrogen, where not only the hydrogen atom but the hydrogen molecule (H2), or the H2+, H-, H,+ ions are bound, the system of N, electrons and Nh holes can also be bound. The latter system is called an excitonic complex. These excitonic complexes, including the charged excitons, have been subject of intensive experimental [105, 1291 and theoretical [99, 130-1351 investigation. The properties and structure of these systems strongly change with the mass ratio, and, by approaching the two limiting cases, one arrives at two completely different worlds. The interest in these systems has =
=
e
=
been intensified when the advance in semiconductor
technology has possible the fabrication of artificial nanostructures with diameters comparable to atomic distances. This restricted geometry has a prominent effect on the dynamics of the excitonic species. In the following we present the ground-state energy and some other properties of 2D excitonic complexes. The general Hamiltonian made
be written
can
IV.
H
as
N
pi,
i=1
2m,
i=IV'+I
N
j>i=N' +l x
TC.
r"Iri
-
e2
+
2Mh
3>z=l N,
e2
E where
Ne
pi,
+
=
N
e2
E E
rj I
i=1
j=N,+1
is the dielectric constant of the
material,
and the position
2D vectors in the xy plane. In atomic units introduced in this chapter, the Hamiltonian becomes dimensionless and the envectors
axe
depends only
(Note
that
do not need to
specify what dependence is hidden in the atomic units.) This is easily understood by introducing dimensionless coordinates and momenta by ri* ri/a* and pi* pil(hla*). Then the ergy
value of
K
on o-.
is used because the
r,
=
Hamiltonian
(10-1)
we
is reduced to
=
10.1 Excitonic N,
H
2
2P'
2
P
2(o-N,
i=N +I
+N
N
+ Z
i>i=l
3
-
N'-
E
+
j>i=N,+l
2
0-
*2
E
+
189
N
/
1
2R*
complexes
Jr
-
'r
7,
3
P'
N,) N
E E
I
i=1
j=N,+l
J'r j*
r 3
I'
At this point, a comment is due. What we consider here is a welldefined quantum-mechanical problem to be solved. This is, however,
experimental situation. In the realistic case there considered, for example the effect of other electronic bands, the possibility that the interaction is different from the pure Coulomb force due to the confinement in the z direction, etc. The discussion of the importance of these effects is beyond the scope only are
model of the
a
many other factors to be
of this book.
As the
problem here is practically the same as that discussed in the chapter of small atoms and molecules (Chap. 8), the same trial function is used (see Eq. (8.1)), except that it is tailored to the 2D case, that is, the position vectors have only x and y components. The results for case
o-
=
0 and
o-
=
1
are
and in Table 10.2 for the 2D
shown in Table 10.1 for the 3D
case.
between the 3D and 2D results is the
The most
large
striking difference binding
increase in the
0) in energy in the 2D case. E.g., the binding energy of R2 (with o2D is about 8 times of that in 3D. The increase of binding has been =
found
experimentally
Table 10. 1.
as
well.
Energies and binding energies (in a.u.) of 3D
of electrons and holes for two orbital
angular
cases
momentum and
of the
mass
ratio
o-.
exciton complexes
L and S
are
The asterisk refers to the states which
are
found to be unbound. The binding
energy is understood with respect to the nearest threshold.
E(o-i)
B(o-=O)
B(o-=I)
-0.500
-0.250
0.500
0.250
-0.528
-0.262
0.028
0.012
-0.602
-0.262
0.102
0.012
-1.174
-OM6
0.174
0.016
System
(L, S)
E(o-=O)
eh
(0,0) (0,1/2) (0,1/2) (0,0) (0,1/2) (0,1/2)
eeh ehh
eehh eeehh eehhh
*
-1.344
the total
spin of the exciton complex, respectively.
0.169
190
10.
Few-body problems
in solid state
physics
Table 10.2. Energies and binding energies (in a.u.) of 2D excito-n complexes of electrons and holes for two cases of the mass ratio o-. M is the z component of the total orbital angular momentum and S is the total spin. See also the caption of Table 10.1.
System
(M, S)
E
eh
(0,0) (0,1/2) (0,1/2) (0,0) (0,1/2) (0,1/2)
-2.00
-1.00
2.00
1.00
-2.24
-1.12
0.24
0.12
-2.82
-1.12
0.82
0.12
-5.33
-2.19
1.33
0.19
eeh ehh eehh eeehh
eehhh
(o-
=
0)
E
(o-
=
1)
B
(o-
*
-6-82
1.50
=
0)
B
(o-
=
1)
10.2
equality 2(r+-)
distances have to fullfil the
=
Quantum
dots
V(r--)2 + (r++)2.
191
Ta-
ble 10.3 shows that this is not satisfied. Moreover, the fact that the uncertainties
Arij
(,ri2j
-
(rij 2
are
large
means
that
no
such in-
terpretation is possible. There is only one case where the uncertainty is in very small, that is the distance between the (heavy) positive charges with accordance in that In can limit. case one the hydrogenic assume, the adiabatic approximation, that the positive charges are fixed at the distance of 0.37 a.u. The equilibrium distance in the H2 molecule in 3D is 1.4
a.u.
Table 10.3. ratio
mass
Average distances in 2D biexcitons (eehh) as a function of the E.g., (r--) is the mean distance between the two negative
o-.
charges. Atomic 0-
(r++) (,r+-) 2 (r _) (r 2+) 2 (r _)
=
0
units o-
are
0.4
=
used. a
=
0.7
0-
=
0.67
1.26
1.55
0.37
1.14
1.49
1.80
0.47
0.93
1.16
1.38
0.59
2.09
3.19
4.33
0.14
1.69
2.95
4.33
0.31
1.29
2.05
2.88
1.80
dots
10.2
Quantum
Rapid
advances in semiconductor
rication of nanostructures called
[136, 137, 1381. electrons,
are
In
I
quanturn dots
technology
have led to the fab-
quantum dots a
few
or
artificial atoms
electrons, typically
2 to 200
bound at semiconductor interfaces. These few-electron
(2D) electron gases (MOS) structures are
systems arise when homogeneous two-dimensional metal-oxide-semiconductor
of heterojunctions
or
laterally confined
to diameters
comparable
to the effective Bohr
ra-
dius of the host semiconductor. The interest in quantum dots arises not only from the prospect of new technological applications but also from the desire to understand the in
physics of a few interacting electrons
external field. Needless to say, we concentrate on the few-electron where the effect of electron-electron interaction and their corre-
an
case
lation
seems
to be very
important.
192
10.
Few-body problems
The axtificial atoms
in solid state
consider here
we
of N electrons confined in 2D
physics are
modelled
by
a
system
by harmonic-oscillator potential. Before starting with artificial atoms in 2D we present a calculation for the energies of 2D "natural" atoms where the "co-ofi-ohn ' potential is a
Coulombic. The results for few-electron atoms and ions Table 10.4. The much
binding energies
are
again, due
are
to the 2D
given
in
geometry,
than in the 3D
case. Otherwise the properties of these laxger systems axe rather similax to those of their 3D counterparts. For ex-
ample,
akin to
3D,
the
H, Li
or
Be atom
can
bind
an
electron,
extra
while the He atom cannot.
Table 10.4.
Energies
of atoms and ions confined in 2D. M is the
nent of the total orbital
units
are
(M, S)
H
(0,0) (0,1/2) (0,0) (0,1/2) (0,0)
He Li
Be
momentum and S is the total
z
compo-
spin. Atomic
used.
System
H-
angular
Energy -2.00000000 -2.24
-11-89 -29-87 -56.77
The
ex6mple of quantum dots considered here is a system of 2D electrons in a quadratic confiniD potential. The Hamiltonian reads as N
N
9
Pi
H
+
_M
2m*
(V__
+
ivy)
M
where v., and v.
by
N
I 2
e2
*WO2,r? +
(10.2)
exp( IiUr),
(10.3)
-
2
are
the
x
and y components of the 2D vector
v
defined
10.2
Quantum dots
193
N V
(10.4)
Uiri.
The parameters ui and A of the basis function SVMDue to the external field
are
determined
(the single-particle potential),
by the
the Hamil-
tonian (10.2) of the system is not translationally invariant. Correspondingly, the basis function (10.3) is written in terms of single-
particle coordinates instead of relative ones. This basis function depends on the center-of-mass coordinate, and the energy obtained will be the total energy of the system, unlike the previous cases where the energy of the center-of-mass motion was always subtracted. The case of the quadratic confinement is very special, because in that case the energy of the center-of-mass can be separated. In general, for an arbitrary single-particle potential, however, this cannot be done. 1 electron is trivial and its solution is a simple The case of N 2 an analytical solution exists for cerharmonic oscillator. For N I a.u., the ground state tain frequencies [139]. For example, for hwo which is 3 is precisely reproduced by the numerical cala.u., energy culation. Not surprisingly, as is shown in Table 10.5, if the quadratic =
=
=
confinement is strong, the spectrum is close to that of a harmonic oscillator. In the realistic cases (hwo < I a.u.), however, the eigen-
strongly deviate from those of the harmonic oscillator. This example illustrates the importance of these systems: by changing the confinement one can "tune" the properties of these artificial atoms.
values
Energies of N-electron quantum dots with the z component of angular momentum M and the total spin S. The numbers parentheses are the eigenvalues without Coulomb interaction. Atomic
Table 10.5.
the total orbital in
units
are
used.
N
(M, S)
hwo
2
(0,0) (0,1/2) (0,0)
3.000
212.198
7.220
525.12
3 4
Figure
=
I
hwo
(2.000) (5.000) 10.600 (6-000)
10.1 shows the
=
654.45
100
(200.00) (500.00) (600.00)
energies of the ground and excited
states
of the energy of a harinonic-oscillator quantum) of two electrons as a function of the inverse square root of the oscillator constant, -1/2 If there would be no Coulomb interaction between the elec-
(in units
(hwo)
.
10.
194
Few-body problems
in solid state
physics
trons, the energy divided by hwo would be given by lines parallel to the horizontal axis., The deviation from the
straight lines is entirely interesting feature is that states but the ground state
due to the Coulomb interaction. Another level
crossings
always
occur
remains M
between the excited
0.
12
10
8
6
4
2 0
2
4
6
8
10
(hcoo )-1/2 10.1.
Fig. potential
Energy
with
an
levels of two electrons confined
by
a
oscillator constant wo. Atomic units
harmonic-oscillator
are
-used.
Quantum dots
10.2
195
3.5 3.0 2.5
2.0 ;-4
t__I (14
1.5 1.0 0.5 0.0
1
Fig.
3
2
4
(a.u.)
r
10.2. Electron densities of two-electron
quantum dots
as a
function
origin. The solid, the dotted and the dashed 5, respectively. Atomic 0.2, hwo 2, and hwo
of the radial distance from the curves
units
correspond to hwo
are
used.
1.2
1.0
0.8
A
0.6
0.4
0.2
0.0 0
4
2
6
r
Fig.
8
10
(a.u.)
(dashed curve), four- (dotted curve), (solid curve) quantum dots. The oscillator frequency of the
10.3. Electron densities of two-
and six-electron
quadratic confining potential
is hwo
=
0.2. Atomic units
are
used.
196
Few-body problems
10.
in
solid state
physics
peak density inreases with the number of electrons, that is, the equilibrium configuration is realized on rings of expanding diameter. In the case of N=4 electrons, for example, the paxticles may move along a ring, while they are situated at the vertices of a square to TniniTni e the Coulomb repulsion.
10.3
dots in
Quantum
magnetic field
A dramatic feature axises when the quantum dots are subjected to magnetic fields: The ground states are stabilized into magic-number states. When the
state
Tn
from
jumps owing
to the combined effect of the electron correlation and
appear
the Pauli
agnitude of the magnetic field is vaxied, the ground magic number state to another. Magic numbers
one
principle [140].
The Hamiltonian of the system is N
N
I
I
H
=
E 2m* (pi + e-A,)2 + -
N
2
us assume
ward in the
z
2
Wd 'ri
e2
E
1' Iri
i>i=l
Let
2
,,
-M
-
C
+
given by
that
a
(10.5)
-,rj I*
homogeneous magnetic
field B is
direction. The above Hamiltonian
can
applied
down-
then be rewritten
in the form
H
P'
2
+
-m w
2
2m*
N
2-
r?-
1 -
2
hw,
Ii,
e2
rdri
-
rj I
where u),
eB/(m*c) is the cyclotron frequency and w
We
the
=
V/W--22. rl4+wo
ignored magnetic interaction energy due to the electron spin. See Complement 10.1 for details. It is shown there that the part of
the Hamiltonian
corresponding to the single-particle motion can easily
be diagonalized by the use of associated Laguerre polynomials. Depending on the strength of the magnetic field, both the single-particle
10.3
Quantum dots
in
motion and the correlation due to the Coulomb
sive roles in
determining
function is the
same as
197
repulsion play deci-
the structure of the quantum dots. The trial
in the
For the two-electron with the
magnetic field
previous
case our
solution
section.
calculation is in
perfect agreement
[1411. The spectrum of two elec0) is shown in Fig. 10.4. See also
in
given analytical spin-singlet state (S Complement 10.1. The harmonic-oscillator frequency wo is taken as 0.01 a.u. The most intriguing phenomenon seen is the level hwo crossing. Due to these level crossings, the ground state changes with the strength of the magnetic field. This is shown in a magnified scale on the right-hand side of Fig. 10.4. If there is no Coulomb interaction between the electrons, no level crossing occurs. trons in the
=
=
10.0 18 9.5
16
9.0 14 +It
+Z
8.5
12 8.0
10 7.5 8
7.0 0
1
2
3
4
0.0
0.5
1.0
1.5
2.0
2.5
(,)C/ ("0
(J)C/(00
10.4. Energy levels of a two-electron quantum dot in magnetic field function of the cyclotron frequency w,. The figure on the right-hand side magnifies the level crossing. The oscillator frequency of the quadratic 0. Atomic units are 0.01. The total spin is S confining potential is hwo
Fig. as a
=
=
used.
Figures
plot the energy levels of a three-electron total spin being S 3/2) against the orbital
10.5 and 10.6
quantum dot
(with
the
=
198
10.
in solid state
Few-body problems
physics
25
20
15
10
5 0
2
4
6
8
10
12
14
COC/ coo 1.0.5.
Fig. z
Energy
levels of
a
three-electron quantum dot
component of the total orbital angular
IVI
=
6
(dotted curve),
and M
=
9
momentum M
(dashed curve)
frequency of the quadratic confining potential S
=
3/2.
Atomic units
are
used,
is hwc)
belonging
=
3
to the
(solid c=e),
orbital motion. The =
1. The total
spin
is
10.3
Quantum
3
4
dots in
magnetic
field
199
15
14--1
13
12
44
10-
9
8
7
6 0
2
1
5
6
7
9
8
10
M
Energy levels as a function of the z component of the total angular momentum, M, of a three-electron quantum dot in a low 3/2. Atomic 1). The total spin is S 0.2, hwo magnetic field (hw,
Fig.
10.6.
orbital
=
units
are
used.
=
=
200
10.
Few-body problems
of these orbits
in solid state
physics
larger (see Eq. (10.16)). The energy of the dots interplay of the single-paxticle energies and the interaction energy. By increasing the strength of the magnetic field, similarly to the case of low magnetic field, the ground state does -not take all possible values of M. By changing the magnetic field the states with M 3,6,9,12,... "magic numbers' become ground states. is determined
axe
by
the
=
40-
35-
30-
25
-
201
10
0
1
2
3
4
5
6
7
8
9
M 1.0.7. The
Fig. hwo
=
1).
same as
Atomic units
in
are
Fig.
10.6 but for
high magnetic
field
(hw,-
=
6,
used.
change of the ground-state angular momentum quantum numbers as a function of the magnetic-field strength is illustrated In Fig 10.5 for a three-electron quantum dot with spin S 3/2. This figure shows the energies of the states with M 3 (solid curve), M 6 M and 9 orbital The calmomentum. angular (dotted) (dashed) culation shows that by changing the strength of the magnetic field the orbital angulax momentum of the ground state of this system is M 3 for w,lw < 5, it is M 6 for 5 < w,lw < 11, and then it The
=
=
=
=
=
switches
=
over
to M
=
9, reproducing the "magic number"
sequence
10.3
(M
3,6,9,..).
Quantum dots
in
magnetic field
201
States with other orbital
angular momentum never ground state and are not included in the figure. The previous examples dealt with spin polaxized electrons (S 3/2). Actually, spin polarized (S 3/2) and spin unpolarized (S 1/2) states compete to be the ground state as the magnetic field changes. This very interesting phenomenon is illustrated in Fig. 10.8. As the magnetic field increases, the ground state changes as (M, S) (1, 1/2) -+ (2,1/2) --+ (3,3/2) and so on. At low magnetic field the lowest energy state is spin unpolarized (S 1/2) and as the magnetic field increases the spin becomes polaxized. The orbital angular momentum (M) changes continuously (in steps of unity). =
become the
=
=
=
=
=
5
4
3
C5
2
0 0.0
0.5
1.0
1.5
2.0
2.5
(j)C/ (1)0 1-0.8. Spin (S) and z component of the total orbital angular momentum (M) quantum numbers of the ground state of a three-electron quantum dot
Fig. as
a
function of
denote M and
magnetic field strength. hwo S, respectively. Atomic units are
=
1. Solid and dotted lines
used.
202
10.
10.4
Few-body problems
Quantum
in solid state
physics
dots in the generator coordinate
method In Sects. 10.2 and 10.3 the correlated Gaussian basis of
Eq. (10.3)
is
to obtain the solution of the
employed not the only
quantum dots. However, th is is Gaussian basis function for the quantum dots. In fact, the
quantum dots offer
(8.9)
that
was
application of another basis function study of small molecules. That basis func-
very nice
a
used for the
by the matrix A which describes the correlated particles and also by the generator coordinates s which represent several peaks of the density distribution of the
tion is characterized motion of the
allow
us
to
system. In this section
consider the quantum dots in 3D and
we
field. The
state of the
quantum dot
apply belong
can ground 3angular momentum (for example, in the N electron case). The application of the basis function (8.9) gives a very simple way to obtain accurate energies of the dots even with nonzero orbital angular momentum, without the hassle of partial-wave expansion or angular momentum algebra. As this basis function has no definite orbital angular momentum, in principle one has to project out the component with good angular momentum quantum numbers. In practice, however, the converged wave function already belongs to the correct ground-state quantum numbers and the energy gain due to an orbital angular momentum projection is negligible. As the Hamil2 tonian and the L operator commute, the optimization of the wave 2 function filters out the eigenstate of L as well. As was mentioned in Sect. 6.1, this type of calculation is called a variation before projection. In the limit that the variational trial function is chosen to be flexible enough, the solution of this type of calculation would reach the 2 eigenstate of the conserved quantity like L to a good approximation.
no
to
magnetic
nonzero
orbital
=
In Table 10.6 the results of the SVM with the basis function of
Eq. (8.9)
is
compared
to that of
The shell model works
a
extremely
large
scale shell-model calculation.
well in this
case
because the
con-
fining potential is a harmonic-oscillator well. By choosing the singleparticle wave functions to be the eigenfunctions of this single-paxticle potential, the diagonalization of the Hamiltonian with the relatively weak Coulomb interaction between the electrons is simple and gives an accurate solution. The results of the two methods are in perfect agreement. This agreement shows that the optimization of the basis of
Eq. (8.9) finds the
correct
ground
state
automatically
even
for the
10.4
case
with
sis size is
Quantum
nonzero
dots in the generator coordinate method
orbital
angular
sufficiently large. An
momentum
orbital
angular
provided
203
that the ba-
momentum
projection
may lead to a faster convergence, but as it can only be carried out by a three-dimensional numerical integration, it would prohibitively slow
down the calculation.
Energies of few-electron quantum dots in 3D calculated by SVM and by a large scale shell-model (SM) basis'. L is the total orbital angular momentum, S the total spin, and -x is the parity. K is the basis 0.5. Atomic units are used. dimension. hwo Table 10.6.
=
N
(L, S, -ri)
(1,1/2,-) 4(0,0,+) 5 (1,1/2,-) 3
a
P.
(SM)
K
E
4.01324
100
4.01324
6.35025
300
6.3506
39 hwo 14 hwo
9.00331
500
9.0032
6 hwo
E
(SVM)
Navratil, private
communication.
SM space
(in hwo)
Complements
204
Complements 10.1 Two-dimensional electron motion in
a
magnetic field
vaxiational solution to the
have
presented a previous section we few-body problems of quantum dots. In this approach a correlated Gaussian basis has been used and the problem has been formulated In a relative coordinate framework. The examples from atomic physics (Chap. 8) convincingly prove that this is a very powerful -way to cope In the
with the correlation between the electrons in the field of
an
attractive
only possible approach. One can, alternatively, use a basis set made up from products of single-particle states to diagonalize the Hamiltonian. The properties of these singleparticle states axe discussed in this Complement. The Hamiltonian of interest is given by center. This is,
H
however,
-2m*
not the
(P eAi)2 +
+
9*ABB
c
-
2
aj
I
+
vori)
2
(10.7)
+ X
i 'i
1ri
-,rj I
where m* is the effective electron mass, AB eh/(2m,c) the Bohr magneton of the electron, g* the effective g-factor of the electron, =
and
r.
is the dielectric constant of the bulk material. The
magnetic
interaction energy due to the electron spin is included in the Hamilltonian. The magnetic field is directed downward in the z direction, i.e. B
B
-
with B
> 0. For high magnetic field of, e.g. T, the value of ILBB becomes 0.5788 meV. The vectors ri
10
and pi
(0, 0, -B)
=
axe
2D vectors, which have
x
and y components. The
one-
body potential V(r) is to confine the quantum dots and is chosen to be quadratic, m* W02T 2/2. A typical value of the confinement en5 meV. By taking a symmetric gauge vector potential, ergy is hwo -
A
-r x
=
(P
B/2 2
e
)
_Ji2' j +
A
+
where A
B(y/2, -x/2, 0),
c
_2B 2
we
e. hB
2
(10.8)
4C2 +
_ey2-
and I-, is the
obtain
c
z
component of the angular
mo-
mentum.
To express the Hamiltonian in terms of dimensionless introduce
vaxiables,
-we
C10.1 Two-dimensional electron motion in
e,B W,
W
=
WC2
=
M
a
magnetic field
205
hI + W02
and
p
.
=
(10.9)
W U)
Here wc is the cyclotron frequency and p is the that the magnetic length is expressed by p
magnetic length. (Note using Bohr and hartree 2R*.) Then the Hamiltonian is expressed by the =
radius a*
- -2R*-1hwa*
dimensionless variable ri: 1
H
=
hw
A,
+
2
1
lr,2 2
hw, 2
E lj ,
-
2
g*AB B
aj ,
i
e2
+XE 1ri
with -
j>i
Here the is
hwc
=
length is
rjI
2R*a*
(10.10)
-
=
P
rIP
in unit of the
(2ra,/m*)ABB.
X
magnetic length. The cyclotron energy
The ratio of m,/rn* is of order 15 for the 2D
electron system in GaAs-Ga,,All-,,As heterojunctions. The strength of the Coulomb potential is of the order of X, while the level spacing of the single-particle energy is determined by hw. Therefore the ratio
Xlhw
-
VI'2--R*-Ihw is
the
quantity which
charac-
terizes the electronic motion. If the ratio is much smaller than
the
independent
tion. As it
the
motion of the electrons becomes
a
unity, good approxima-
increases, the correlated motion of the electrons overwhelms
independent
motion.
Recall that the second and third terms of the above Hamiltonian are
consistent with the well-known fact that
mass
M and
charge
qe in the
magnetic
a
spin 1/2 particle with magnetic
ffied B has the
interaction energy -IPB, where the magnetic moment tL of the particle is in general expressed by tL g1l + gs in units of ehl(2Mc). Here g, and g, are the respective gyromagnetic ratios for the orbital and =
spin angular momenta. The gi value is equal to q, while the Dirac 2q for a point charged paxticle like the equation would give g, 2.0023 ILB for the The measurement electron. gives I g, I precision electron and the deviation from 2 can be very accurately computed by the higher order corrections of quantum electrodynamics. On the other hand, the experimental values for the nucleons are far from the 5.586 for the proton Dirac value expected for point paxticles: g, =
=
=
and g, AN
=
=
-3.826 for the neutron in units of the nuclear
eh/(2mpc).
nucleon.
magneton
This fact is ascribed to the internal structure of the
206
Complements
The eigenvalue of the single-particle Hamiltonian, Ho hw (-,A/2 +r2 /2) -(hw,12)1., -(g*p.BB12)o-,,, is obtained easily. Rewriting the spatial coordinate as a complex variable z (x iy)/-\/-2, we defme =
-
I
at
72= (
=
0),
*
Z
-
'0
I
--.,f2- (z- OZ* ),
bt=
(z
a=
OZ
=
[at, b]
Hamiltonian
Ho
=
=
[at, bt]
=
OZ*
(z*
b
0 +
OZ
[a, at]
with the commutation relations
[a, bt]
+
0. We
can
[b, bt]
=
=
[a, b]
1 and
then express the
single-particle
as
hw(ata + btb + 1)
Ihw,(ata
-
-
2
btb)
Ig*ttBBc-z-
-
2
(10.12)
The operator at creates a right circular quantum of counterclockwise rotation about z axis, and bt creates a left one. Let a non-negative
eigenvalue of the total oscillator quanta ata+btb at a b b. and m denote an eigenvalue of the angular momentum 1,, Then the eigenvalues of A and btb are given by (N + m)/2 and (N m)/2, respectively. Since they should be positive integers or zero, the possible values of m for a given N are m -N+ N, N-2, N-4, 2, -N. The single-particle energy becomes integer N denote
an
=
-
-
=
...,
I
E,,,,,,
=
hw(N+1)
hwm
-
-
2
g*ABBS,
I =
with Sz
quanta
hw(2n + Iml
being
either
N is set
1/2
For
of
hw(N + 1),
1) or
to N
equal given N, possible
n.
a
+
hwm
-
-
2
-1/2, =
2n +
values of
(10.13)
g*ABB$,
and where the number of total
Iml
with
n are
0,
1
a
non-negative integer [N12]. The energies
....
with N + I-fold
degeneracy, corresponding to the 2D are independent of m conforming to N. The singleparticle energy is shown in Fig. 10.9. In the case of no magnetic field or very weak magnetic field (w,lwo < 1) the single-particle level shows a shell structure with degeneracy of N + 1. The single-particle level is then in the order of haxmonic-oscillator
(n, m)
=
(0, 0); (0, 1), (0, -1); (0, 2), (1, 0), (0, -2);...
The level
cross-
V/-I/-2
ing of the low-lying single-particle levels occurs at W,/WO 0.71 and the level with (n, m) (0, 2) becomes lower than the =
=
with
(0, -1).
Z-'
one
CIO. 1 Two-dimensional electron motion in
a
magnetic field
207
If wo 0, that is, there is no confining potential present or the magnetic field is very high (wc.Iwo > 1), w is (nearly) equal to (112)w, and the energy becomes E,,,,, hw,(n+(Iml -m)/2+1/2) -9*ABBS,, de=
=
(Iml
pending only
on n +
dent of m for
positive
-
m.
m) /2. Clearly the energy becomes indepen-
This
gives
rise to
infinitely degenerate levels
called the Landau levels with the Landau level index The fact that the
levels with
an
single-electron degeneracy
infinite
motion is
quantized
number in the
+
n
case
(IM I
-
m) /2.
to the Landau
of
no
confining
potential reflects the fact that the energy is independent of the center of the cyclotron motion. In the presence of the confining potential, the
single-particle energy increases with m as seen from Fig. 10.9. By using the polar coordinates r and 6 the eigenfimetions single-particle Hamiltonian are expressed as
rp2
ni n!
(n-t TP 2(n+lml)!
(r)
1-1
L (I ' 1) n
P
r2
(p2
-
r
ey-P
of the
2
2p2
(10.14) where
Ln(c) (x)
is
an
associated
Laguerre polynomial defined by
d L,(,o I (X) =ex-' (e-XX n+a n! dXn n
E k! (n
'k=O
F(n+a+l) (-X),. k)! -V(k + a + 1)
(10.15)
-
parity operation changes V to V + 7, so the parity of the singleparticle eigenfunction is given by (-l)'. For the lowest Landau states (n 0, m > 0) the larger angular momentum state corresponds to a wave function which is more extended from the origin:
The
=
2
(0o,jr loom
(Iral
The Han-d1tonian
+
1)P2.
(10.10)
(10.16)
contains the Coulomb interaction be-
electrons, which leads to interesting effects on quantum interplay between the magnetic field and the Coulomb interaction is an essential ingredient for studying the dots. The relative importance of various pieces of the Hamiltonian depends on m*, K, g*,
tween the
dots. The
and B. For two electrons the Hamiltonian
can
be
separated in the relative By introducing
and center-of-mass coordinates of the two electrons. the coordinates
7 C)
Is 6
5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
10
O)C /0)0
Fig. 10.9. The single-particle energy E,,,,, of a quantum dot as a function of magnetic field strength. The Zeeman spin splitting of the energy is not included.
C10.1 Two-dimensional electron motion in
I
R=
-\f2-
('rl +r2),
Ir
a
magnetic field
I-(Irl -r2),
=
209
(10.17)
A/2
the Hamiltonian is reduced to H
=
hwf
I
,AR
-
+
2
1R21
'&V,, IR ,
-
"A'r + 21,r2 2
+ hW
hWcl
,T Z
2
-
C
.
2
2
g*[tBB(Sj,,
+
S2 ,)
X 1 +
(10.18)
-
vF2 r
by exactly the same Hamiltonian as the single-particle Hamiltonian and its energy and eigenfunction are given by Eqs. (10.13) and (10.14). The eigenfunction b(T) of the relative motion Hamiltonian may be obtained in polar coordinates. By separating the angular part as b(r) u(r)e " (m and the the function radial eigenvalue E, are obu(r) 0, 1, 2, ...), tained from the following equation The center-of-mass motion is determined
=
d2u
1 du +
-
.-
r
where
u(r)
r2
+
hW
dr
has to
M2
V2-X 1 +
satisfy
=
2Er
W, 7n
U
-
=
0,
(10.19)
r
the condition that its norm,
fo' drru(r)2, is
finite.
analytical solutions are available provided Xlhw belongs to a certain enumarably infinite set of values [1411. For a general case we can obtain the eigenvalue by numerical integrations. The eigenvalue 0, 1, 2,... for each m. We may be labelled by a quantum number n note the symmetry property of the eigenfunctions. The interchange of the position coordinates ri and V2 leads to r -r, which is # + 7r. Therefore the eigenfunction equivalent to the change of 6 and labelled by n m receives a phase change e"' (-I)m, so that it is symmetric for even m and antisymmetric, for odd m. Figure 10.10 displays the energy E, of the relative motion as a function of the cyclotron frequency. The most striking feature is the level crossing in the lowest level (ground state). With increasing magnetic field the so (0, 0) to (0, 1), (0, 2), (0, 3), ground state changes from (n, m) that the angular momentum m of the ground state increases one by one. (This figure is basically the same as Fig. 10.4 which is obtained by the variational calculation, but included here to show the significant Paxticular
=
---).
--*
=
=
role of the Coulomb
Finally
we
electrons in
a
..
-,
interaction.)
importance of the correlated motion of strong magnetic field has already been recognized in
note that the
the quantum Hall effect
[1421.
The
broadening of the Landau levels
210
Complements 12-
11
10I f I I
9-
8 It
7
*11
IA 41/
*, -
3-
2
1
0
-
1
0
1
2
3
4
5 0) C
Fig. dot
6
7
8
9
10
/ (00
10.10. The energy E, of the relative motion of a two-electron quantum function of magnetic field strength. The Zeeman spin splitting of
as a
the energy is not included. The solid curves are for states with even m and the dashed curves for states with odd m. hwo = 1. Atomic units are used.
CIOA Two-dimensional electron motion in is caused
by
scatterers such
as
a
magnetic field
211
impurities and lattice defects. The repulsion between many electrons
correlations due to the Coulomb
lead to the Anderson localization and the fractional quantum Hall effect. Laughlin [1431 successfully used a correlated basis of Jastrow
type
to describe the electron motion.
11. Nuclear
few-body systems
originally developed to solve physics. In this chapter we collect a
The stochastic variational method
was
few-body problems in nuclear few possible applications of the SVM for nuclear systems. The nuclear force, due to the effect of the underlying quark structure and relativistic motion, is a very complicated interaction. h addition to the spinisospin dependent central part, the two-nucleon interaction contains spin-orbit and tensor forces, and L2- and (LS)2_dependent terms. The two-body interaction designed to fit the nucleon-nucleon phase shift and the properties of the deuteron fails to reproduce the binding energies of the three- and four-nucleon nuclei, and a three-body interaction has to be introduced to get the correct binding energy. As the nucleons are not structureless point-like entities, the spatial part of the nuclear force contains a strong repulsion at short distances. These circumstances altogether make the solution of the nuclear few-body problem a
formidable task.
application is restricted to central forces. These schematic forces are used in simple model calculations and serve for benchmark tests. Examples of more realistic problems are shown J-n the last secThe first
tion.
Introductory remark
11.1
on
nucleon-nucleon
potentials The nucleon
(N)
has
spin and isospin degrees of freedom
in addition
of Eq. (2.3) is a product of these spatial one. The operator 0?. V factors, one acting on each coordinate of the three types:
to the
0?. 73
=
(0,,ij (space) 0,.ij (spin) -
Oij (isospin),
Y. Suzuki and K. Varga: LNPm 54, pp. 213 - 246, 1998 © Springer-Verlag Berlin Heidelberg 1998
214
11. Nuclear
where,
e.g.,
few-body systems
0,,,,j (space) (IL
=
n,
r.
-
-K)
1,
is
a
spherical tensor a scalar product
spatial part, and product (-I)"A/2__K+ I[T,,, x U,,Ioo, which (T,,,.U,,) is defined with Clebsch-Gordan coefficients by operator of rank
acting
K
in the
stands for the tensor
[Tr-1
X
Ur-2]r-393
=
E
(11.2)
rv1jL1K2A2jK3A3 >TK,jL1 Ur-2/L2'
<
IIIA2
e.g., [2, 70, 71, 72] for angular momentum algebra. The NIV interaction is a typical example of the strong interaction
See,
and is still not known in fiiU detail. There are, however, several versions of the nuclear potential whose parameters have been determined fairly
accurately so
to
as
reproduce two-nucleon
bound state and
scattering
data. The nuclear force reaches at most up to 2 fla (I fin = 10-13 cm). An NN potential which is consistent with the scattering data is called
"realistic". The most important ingredients of the realistic include central, ,tensor, and spin-orbit components. The central part of the nents:
0-'12
is thus
==
Ii
TTT27 0'1*(7'2,
expressed
V12
=
potential
is characterized
and(71'72) (o-j-a2).
potential
by four compopotential
The central
as
+ V' (,r) -rl'72 + VCO_ (r) 01 0'2
V(,r)
*
-
+ VcTO
(r) (Irl 72) (Ol U2) *
*
expressed by the space- (Pj'2), spin1+'rr72 and isospin-exchange (P172) operators. (P12 2 2 The generalized Pauli principle requires the relation These
components
can
also be
=
Pi2P12'P12_
=
Therefore the central force
VI 2
=
V12
=
(11.4)
_1* can
also be written
VB H Pl' 2
VW H
+
VM (r) PJ'2
(V-t (r)
+
V` (r) 7, '72) S12)
+
+
as
VH (r) PI'2 PI 2,
(11.5)
(11.6)
11.1
remark
Introductory
nucleon-nucleon
on
potentials
215
with the tensor operator 3 (o-1
S12
*
f6) (0`2 f6) *
U1 Cr2 *
/E4T (Y2
"
V
The deuteron
X
5
consisting of
bound through the force. Here
-
a
0212) (p)
proton
and
coupling of the 3SI and 3D,
2s+'Lj
neutron
a
waves
(n)
becomes
due to the tensor
denotes the two-nucleon channel with the
spin S
and the relative orbital
angular momentum L being coupled to the However, no such coupling occurs in the case of two protons or two neutrons because the Pauli principle (11.4) forbids two like nucleons with L 0 or 2 to be in a spin-singlet state. The interaction between like nucleons is just too weak to produce a total
angular
momentum J.
=
bound state. The
spin-orbit force has non-vanishing matrix elements only spin-triplet states, too. It is parametrized in the form V12
=
in
(Vb (r) + Vb-r (r) -ri '72) (L S) 12 -
with
(L-S)12
r X
2h
h
where p
=
-ih-L, ar
(r
X
hl
(PI
-
P2))
P)' (Ol + 2
=
(r
x
p),
2
(Ol
+
6'2))
and
(
Cr2)
2 x
)
(Ol
472))
+
indicates
a
product. The form factors of the various components pressed in terms of Y(x) e-xlx and its derivatives.
vector
are
(outer)
usually
ex-
=
As is evident from the above
discussion, one of the characteristic features of the realistic NN potential is its strong state-dependence, namely it differs in the four states of singlet-even, triplet-even, singletodd, and triplet-odd character, where even and odd refer to the parity of the orbital motion.
The tensor force makes it necessary to introduce at least D-waves in the calculations. Another important feature of the most commonly used
family of NN potentials is the strong repulsion at short distances, say within 0.5 fi-n. This requires due care in so far as short-range correlations must be properly taken into account in theoretical models. All of these aspects of the NN interaction really make a variational calculation for a nuclear system very challenging. In these circumstances
11. Nuclear
216
few-body systems
frequent to use some "effective" potentials which axe not necessarily designed to reproduce the scattering data but determined to fit some bulk properties of nuclei such as the binding energy and the size. Although relativistic effects should be fairly large in nuclei ((V/C)2 0.08, which is estimated from the kinetic energy TF 40 MeV belonging to the Fermi level), the ambiguity of the nuclear force makes it rather difficult to extract these clearly from experimental it is rather
-
-
data. The
complexity
of the NN interaction is due to the
substructure of the nucleon
as was
compositeness of mesons exchanged
11.2 Few-nucleon
discussed in
Chapt.
quark-gluon 9 and to the
between the nucleons.
systems with central forces
performance of the SVM by performing model calpotentials, e.g., the spin-averaged Malffiet-Tjon (MT-V) potential [1441, the Volkov no. I (Vi) "supersoft" core potential [1451, the Afnan-Tang S3 (ATS3) potential [1461, the Afinnesota potential [1471 and the Brink-Boeker (BI) potential [1481. Parameters of these potentials are given in Table 11.1. Some of these model problems have already been solved to high accuracy by various methods and therefore we can directly compare the solutions. The results presented in this section do not include the contribution from the Coulomb potential. Each calculation has been repeated several times staxting from different random points to check the convergence. The energy as a function of the number of basis states is shown in Fig. 11.1 for the case of 6Li with the V1 potential. The energies on different random paths approach each other after a few initial steps, and converge to the final solution. The energy difference between two random paths as well as the tangent of the curves give us some information on the accuracy of the method for a given size of the basis. The root-meansquare (rms) radius is calculated in each step and found to be rapidly convergent to its final value. By increasing the basis size the results can be arbitraxily improved when needed. The number of basis states required to reach energy convergence increases with the number of particles, but it depends on the form of We have tested the
culations with different NN central
well. This latter property is illustrated in of the alpha-paxticle. The soft-core Volkov (Vi)
the interaction for the
case
as
Fig. 11.2 potential
11.2 Few-nucleon
systems with central forces
217
Table 11J. List of the parameters of the nucleon-nucleon central po-
tentials. The potential consists of a few terms; each is expressed as Vf (IL, r)(W + MP' + BP' + HP'P'). See Eq. (11.5). For the Gaussian
potentials (VI, ATS3, Minnesota, Bj), the form factor f ([L, r) is eXp(_/tr2)' the potential strength V is in units of MeV, and the potential range p is in units Of fM -2, while for the Yukawa potential (MT-V) f (A, r) is defined by exp(-[tr)/r, V is given in units of MeVfin, and IL is in units of fin-, respectively. The Majorana, mixtures m and m' are set equal to zero in the calculation. For realistic values of m and n see the original papers. The parameter u of the Minnesota potential is set equal to unity. V
A
W
M
B
H
MT-V
1458.05
3.11
1
0
0
0
[1441
-578.09
1.55
1
0
0
0
VI
144.86
0.82
[1451
-83-34
1.60
ATS3
1000.0
[1461
-326.7
Potential
-2
M
M
0
0
1-M
M
0
0
3.0
1
0
0
0
1.05
0.5
0
0.5
0
-166.0
0.80
0.5
0
-0.5
0
-43.0
0.60
0.5
0
0.5
0
-23.0
0.40
0.5
0
-0.5
0
u/2 u/4 u/4
(2 (2 (2
-2
Minnesota
200.0
1.487
[1471
-178.0
0.639
-91.85
0.465
Bi
389.5
0.7
[1481
-140.6
1.4-2
shows
-2
rapid convergence, requires more basis states
1
-
I-M 1
-
n
while the to
get
an
-
-
-
u)/2 0 u)/4 u/4 u)/4 -u/4
0
(2 u)/4 (-2 + u) /4 -
M
0
0
MI
0
0
repulsive-core ATS3
interaction
accurate solution. The
relatively
fast convergence for the MT-V potential of a strong repulsion can be explained by the simplicity of the spin-independence of this interaction.
results, together with results of others, potential to N=2-7-nucleon systems. application 41.47 MeV fi-n2. In the table the ground-state enerWe chose h2/m E and matter rms radii (r 2)1/2 are given. The basis dimenpoint gies sion K of the SVM listed in the table is such that, beyond it, the energy and the radius do not change in the digits shown. For three-body systems, the solution of the Faddeev equation is known to be the method of choice (see Sect. 5.3), but the SVM can easily yield energy of the Table 11.2 shows the SVM
of the MT-V
for the
=
11. Nuclear
218
few-body systems
-65.5 -65.6 -65.7 -65.8
5
-65.9
0)
-: W
-66.0
-66.1 -66.2 -66.3
-66.4 -66.E 0
50
100
150
200
250
300
350
40(
dimension of the basis
Fig.
11.1.
Volkov
Convergence
of the
(VI) potential [145]
6Li
energy
on
different random
paths.
The
is used.
-29.0
-29.5-
Minnesota
................................................................. -30.0-
ATS3 2
Vo-lkov
-30.5-
W
-31.0
MTV -31.5
-32.0
0
20
40
60
80
100
120
140
16(
dimension of the basis
Fig. 11.2. Convergence potentials
of the
alpha-particle
energy for different central
11.2 Few-nucleon
systems with central forces
219
Energies (in MeV) and root-mean-square (rms) radii (in fin) of interacting via the Malfliet-Tjon (MT-V) potential [14 . The value of K denotes the basis dimension beyond which the energies and the radii of the SVM calculation do not change in the digits shown. Table 11.2.
N-nucleon systems
N
(L, S) J'
Method
E
(T 2)1/2
2
(0,1)1+
Numerical
-0.4107
3.743
SVM
-0.4107
3.743
3
4
(0,1/2)1/2+ Faddeev[150, 1511
(0,0)0+1
(0,0)0+ 2 5
6
7
(1,1/2)3/2-
(0,0)0+ (1,1/2)3/2-
5
-8.25273
ATMS[121 CHH[1521 GFMC[411 VMC[1531a
-8.26
SVM
-8.2527
FY[1491 ATMS[121 CRCG[261 GFMC[411 VMC[1531a
-31.36
SVM
-31.360
CRCG[261
-8.50
2000
SVM
-8.49
150
ZLO-01
1.682
-8.26
0.01
1.682
-8.27
0.03
1.68
-8.240
1.682
80
1.40
-31.36 -31.357
1000
-31.3
0.2
-31.3
0.05
1.36 1.39 1.4087
150
VMC[153]a
-42.98
SVM
-43.48
VMC[1531a
-66-34
SVM
-66.68
1.52
800
SVM
-83.4
1.68
1300
'Calculated with Coulomb tracted
K
perturbatively.
The
0.16
1.51 1.51
0.29
500
1.50
potential, the Coulomb contribution then subpotential strength used in the VMC [153] cal-
-578.17 MeV, which is culation is Vi=1458.25 and V2 from that used in the calculation. See Table 11.1. =
slightly different
220
11. Nuclear
few-body systems
accuracy. As the MT-V potential is a preferred benchmark test of the few-body calculations, there are numerous solutions available. same
Table 11.2 includes
a
few of the most accurate results. The nice agree-
ment for the four-nucleon case corroborates the fact that the SVM is as
accurate
as
the direct solution of the
Faddeev-Yakubovsky (FY) Amalgamation of Two-body corre-
equations [1491, Multiple Scattering (ATMS) [121 the method of the
lations into
Carlo
[411
(VMC) [153]
methods. The basis used in the
Gaussian-basis the
SVM,
or
the variational Monte
and the Green's function Monte Carlo
(CRCG)
(GFMC)
Coupled-Rearrangement-Channel.
variational method
[26]
but the Gaussian parameters follow
is similar to that of
geometric progressions. ground
The fact that the basis size needed in the SVM for both the
and first excited states of the four-nucleon system is much smaller proves the efficiency of the selection procedure. See Chap. 4. The re-
sults of the VMC calculation for the five- and six-nucleon systems are also in good agreement with the results of the SVM- The MT-V potential has no exchange term; therefore, unlike nature, it renders the
five-nucleon system
bound,
and the nucleus tends to
collapse
the
as
binding energy increases with the number of particles. Neither the MT-V nor the V, potential contains the full set of change components of the central potential, rather simple. The Minnesota potential, spin-,
and
isospin-exchange
realistic. In fact it has been
so
ex-
the calculation becomes
however,
contains space-, and is considered to be more operators,
successfully
used in
microscopic cluster-
model calculations tential
was
[28, 29, 301. An extensive calculation with this pocarried out for N=2-6 nucleon systems in [31], where all
possible spin and isospin configurations were allowed for and all angula.r functions that give non-negligible contributions were included. The agreement with experiment was surprisingly good. The energies and the'radii of the triton and the alpha-particle converge to realistic values. The Minnesota potential, correctly, does not bind the N=5 system, but it binds 6He and slightly overbinds ILL The radius of 6He is found to be much larger than that of 4He, consistent with the neutron-halo structure of 6He
[29, 301.
The SVM calculation of
[311
has
made it
possible to test for the first time the Minnesota force without assuming any cluster structure or restricting the model space by any
other bias.
The last
example
is the
application
to Efimov states
how well the correlated Gaussian basis describes the totics. A
three-body system
with
short-range
[1541
few-body
to show asymp-
forces may have several
11.2 Few-nucleon systems with central forces
221
bound states. If the
two-body subsystem has just one bound state (close to) zero, the three particles can interact at long
whose energy is distances and an infinite number of bound states may appear in the three-body system. These "Efimov states" are extremely interesting
experimental and theoretical points of view because of their distinctive properties. These states are very loosely bound and their wave functions extend far beyond those of normal states. By increasing the strength of the two-body interactions these states disappear. from both
A system of three identical bosons of nucleonic mass is considered for simplicity. The potential between the particles is taken as the Poeschl-Teller interaction because this
can
be
easily
V(r) Here
r
potential
is
analytically
solv-
body case, and therefore the two-body binding energy
able for the two-
set to zero;
625.972
1251-943
sinh(l.586 r)2
cosh(I.586 r)2
(11.10)
in MeV.
=
is in units of fi-n. A
spatially
very extended basis is needed to
represent the
bound states. In this
basis. The
basis is created
predefined diagonal elements of the matrix A. These diagonal elements are taken as geometric progression:
wealdy predefined
case we use a
by using only
the
1
A,-,
(i
-
('
-
(aoq6
1))2
=
1,
...,
n),
1
A22
(j
-
( ao q(j-'))2 10
This construction defines
(i, A 0.1,
(I, I),-, (n, n).
=
q0
2.4. One
=
an n
can
=
reveal
2-dimensional basis corresponding are chosen as n 20, ao =
to =
find the first three Efimov states with this
can
expJ-2vJ
more
(11.11)
The parameters
basis size. The ratios of the
Ei+IlEi
n).
rule
energies of the bound states follow the [1541. By increasing the basis size one
bound excited states. Note that the value of
roughly corresponds to the spatial extension basis in this example goes out up to aoqon-I
aoqo'-'
of the basis. The present =
0.1
x
2.419
=
1674990
This extension is necessary to get the third bound state. For a comparison, to calculate the ground-state energy of -4.81 MeV it 7 and the basis covers only the [0, 101 (fi-n) is enough to choose n
(fin).
=
interval. The
rms
radii of the
ground, first and
second excited states
are about 1.5, 25 and 6000 fin, respectively. As shown in Fig. 11.3, these facts illustrate the tremendous spatial extension of the Efimov
222
11. Nuclear
few-body systems
Ground state
.0 .0 0 10.0
Efimov state
0.1
0.0 M
x
Fig. IXM,
Wave fimetion
amplitudes
of the
ground
state and the first
Efimov'stAte. Note the different scale (in fin) of the lengths of the Jacobi coordinates
x
and y.
11.3 Realistic
223
potentials
states. These results indicate that the extended correlated Gaussian
basis
gives
a
practical
to describe the
means
tics of the Efimov states. See
Complement
asymptotics characterispossibility of
11.4 for the
Efimov states in the 12C nucleus.
11.3 Realistic
potentials
The trial function is
given
as a
combination of the correlated Gaus-
sians:
'O(LS)JMTMT(-'C 14) 1
Ajexp(- 1.+,Ax) [OL (X)
=
XSJ jM
X
2
where
x
=
jX1i
...
XN-11
is
a
TITMTli
set of relative
(11.12)
(Jacobi) coordinates,
the operator A is an antisymmetrizer,, XsMs is the spin nTmTis the isospin function. The matrix A is an (N
-
symmetric, positive-definite
function, and
1)
x
(N
-
1)
matrix of nonlinear variational parame-
angular part of the wave function. It is taken as a vector coupled product of partial waves i3)]L123 I LML This is re[[[Yli 41) X Y12 (ZAL12 X Yls (X3 OLML (X) ferred to as the method of partial-wave expansion. The spin and isospin functions are also successively coupled, for example XSM,
OLML (x) represents
ters. The function
the
:`
...
.
=
X(E12SI23 )SMs ...
we
L
used the
=
0, 1,
satisf3 g
2.
=
[[[XI/2
X
XI/21SI2
partial-wave channels For the alpha-particle,
the condition 11 +
12
+
X
X1/21S1,23 ...J,sms
with
(11, 12)L,
For the
triton,
where 1i < 2 and
partial waves ((11, 12) L 12 13) L 13 :! 4, 1i :! _ 2, 111 121 <
all
1
-
0, 1, 2 have been tried. As the triton has J 1/2, there are three spin and two isospin chan1/2 and T 0, 1. For the alphanels; (S12)S (0)1/2,(1)1/2,(1)3/2 and T12 0, T 0) there are six spin and two isospin chanparticle (J nels unless very small isospin mixing is taken into account; (S12 S123) L12 !
11
+
12 and L
-
=
=
=
=
7
(0, 1/2)0,(0,1/2)1,(1,1/2)0,(1,1/2)1,(1,3/2)1,(1,3/2)2 T123) (0, 1/2), (1, 1/2). =
and
(T12,
=
dependence of realistic nucleon-nucleon interaction is extremely important. To extend the application of the SVM calculations to nuclear systems interacting with realistic potentials, in addition to the previously applied random selection of the nonlinear parameters of the correlated Gaussian, a random selection of the spin and orbital components of the wave function The
spin, isospin and orbital angular
momentum
11. Nuclear
224
few-body system
is introduced. The main motivation for the random selection is that
the numbers of channels and nonlinear parameters are prohibitively large, therefore the calculation of all of the matrix elements and di-
potentially important basis states is out alpha-particle, for example, there partial-wave channels even in our truncated-partial wave expansion. Just to show up an alternative, it is worthwhile assessing what dimensions a deterrni-ni tic optimization would require. The simplest choice of the nonlineax parameters of the Gaussian basis is to use a diagonal matrix A. One of the best deterministic choices for the diagonal elements is a geometric progression [251. To reach good accuracy, at least five values for each of AJ-1 A22 3 125 functions in a given and A33 have to be used, requiring 5 channel. The spin-isospin and space paxt, therefore, would result in a all
agonalization including
of the question. In the case of the are 12 spin-isospin channels and 32
7
=
basis size of about a
one
hundred thousand
"direct" calculation for the
(2
125
x
x
12
x
in such
32)
alpha-particle.
The literature is very rich in papers devoted to few-nucleon prob3 nuclei it is possible lems [155, 156, 157, 12, 25, 158-1611. For N =
to include
enough
performed, N
4 nuclei
=
channels and
but other methods some
results of the
an
exact Faddeev calculation
the
give essentially
of these methods become too calculations agree
existing
only
same
complicated,
within
a
can
be
results. For
and the
few hundred
keV.
To test the SVM
used two, from the
we
form,
rather different interactions. The
(AV6
and
ond,
AV8) [1621,
the Reid
is
a
the
first,
view of
spatial Argonne potential
well-behaved smooth function. The
potential (M) [1631
linear combination of
point of
is
more
singulax
exp(_/tr2)/,rk-type terms.
sec-
and includes
a
The latter
potential AV6 The includes can certainly cause some problems. potential only central and tensor components and serves as a good test case for the inclusion of non-central forces. The AV6 and AV8 potentials axe used without Coulomb potential, while in the case of the RV8 potential the Coulomb interaction is added. To
keep
the number of nonlinear parameters
matrix A to be
diagonal
in
one
of the
possible
low,
we
restricted the
Jacobi coordinate sets.
of the three-nucleon system we have only one possibility of the Jacobi coordinate, (NIV) + N. For the alpha-particle we use
In the
case
possible coordinate sets, (3N) + N ("K") and (2N) + (2N)type ("IF) Jacobi coordinates. Note that the basis function is fuHy both
antisymmetrized.
This choice is also dictated
by physical
intuition
as
11.3 Realistic
it is natural to consider
potentials
225
3H+P 3He+n and d+d-type partitions in the ,
Test calculations show that tbis choice
alpha-paxticle. satisfactory results.
already gives
As is shown in Table
11.3, the convergence is relatively fast both for the triton and the alpha-particle with the AV6 and AV8 potentials, but in the case of the RV8 potential, due to its singular nature and stronger repulsive core, the convergence is slower. About 50 basis states for the triton and about 200 basis states for the alpha-particle give fairly good binding energies. The energy of the alpha-paxticle with the basis size of 200 is already within 0.5 MeV of its final value. It is very satisfactory that the SVM has succeeded in reproducing a reasonably good energy with just 400 basis states.
Table 11.3. The convergence of energies alpha-particle for different potentials, AV6
(in MeV)
[1621,
for the triton and the
AV8
[1621,
and RV8
[163].
K is the basis dhnension.
'He
311 K
AV6
AV8
RV8
K
AV6
AV8
RV8
25
-6.63
-7.36
50
-7.04
-7.69
-6-53
100
-24-37
-25.15
-23-35
-7.41
200
-25.05
-25.50
75
-7.11
-24.15
-7.74
-7.54
300
-25.28
-25.60
100
-7.15
-24.35
-7-79
-7.59
400
-25.40
-25.62
-24.49
The results of the SVM and other calculations Tables
11.4 and
11-5. We show and compare
compared in the results not only are
for the energy but for the rms radius and average kinetic and potential energies as well. The results remain the same by repeating the calculations several times
staxting from different random values.
comparison for the triton with RV8 demonstrates that the SVM are in very good agreement with the Faddeev results. Since Faddeev calculations are considered exact for three-body systems (see The
results
Sect.
this agreement confirms that the basis selection of the SVM performed efficiently even for the singular potential of RV8.
5.3),
has been
The correlated
hyperspherical
harmonics
(CHH) expansion
method
(see 5.2) also agrees with the SVM for the triton. The results of the SVM and GFMC are very close to each other. Except for the case Sect.
of the
alpha-particle
the statistical
errors
AV6, the energies of the SVM axe within of the GFMC, though the expectation values of with
11. Nuclear
226
few-body systems
Table 11.4. Energies (in MeV) and radii (in fin) of the triton calculated by different methods and with different interactions. (T), (V), and (V6) are the expectation values of the kinetic energy, the total potential energy, and the of the central and tensor components, respectively. (, 2) 1/2 denotes the root-mean-square radius. The value of PL shows the percentage of the component with the total orbital angular momentum L.
potential energy
SVM
Faddeev
GFMC
CHH
VMC
AV6
(T) (V6) (r 2 1/2
44.8
PS PP PD
91.2
E
-7.15
-51.9 1.76
-43-7(1-0) 1.95(0.03)
-52.0(3.00) 1.75(0.10)
< 0.1
8.7
-7.22(0.12)a
-7.15'
-6.33(0.05)'
AV8
(T) (V6) (VLS) (T 2)1/2
46.3
PS PP PD
91.1
E
-7.79
-52.9 -1.2
1.75 < 0.1
8.9
_7.79d
-7.79'
RV8
(T (V) (T 2)1/2
52.2
PS PP PD
90.3
E
-7-59
aRef.
-59.8
1.75
54.0(0.20) -62.0(0.20) 1.68(0.07)
52.2
-7.54(0. 10)b
-7. 59b
-59.8
1.76
< 0.1
9.7
[1641 bRef. [1651. .
'Ref.
-7.44(0.03)'
[1571. dRef. [1591.
'Ref.
[1611.
-7-60'
11.3 Realistic
potentials
227
Table 11.5. Energies (in MeV) and radii (in fin) of the alpha-particle by different methods and with different interactions. The isospin mixing due to the Coulomb potential is not taken into account. See also the caption of
Table 11.4. SVM
CHH
vMC
FY
GFMC
AV6
(T) (W) (r 2)1/2
100.1
-125.4 1.49
PS PP PD
84.3
E
-25.40
-122.0(3.0) 1.50(0.04)
-122.0(1.0) 1.50(0.01)
-25.50(0.20)a
-22.75(0.01)a
0.5 15.1
AV8
(T) M (W) (VLS) (r 2)1/2
98.8
PS PP PD
85.5
0.3
E
-25.62
-25.75(0.05)
111.7
109.2(0.20) -137.5(0.20) 2.45(0.23) 0.71(0.02) 1.53(0.02)
-124.4
-124.20(l.0)
-121.5
-2.9 1.50
1.51(0.01)
14.2 ,d
-25.60'
-25.31-
RV8
(T) (V6) (VLS) (KOUI) (T 2)1/2
-139.1 2.1
0.75 1.51
84.1
PS PP PD
0.4 15.5
15.5(0.20)
E
-24.49
-24.55(0.13)b
aRef.
[1641. bRef. [165].
cRef.
-23.79f
[1571 dRef. [1591 .
-24.01e
.
eRef.
-23-9c
[1611. fRef. [1551.
11. Nuclear
228
the kinetic and the
few-body systems
potential energies
VMC and GFMC have
a
integration. Though the of the trial function
statistical error
well,
as
are
error
somewhat different. Both
bar of VMC is influenced
the
error
by the
choice
of GFMC is considered
purely
statistical in the limit of infinitesimal time step (see Sect. 5.1). Variational calculations were considered to be inappropriate
realistic
not be able to
for
the
reproduce large potential energies [1581. Our calculation shows that, with the careful optimization of the nonlinear potentials,
as
they might
of
involved in the Monte Carlo
cancellation between the kinetic and the
variational parameters, this is not the case. We energies even in the case of the RV8 potential.
can
obtain accurate
experimental energies of the triton and the alplia-particle are MeV, respectively. We see that the calculation using the realistic potentials underbinds; about 0.9 MeV for the triton and about 3.8 MeV for the alpha-particle. This indicates that the binding energies of few-nucleon systems cannot be accounted for in two-nucleon potential models but call for other mechanism to get more attraction. One possibility for the mechanism is a three-body potential which can be produced by two-pion exchanges among three nucleons. Once we have a basis for a given potential, one naturally expects that the same basis may give fair ground-state energy for other potentials of similar nature as well. We have checked if it is really the case. The basis optimized for the RV8 potential gives excellent results (within 100 keV) for AV6 and AV8 as well. Due to the singular nature and stronger repulsive core of RV8, however, the basis optimized for AV8 gives about 500 keV less binding for the alpha-particle with RV8 than the basis optimized for RV8 itself. This result is still not so bad and can be easily improved by refining the nonlinear parameters as explained in Sect. 4.2. Therefore, one does not have to look for an entirely new basis set for another interaction, but can use the same basis for a given system, with some "fine tuning" if necessary. It is obviously important to calculate accurate binding energies of light nuclei with realistic forces. For example, the two-neutron separation energy of 6He is about I MeV, so that a few hundred keV less binding is thought to change its neutron "halo" structure significantly. One of the advantages of the SVM is that it is relatively easy to extend it to N 5-, 6-, 7-nucleon systems [31]. The low dimension of the bases required to solve the N 3- and 4-nucleon problems The
-8.48 and -28.295
=
=
confirms that the SVM is suitable for
treating larger
nuclei with
alistic forces. The formalism and the computer code itself is
re-
general
11.3 Realistic
[311,
so
potentials
229
applicability is mostly limited by the memory and speed of no dffficulty in including three-body
the
the available computer. There is forces if necessary.
utility of the global vector representation of Eq. (6.4) in describing the angular part of the system interacting via realistic potentials. We calculated the ground-state energies of the triton and the alpha-particle by using the AV6 interaction [162]. The results by the global vector representation and by the partial-wave expansion are compared to each other in Table 11.6. The model space allows for natural parity states only. The partial-wave expansion is restricted for partial waves with orbital angular momentum 0, 1 and 2. These partial waves give a good description of the ground states of these We show the
nuclei the
even
The table shows that
[321.
global
vector
in the
more
different orbital
one can
reach the
same
results with
representation as with the partial-wave expansion complicated case when the tensor force couples the
angular
momenta. The
nice illustration of the main
case
of the
alpha-particle
is
a
of the
global vector representaadvantage partial-wave expansion: Even in the expansion truncated to the natural parity states, one needs 24 partial-wave combinations due to the large number of combinations of the individual partial waves, while in the global vector representation only 4 sets of (K, L) ((K, L) (0, 0), (1, 0), (0, 2), (1, 2)) were used to reproduce the same result [1661. By increasing the number of particles the partial-wave expansion would contain a prohibitively large number of combinations. The number of sets (K, L), on the contrary, does not depend on the number of particles. Only the number of parameters ui increases (linearly) by adding particles to the system. tion
over
the
=
energies of the ground states of the triton and the alphaparticle [1621. Only natural parity states have been taken into account for the orbital motion. Two types of angular functions are used; the global vector representation (GVR) and the paxtial-wave expansion (PVVE). K is the basis dimension. Table 3-1.6. Total
with the AV6 interaction
3
411
11 Method
K
GVR
100
PWE
100
Method
K
E
-6.80
GVR
300
-23.29
-6.80
PWE
300
-23.27
E
(MeV)
(MeV)
230
Complements
Complements 11.1 Correlations in
few-nucleon systems significance of the correlation we calculate the binding energies of the triton and the alpha-particle by using the correlated Gaussians and the correlated Gaussian-type geminals. The spinisospin parts of the triton and the alpha-particle wave functions are considered totally antisymmetric and, as the nucleon is a fermion, their spatial parts must be totally symmetric. The spatial motion in the ground state of the N 3- or 4-nucleon system may thus be To elucidate the
=
treated
as
motion of bosons. We obtain variational solutions for the
triton and the
alpha-partile interacting via a spin-isospi-n independent two-nucleon potential. The orbital wave function is assumed to be given by combinations, of the correlated Gaussians with L 0, see Sect. 7.2: =
Tf
CkS eXP
(_2
,7c A k X
k=1
Here
x
is the Jacobi coordinate set and S is
positive-definite symmetric
matrices
Ak
a
symmetrizer. Each of the
contains nonlinear variational
parameters, three for the triton and six for the alpha-axticle. The
paxameters
can
be determined
with the code of chosen
so as
[811.
was
done
The number K of the basis functions has been
1V =
function,
The function
Ffj (-v)'Iexp( P,
a
(Gaussians),
one
may
try the following
corre-
3
-
product
1Vri2).
2
IF
tions
The calculation
to reach convergence.
As another type of trial lated wave function
Tl'= F P
by the SVM.
of the nodeless harmonic-oscillator func-
describes the
independent single-particle motion,
whereas the function F takes into account the correlations. The
Jastrow-type correlation factor [471
is
taken,
as
IV
F=
(I+aexpj-b(rj
fj
_
ri)21
i>i=l KN
E ak exp (__2 i;BkT') k=1
in
a
single Gaussian form,
CILI Correlations in few-nudeon systems
with KIv
231
1, where ak is given by an integer power of a, and Bk is an N x N symmetric matrix of type, 2b Ei uiiii, where ui is an N-dimensional vector which has only two nonzero elements. =
2
2
Therefore the trial function TJ- becomes lated
Gaussian-type geminals (6.26)
needed formulas for matrix elements
a
with
combination of the
n
=
0 and R
=
corre-
0, and the
involving such special geminals
in that section. Three parameters v, a, and b can be varied to minimize the energy. Note that Tf is totally symmetric and that were
given
the center-of-mass coordinate x1v can be factored out from the intrinsic motion. This makes it possible to calculate the
its
dependence
on
intrinsic energy. We also consider contains
only
the
a special type of the correlation factor which pair correlation
K
F2
=
I +
N
1: ak
exp
bk (ri
_,rj)2)
k=1
Here the
pair correlation
can
be better described with combinations
of K Gaussian functions than with
Jastrow-type
correlation becomes
correlations of ear
more
than two
parameters bk and
v
Eq. (11.15). more
It is
useful in
particles play
a
where various
major role. The nonlin-
and linear parameters ak
minimize the energy. We use three potentials, the MT-V
expected that the
cases
are
[1441, V, [1451
determined to
and B,
[1481*
potentials listed in Table 11.1. The space exchange operator PI of the potential can be set equal to unity because of the bosonic symmetry of the
spatial wave function. The Majorana mixture parameters.
energy is thus
independent
Table 11.7 summarizes the results. The result for N
[311
=
of the
3 and 4
calculated with the correlated Gaussian basis is known to be in
good agreement with other accurate calculations available in the literature. It is interesting to note that, except for the alpha-article with the MT-V potential, the pair correlation with K 2 terms-already gives lower energy than the Jastrow-type correlation described with a single Gaussian function. (The alpha-paxticle has more compact structure than the triton. Because of this property, the strong repulsion of the MT-IT potential has to be fully taken caxe of in order to obtain the ground-state energy particularly for the alpha-particle. The correlation factor of Jastrow type appears to be more suited than just the pair correlation of Eq. (11.16) in this regard.) The small difference =
Complements
232
between K tion
can
=
2 and K
pair correladescription of
3 calculations indicates that the
=
well be described with two Gaussians. A better
pair correlation appears to be important. The trial function with K 0 is nothing but a product of the single-particle wave functions the
=
and contaiins
no
lot of energy. This potentials such a.-, the MT-
correlation. This function misses
evident in
a
singular significant energy difference between potential. MT-V potential. This suggests that with the results the CG asid CGG at least a triple particle correlation, e.g.,
becomes
more
more
Note that there is
V
a
IV
E
exp(
-
b(Tj
-
Tj)'
-
b(Tj
-
Tk
)2
-
b(Tk
_
T,)2),
(11.17)
k>j>i=l
has to be taken into account in the correlation factor for this potential to obtain accurate energy. This problem was investigated in [1671, which takes into account the two-, three- and even more-particle correlations and shows the importance of triplet correlations in producing accurate energy.
Table 11.7.
Energies
thealpha-particla in
of the triton and
the correlated
(CG) of Eq. (11-13) and the correlated Gaussian-type geminals (CGG) of Eq. (11.14), which axe chosen to represent correlations in one of two forms. The correlation factor F of CGG is defined by Eq. (11.15) for Jastrow and by Eq. (11.16) for the number K of pair correlations. See Table Gaussians
11.1 for the nucleon-nucleon
potential.
The
am
in units of MeV.
CGG
CG
MT-V
energies
Jastrow
K
=
0
K=I
K=2
K=3
[1441
triton
-8.25
-6.00
unbound
-5.78
-7.11
-7.27
alpha
-31.36
-29.04
-6.41
-27-03
-28.47
-28.93
triton
-8.46
-6.99
-6.66
-6-99
-8-11
-8-11
alpha
-30.42
-29.14
-27.92
-29.13
-30-06
-30-13
triton
-11.64
-9.95
-7-00
-9.87
-11.20
-11.20
,alpha
-38.34
-36.73
-28.19
-36.02
-37.60
-37.60
V1
B1
[1451
[1481
spin-isospin dependence, the correlation state-dependent. The evaluation of matrix
When the interaction has factor is also
expected
to be
CII.2
of
Convergence
partial-wave expansions
elements with such correlation factors would then be
Refer to related
11.2
[1681
for
a
recent
development
on
the
more
application
233
involved.
of the
cor-
functions to heavier nuclei.
wave
Convergence of partial-wave expansions
Here we examine the convergence of the partial-wave
expansion (PVVE)
bound-state solution for
in
obtaining few-body systems. We consider examples of three-body systems consisting of p, n, and n (triton) or of p, n, and a hyperon A. The A is a neutral baryon with mass mAO 1116 MeV. It has spin 1/2, isospin 0 and strangeness -1. The NA interaction is not strong enough to form a bound state between a nucleon and a A. When one or few A's are embedded in a nucleus, they form a bound system called a A hypernucleus. The simplest hywhich is called a hypertriton pernucleus is the (p, n, A) system 3H, A and its ground state has P 1/2+. The binding energy of 3A H is 2.4 MeV from the three-particle threshold. As neither subsystem of pA nor nA forms a bound state, A couples weakly with p and n. In fact the A separation energy of 3A 1-1 is only about 0.2 MeV. Most of the binding energy of 3A 11 thus comes from that of the pn subsystem, namely the ground-state energy is just =
=
0.2 MeV below the threshold of deuteron to
attempt
to describe the
the coordinates
xI
=
rp
-
ground-state r, and X2
(d) +A. wave
(I'P
=
+
It would be natural
function Tf in terms of
Tn)/2
-
rA.
simplify the analysis we assume that the pn subsystem has the same spin and isospin state as d, that is spin 1 and isospin 0. This assumption, known to be a rather good approximation, leads us to To
the condition that Tf must be we assume
an even
that the total orbital
angular
function of xI. In addition momentum of the system is
zero, and that the three 1/2 spins are coupled to J 1/2. Under these in Tf be follows: can general expressed as assumptions =
Tf
=
f (XI
I
X2, XI
*
X2) [xi (pn)
x
X.L
(A)].LM,
2
2
where the space part f is a function Of X1, X2 and X1'X2. For Tf to be an even function of xI, f has to be an even function Of Xl'X2The Hamiltonian of the system is
given by
H=T+V
h2
a2
h2
192
---
21L, ax,
2
21L2 C9X2 2
+
Vpn
+
VpA + VnAI
Complements
234
where IL1
and A2 2MNMj1(2MN + IVIA). Only the central is included in the Hamiltonian. The pn interaction VP" in
=
potential the triplet u
1
=
MN12,
even
state is taken from the Minnesota
Table
(see
=
11.1).
The NA central
potential
potential [147]
with
is assumed to have
the form of Minnesota type
VNA=
x
I VR+I(I+Po')Vt+-(I-P -)V 2 2
(lu+ 1(2-u)P')
(11.20)
2
2
with
VR The
u
=
2
VORe-ISRT2,
value is set
Vt
equal
they
are
-Vfote-""
to 1.5. We
-Vfoe-S' 2. (11.21)
V,
1
assume
that the
pA
interaction
the same, though there is some evidence hi fact slightly different. The potential parameters are
and the nA interaction
that
=
are
3, 4 binding-energy data: The strength 109.8, and Vo, 200.0, Vot parameters in units of MeV axe VOR h3j-2 of in units the and are 0.7864, and 1.638, nt r,-,R 121.3, ranges 0.7513, respectively. This NA potential will be used in the next ms determined
so as
to fit A
=
=
=
=
=
=
=
Complement to calculate the binding energies of s-shell hypernuclei. The equation of motion for f (XIi X27 XI *X2) is obtained by substi0 and tuting Eq. (11.18) into the Schr6dinger equation (H E)Tf the to coordinates. the Owing symmetry propspin integrating over of and the function of the wave assumption VpA VnA, we only erty element matrix the evaluate need to of, say VnA. The matrix elspin ement of PI exchanging the spin coordinates of the nA pair can be calculated by rewriting the spin function of Eq. (11.18), expressed in (pn)A coupling order, to that of the p(nA) order by using the Racah coefficient in unitary form (see Complement 6.3): =
-
=
[Xj(pn)xX.j(A)j.iM= 2
2
1: U(-!-!.!1;IS)[X.L(p)xXs(nA)].jM 2222
2
2
S=0'1
Vf3_ 2
1
[X.L (p) x Xo (nA) I.LM + 2
2
2
[Xi(p)xXj(nA)jjM. 2
2
Then the desired matrix element is
&1(pn) xx.L(A)ji].LMjP'j[xj(pn) 2
2
2
x
X.L(A)jfljm) 2
2
2
(11.22)
C11.2
Convergence
of
235
partial-wave expansions
1 .1 .1 .1; is) 1.1.1; IS) U( 2222 1) S+I- U(I:2222
.1.
(11.23)
2
S
The
equation of motion for f reads h2
a2
h2
a2
-
2A2 19X2 2
21L1- axi 2
I
+V,A(IX2
x,
-
2
+
1, P)
Vp. (XI)
-
E
boundary condition that f VpA(r, P') is given by
with the Here
VpA (r, P')
VR +
as
f (XI
One has to note that P'
i
X2 i
(I
1 X2 +
2
XI'X2)
has to vanish for
1Vt + 4
4
VpA
+
3V,) (lu
=
0,
large
(11.24)
xl_ and X2-
1(2 u)P')
+
-
2
2
exchanges rp
XII,P
(11.25)
and rA, which induces the
transformation of the coordinates: x, --+ x1/2 X21 X2 --+ -3x,/4 X2/2. The potential V,,A(r, P') is defined in exactly the same manner,
-
-
where pr
now
and
3xj/4
ground-state energy and the solution
of Eq.
transforms x, and X2 to
xl_/2 + X2
-
X2/2,
respectively. We obtain the
The variational trial function may be chosen correlated Gaussians (CG) of Eq. (7.3):
as a
(11.24).
combination of the
K
f(XliX2iXI*X2)
=
ECkfFAk(X) +FPAkP(X)II
(11.26)
k=1
1
where the second term with P
function is
an even
(0
0
-1
)
function of xj, Le.., for A
assures
All A12
that the trial
A12 A22
PAP
-A12 All A22 -A12 It is interesting to analyse the paxtial-wave contents of the solution. For this purpose we calculate the quantity becomes PAP
2
C1
=(f(XliX27Xl*X2)lpllf(Xl7X27Xl*X2))
(11.27)
with
PI
=
I[yl(XI)
X
YI(X2)j00)([YI(Xl)
X
Yl(X2)1001-
(11.28)
236
Complements
The result is
given
in Table 11.8. The admixtures of
D, G andhigher
waves axe small and the PWE is very effective in the present case. The root-mean-square (rms) radius of the hypertriton is calculated to be
4.87 fm. The
fm,
distance between the proton and the neutron is 3.61 which is slightly smaller th an the corresponding distance (3-90 fin) rms
obtained for the free
deuteron,
and the
distance between the A
rms
and the center-of-mass of the proton and the neutron is as large as 9.98 fin. Therefore the hypertriton is barely bound by polarizing the deuteron slightly in order to increase the binding between the deuteron
and the A.
Table 11.8. Partial-wave
decomposition
1
0
2
4
C2
0.99913
0.79 X 10-3
0.6x 10-4
The trial function from the
of the
hypertriton
(11.26) gets higher paxtial-wave
wave
function
contributions
term X1'X2. The
parameters A12 determine the appropriate weights of various partial waves. It is of course possible to solve Eq. (11.24) by expanding f in partial waves: cross
f(X1iX2iXI'X2)
E
"'
A (XI X2) [YI (ill) i
X
YI (F2)] 00
.29)
I=even
Eq. (11.24) reduces to a coupled equation for fI(X1, X2)'s. Refer to Appendix A.5 for a method of evaluating the matrix element with this type of basis functions in the case that A (XI X2) is expanded as Then
7
a
(XlX2)le
combination of Gaussians
"
2 1
-bX2
2.Table 11.9 shows the
energy convergence in the partial waves included in the calculation. fin conformity with Table 11.8 the inclusion of S wave alone is a good
approximation to the hypertriton and bigher paxtial waves give modest contributions to the binding energy. Noncentral forces necessitate the inclusion of higher paxtial waves and the PWE will be
slowly convergent
in
general.
Even in the
case
of
potentials the energy convergence in the PWE in one particula,r Jacobi coordinate set depends on a system and an interaction. To central
examine the contribution of forces
we
tightly
bound than the
higher partial waves
in the
case
of central
example of the triton (nnp), which is more hypertriton. The energy has been calculated
consider another
C11.2 Table IL9.
Convergence
of
partial-wave expansions
Convergence of the hypertriton
energy in the
237
partial-wave
expansion (PWE). The CG indicates the result obtained with the correlated Gaussian of Eq. (11.26). The energy is from the three-particle threshold. The deuteron energy is -2.202 MeV.
CG
PWE
1.. E
(MeV)
by using the
0
2
-2.305
-2.358
wave
-2-378
function of type
(11.29),
where x, is the relative
distance vector between the two neutrons and X2 is the relative coordinate between the proton and the center-of-mass of the neutrons. The
spin-singlet state. Two potentials are used: One is the soft-core Volkov (VI) potential [1451 and the other is the repulsive-core ATS3 potential [1461. Table 11.10 compares the PWE results with other methods, the hyperspherical harmonics (HH) expansion method (see Sect. 5-2) and the CG calculation. The PWE gives fairly fast convergence for the V, potential and the inclusion of S and D waves already produces the energy that is close to the accurate value by the HH method. However, for the ATS3 potential the 6 are not convergence is rather slow and the partial waves of up to I
two neutrons
are
assumed to be in
a
=
sufficient to get the energy which is close to the value calculated with the CG basis (11.26). (The two neutrons are assumed to be in a spin-
singlet state, but, as the ATS3 potential is spin dependent, to allow them to be in a spin-triplet state as well leads to lower energy than the result of Table 11.10. The CG calculation including this possibility 100 dimension, gives -8.765 MeV (the rms radius is 1.67 fin) in K which is in perfect agreement with the energy [170] obtained by the =
Faddeev method
(see
Sect.
5.3).
vergence
explain why the ATS3 potential leads to the than the V, potential. The matrix element for
potential
between the two neutrons vanishes for the functions with
Here
we
slower
con-
the central
values, so the potential is not responsible for the mixingin of different partial waves. The potential between the neutron and the proton, however, induces the I-mixing. To see this, we note that the latter potential has the form V(Iax, + bX21) (for the triton case a 1, but it is extended to a general case.), which is a :E(1/2), b function Of X17 X2 and the angle,& between x, and X2. The potential can be expanded in terms of the multipole operators as follows different I
=
=
Complements
238
Convergence of the triton
Table 11.10. sion
(PWE).
The two neutrons
are
See Table 11.1 for the nucleon-nucleon
V,
energy in the
assumed to be in
partial-wave expanspin-singlet state.
a
potential.
[1451 HH[1691
PVVE
1niax E
(MeV)
ATS3
0
2
4
6
-8.005
-8.390
-8.447
-8.460
-8.4647
[146] CG
PVVE
Imax
E(MeV)
0
2
4
6
-2.948
-6.215
-7.210
-7.484
-7-616
00
V(Iax, +bX21)
=
EV1(XI,X2)P1(COS'6)
(11.30)
1=0
with VI (X1 i
X2)
=21+1 2
7r
fo
V(Iax,
+
bX2 I)PI (COS 6)Sin'0d#-
example, for the Gaussian potential of V(r) (see Eq. (6.53) or Eq. (A.81))
For
VI
Voe-t2T2
we
have
(X1 X2) i
=
where
=
(11-31)
Vo (21 + 1) c(l, ab) il (2 M2 jabjX1X2)
E(I, ab)
=
I for ab >
0, E(I, ab)
=
2 2 2) -g (a 2 xj+b X2,
(- 1)
1
(11.32)
for ab < 0. Since the
Legendre polynomial PI(cosO) is proportional to the tensor product [YI(i-1) x Yj(i2-)joo (see Eq. (6.54)) and has non-vanishing matrix elements between the functions of different partial waves, we find that the various multipole components contained in the potential bring about the mixing of the higher partial waves. The relative strength of the multipoles thus determines the extent to which each partial wave is contained in the solution. For a long-ranged potential (small p) the magnitude Of V1 (X1 X2) is determined by the factor IL21/ (21 1) 11, the soluto contributions main low that so partial waves give only tion. However, this is not the case for a short-ranged potential. The -
7
is, the larger the contribution of thehigher in the extreme case of V(I ax, + bX2 1) E.g., inultipole components. shorter the
potential
range
-
Quark
C11.3
Pauli effect in s-shell A
hypernuclei
239
(fk2/7)3/2 and letting 1L oc in Eq. J(ax, + bX2), by putting Vo (11.32) We get VI(X1, X2) ((21+ 1)/(47))J(Ialxl jbjX2)1(jabjXIX2), ,
=
-
-
which indicates that the
multipole strength is equally strong for arbi-
trary 1. The result of Table 11.10 the fact that the ATS3
repulsive-core part. Clearly the matrix
be understood in this way from contains the strong short-ranged
can
potential
element for the central
potential
between the
neutron and the proton vanishes for the functions of diffrent I values if they are expressed in terms of the different Jacobi coordinate set where
the first coordinate is chosen to the relative coordinate between the neutron and the proton. This consideration leads us to the following ansatz for the solution
(X(I), X(1)) [Y,(X(,)) YJ(X(1)) 100
f
1
1
2
X
2
1
+
X
fl(3) (X (3), (3)) [yJ(X(3))
X
1
+
where
fl(2) (X (2), (2)) [yJ(X(2)) 1
x('), x(') (i 1
=
2
X
X
1
2
1
2
1
1
y1I
(X(2))j 2
Yj(X 2(3))
00
100
(11-33)
1, 2, 3) stands for the ith Jacobi set (see Sect. 2.4).
This type of functions is used in the Faddeev method and the CRCG [26] method. It is known that this expansion converges much faster than the one using one particular Jacobi set, and the inclusion of low
partial waves
is
be sufficient to get accurate solutions (see In contrast to this approach, the CG basis
expected to
also Sect. 4.2.1 and
[30]).
particular Jacobi coordinate but takes account of the contribution of the high partial waves by the cross term 1412XVX2 in the exponent of the trial function. This ensures accurate solutions in the correlated Gaussian approach. It is noted that the calculation of matrix elements with high I values is computer time consuming (see Appendix A-5), whereas in the CG basis no such PWE is employed, which makes it possible to calculate matrix elements very quickly. of
Eq. (11.26)
uses
only
one
Quark Pauli effect in s-shell A hypernuclei 4 3 and 5A Ije are Called Among a few known A hypernuclei AH, 4jA 1, AIIe, s-shell hypernuclei because in the simplest version of the shell model 0 all the constituents can reside in an s orbit. By applying the L 11-3
-
=
240
Complements
correlated Gaussians of
Eq. (7.3),
of s-shell A
in order to reveal the
hypernuclei quark substructure of baryons of
in
we
calculate the
resolving
the
binding energies significant role of the long-standing problem
'He. Since the
pioneering work of [1711, the S-shell A hypernuclei have intriguing problems, among others, the anomalously small binding of 5AHe. According to a recent survey of hypernu5 clear physics [1721, "The anomalously small binding of AHe remains model calculations an enigma. Simple based upon AN potentials, parametrized to account for the low-energy AN scattering data and the binding energy of the A 3, 4 A-hypernuclei, overbindS A5He by offered several
=
2-3 MeV."
The trial function is
given as
a
combination of the correlated Gaus-
sians:
CkAjFAk (X)XkJM?7kTMTli
TfJMTMT
(11.34)
k
where the operator A
antisymmetrizes the nucleon coordinates, XkjjU (Since L 0, the spin S is equal to the total angular momentum J), and 77kTMT is the isospin function. The spin and isospin functions are obtained by successive couplings, for examPle, XkJM= X(S12SI23 )JM [[[Xl/2 X XI/21SI2 X X1/2JSI23-IJTM with k representing a set of the intermediate spins S12, S1237 The optimal set was chosen for each matrix Ak. The matrix Ak containing is the
spin function
=
:::--:
...
....
nonlinear parameters
was
selected
by
the SVM.
The NEnnesota potential [1471 with u I was used for the NN interaction. The MA potential is the same as was used in the previous Complement. Table 11.11 lists the results of the calculation. The calculation reproduces the data well except for the case Of AHe, for which the theory overestimates the binding energy by about 1.9 MeV, =
consistent with what is mentioned above. Another is that
4 He A
is
bound than
4 H. A
thing to be noticed
strongly introducing a charge-dependent component as the IVA interaction, because then 4AHe would be less tightly bound because of the Coulomb repulsion. We have so far treated a baryon as a structureless particle and distinguished A from N. What happens if we take their quaxk substructure into consideration? Recently it has been shown in [173] that the anomalous binding problem of 5A He can be, at least partly, resolved more
It would be difficult to
understand this without
by considering
the
quark substructure
of the
baryons.
C11.3
Quark Pauli effect
in s-shell A
241
hypernuclei
Binding energies (in units of MeV) of s-shell hypernuclei. The separation energy BA of the hypernucleus AAX is defined by the difference of the binding energies, B(-A'1X)-B(-'1-1X). Table 11.1-1.
A
3
AH
A411
4AHe
5 He A
B(AX) A B(`1-1X)
2.38
10.62
9.91
34.93
2.20
8.38
7.71
29.95
BA(Cal.) BA(Exp.)
0.18
2.23
2.20
4.98
0.130.05
2.040.04
2.390.03
3.120.02
Let
us assume
for the sake of simplicity that
baryon is (u, d, d), A a
a
coinpos-
(u, d, s), (u, u, d), n particle of three quarks, p well. harmonic-oscUlator the in orbit in the Os and each quark moves with size parameter b. The spatial part of the three-quark baxyon is described by the function ite
=
=
OB
-`
(7rb2)
1
-1 4
exp
Z3
b2
( '3
(_ (P2 P2 P2)) (_ _b 2) O(int) 2b2
1
+
2
+
3
3
exp
-
where pi is the
=
2rI
(11.35)
B
quark's position vector,
rl
=
(P1
+ P2 +
P30
is the
is the spatial part of the intrinsic baryon's position vector, and 0(int) B of the function wave baryon depending on p, P27 (PI + P2)/2 P3* The value of b can be estimated to be about 0.86 fin by requiring that the function (11.35) reproduce the charge radius of the proton [1741. When the spatial part of the two-baryon wave function is described we interpret it as with the configuration of expf -3/(2b2)(r21 + r2)1, 2 the Os in all six orbit, exp(-p 2/(2b 2)), quarks move indicating that of the common harmonic-oscillator well. with size parameter b. By increasing the number of baryons, we thus expect that the manybaxyon wave function may receive a special constraint arising from the quark Pauli principle that any single-particle orbit can accommodate at most six quarks (three colors and up-down spins) for each flavor. It is easy to see that we have no apparent quark Pauli-forbidden 4 hypernuclei. However, this is not the case for states up to A 5 5He: Four nucleons of A He, when they are on top of one another, A have already six u-quarks and six d-quarks in the Os orbit, so neither u nor d quark of A can take the same Os orbit. This leads us to the conclusion that the five baryons cannot take a configuration of -
=
-
242
Complements
exp(-3/(2b 2) 1:5i=1_ r i2) forbidden state for
rated,
is
.
To be
5AHe,
more
specific, the (normalized) Paulibeing sepa-
with its center-of-mass motion
given by 5
TfPF (X)
=
JV
'(7b2)-3eXP(_ 4
-
3
3
_b2 Dr'
X5)2
(11.36)
i=1
where X5 is the center-of-mass coordinate of the five baryons. The normalization constant is given by JV (4M2 + M2)/(4MN + MA)2' =
N
A
where MN is the mass of N. Other Pauli-forbidden states sibly exist, but this is the simplest and most evident one. The solution TI obtained above for 5He A
Pauli-forbidden component of Eq. quence is very
was
might
pos-
found to contain the
(11.36) by only 0.44:'YD, but its conse-
If this Pauli-forbidden component is simply subtracted from the wave function Tf, the BA value would have been
from 4.98 MeV to 2.74
changed of
A5He!
The reduction is
pectation may be
significant:
a
principle
MeV,
anomalously
small
binding
because the reduction in the
calculation of the so-called variation
6.1),
no
due to the fact that the energy exvalue in TfF(x) is very large, i.e., 513.6 MeV. However, this premature conclusion from the viewpoint of the variational
mainly
binding energy is based on a before projection type (see Sect.
that
is, the variation has been done before the elimination of the Pauli-forbidden component is made from the trial function. To calculate the
binding energy more precisely by taldng into acquark effect, we have repeated the calculation by replacing the correlated Gaussian in Eq. (11.34) with the one that 'has count the
no
Pauli
Pauli-forbidden component:
FAk (X)
11.4 The
FAA: (X)
C nucleus
-
TfPF (X) (TfPF (X) IFAI, (X))
-
(11.37)
system of three alpha-particles example application of nonlocal potentials, we take up simple model for the 12 C nucleus, the 3a model. In this model 12C
As a
12
=
an
of the
as a
C11.4 The
12C
nucleus
as a
system of three alpha-particles
243
5
>4
3
Exp.
Pq
2
0.6
0.4
0.2
0.0
b
Fig.
[fml
separation energy BA of 5H A three-quark baryons
11.4. The A
parameter b of
containing
1.2
1.0
0.8
six protons and six neutrons is
as
a
function of the size
approximatedas
bosonic
a
system of three alpha-particles. The physical reason behind this picture is in the fact that the alpha-particle is a strongly bound, very stable system compared to light neighboring nuclei. One needs an energy of about 20 MeV to excite the of the
alpha-particle.
Because of this
nuclei tend to form
unique
consist-
feature, light subsystem ing of two protons and two neutrons, which is called an alpha-cluster. The alpha-cluster would never be identical to the alpha-particle but it may happen that a model descriptioii assun-Ang such subsystems can explain many properties of the light nuclei [60, 175, 59]. To calculate the binding energy of 12C in the 3a model, one needs to know the potential between two alpha-paxticles. It is hard to derive a potential which acts between composite particles through the underlying interactions of the particles composing the composite particles. Thus we use a phenomenological potential which successfully reproduces the alpha-alpha scattering phase shifts. The one employed here is an I-independent local potential consisting of both the nuclear some
paxt of Gaussian form and the Coulomb part
V(r)
-Vo
e
_P,
[1761:
4e2
2 =
a
erf (,3r),
+ r
(11-38)
244
Complements
where VO
122.6225
=
MeV, p 0.22:ftn-2, and scattering data very well
0.75 fin-'.
=
fits the
this
Though
to about Em
30 potential MeV and reproduces the 0+ resonance at 92 keV quite well, we have to note that it predicts "redundant bound states", two (-72.625 and
-25.617
I
=
2
in the I
MeV)
state, these bound states ftom the
0
=
Since the two
wave.
wave
and
one
(-21.999 MeV)
known to form
in the
bound
alpha-paxticles considered spurious and must be excluded are
no
axe
space. The existence of
configuration
=
system of two nuclei is due understood in the
to the Pauli
spurious
principle
and
states for
can
be
a
easily
version of the nuclear shell model. This is
simplest orthogonality condition model [1771 which succeeded to give a foundation to the deep local potential of type (11.38) from the microscopic theory of scattering between composite particles. Let us denote the spurious states as 0, 1 for I (r), (n 0 for I 0, and n 2). We require that the wave function TI, for the 3a system be free from the spurious components in the alpha-alpha pairs, that is, the basis of the
=
=
=
( Pnim(ri
-
rAff)
=
(11.39)
0-
Here ri is the position vector of ith alpha-particle. Unless this condition is imposed, the ground-state energy with the potential (11.38)
strongly overbound because
would be
the
alpha-clusters,
as
bosons,
would occupy the lowest possible states. An alternative, convenient approach to eliminate reasonably well the spurious components is to use
the nonlocal
Fini
1M ==
potential of projection operator type
I
(ri
rj)) (W,,,M (ri
-
and include it in the Hamilto-nian
-
as a
kind of
N
I
=
T
-
Vij(lri-rjl)+A
Here A is
a
wave
positive
E EI'i'!
(11.41)
j>i=l n1m
j>i=l
that if the
"pseudo potential" [178] N
E
Tn, +
(11.40)
rj) 1,
constant chosen to be very
large. The idea
is
function contains the spurious components, then the energy would be comparatively high. Therefore, the variational lowest solution would approach a state that has a negligible overlap with the
spurious components. The spurious bound
state
lar
O,am (r)
Cae- 2 a
can
be well
approximated
in the form
2
Y1. (r)
(11.42)
CIIA The
12
Then the nonlocal
C nucleus
potential
as a
245
system of three alpha-particles
summed
over m
takes the form
_Vnlra
f
ij
M
21+1
(rr') -47r
R
(cosv) E ca*ca, e-la 2
r2_ 1 af rf2
(11.43)
aa,
where V is the
between
angle
r
and
r.
The trial function for the
system was chosen as a linear combination of the correlated Gaussians, FA(x), of Sect. 7.2. The matrix element of the nonlocal potential (11.43) in this basis can be calculated with the use of Eq. (7.46). 3a
high sensitivity of the energy on A calls for numerical calculations with a high accuracy in order for the pseudo potential to play the role of the Pauli projector. It was found [1791 that the pro0 and 2 do not jection effect is so strong that the partial waves 1 The
=
contain bound states, and a contribution of more than 95 % to the ground-state energy is due to the partial wave I 4. Table 11.12 lists =
the
energies
of the
ground
state and the excited states obtained in
a
[1801 using the basis function of Eq. (11.33). h (X1 X2) is expanded as combinations of Gaussians 2 The calculation does not reproduce the experix11 x12 exp(-ax 1 _OX2) 2 mental energies very well. In [1801 the energies of the 2+1 and 41+ states
variational calculation The radial part
are
also
given. They
are
-3.77 and -2.25
MeV, respectively, which
compared to the experimental energies of -2.84 and 6.80 MeV. Here the discrepancy between theory and experiment is more serious: state is lower than that of the 0+1 state and the The energy of the 2+ , energy of the 4+1 state is much lower than the experimental energy. are
Note that there is
no
Pauli-forbidden state in the 1
makes the contribution of the
high partial
> 4 waves,
waves too much
which
important.
This calculation suggests that the 3a model with the local aa potential has only limited success and is not very realistic to reproduce the experimental energy spectra. This doe not, however, exclude the
Table 11.12. Energies of the 0+ states of 12C calculated in the 3a model [180]. The energy is measured from the threshold of the 3a breakup.
0+ 2
0+ 3
-3.38
-1.43
3.70
-7.27
0.38
3.03
0+ 1
(MeV) Exp. (MeV) Cal.
246
Complements
a
+
8Be,
a (
and
'Be(a,-y)12c.
Appendix
Matrix elements for
Gaussians
general
The matrix element for the correlated Gaussian with
arbitrary physically important potentials including central, tensor and spin-orbit components. A simple and straightforward method is presented to calculate the matrix element for the general correlated Gaussian-type geminals.
angular
momentum is obtained in
a
closed form for most
A.1 Correlated Gaussians We start with
Eq. (7.2)
to evaluate the matrix element for
a general generating functions g is given perform the operations prescribed in Eq.
The matrix element between the
case.
in Table 7.1 and
one
has to
(7.2). A.1.1
The
Overlap
overlap
of the basis functions
matrix element
can
be obtained
through
(fK" LM (u, A", X) 1 fKLM (u, A, x)) I
BKLBK-'L
ff
_ d9de'YLm(i;_)YLm(e')
d2K+L+2K+L
dA2K+LdAI2K'+L
3
X
detB
exp
[q A2 + q"A/2 + PAA/e. er]
where
Y. Suzuki and K. Varga: LNPm 54, pp. 247 - 298, 1998 © Springer-Verlag Berlin Heidelberg 1998
\,=0
,
(A. 1)
Appendix
248
B=A+A',
q=
2
fiB-lu,
B-lu ,
q
2
p
=; B-lu.
(A.2) To
perform
the
operation prescribed
Eq. (A.1),
in
we use
the
ex-
pansion
[qA2 + q, A/2 + pAA/ e. el]
exp
CO
00
CO
E 1: E H(n, q).ff(i , q) H(m, P) Vn+m y2n+m
fn
n=O n'=O m=O
(A-3) where
H(n, x)
is introduced to Xn
10
H(n, x)
for
n!
simplify the
notation and defined
non-negative integer
n
by
(A.4)
otherwise.
Differentiating with respect to A and Y, followed by A 0, 2K + L and gives non-vanishing contribution only when 2n + m 2K' + L, while the integration over the angles of e and e' 2n + m becomes nonzero if m is equal to L + 2k with non-negative integer k (see Eq. (6.18)): =
=
if
d iYLM(e-)yLM(,
di
1
j)*(e.eI)2k+L
K k and n Rewriting n overlap matrix element =
-
=
K'
-
k,
=
we
BI.-L
(A.5)
obtain the result for the
(fKIILM(UI, A', X) IfKLM(u, A, x)) 3
(2K+L)!(2K'+L)i
(27)N-1-
BKLBKIL
detB
2
min(K,K') X
E
H(K-k,q)H(K-k,q)H(L+2k,p)BkL- (A.6)
k=O
The B,,I value is given in Eq. (6.9). Note that the values of K and K' usually be chosen to be small. in practical cases, and then the sum
can
over
of K
k is limited to =
K'
=
just
a
few terms. In
0 the above result
particular, simplifies to
for the
special
case
A. 1 Correlated Gaussians
249
(ALM(Uli A i X) IfOLM(Ui A X)) 3
(2L + 1)!!
(2v) N_I
(
4v
detB
2
)
L
(A.7)
P
A.1.2 Kinetic energy
The kinetic energy with the center-of-mass kinetic energy subtracted is given in Eqs. (2.10) and (2.11). It is simply written as FrAn-/2. The matrix element of the kinetic energy is then
Table 7.1
easily obtained by using
as
I
(fK"LM(UIjXjX)j ?rA7rjfKLM(UjAjX)) h2
(2r) N-I
2BKLBKIL
detB
(
d2K+L+2K+L
[R
X
dA2K+LdA/2K'+L
X
exp
3 2
+
if
de-- do
pA2
+
YLm (&) YLM (4 )
p/A/2
+
QAA'e-.e-f
[qA2 + q' A/2 + PAAte.e/
(A.8)
where R
M(AB-'A!A),
=
P'=
P
=
-i B-'AAAB-lu ,
-i!B-1A!AA!B-'u, Q
=
2
The matrix element of the kinetic energy manipulation similar to the overlap case:
B-'AAXB-lu. can
be obtained
I
rA7rjfKLM(u, A, x))
(fKfLM(Uf7A!jX)j 2
3
(2r)N-1
h2(2K+L)!(2K+L)!
x
detB
BKLBK"L
2
E f RH(K
k, q)H(K'
-
2
-
k, q)H(L + 2k, p)
k
+PH(K
-
k
-
1, q)H(K'
-
k, q')H(L + 2k, p)
(A.9) by using
a
250
Appendix
+P'H(K
k, q)H(K'
-
-
k
-
1, q)H(L -IF 2k, p)
+QH(K-k,q)H(K'-k,q')H(L+2k-l,p)lBkLThe
case
K'
with K
=
0 reduces to
a
(A.10)
simple result
very
(ALM(U17A x) I --iAw I fOLM (u, A, x)) 2 ,
h? 2
A.1.3 Next
(2,x)'V-1
(2L + 1)!! (R+LQp- )-
detB
4-x
3 2
)
L
(A.11)
P
Two-body interactions
we
derive the matrix element for the interaction of Eqs.
(2.3)
and
(11.1). To evaluate the matrix element of the operator expressed as a tensor product, it is convenient to make use of the famous WignerEckart theorem
spherical
[70, 71, 721, which states that a matrix element of a operator 0.-, between states with angular momenta
tensor
JM and XIW
coefficient and
be
can a
expressed
as
product of
a
reduced matrix element which is
z-components of the
(J'M'JO-AIJM)
angular
a
Clebsch-Gordan
independent of the
momenta:
(JMrILJYM-) =
,V2J'
V 110. 11 J).
(A.12)
+ I
Here the reduced matrix element is barred matrix element. The factor
expressed by the so-called double-
I/V -2-Y+I is factored out because
then the reduced matrix element is
symmetric
to the bra and ket
interchange
IEV 110. 11 X) provided of
OKIL
is
=
(-I)J+r.-J,(y 110. 11 J),
(A-13)
that the matrix element is real and the Hermitian
(0'1')
t
=
E(- 1)
"L
Or,, -i-L
with
a
phase
factor
E(E2
conjugate =
1).
By applying the Wigner-Eckart theorem we can express the matrix element for the orbital and spin angular momentum coupled wave function
as
follows:
(Tf(LI SI) JM IV(I'ri
-rj 1)
(Or ij (space) 0,,ij (spin)) ITf(LS) JM)
U(LnJS; L'S) V,f(2L' + 1) (2S + 1)
-
A.1 Correlated Gaussians
(L' 11 V(Iri
x
rj J)0,,jj (space)
-
251
11 L) (S' 11 0,,jj (spin) 11 S). (A.14)
potential V(r) is assumed to be a function of r only. Eq. (A.14) and its extension to a general coupled tensor operator is given in Exercise A.I. The reduced matrix element of the spatial part is obtained through Here the form factor of the
The derivation of
(LMn1ijL'M') (L' II V(I ri v/2--L'+I (L'IWIV(Iri
=
jV(r)(LM'jJ(rj case
uncoupled by
where
wave
-rj
(_1)r. 2S'
one uses
-
I I L)
-
r)O,,,,,j (space) ILM)dr.
the orbital and spin matrix element
(A.15)
angular momentum
(A.14)
can
be obtained
(0,,,j (space) 0,j (spin)) lTf(LS)JM)
ri 1)
-
U(LKJS'; LIS)
+ I
2S+1
x
-rj
function, the
R(L"S")JM IV(Iri
rj 1) 0,,i, (space)
(space) ILM)
=
In the
-
(LMLts1-tjLIML')(SMSrvjS,MS)
(TfL,MLS,ms, IV(Iri -rj J)0,st,,j (space)Oc,jj (spin jTfLmLsms) (A.16)
Since the spin matrix element is easily calculated as will be shown in Appendix AA, we will focus on the spatial matrix element and evaluate it for the most important components of the nuclear potential, that is, central, tensor and spin-orbit components.
(i) central
and tensor interactions
The operator
Or-A,j (space)
can
be I for the central
potential
and
Y2,,(r-i---r-j)
for the tensor operator. See Eq. (11.7). Thus both the central and tensor components can be treated by assuming the form
---rj) for 0,,,,,3. (space). Expressing ri
of Y,,,,, (,r-i
(N 7.1,
-
1)
we
x
1 column matrix
-
rj
as w (ij) x
Oj) defined iii Eq. (2.13)
and
with the
using Table
obtain
(fK'L'M'(UI74 x)IS(ri
-
rj
-
r)Y,,j,(rj
-
rj) IfKLM(U7 -47 X))
Appendix
252
'::::-
6)(fKLM(UIi W 7x)IJ(W(ij)X-r)jfKLM(u,A,x))
Yrg(
(27r)N-2C
1
2
1
e-fc7'Y-"(i )BKLBKL' X
if
de^- d,
detB
3 2
)
d 2K+L+2K'+L'
iYjLM(e,-) YJL, M,
dA2K+LdAI2K'+L'
XeXp[q,X2+ A/2+ Aye.e/+7Ae.,r+,Y/,XIe/.,r] (A-17) where C
w(ij)B-lw (ij)
c
I
^/2
q=q-
712
q'-
2c
w(ij) B-1u, =P-
2c
cw(ij)B-1U',
7
1'7YI.
(A.18)
C
All of these quantities
depend on i and j but we omit the labels i and j to simplify notations. The integration over the angles of e and e' can be performed by expanding exp (pAVe- e' + -/Ae- r + -l'.Ve,- r) in -
power series and
-
using the relation
(e.e )nj(e.T)n2(eI.,r)n3 n2 +n3
Lel=l
njn2n3
RL Lfn
[[YjL( )
X
Y
X
(A.19)
Y. 00
LL'r.
with Rnjn2n3 2 LIts
=
(_I)nl+n2+n3
Bnl-LI7 Bn2-12 12 Bn,3 -13 2
'1
2
2
3
111213
E(2L+:1:1 ) (2
V
r.
+
212+1
1)
C(1112; L)C(1113; L)C(1213; K)
U(IIIILK; L'12).
(A.20)
See Exercise A.2 for the derivation of this relation. In the
defining
I or 0, and 12 equation for R the sum over 11 is limited to ni, ni 2, and 13 have similar ranges, respectively. Possible values of L, L", and r, -
...'
A.1 Correlated Gaussians
for
a
given set of values of ni, n2, and
n3
limited
axe
253
by the conditions
that L takes the values nj +n2, nj +n2 2, ..., I or 0, and L' and r, take similar values given by ni, n3 and n2, n3, respectively. In addition, the -
is restricted to
sum over
L, L,
for
values of
given
and n2 + n3
-
would vanish.
K
r.
even
values of L + L+ K.
Conversely,
L, L', and K, all of the nj + n2 L, nj + n3 L, have to be non-negative and even, otherwise Rnin2n3 -
-
LL'r.
R has the symmetry: The reduced matrix element becomes
Clearly
(fK'L'(UIi -4 i X) 11 V(17'i
-
ri Dyt#
njn2n3
RLLII,,
pnjn3n2 "L'Ln
i -rj) 11 fKL (u, A, x)) 'R
Lf
(2K + L)! (2K'
+
L') 1
BKLBKILI
E
x
(
(27) N-2 detB
C.)2
H(ni, #)H(n2, -y)H(n3, 71) I(n2+n3+2,c)RnL,-n2n3 LIn
n,n2n3
2K + L
-
XH
nj
-
2K+L'-ni-n3
n2
2
2
(A.21) where the
I is defined in
Eq. (7. 10) potential for different pairs of particles be calculated with the above formula by changing only 01) in
integral
The matrix element of the can
Eq. (A. 18). We note
some
useful
applications
of
Eq. (A.21). For example,
calculate the matrix element of I ri-rj I' simply by putting r' and K 0. The two-paxticle correlation function can
one
V(r)
=
(fK-'Ll (u, A', x)
J (I ri
-
rj
a) Y,, (ri
-
rj)
fKL (u, A, x))
(A.22) easily calculated by taking J(r a) for V(r). K' We note again that the formula (A.21) simplifies for K 0, that is, the triple sum reduces to a single sum over, say ni (nl_ 0, min( L, L', (L + L' r,)/2)) and n2 and n3 have to satisfy L and nj + n3 L, respectively. In addition, in this nj + n2 njn2n3 calculated is R case simply by a term with 11 special nj, 12 n3 alone in Eq. (A.20). This simplicity will be used to obtain n2, 13 the matrix element of a density multipole operator. See Complement can
also be
-
=
=
=
-
=
=
=
=
A.2 for the details.
254
Appendix
(ii)spin-orbit The operator
angular
interaction
0,,,,,, (space)
momentum
((ri
rj)
-
1-
x
2h
and
and
(2.14)
trix element in Table 7.1 into
(fK"L"M"(Ufi Al7X)jV(jT'i
substituting
pj))
See
,.
x
and
Eq. (11. 9). -r,
with the
we
the
corresponding
ma-
i
obt
I'jj)1jzjjjfKLM(u, A, x)) (27)N-2
.1 Cr2
drV(r)e-2
-
is the orbital
1,
Eq. (7.2),
-
(pi
in terms of
x
Eqs. (2.13)
of
=
spin-orbit potential
Fij)x W)7r)
Expressing 1,_,,j use
1,1i,
for the
3 2
C) I
detB
BKLBK'Ll
d2K+L+2K'+L'
dA2K+LdA/2K'+LI q
x
exp
X
ifn*
+
X
qf A/2
r)t,
-
+
PAy e. ef + yXe.,r + -Y/Ale/. -
q'Af (e,
X
(A.23)
r),,
with 77
=
(MXB-lu,
Note that to derive
an
arbitrary
=
Eq. (A.23)
V(r) e-c('-a)2 (r for
q'
vector
x
a
a)dr
(MAB-luf. use
=
(A.24)
is made of the relation
(A.25)
0
provided V(r)
is
a
function of
r.
See
Eq.
(A-163). Before
performing the operation prescribed
that the matrix element vanishes in the has
parity
(_I)L
and because the
case
Eq. (A.23), we note 0 L. Because fKLM
in
of L
spin-orbit potential
does not
change
A. 1 Correlated Gaussians
parity, IL
-
L'I
has to be
even.
On the other
hand,
255
the tensorial char-
spin-orbit potential imposes the condition IL LI :5 1. Both the conditions are met only when L is equal to L'. This special result is entirely due to the unique feature Of fKLM and does not always hold for general wave functions. Combining Eq. (A.19) and the relation acter of the
i
fqX(e
-
r) 0
X
4vf2--x 3
-
77W (e'
rjqA[Yj(ii)
X
x
r)
Yj(i )Jj, -,qA[Yj(( )
x
YI(,P)JI, (A.26)
yields
the reduced matrix element
(fK,ILI (Ul A! X) I I V(I'ri I
I
as
-rj 1) Iij
I I fKL (u, A, x)) '3
4 V2--x
JLLI (-1)
3
x
E
q
L
(2K + L)!(2K' + L)!.
(27r)
( N-2C.)2 detB
BKLBKIL
l(n2+n3+3,c)H(nl,p)H(n2,,y)H(n3,-y')
n,n2n3
2K+L-nl-n2-1 H
xH 2
(2K'+
L
-
nj
-
n3,,,)
2
n., n2 ns
C(AI; L)U(LAII; 1L)RAL1
x
A
I(n2+n3+3,c)H(nj,#)H(n2, -y) H(n3 7f)
+77'
,
njn2n3
H
x
(
2K+L-nl 2
-n2
q)H(
2K+L-nj
njn2n3
C(AI; L)U(LA11; 1L)RLAI
-n3
-
1
2
(A.27)
straightforward and easy to follow. As a simple check of the above formula, the matrix element of the orbital angular momentum is calculated in Exercise A.3. It is again possible, however, to get a simpler formula by performing the'P integration first, as was done in Complement A. 1. This task is reserved for Exercise A.4. The above derivation is
256
Appendix
A.1.4
Density multipole operators
The matrix element of
density multipole operator plays a substantial role in investigating the properties of a system, e.g. the density, the deformation, the electromagnetic transition rates and the electron scattering form factors. The basic element of the Multipole operator a
takes the form
0,,,L, (space)
:---
f (Iri
xm
-
I)YI,
(A.28)
Note that the argument of the
density multipole operator is not ri but is correctly taken as ri xN, which is the single-particle coordinate measured from the center-of-mass coordinate. Comparison of Eqs. (2.12) and (2.13) immediately suggests that the matrix element of the density multipole operator ought to be calculated in exactly the same manner as that of the two-body potential. In fact the reduced -
matrix element
(fK"L-'(U/7 X7 X) 11 f (ITi
-
XNI)yn(Tii
-
N)
fKL (u, A, x))
XN
(A.29) be calculated
can
replacements
of
by Oj)
the --+
same
formula
as
0) inEq. (A.18)
Eq. (A.21) and of
with the trivial
V(r)
--+
f (r)
in
Eq.
(7.10). Just
as
the matrix element for the
lated from that of relation function
As
we
J(ri
can
xN
-
-
r),
and ri
be calcu-
the matrix element for the
also be derived ftom. that of
J(ri
-
rj
-
-
-
reduced to the
are
r)jfKLM(u,A,x)).
in fact obtained in the derivation
one
(fKfL'M" (U , X7 X) IJ(17VX (2K + L)! (2K'
+
BKLBK'L'
-
L)!
following Eq. (A.17),
r) IfKLM(u, A, x)) (2T,
)N-2C)
32
detB
E(LMr.M -MjLM)Y,,m, -m(i )* M
of type
This matrix element
the fi-nal result:
x
r).
-
(fK'L'Mf(U'jA!jx)jJ(iv-x give
cor-
-
already noted in the above derivations, both of ri xN rj are expressible in terms of the relative coordinates x as iv- being an appropriate 1 x (N 1) row matrix. There-
fore all of these matrix elements
we
can
have
Cvx with
was
density operator
L'
-1) -e -v/-2Ll + I
2
so
that
A.2 Correlated Gaussians with different coordinate sets
E
X
257
rn2+n3 Rnin2n3 H(nl, P) H(n2, -y) H(n8, -y') LLIn
nin2n3
2K+L X
H
-nI -n2
2K' + L'
-
H
2
n,
-
n3
2
(A.30) 7/,
given in exactly the same manner as in Eq. (A.18) with w(W being replaced by w. Here, r, takes the values
where c, 7,
p
are
L+L', L+L'-2,..., IL-L'I,
and it is of course restricted
by IAF-MI :!
The above matrix element becomes very simple for the special case K' of K 0 as was noticed in the previous subsection. Since the is.
=
=
applications, we show in Complement A.2 simplified to just a double sum.
matrix element has useful
that
Eq. (A.30)
can
be
A.2 Correlated Gaussians with different coordinate sets As the correlated Gaussians treated in the previous section have a very simple transformation property implied by Eq. (6.29), we as-
they are all expressed by a particular set of the relative coordinates x. However, the correlated Gaussians with the angular function OLM(X) given by Eq. (6.3) do not have such a nice property. Moreover, the use of different sets of the relative coordinates for these correlated Gaussians leads in general to a faster convergence because they allow us the possibility of describing naturally different types of correlations and asymptotics. The lineax transformation of the coordinate sets, however, leads to a formidable task even for a system of only four particles, because the function OLM(X) shows no simple transformation rule. See Appendix A.5 where the matrix elements for a three-body system are explicitly evaluated by transforming the corsumed that
related Gaussians from
one
Jacobi coordinate set to another. The aim
of this section is to outline ments for are
we
a
method of
calculating
the matrix ele-
the correlated Gaussians which
N-body system using expressed by different sets of the coordinates. By making use of Eq. (6.22) to generate the correlated Gaussian, have to evaluate the following matrix element for the operator 0: an
N-1
exp,
2
x-fXx)
11 Yij, j, (x ) j=1
Appendix
258
N-1
-1:Z.Ax)
1
0 exp
x
N-1
Yli.i (xi)
2
Climi,9ai
N
ali-Mi
01i
-
1
01j,i +Mi'
(-')Mj 01j,
3
C11 -mlj aal - ar'1j'+mi' j j j
97i j=1
(A.31)
(g(a'Jt';A!,x')J0Jg(aJt;A,x))
x
ri=O,-r =O 3
(-1)'YI-,,,(xi) to identity (Yl,,,(xi))* take care of the complex conjugation in the bra side. Note that the vector ti is defined by 2, i(l + i 2), -2-Fi) and likewise t!j by Here
(1
is made of the
use
T,
/2' i(I + T/2), -2,Tj j
complex conjugated.
transformation T trix element
x'
as
(A.31)
of g in the bra side should Assume that x' is related to x by a linear
The vector
j
not be
=
=
tj'
Tx. With the
between two
g's which
different sets of relative coordinates
Eq. (6.28) expressed in
of
use are
the
ma-
terms of
be reduced to
can
(g(a'lt'; X, x')J0Jg(aJt;A, x))
(g(Ta'lt';i XTx)101g(alt;A,x)), and
can
(A-32)
be obtained from Table 7.1 for most operators. 1, the matrix element
(A.32) takes, where v alt+ except for a trivial constant factor, the form e2 B' is matrix the and x given by (A+ TXT) (N 1) (N 1) Ta'It' For the unit operator 0
=
-IbB'V
=
,
-
-
To
simplify the
SN-1,
a.el
=
aNtIV
=
we
SN,
---,
rename
alt,
afN-,eN-l
=
expressed by a (2N 2) by 9Bs, where B is expressed in
6B'v
Then
notation
matrix B
can
be
-
aN-1tN-1 a2N-2t2N-2 82N-2-
-=
si,
x
(2N
-.-,
-
2) symmetric
terms of B' and T
as
follows
B
=
Thus the
(
B'
TB'
Bi TB'i
(A.33)
operation prescribed
in
Eq. (A.31)
can
be reduced to the
form N-
Oi",`
exp,
(2IBs
+ ibs
))
(A.34) .1=0'---'12M-2=0, 1=0,---,'2N-2=0
A.2 Correlated Gaussians with different coordinate sets
259
with
(A.35)
i9ail 0-Til-m Here the factor iv-s is included because it is needed in the
case
of the
potential energy matrix elements. See Table 7.1. It is worthwhile noting that the present formulation leads to a unified prescription of Eq. (A.34), which is independent of the choice of the relative coordinate sets but requires only a very simple calculation of the matrix B. It would be extremely tedious if one were to try to rewrite the anguN 1 lar function of fL'=I Yj . (x'.) in terms of those angular functions 3 conforming to the coordinates x. Using Leibniz's formula we can express Eq. (A.34) as 3
2
2
T
3
=1
Aj! (Ij
'XiAi
-
Ai)! (Ai
(1i Mj)! (1i
-
-
-
Mi)! mi
-
Ai
+
Ai)!
2N-2
Oj"
X
exp
(2 Os) -ri=O
2N-2
ai
X
ie,
(A.36) aj=O -ri=O
The last factor is
easily evaluated by using Eq. (6.58):
2N-2
11
ali-'Xi'mi-Ai e-vs i Ui=O -ri=O 2N-2
(A-37)
CIi-'XiMi-/-tiY1i-)LiMi-Ai(Wi)-
Possible values of Ai and [ii are determined by the conditions that O
vanish. Because of condition that Ai
-
assures
do not
that the coefficients
a special form of si-sj mentioned pi < 2Aj has also to be met.
below,
another
remaining factor, we note that ABs is quadratic both in the ai's and -ri's because of si sj -2aiaj (ri -r.j) 2. (S ince, for each aj, -ri appears at most in quadratic power, the number of To evaluate the
-
=
-
260
Appendix
differentiation with respect to -ri cannot exceed two times the number of differentiation with respect to aj, that is, /Xi pi :! 2Aj.) Therefore, -
.!Bs 2
when
e
these
are
contributes Thus
we
expanded in power series in both ai's and -Fi's, and when zero after the differentiation, only the term (.IgBs)Q 2
is
set to
2N-2
provided that Ej=j
2Q and
Ai
2M-2
2Q.
i=1
(-2)Q
19Bs a,\"/L'exp i
(2
Q!
Cti=O -ri=O
Q
2N-2
EBjjajaj(Tj-T.j)2
X
i=1
To have
1:2N-2 (Ai ILi)
have
(A-38)
i<j
aj=O -ri=o
non-vanishing contributions,
each of
Q
terms must be dif-
ferentiated with respect to aj, aj, and either twice with respect to 7i or -Fj, or once each with respect to both 7-i and -rj. This operation
yields 2Bij,
where the miniis
to -ri and -Fj. This
respect
sip
sign comes from the differentiation with is
denotedEij.
Therefore
we
obtain
2N-2
a,"' '14 exp
(2sBs)
ai=o -ri=O
(-4)Q Q! Q
(A-39)
Ejj, Bi,j,
x k=1
'k<jk
where the
extends
all the
possible combinations distributing the given multiple differentiation into pairs of differentiations. In fact the operation prescribed in Eq. (A.38) can be done easily by using the software Mathematica [1841. The case of a three-particle system is given in Co mplement A .3. sum
over
The matrix element for the kinetic energy can be obtained similarly. In fact the matrix element (A.32) for the kinetic energy takes the form relation
(9Cs)e12'BS,
but this
can
be dealt with
easily. Using
the
A.2 Correlated Gaussians with different coordinate sets
(Ws) exp
d
1Ms
(2 )
2
dp
( 19 (B +pC) s)
exp
261
(A.40)
2
P=O IgBs
(112N-2 aj " "Ws ing
one
of the
)
e2
j=1
Bi,,j,,'s
Eq. (A.39)
in
be obtained
can
by simply replac-
ai=O,ti=O
with
Ci,j,.
The
potential energy matrix element for the spin-orbit interaction requires a slight modification. Noting that the general form of the matrix element (A.32) for this case can be expressed as + ivis) (see (Sk)yeXP (19B.9 2
(2IMs
(Sk)jzeXP
7.1),
Table
+ iv-s
( -1)"
we
make
use
exp
ly(Wk)-M
(OWk)jj (21ABs ) exp
where the a,
spherical components a,,
- 2=(a.,
=
+
iay),
ical components a
( aa) .1
V-2
1 =
-72=
1
(
(
ao
of
aa
a
+i
-9a.
=
Da,
-67a--,
'9
aal
-!9Bs+fvs
e2
)ai=O,-ri=07
Eq. (A.37) with
lated ,r
by -
the
can
.
be obtained
a
-
(A.41) are
iay),
(a
defined
and the
-5a I o
=
Da-,
by spher-
given by
are
la
a
7Eo
I
"&a )-1
=
2N-2
( fjj=j 19i1Xi'lLi (Sk)g
by replacing YIk-Ak -k-Mk (Wk)
YIk-AkMk-Ak(Wk),
formula
which is
easily
calcu-
[70, 71]
Vaf (a) Yjm(et)
-Ti -1 (f(a)
I+ I
particular
we
f(a)
f (a)
+
[T
x
Yj-j-(et)]j,,.,
a
+ I
:21EI++11: In
a
spin-orbit interaction,
( aTwul,
gradient
vector
+ fvs
By using Eqs. (A.41) and (A.34)-(A.37),
-
-
aa,
I
differential operator V,, a
the matrix element for the
in
a
a
(219Bs
+ iv-s
!,, (a-v 2-
=
a
'9
aax
a,,, a-,
of
of the relation
-
1f (a)
[r
X
Y
(A.42)
a
obtain
Yjm(a)
aa A
-(21 + 1)
V I--11 (1/-t1mjI-Im+tt)Yj-jm+, ,(a).
(A.43)
Appendix
262
A.3 Correlated
Gaussian-type geminals general correlated Gaussian-type geminals calculated from the equation
The matrix element for the of
Eq. (6.26)
be
can
(W, BI k r) 10 1 f,,, (A, B, R, I
7
N
N
3
3
i=1
exp(
iP
IiZBR
t,. n ,,
p
p=1 j=1 q=1
-
qn,',
anip
ff H H H yt-i
3q
-
iR
-
2
liiBk 2
-
Pk)
(g(t+B'k;X+B',r)101g(t+BR;A+B,r)) tj=O' -=O
(A.44) It is crucial to know the
generating functions on i multiple differentiations. As
(tj,...,tN)
=
polynomial
as a
most
of the matrix element in the
dependence
of at most
can
be
degree
important operators 0
=
seen
2 in
=
from Table
exponential
J(fvr
I and
P
and
-
S).
%,...'t') 7.1, they
to do
appear
functions for the
The matrix element
for the Idnetic: energy has an additional factor which is again quadratic. The basic elements of the t and t' dependence include ti-tj, -ej--ej, ti.-ej,
and
S-ti, S-tj. As
the differentiation
the Cartesian coordinates no cross as
tip
and
can
%
,
be made
and
as
separately in each of
the basic elements have
terms in different directions of the Cartesian coordinates such
tiltj2,
x, y, and
possible
it is z
to do the needed differentiation in each of the
components.
The above considerations reduce the
operation prescribed in Eq. problem given in Exercise A.5. The for(A.44) mula developed in Exercise A.5 can be applied in each of the x, y and z components to complete the operations needed in Eq. (A.44). It is not difficult to extend the formula given in Exercise A.5 to get the matrix element for the Idnetic energy which has an additional polynomial of degree 2. In fact, Eq. (A.169) can be generalized to a relation which is valid for a general function: to the mathematical
( cNf (X1
,X2,...,
XK))
xl=o,---,XK=O
AA
Spin
matrix elements
263
N
(W#-N f (AA AA N!
f (AlZi A2Z,
27ri
A.4
Spin
AK
...
'O))'&=O i
AKZ)
(A.45)
dz.
ZN+I
IZ1=1
matrix elements
The interaction may depend not only on position coordinates but also on intrinsic degrees of freedom such as spin and flavor.. By represent-
ing the single-particle spin function by Im element is given by
(MI I o't, I m)
=
.1
V_3 (.12 m I IL 1
The matrix elements for
2
m!)
=
1/2),
a
basic matrix
(A.46)
-
spin operators, (S'AF 10,,,,, (spin) I SM),
axe
by using the Wigner-Eckaxt theorem (A-12). Here a usually reduced matrix element which is needed to evaluate the matrix element evaluated
is the
one
for Pauli matrix and it is
o-
2
given by
(A.47)
2
spin function discussed in Sect. 6.4 no technique involving angular momentum algebra is needed to obtain the spin matrix element. An elementary, direct method enables one to calculate the matrix element. We list the spin matrix element that appears in evaluating the nucleon-nucleon potential energy. For the
(Tr l M12 1 0"1 X
*
a2
I M1 M2 )
=
3(-1)M'1"_M1jMI+M27M/j+Mf2
("2 MI I MI1-?nl I "2 Tdi) ("2 M2 1 MI2 --M2
X a212m I MI M2 (MIMf 2 1 [01 1
("2 MI I Tnf1-MI
X
(1 R I_Ml "rnf2--M2 12 m).
X (Ul (M /M-I 2 I [r 1
+
2
a2)] 1M IM1 M2
Mf2
(A.48)
JM,MII+M2I-MI-M2
Tn 1) (-1 M2 1 Tnf2-M2
X
2
3
I 2
2
MI2
(A.49)
V3
E E qj=-I q2=-l
r
qj.
Appendix
264
IJ,,,,,n, (' 2
(I q,
I q2
2
ml_ 1
q2I
" 2
M) 1
+
Jmjm,('I2 M2 1 q2I .12 m) 2 1
I I m).
(A-50)
(IMIMI 1 2 i(T*0'2)(al)m+(r*0"1)(0'2)-raiMlM2)
f (-l)M2-M2 JM?M'j-MIrM2-M2
3
+
(-I)M,
1
-MI
JM7M2-M27MI-M'I'
(I2 mi I m---7n, 1
x
(al M12 I (r'0'1) (r 0*2) *
X
*1 2
Tn/1) (I2 M2 I m'--Tn2 2
MI
Tn2)
=
3
1 7n/1 -'rnl rMl-M"1 rM2-M'2 (I 2 TnI
2
7nf2)
(A-51)
(-J)MI-MI+M2-M2 2
Mf1 )(1
2
Tn2 1
TnI--Tn2j 2
I 2
MI)2
(A.52)
(al M2 3
[T X O'll I
[r X O'21112m IMI M2
X
(.12 mi 1 m--7n,
2
7nfl) ("2 M2 I Tnf2--Tn2
2
7nf2
7'ql 7q2(l qj+m'j-mj 1 q2+rn"2 -M2
X
12 m)
ql.=-l q2=-l x
(I q, I m'j-m,
ql+TWI--rni) (I q2 I m'2--M2
q2+m/2-M2) (A.53)
Here
r
is
a
3-dimensional vector and rn stands for its
spherical
com-
ponent Matrix elements for the isospin part same manner.
can
be obtained in
exactly the
A.5
with
Three-body problem
A.5
Three-body problem spin-orbit forces The matrix element for
tensor and
central,
with
spin-orbit forces
central,
265
tensor and
N-body system can be derived from the previous sections. In order to illusthat there alternative trate exist possibilities, here we use a sligthly different way of calculating the matrix elements for three-body problems with central, tensor and spin-orbit forces. In this approach the radial and angulax integrations are done separately and this involves the extensive use of angular momentum algebra. Needless to say, this approach cannot be easily pushed far. One may try to attempt to derive similar formulas for the four-body case, but to go beyond that general
formulas
in the
to be too tedious.
seems
choice
(i) a
an
presented
of basis functions
The trial function is taken in the form
TfJM,TTMT(123)
C,,,,A p ac(1, 23)
(A.54)
ac
with
expansion coefficients Q, and functions A o,,(I, 23) that are fully antisymmetric under the exchange of particles:
constructed to be
Aw,,, (1, 23)
Wc (1,
23)
oac(l, 32)
(the
construction of
+ W,,, (2, 3 1) + W,,c (3, -
(pac(2, 13)
-
12)
(pac(3, 21),
(A-55)
function is the same, but then the last three terms should have positive sign in the above
equation). function in
totally symmetric
a
The basis function a
Jacobi channel k
stands for
wave
W,,(k,pq) represents a specific (k 1, 2,3) (see Fig. 2.2). The =
wave
nota-
system where the particles p and q axe first (k, pq) connected and then the center-of-mass of the (pq) pair is connected
tion
with the nel, X (k)
particle ==
a
k. Thus the coordinate used in the kth Jacobi chan-
JXk, Ykb
is
given by MpT'p + Tnqrq
Xk
=
Tp
-
rq,
Yk
_
(A-56)
rk.
Mp +rnq Yk I in this section, instead of denote the first and second Jacobi coordinates.)
(Note
that
we use
f Xk
The basis function and isospin parts:
,
o,,,,(k,pq)
is
given by
a
(k)
Ix 1
product
I
X(k)I, 2
to
of space, spin
Appendix
266
'
FL'(kpq)
W,,r (k, pq)
where the
x
X'(k,pq)]jMj?7TtM,(k,pq), S
correlated Gaussian of the
is chosen to be
a
(-2 ;(-k)
Xk
spatial part
(A-57)
form
.FL'ML (k, pq)
=
exp
Ax(k)
2v+,\ 2n+l
Yk
YL' ML ") (iik, Vk) (A.58)
witil
'3
(
A=
62
6
0)
(A.59)
YI (Vk)ILML
(A.60)
-Y
and
L('ML Oik
-
k)
Y
The index
a
=
[YX(ik)
stands for
a
X
set of discrete labels
a
=
(Y, n, A, 1)
and
for the matrix A which contains the nonlinear paxameters 0, -y7 J. Here Y, n 0, 1, 2,..., while A is the orbital angular momentum of =
and q, and I is the orbital momentum between the center-of-mass of the (pq) pair and
the relative motion between the
angulax particle
the
particles p
k.
The index
c
denotes the labels
(L, s, S, t),
where L is the total
angular momentum, S the total spin, and s and t are the spin and the isospin of the (pq) pair, respectively. (The total isospin T
orbital
quantity, so that no T-mbdng is the trial function.) The spin and isospin parts axe thus
is here assumed to be
considered in
a
conserved
given by S
X S M, (k, pq)
=
,qTtMT(k,pq)
=
(k)] sm,
(A.61)
[?7t(pq) xqi(k)1TMT,
(A-62)
[X, (pq)
x
Xi 2
2
where Xsm (pq)
,qtm (pq)
[Xi (p)
x
X.1 (q)
2
[77.1 (p) 2
2
x
(A-63)
q.1 (q)jtm. 2
The function o,,, (k, qp) is obtained from p,,, (k, pq) by changing the sign of xk and the order of couplings in the spin and isospin functions. The sign change of xk results multiplication by the phase factor functions change as follows:
in
a
change of J to -J and spin and isospin
and the
A.5
Three-body problem with central,
XsrrL(qp)
(-1)1-'X ,,,(pq),
iltm(qp)
(-1)1-lqtm(pq). p,,,(k, qp)
Hence the function the
sign
of 6 and
267
spin-orbit forces
tensor and
(A.64)
V,,,(k,pq) by changing
is obtained from
by multiplying by
the.
phase
factor
(-I),\+'+t.
The kinetic energy operator with the center-of-mass kinetic. energy subtracted can. be written in terms of the relative coordinates as 3
2
0
0
A
-Tein
Trel
'k
k)
2rni
(k)
Yk
(A.65)
=Tk,
21j,2
where
(k) III
This
Mprn,
(Mp + Mq)Mk
(k) 2
mp + ni., +
Mp +Mq
(A-66)
Mk
is valid in any of the kth Jacobi coordinate set.
expression
basis
(ii) transformation of the
function
We, have to calculate the matrix elements between the basis functions To expressed in terms of different coordinate sets (1,23), (2,31), -
achieve this
one
has to transform the basis function from
coordinate set to another. This
can
be
one
..
Jacobi
accomplished by using
the
transformation Xk
Yk
)
T (kq)
where the matrices
(
X
(A.67)
q
Yq
connecting
the different Jacobi sets
U12
Tn2+rn3
T (21)
=
MS(7nl+M2+M3) (M2+M3)(tnl+M3)
MI
Mj+M3
M3
7n2+Tn3
T (31)
M2(Ml+M2+M3)
M1
(1712+M3)(M1+M2)
Tn1+M2
M3 __
tn I
I
+rn:3
T (32)
mj(mj+Yn2+Tn.3) (tnj+mj)(Tnj+tn2)
M2 "Ll
+TrL2
are
given by
268
Appendix MI -
M3+Mj
T(12)
TrIS(Ml+M2+M3) (7n3+7nI)(M2+7n3)
7
M2
M2+M3
MI
MI+M2
T(13) 1112(IILI +IIL2'1-11L.1)
?IL'j
(7rII+M2)(?tL2+7J'L.3)
"12 +7,11.3
_M2
MI+M2
T (23)
(A.68) MI(MI+M2+M3) (Tnl+Tn2)(M3+Ml)
M.4
M3+?nI
The space function TLc'M,, (k, pq), expressed in the kth Jacobi coordinate set, can be transformed to the one expressed in terms of the
qth Jacobi
coordinate set
as
follows:
TL'M (k, pq)=1: L3(kq) (ad),TLagr (q, kp), L 11
(A.69)
,
ry
where
,
MI,
(q, kp)
=
exp
(_2
)
and where a stands for 2P +
+ 2h +
The matrix A is
(A.67)
F
=
a
set
X
2V+X q
Yq2fz+Ty(Al) LM ,
2v + A + 2n + 1.
uniquely
(A.70)
(T/, ii, , 1), which has to satisfy the relation
(A.71)
determined from A
by
the
transformation
as
j_(kq) AT(kq) The the
Q, (X q, Yq
6(kq) (ad) L
in
(A.72) Eq. (A.69)
is the transformation coefficient of
polynomial part: 2v+.k X
k
Yk2n+Iy(AI) LML, (Xki Yk
EB(kq)(ad)X2F/+XY2Ft+Ty(XO (XV Yq -
=
L
q
LML,
(A.73)
a
To get this expression one has to make. use of Eq. (6.7) and to recouple, the product of the two angular functions (see Complement 6.3) by
A.5
with
Three-body problem
[y(1112) (:j,
)
12
X
central,
tensor and
spin-orbit
forces
269
y(1314-) (:j, )JLM 134
Ef 1121121314-134L y(113124) (:j LM
(A.74)
113124
113124
where the coefficient E is ficient C of
Eq. (6.11)
given by
a
unitary 9j symbol and the coef-
as
11 13 113
El'- 121121314134L 113124
12 14 124
112 134
Q11 13; 113) C(12 14; 124) (A-75) -
L
Note that the coefficient E may be defined
by the reduced
matrix
element -
-
121121314134L N 2L T 1 Ell 113124
(k '/2L + I (y(113124) LM
y(1112) (60, ) I 712
C
X
y(131A) I
)JLM)
,34
=
C ' ) 11 y(1314)( (-j)112+134-L(y(113124)(XA' ) Y(1112)(:j 134 112
=
(-1)13+lA (y(1112) (:j, 1
L
y(1314) ( C'
y(113124) L
12
34-
(A.76) which is
(YI
easily verified by using Eq. (A.147)
" 1) v/2k + I QW; k)
Y1,
Yk
and the relation
V21 + I C(kl'; 1). The transformation coefficient
(kq) BL (aa)
B(kq) (Cla) L
(A-77)
is then
expressed
as
nI &I- X2,XI1121L D"X D V1XIV2-X2 nj1jn212
X[
vl"*-1"2-"2 njIjn2I2
X
X
T(kq))21/1-+Ai(,p(kq) -
22
where the coefficient D is is constrained
)
kq) 12
21/2+1\2
(T(kq))
2n1+11
21
2n2+12
(A.78)
I
given in Eq. (6.8) and where the summation
by the conditions
2vl_ + A, + 21/2 + A2
=
27/ +
A,
2n, + 11 + 2n2 +12
=
2n +
11
Appendix
270
2vi + A, + 2n, + 11
=
X,
2 i+
2v2 + A2 + 2n2 +12
=
2Tz +
17
(A.79) and
the
triangular inequality for the angular momentum among (/XI7'X2iA)7 (11712 07 (Al 117 X) and (A2 12 1), respectively. The transformation to the pth Jacobi coordinate set can be per-
by
7
formed
well in
as
a
similar
The spin part of the
1
.
7
manner.
wave
function transforms
as
(-l)'21+'-SU(.! ISI; gs)X SIVIS (q, k-P)
X8S Ms (k, pq)
22
E(-I)'!+-SU(.!
-IS-!; 2 sft S Ms (p, qk).
2
The transformation of the
2
22
isospin part
is
given
in
(A.80)
exactly the
same
way-
The radial and
However,
no
such
appears in the
integration
angular integrations are separated in this approach. separation is made hi the factor e-6xk-yk which
exponent of the has to
one
function.FLIM,: (k, pq).
expand this
term into
Therefore in the
partial
waves
(see Eq.
(6.53)): ,-wx-Y
where
ii(x)
(6.48)
and
radial paxt
=
v"2-1-+I e(l, w) il (I w I xy) 01) 100 (--
4r
(A.81)
7
is the modified
E(I, w) one
=
I for
encounters
spherical Bessel. function of the first kind w rel="nofollow"> 0, 6(1, w) (-I)' for w < 0. In the of the form an integral =
cc
fo
I(n, 1, v, I w
2n+1+2
Y
Ivy 2
e-2
r-(2n)!! w I' -2 vn+l+a2 where
n
guerre
polynomial (10.15).
is
a
eXp
ij(jwjy)dy
(W2 ) L('+'!) ( W2) 2
n
2v
non-negative integer and
2v
Ln(')(x)
We define the
(A.82)
is the associated La-
following integral
I(v, n, 1, u, v, IwI; V) 0
fo JO
CO
X
2v+1+2
2n+1+2
Y
1
e-
UX2
_
IvY2
V(x)il(lwlxy)dxdy
Three-body problein
A.5
with
central,
tensor and
spin-orbit
forces
271
00
1
x
2u+1+2
IUx2 F
--
e
(A.83)
V(x)l(n, 1, v, lwlx)dx.
0
The one-dimensional
integration
the functional form of
ated. In the
special
V(x),
case
can
when
be
V(x)
I(v, n, 1, u, v, IwI; V=1) n
X
equation, depending on analytically or numerically evalu-
in the above
=
1
we
obtain
(2n)!!F (n + I +
8
V
F(k + U + + 1) 2 kl.(n k)!F(k + I +
2v
w2)
2 2
where F is the Gamma function
given
in
we axe
(k I q)
,
(A.84)
Eq. (7.8).
ready
to calculate the matrix elements. The
tween the functions in different Jacobi sets k and q
separately
k+v+l+A2
matrix elements
(iii)overlap Now
n+l+ 2
(w2)k
-
k=O
3
2
in the space,
=
spin and
can
overlap be-
be calculated
isospin parts:
( pc(k, pq) I Wycl (q, kp)) "I" 0
=
X
JLLI JSSI (.FL'ML (k, pq) I )7LMI (q, kp))
(X'SMS (k,pq)IX" SMS (q,kp))(,qtTm,(k,pq)lqt
m,(q,kp))(A.85)
The
spin part
is
easily obtained by using Eq. (A.80)
(X'SMS (k, pq) I X" SMS (q, kp))
=
(- 1)
" +'-S 2
as
.1 S U (.1 .1; 2 22
s's) -(A.86)
Likewise, the isospin part becomes
(,qtTM, (k, pq) lqt'TMT (q, kp)) The
ing
integration
=
( _j)-!+t-TU(I22ITI; 2 tt). 2
of the space part
the function in the bra to the
performed by transformdepending on the qth Jacobi
can
one
be
By substituting Eqs. (A.69) and (A-70) we obtain (the label the coordinate set is suppressed in the integral)
coordinate.
suffix q to
(A.87)
(-'FL'ML (k, pq) I
"
(q, kp))
=
E L3(kq) (aa) L
272
Appendix
X
if
x
2P+X+2Y'+)L" 2fz+[+2n'+I'- 2IUX2_lVy2_WX.,y e
y
(Y'X) oi, m) *Y(""/
X
LML
dx dy
LML
(A.88)
with
U= +o
=!+-t
V
The
angular integration
and
(A.74)
(YL(
easily by
(A-89) the
use
of
Eqs. (A.81)
can
ML
( C' m
d
L ML
_X-+ +I1,E(r w) Ety.OX'1"LL i.(Iwlxy). V2_x
4w
One
be done
=S+j'.
as
e- wX-Y
and then
can
w
(A.90)
easily verify that 1+1'-r, has to be even and non-negative, the integral formula (A.84) to obtain
one can use
(.FL'ML (k, pq) I
"' -1
LML
(q, kp))
4y"13(kq) Ir
,
X
I
X + 16 (M' W) EX"jj'-0,X'1 K Vf2-1
L
(P+vl+
"
LL
A+AI-K 2
2
'K,U, V,
IWI;
V=I) (A.91)
(iv) matrix elements of kinetic energy operator The next step is the calculation of the matrix elements of the kinetic energy operator
(k ITrel I q)
=
(W,, , (k, pq) ITq I W& d (q, kp)).
In the above
erator
can
(A.65)),
expression be expressed
and thus
we
(A.92)
used the fact that the kinetic energy opin different relative coordinate sets (see Eq.
we
defined Tr,,l in
conformity
with the coordinate
set in the ket. To calculate the matrix elements of the kinetic energy
operator
one
(X
first notes that
2v+.X
2n+l
Y
e-.21 ,6.2_:I_YY2_,E._y YLM
A.5
with
Three-body problem
30X2 X X
-2(
x
central,
02X4 + j2X2Y2
+
21/+)L-2
2n+1
Y
e
x
20(2v + A)X2 LM
e
1))
4 M 7F
13X2)
I
LM
+
+I
(2v
-
Ox 2)
x
proof of this
2Y(2z/ + 2/X +
EU(A-I1Ll;AK)C(11;K)Y('X-I
-V2A A
+
273
F3
1, _.L,3X2_ YY2_,5x.y 2
(2v + 2A +
X
-
spin-orbit forces
2_1 --Ox ilyy 2_,SX.y Y. (,Xj) 2
2v+X-1 2n+l+l
Y
tensor and
(A.93)
LM
relation is
given
in Exercise A.6. A similar
expression
holds for the second relative coordinate y. By the help of the above equation the effect of the kinetic energy operator applied on the basis function is
given by
a
linear combination of terms akin to the basis
function. The matrix element of the kinetic energy can therefore be given by repeatedly using the expression obtained for the overlap of the basis function.
(v) matTix
elements
The central V
=
=
of central potentials two-body interaction can be
V(I'r2
V(Xl-)
that is, the
7'3
-
+
1)
V(Irl
+
V(X2)
potential
is
+
-
T3
1)
+
V(jrI
as
-r2
1)
V(X3)
a sum
of terms
coordinate in different Jacobi sets. In for the
written
(A.94) depending on the first relative calculating the matrix element
case
(k I V(xp) I q)
=
( O,, (k, pq) I V(xp) I Wa,,, (q, kp)) -
JLLIJSSI(-'r-7LaML (k, pq) I V(xp) I.F&L (q, kp)) X
(X'SMS (k,pq)IX" SMS (q,kp))(TItTm,(k,pq)lqt
m,(q,kp)),
Appendix
274
(A.95) the
spin and isospin paxts
can
be calculated
as
before but the
spatial pth
functions in the bra and the ket have to be transformed into the
Jacobi coordinate set in which the potential is defined. The space paxt of the matrix element is calculated transformations k -+ p and q --+ p:
by using
the
"'
(FL'M,,(k,pq)IV(xp)l J,w, (q,kp)) L3(qp) (ala/) L
B(kp) (aa) L
(-'FL'M.,,(p,qk)IV(xp)IFL'
x
Here the matrices
corresponding
(A.96)
(p, qk)).
to the transformations
axe
defined
by T(kp) AT(kp)
A'
T(qP) A!T(qP)
=
(A.97)
respectively. By introducing +
U
one
+7
V
j+
W
(A.98)
&I
obtains
(.FL5'ML (p, q k) I V (xp) 1.FL 4 jr
X
IL
(p, q k))
/-2-r.+1,E(n, w) Et nOXI'P'LL
I(P+Vf+
+
ii+?P+ 2
2
K) U, V,
IWI;
V)
-
(A.99) (vi) matrix
elements
of tensor potentials through
The tensor interaction is defined
SP
2 /-4r
V -5 (Y2 (ip ) [O'q '
X
Crk] 2)
-
the tensor operator
(11.7) (A.100)
A.5
Three-body problem
with
central,
tensor and
275
spin-orbit forces
To calculate the matrix element of the tensor interaction
one
has to
same steps as for the central interaction, except that now part of the wave function has to be taken into account, and the
follow the the
spin spin part should as
the
one
potential.
also be transformed into the
The
use
of the
coordinate system theorem (A. 14) enables
same
Wigner-Eckart
to obtain
(kjV(xp)Spjq)
-=
(W,,(k,pq) IV(xp)Spl W,,,e(q, kp))
U(L2JS; LS) ',E47 VF(-2L+ 1) (2S' 1) 247r 7r r 5
+
x
II (,)7L' (k, pq) II V(xp) Y2 (x-) P
x
(X'S (k, pq) II [Cq
x
(nTtMT(k, pq) Iq tm, (q, kp)).
X
FL',f (q, kp))
O'kj 2 11 X'S/ (q, kp))
(A.101)
spatial part of the matrix element is calculated as before by transforming the basis functions to the pth Jacobi coordinate set Here the
(Jc'L'(k,pq) 11 V(xp)Y2( p-) Ij.FL,(q,kp))
13() (a& ) L'
13(kp) (ad) L
x
II FL',f (p, qk)), (YL' (p, qk) I I V(xp) Y2 (x-) P
where the last term is obtained
(JC L"(p, qk)
V( Xp) Y2 (Xp) ip
4ir
X
by using Eqs. (A.81) i
(A.102) and
(A.82)
as
(p, qk))
V2- r, + 1 c (r., w)
(YL' 1)
YO
0
Y2 (-
O)
YL
ii+?P+
X
2
2
r" U, V,
IWI;
V) (A.103)
Here u, v, w are defined in even and non-negative.
Eq. (A.98).
Note that
f+
r,
has to be
276
Appendix The reduced matrix element of the
using
6.3)
angular paxt is calculated by recoupling technique (see Complement
the
and
angular momentum Eq. (A.76). One obtains
YO (I
( Y.,L
K
YL(XI F) ( c
Y2 ( 0)
I
U(2n2r,; WO) Q2K; W) Kf
X
(YL
Y2
01
Ll
/2r,'+ 1
EV
2K+I
C(2r.; K)EX LXII,-'L2.
(A.104)
The
spin matrix element is also easily calculated by transforming the spin functions to the pth Jacobi system and by using Eqs. (A.147)
(A. 149)
and
as
(X'S (k, pq) 11 [Crq =
X
Uk]2 11 X" S" (q, kp))
E(-l).2k+g-SU(-nS-1; S) E(-1)-!+""-S'U(.! 2
22
X
(X S (p, qk)
[G'q
X
2
Ok] 2 11
22
XS P, (p, qk)),
2
(A.105)
where
(X S (p, qk) 11 [Uq
X
Uk12 11
XS P, (p, qk))
(2S + 1
(_I)s,,-St+s-g V X
(("221) 9 11 [O'q
X
29+1
U(Ii S2; S'9) 2
UkI 2 11 ("22'0 i
(A.106)
with
(('1'1)9 11 [O'q 22
X
4+ (vii) matrix The
Uk12 11 1
elements
spin-orbit force
22
2
2
1
1
1
1
2
2
8
2
(,V6) 2 =2v/_5JgjJ pj.
9
of spin-orbit potentials given through the operator (11.9)
is
(A.107)
277
A.5 Three-body problem with central, tensor and spin-orbit forces I
with
LV Sp -
As in the
Lp
=
-ixp
Sp
Vp,
=
2
(0'q+ffk)- (A.108)
of the tensor interaction, with the
case
Eckart theorem
(A.14)
(kjV(xp)Lp-Spjq)
one
_=
use
of the
Wigner-
obtains
( o,,,(k,pq) IV(xp)Lp-Spl W,,,,(q, kp))
U(L'lJS; LS)
r
+ 1) + -1)(2S' V-(2L x
x
(FE'(k,pq) 11 V(xp)Lp 11 FL, (q, kp))
(X'S (k, pq) I I Sp I I X'S, (q, kp)) (,qtTMT (k, pq) lqt'TMT (q, kp)).
(A.109) spatial part of the matrix element is calculated as before by transforming the basis functions to the pth Jacobi coordinate set
The
V (xp) Lp
(.FL' (k, p q)
L3(qp) (af&i) LI
L3L(kp) (aa) x
YL, (q, kp))
(.FL5 (p, qk) 11 V(xp)Lp 11 Jc'L",(p, qk)).
When the orbital
angular
momentum
(A.110)
operator
acts
on
the basis
(provided both the basis function and the operator in the same coordinate system) one obtains
function
pressed L
(x
21/+X 2n+1
Y
ij(x
X
A0 e--21- '3X2 _!IY _IYI _,5X_y YL(M
Y)X" +A Y2n+l e_21,3x
2v+,\ 2n+1
+X
Y
Note that the outer
vf2-[x
x
yll_,,
=
axe ex-
1)3X2
1 _
e- 2
'f
YY2
product i(x
X
(4-\,F2-x13)xyYj(,',1)
2_1
'YY
2_dX.y
YLI'Xj) (ic M
A0 -6"YLYL(M
y),, ).
is
equal
(A.111) to the tensor
Therefore the
product
spatial part
of
tained in
element, (.FL6(p, qk) 11 V(xp)L, 11 J:L` ,(p, qk)), can be obexactly the same way as in the central potential. It is given
by
of two terms: One is
the matrix
a sum
4vf2-7 J
47r 3
X
vf2-_K+I c (K, w)
(YL` r) ( C' m I I YO(Kn) ( C' m YI(I-I-) ( C' m I I YL(" F") (:t' o
278
Appendix
(P+1/1+
X1
n+W+ 2
K) U, V,
2
IWI;
V)
(A.112) and the other is
v/2 r. + 1 E (r,, w) (YL(
4-,T
YO(MM) (i
,
Fj+P+
X
fi+??+ 2
L
KI U1 V,
2
II
YL(fk'[')
IWI; V (A.113)
Here u, v,
defined in
Eq. (A.98). Note that in the first term and non-negative, and likewise in the second term I + 11 r. has to be even and non-negative. The angulax part in the first term can be obtained by using Eqs.
T+ P
+ 1
w
-
axe
r,
has to be
even
-
(A.74)
(A.76)
and
as
YO( MIS)
(YL
YI(l 1)
YL
(A.114)
pq
pq
The
angular part
(YL(
( b,
in the second term becomes
YO(K n)
YL(',"
L
i, L
L
YL(, "")
(A.115)
where
(YL(X"")
L
(_,)L-L" V
YLI 2-L+ 1
U
2XI +I
L 1; L'XI) (I
Xi,
i
L
YX, ( b)
(A.116) with
(Yl(;k) 11
L
11 Yl(.-b))
=
0(1 + 1)(21 + 1).
The spin matrix element action. The result is
can
be calculated
(A.117) as
in the tensor inter-
A.5
Three-body problem with central,
(x'S (k, pq) I I Sp I I
tensor and
spin-orbit
X'S'/ (q, kp))
-21+-SU('.!S.1; Sg) E(-I) 12+S'-S'U(.1 22
1
22
2
x
279
forces
(X S (p, qk) 11 Sv 11 X S-', (p, qk)),
S' 1;
(A.118)
where
(X'S (p, qk) I I Sp II X'S'f (p, qk)) S+I
S+IU (Iksi; Sg) =29+
-S,+S-
X
2
1
2
(A.119)
(("I22'1) 9O'q + Uk I I (I22'I) k)
with + O'k ((*"1) 22 9O'q
22
R)
8+1 E: 122 U(-! -191; '9
2+
206-JgJi j.
06-1) 2
((- 1)
-j;
+
1) (A.120)
280
Complements
Complements A. I Mabix elements
central
o
potentials
The matrix element of the central and tensor
potentials
can
be ob-
tained
through a step which is slightly different from the procedure presented in Appendix A. 1.3. Then the formula becomes simpler than Eq. (A.21). Here we outline the step which leads to this simpler result. Instead of using Eq. (A.19), we first perform the integration over the angle i6 in Eq. (A. 17), which leads to Tj I r2
(fK'LIMI (7Z, X, X) [
r)
YxIL ( ri
-
rj ) I fKLM (u,
A, x))
3
47r
(2r)N-2
BKLBKILI
detB
_.LCT2 e
X
x
[
A2
+
q/A/2 +
c) if
d& d
'
d2K+L+2K+L"
YLMWYLIMfO )*
exp
2
dA2K+LdA/2K'+Ll
AA/e. et] in (rV)Y"'j#Y)
(A.121)
Wi0 V
=
-tAe + -Y'Ye'.
(A.122)
If either -y or -y' vanishes, v is independent of the scalar product e-e', and the integration and the differentiation prescribed in Eq. (A.121) becomes very easy. When neither -y nor 2 exponent can be expressed in terms of v
qA2 + qiA/2 + pAA'e P
2 V
+
2-y-y' One first has to
-
-y'
is zero, the term in the
as
e'
(q_ 2-y
expand
A2
+
(qf
A 2) i,,,(rv) exp(2-y,,1V
A/2. 2-/
in terms of powers of
(see Eq. (6.48) for the expansion). Secondly, multiplied by the spherical haxmonic the
coupled
(A.123)
its
general term v expanded
must be
v
2n+n
in
The second step can [YL ( -_) x YL, (,E )] In this derivation one can avoid Eq. (6.7).
tensor form of
be done with the
use
of
recoupling the angular momenta, and hence would get the simplest possible formula for the matrix element.
CA. 1 Matrix elements of central potentials In the
with
x
make
following we work out the details for the central potential However, we shall not follow the above route exactly but
0.
=
a
281
little detour in order to make transparent the relation to the To this end we rewrite the left-hand side of
overlap matrix element. Eq. (A.123) as follows:
A2 + I A/2
+
pAA/ e. e/
qX2 + q'A12 + PAA/ e. e
=
2c
V2. (A.124)
V2
The term Hermite
2c
e
jo (VV) is expanded in
powers of
v
with the
use
of the
polynomials: V
V
2c)
exp
-
i o (r v)
=_
2c)
exp
1
-
2rv
(e'r
V
-rv)
e
-
00
V22 r) (V2c2)n. C
,,F2c Next
we
expand
exp q A2 +
q' A/2
n=0
exp
+
(2n + 1)!
H2n+l
c,
(qA2 + q'A/2 + pAAI e. ef)
p,\A'e
H(p -ra, qA2)H(pI
e'
-
-
) (2c V2)n -
2c
as
follows:
ni _
(2
n , q'A/2 )H(I
H(M' 2A2 )H(a, 7/2,X/2 )H(n
(,,2,)n
(A.125)
m
PPIIMMI
-
-
n
+
+
m
m', pAA'e-e)
a, 27-yAYe-e)
1: FPnPj (c, q, q, P, t,,Yf),X2p+'A/2p'+l (e-e')'
(A.126)
PPfj
with n!
Fpnp,,(c, q, q, p, 7,7')
1: H(p
=
-
m,
q)H(p
I -
M
Iq
MW
7
x
Here p, p, I,m, and m' are all of their values are determined m <
is
p, ne <
now
n+m-nz!
H(I -n+m+m',p) 2m+m'm!
(fK' L' MI (Uf, X) X) I V(I Vi
(n
-
m
-
ne)!'
(A.127)
non-negative integers and the
ranges
the conditions p + p' + I > n and + m! < n. The operation in Eq. (A. 121)
p', n I < m easily done, leading to -
ml 1.
in-m+m'
7
by
the result
-
'rj 1) 1 fKLM (U, A, X))
282
Complements
(2K + L)! (2K'
JLLIJMM'
+
L)!
BKLBK-'L
K+K+L
2
(2
E
detB
J(n, c)
n=O
min(K,K")
E
X
Fk7-k KI-k L+2k (c, q, q
A'Yi
^//) BkL
(A.128)
i
k=O
where
J(n, c)
is the
integral defined by I
J(n, c)
= -
-V/7-r-(2n + 1)! "0
f V(jx)e-X2HI (X)H2n+l(X)dX-
x
In the
overlap
case
of
V(r)
=
matrix element
comes nonzero
(A.129)
C
0
if and
I the formula
(A.6),
only if n
(A. 128)
reduces to that of the
because then the
=
0 and
integral J(n, c)
be-
FF2k Kf -k L+2k (c, q, q, p, -Y, -y')
k, q)H(K' k, q')H(L + 2k, p). Though this relationship is not apparent in Eq. (A.21), both equations should give the same result for the central potential. For the special case with K K' 0, Eq. (A.128) reduces to a much simpler form which essentially includes a double sum: reduces to
H(K
-
-
=
(fOLM (U =
I
=
A! 7 X) I V(ITi -rj 1) 1 fOLM (u, A, x)
(fOLM(W,X,X)jfOLM(u,A,x)) L
X
1
D
77
n
E (L-n)! ( cp )
J(n, c),
(A.130)
n=O
where the
overlap (fOLM(UIiWiX)jfOLM(u,Ax)) is given by Eq. a particularly nice feature for power (A.7). k law potentials V(r) because then the c-dependence of J(n, c), r The above formula has =
,
now
denoted Jk (n,
c),
is factored out
as
follows:
k
Jk (n C)
=
(2)
2
Vir(2n + 1)!
C
(2) C
00
-2
k 2
n
1
7,=, E
M=0
fo
X
k+I
ai2n+l
( dX2n+1
2% -X
-e
(-l)m22n-2m+l mI(2n-2m+I)!
-V
n-m
+
dx
k+3). 2
(A.131)
CA.2 Matrix elements of
283
density multipoles
j are contained only in c and -y-y'lcp, this factorization makes the summing up of Eq. (A.130) over i and j much faster. Especially for the Coulomb force (k -1), we get Since the
indices i and
particle
=
2c
J-1- (n, c)
(-l)n (2n + 1) n!
(A-132)
-
-x
potentials the route given below Eq. (A.123) leads following simpler result including only a single sum:
For other types of us
to the
(f0LM(UIiWiX)jV(jri
-
rjj)jf0LM(UiA7X))
(2L + 1)!!
(27r)N-1
4v
detB
L
)3 1: 2
n=O
I
L!
n!(L
77
-
n)!
n L-n
c
00
2n+2 X
V'r(2n + 1)!!
10 V(eX) X2n+2
e-'2dx.
(A.133)
C
particular simplicity occurs for a Gaussian potential as e-'2, we obtain perform the n sum easily. For V(r) A
one can
then
=
(ALM (U X X) I eXP I_ X(r, i
i
_,rj)21 IfOLM(U7 Ai X)) 3
(2L + 1)!!
(21r)N-1,
4v
detB
3
f
-I
)2
+
2r,) (P C
L
7,y + C
i+
(A.134) of density multipoles Here we Eq. (A.30) takes a simple form for the special case K' 0. As we have seen in the chapters of applications, the of K 0 is, with the use of its fall general A correlated Gaussian with K matrix and u vector, a very useful basis function. The simplified form A.2 MatTix elements show that
=
=
=
of the matrix elements
was
in fact used in
our
calculations. TrI the
and
case
in order to have
even 0, L nj n2 has to be non-negative the term from contribution -n2) /2, q). H((2K+L-ni non-vanishing the other On and even. Likewise Lf n3 has to be non-negative nj of Eq. (A.20) becomes nonhand, the recoupling coefficient Rnin2n3 LLIr.
of K
=
-
-
a
-
-
L' and n2 + n3 K vanishing provided that ni + n2 L, n, + n3 are all non-negative and even. Therefore we are led to. the result that L the summation in Eq. (A.30) is restricted to the case of nj + n2 L n and L. By renaming nj as n, we obtain n2 and nj + n3 -
-
-
=
=
=
-
284
n3
Complements
=
L'
n, where
-
from 0 to
n runs
min(L, L').
limited
by the condition that n2 + n3 non-negative and even and also K > max(IL are
the
-
triangular relation
n
=
-
Possible values of L + L'
LJ, IM
2n
-
-
of the Clebsch-Gordan coefficient
-
is
is
be
M'j) (L M K AV-
due to
MIL'IW). Furthermore, L-n L-n RLL'r. n
n2
=
L
11 :!
-
12
note that under these conditions
we
is contributed
13
n,
n3
=
by
=
single
a
L'
-
n
in
term with
11
=
RnLjn2n3 L-lis
ni
Eq. (A.20). This
=
n,
12
=
=
is because
13 :5 n3 and 11 + 12 > L, 11 + 13 ! L. Using the ni, well-known formulas for the Clebsch-Gordan coefficients (10 P 0 1 L 0) < n2,
and the Racah coefficient L-n L r,; L'
U(n
[751
L-n)
(2L 2n+ 1)! (2L' 2n)! (L +L' is) 1 (L +L' + n + 1) 1 (2L + 1)! (2L')! (L + L' 2n is) 1 (L +L' 2n + r, + 1)! -
-
-
-
-
-
(A-135) we
obtain the
following
result:
(fOLIMI'(Ul X, X)IJ(17VX -r)jfOLM(U, A, X) (2T) M-2
(
=
min(L,Ll)
2
3
2
1
E pnL-nI/ IL-nrL+L'-2n
cl e-'Ycr
detB
n=O
L+L'-2n
1:
x
ZLL'nn (L M r, M
-MJL'M)
r.=max(jL-L'J, IM-M,'I) x
where
r.
Y. M'
--M
('0 *'
(A-136)
+ L + L' has to be
ZLLfnr.
-
(-1)
2
even
(L-L'+x)
and the coefficient
F 2L
ZLL'nr.
is
given by
(2 +1) 47r
T L, + L' + -+I)! JL-K)!(L K
-
x
(2n)!! (L + V
F L-(LL --+L"5; L
x
-
2n
-
n)!! (L + L'
-1i ((L ---LL++-'KS
-
2n +
x
+
1)!! (A-137)
CA.3 Overlap matrix elements for
With one can
an
three-particle system
a
appropriate choice of the (N
-
I)-dimensional
285
vector
w
calculate various matrix elements. To calculate the matrix
J(ri
element of
-
xN
-
r),
one
only needs
to set
w
equal
to
w(') which
Eq. (2.12). The values of c, p, -y, -/' are obtained from Eq. w(ij) with being replaced by 0). In exactly the same manner,
is defined in
(A.18)
one can
also calculate the matrix element of J(ri -rj -r)
the vector
w('j),
defined in
Eq. (2.13),
for
w.
By using
by employing
the well-known
formula
J(iv-x -,r)
J(Ifvxl
-
r)
Y"11
r2 r.
WX
(A.138)
Y"tt
it
Eq. (A.136) immediately enables one to obtain the matrix element of the density multipole operator J(ji7vxj r)Y,,,,(w` _x). The matrix element for the central potential, V(j ri -rj 1), is obtainable from Eq. (A. 136) as well. By integrating the equation multiplied 0 contributes to by V(r) over r, we see that only the term with n the matrix element of the central potential. It is easy to see that the integration leads us to the same result as Eq. (A.133). -
=
Overlap matrix elements of the correlated Gaussians for a threeparticle system We tabulate in Table A.1 results of the calculation for the quantity A.3
/2N-2
61, i9aili (9-rili-mi
exp
( -2IS-Bs)
(A.139)
three-particle system (N 3) assun-Ang li < 2. The vector si is defined by aj(1 2), -2-ri). The calculation was done with i 2, i(l + Mathematica [1841. The quantity apparently has a symmetry property with respect to the interchange of li, mi +-+ 1j, mj. E.g., if one calculates this quantity for a given set of angular momenta, then the quantity corresponding to a set (111 MI 13 M3 12 M2 14 M4) can easily be obtained from it by simply interchanging the suffices 2 and 3 of the B matrix. Note that the table can be used for the four-paxticle system as well if one of the angular momenta is restricted to zero. for the
=
1
,
,
1
1
1
286
Complements
Table A.I. Tabulation of vector si is defined
Eq. (A.139) for
the
three-particle system. The
*Tj 2, i(l +7,i 2), -2-ri).
The matrix B is a 4x4 by ai (1 symmetric matrix and its elements are denoted a B12, b B137 C B34. The tenth to twelfth colunu3.s give the B23, eB241 f B14, d types and the coefficients of terms needed to construct the solution, where each coefficient must be multiplied by a factor given in the ninth column of the corresponding entry. For example, in the case of 11 12 MI M2 -2, 14 2, 13 2, M3 2, M4 -2, the quantity of Eq. (A. 139) is given by 12288 x (3b 2e2 + 3c2d2 + 12bcde). -
=
=
=
=
=
=
=
11
MI
12
M2
=
=
=
13
M3
=
=
=
=
14
M4
-4
f
--
0
0
0
0
1
1
1
-1
0
0
0
0
1
0
1
0 192
f2
0
0
0
0
2
2
2
-2
1
0
0
0
0
2
1
2
-1
-1
0
0
0
0
2
0
2
0
1
32
ef
0
0
1
1
1
1
2
-2
6
0
0
1
1
1
0
2
-1
-3
0
0
1
1
1
-1
2
0
1
0
0
1
0
1
0
2
0
2
0
0
1
0
1
-1
2
1
-1
0
0
1
-1
1
-1
2
2
1
16
I
1
1
0
1
0
1
-1
1
0
1
0
1
0
1
0
af
be
cd
1
1
1
(Continued
on
the next
page.)
CA.3
(Continued
Overlap
matrix elements for
a
three-particle system,
-
from Table A. I.)
1536
def
0
0
2
2
2
0
2
-2
-2
0
0
2
2
2
-1
2
-1
3
0
0
2
1
2
1
2
-2
2
0
0
2
1
2
0
2
-1
-1
0
0
2
0
2
0
2
0
2
256
af2
bef
cdf
I
1
1
2
0
2
-2
-6
-6
1
1
1
2
-1
2
-1
9
9
1
1
0
2
1
2
-2
1
1
1
0
2
0
2
-1
6
1
1
1
-1
2
2
2
-2
-3
1
1
1
-1
2
1
2
-1
3
1
1
1
-1
2
0
2
0
-3
2
2
-2
3
3
-6 -6 3
-1
-1
1
0
1
0
2
1
0
1
0
2
1
2
-1
-3
-3
-3
1
0
1
0
2
0
2
0
3
4
4
1
0
1
-1
2
2
2
-1
1
0
1
-1
2
1
2
0
1
-2
-1
1
-1
2
2
2
0
-1
-1
-1
1
-1
2
1
2
1
1
1
b2e2 acdf
C2d2
3
3
3
12288
a2f2 abef
bede
2
2
2
2
2
-2
2
-2
2
2
2
1
2
-1
2
-2
-3
2
2
2
0
2
0
2
-2
3
2
2
2
0
2
-1
2
-1
2
1
2
1
2
0
2
-2
2
1
2
1
2
-1
2
-1
12
-6 2
2
2
-3
-3
6
-2
4
3
3
3
3
6
4
-5
-5
3
3
3
6
6
6
-2
2
2
1
0
2
2
0
0
2
2
0
0
2
2
-3
-1
0
287
Exercises
288
Exercises
Eq. (A.14).
A.I. Derive
Noting the definition of the scalar product (11.2), expanding the angular momentum coupled ket functions, e.g., Solution.
(LS) JM)
E
=
in
Eqs. (11.1) and
states of bra and
(Lm, SM2 I JM) Lrni) I SM2)
(A.140)
TaIM2
and
using the Wigner-Eckart theorem (A. 12),
((L'S') JM 1
we
have
(0,, (space) 0,,, (spin)) (LS) JM) -
1) 1'(Lm, SM2 I JM) (LW1 SW2 I JM) AMIM2M'MI 1 2
X
-vF2-L'+1 X
The
use
(L' 11 0,,(space)
(SM2K --PISIMf2) (S' 11 0,,(spin) V12--S'+1
of
a
L)
(A.141)
S).
symmetry property of Clebsch-Gordan coefficients +
(SM21S --AISIMI) 2 enables
one
(-1)IS
V
2S+1
(KILSIMf2 I SM2)
to rewrite the above matrix element
((L'S') JIVI 1 (-W
X
=
(A.142)
as
(0,, (space) Q, (spin)) I (LS) JM) -
(V 11 0,,(space) 11 L)(S' 11 0,,(spin) 11 S) + + 1) -1)(2S V/(-2L'
E
(Lm1r,/LJL'm'I)(L'm'IS'm'2JJM)
JL'ra1M2?-afIM2r x
(Lm1SM2JJM)(K.ASITnI2JSTn2)-
(A.143)
sum over A, Tn1 , M2 , MfJ , TnI2 of the product of the four ClebschGordan coefficients is just the recoupling coefficient U(Lr.JS';L"S)
The
(see Eq. (6.67)).
This
complets
the derivation.
Exercises
289
A generalization to the matrix element of a coupled tensor operator be made similarly. Let us calculate the matrix element
can
((L',5 )XM'j [01(space)
x
Ox(spin)j,,, J(LS)JM
(LIMI1 SIM/2 I JM) (Lm, SM2 I JM) (1mAp I rw) MIMMIM2MA 2 1
(Lmj1mjL'm ,f2--L'+
(L'
01(space) 11 L)
(SMALI S/ M2I) (S' V -2-Sl+1
OX(spin) 11 S).
11
The
sum over
?n j, m'2
Tn1 , M2 i M7 A
,
of the
(A-144)
product of the five Clebsch-
Gordan coefficients
(LmISM2 I JM) (1mXMjnv) (LmIlmILal) MIM2MtlMflM'2 X
can
be taken
by
(STn2l\l-tlS'M2)(VM'IS'M'21XM')
means
(JMr,vIYM')
of the
unitary 9j symbol
L
S
J
I
A
K
L'
S'
X
(A.145)
as
(A-146) _
_
multiply Eq. (A.145) by (JMr'VjYAF) and sum over M and v with M' being fixed, then what one has is a sum of the product of six Clebsch-Gordan coefficients involving nine angular momenta, and it is nothing but the 9j symbol in unitary form given in Eq. (A.146) (see Eq. (6.78)). Then with the help of the orthogonality relation of the Clebsch-Gordan coefficients Eq. (A.145) itself has to be equal to (A.146). Substituting this result into Eq. (A.144) and using To derive this result
the definition of the reduced matrix element
((L'S')X 11 [01(space)
x
one
obtains
0,\(spin)l,, 11 (LS)J) L
S
J
2JI + 1
(2LI
+
1) (2SI + 1)
1
A
P
S,
-
x
(L'
01 (space)
L) (S' I I 0,\ (spin)
S).
(A.147)
Exercises
290
In and
special by using
cases one can
(S' 11111 S)
=
set A
=
0
or
1
=
0 in the above formula
JSS, A/2S +I
derive useful formulas
as
(A.148)
given below:
((L'S')J' 11 01(space) 11 (LS)J)
jssl(_I)L-J+J-L' V1-2LI-j,-+1U(SLXl;JL) +I
(L' 11 01(space) 11 L .
x
((L'S')X 11 OX(spin) JLLI
(A.149)
(LS)J)
V TS-,+-, U(LSJA; JS) (S'
Ox (spin)
S).
(A.150) A.2. Derive
Solution.
Eq. (A.19).
Using Eq. (6.18),
we
have
(e. e ) ni (e. r) n2 (ef.,r) n3 rn2 +n3
Y,
Y,
E
BkllBk2l2Bk3l3
2ki+1,,=nj. 2k2+12=n2 2k3+13=n3
(-1)11+12+13-v/'(211
I)T(2172 1) (2K3 +T) [Y11 ( -)
[y12 (' -) Xy12 ('0100 [Y13 0 ) As
was
done in Exercise 6.1
IY12 0
X
Y12 (f9]OO [y13 0 12 13
12 13
0
X
0
X
Y13 ('06'
(see Eq. (6-106)),
)
X
y13 0
0100
(A.151)
we
recouple the product
0
Q12 13; "0 11Y12 (0-)
2K+1 +
Y711 (
)100
-
212
X
1) (213
+
1)
Q12 13; K)
X
Y13 0
)Ir-
X
Yr-(f6)100
Exercises
1[Y12 (P-)
X
X
Y13
0 )I
r-
X
YK M 1 00'
(A.152)
The last step is to combine the two scalar
1YII (10
X
Y11 (0
IRY11 (P-)
X
Y13 (
)100 RY12 ( -)
X
0 )10
[Y12
Y11
X
EELI111101213r-r- [[YL(&)
X
I)JrX
291
products X
Y13
Lf( I&
Y
follows:
YK00100
0 )]KlrX
as
YIs (j
X
Yr-00100 (A.153)
)100,
LL'
given by Eq. (A.75). When one of the angular momenta is zero, the 9j symbol is reduced to the Racah coefficient (see Eq.(6.86)). Using this simplification in the coefficient E and substituting the above results completes the derivation.
where the coefficient E is
A.3. Calculate the matrix element of the orbital
IAjj
angular
momentum
for the correlated Gaussians.
Solution. The matrix element of tween the
gration
lAij
be obtained from that be-
can
generating functions of Table 7.1 or by performin the intein Eq. (A.23) with V(r) being set to unity. The result
over r
is
(fK'L'M'(Ul
i
A
10) 1 IjLjj I fKLM (u, A, x)) 3
BKLBKILI
ff
(
(2v) M-1 detB
2
)
1
*Y, +,q,-Y) c
d2K+L+2K+L'
dMi YLm ( _) YL, m, (o )
AAexp
dA2K+LdX12K-+L'
[q A2 + q1 A/2 + pAYe-e'] i(e x e),, (A.154)
The
integration
over
_ and
o
can
be done
by using the formula
ff de^_d4! YLm( -_)YL,m,(, )*(i(e e),,(e-e')') x
_2 4
1
e=I,eI=1
110111, d,T V '2__Ll+1 (LMIjLjL'M) E Bklv _21+IEW 2k+l=n
Exercises
292
(A.155) Clearly I has to be L I and also L' 1. In order that a possible value L' or IL L'I 2 has to be satisfied. The latter of 1 exists, either L case is obviously impossible because the condition IL L'I < I has to =
=
-
-
be met. Thus
we
equal, otherwise
reach the
the
conclusion that L and L' must be
same
would vanish.
integral
7r(2L
Using
3(L 1) 3(L 7(L2L 47(2L 7F(2L 1)'
3L
QL-I 1; L)
+
QL+1 1; L)
____
-
47r(2L + 1)'
+
-
2+L
U(L-llLl;Ll)=
V:2LL+
U(L+l IM; L1)
2
(A.156) together
with the relation 2k + 2L + 1
BkL-1
=
2k+L we can r
2k
BkLi
Bk-I L+I
Bk-Li (A.157)
show that 11
JJ dM4 YLm( -)YL,Iv,(i )*(i(eJLLI VFL (L In order for the
and
2k+L
+
x
1) (LMly I LM)
integral not
to
vanish,
e'),(e-e)' (A.158)
Bn+I-L
n+I n
+1
2
-
L must be
non-negative
even.
Employing this leads to
Eq. (A.6)
result in a
Eq. (A. 154)
and
compaxing its result with
solution
(fKILIMI (u!, X, X) 11pi, IfKLM(u, A, x))
05LL'1\1L(L + 1) (LMI[tILM) 1 X
-
('Y?71 + 7177) (fKI LM (W, X, X) I fKLM (u, A, x)).
(A.159)
PC
As
a
check of the above
formula,
one
the matrix element of the total orbital 2
7
I:i<j IlLij (see
^11,q)
can
be
Therefore,
Eq. (2.22)).
easily
we
The
done with the
sum-ma
use
of
may
use
angular i n,
this to calculate
momentum
over
i <
LIL
=
j, of .1C (-y?7' +
Eq. (2.15), yielding just
expect that the desired matrix element becomes
z2 Lp.
Exercises
(fK"LfM' (ul A!, x) ILIIfKLM(UA, X)
--`
,
x
JLLI -
fL-(L + 1)
(LM11-ilLM)(fK'LM(WiA iX)IfKLM(UIAIX))-
On the other
hand,
reach the
we can
same
result
293
(A-160)
through the Wigner-
Eckart theorem:
(a'L'M'IL,IaLM) JLLf
:::
JLLI'
(LM IAILM) (aL 11 ,F2L + I
L
11 aL)
(LM 1/ilLM) VIL(L + 1) (aL 11111 aL) ,V -2-L+1
+ 1)(aLMIaLM), JLLI (LM lpILM) VFL-(L
where the labels tions. Here
the
and a' stand for the other labels of the
used the fact that
we
angular
a
(A.161)
momentum of the
L,,
wave
does not
wave
func-
change the magnitude of
function because it is
a
generator
of the rotation group. A.4. Calculate the matrix element of the
the method of Solution. The
spin-orbit potential following
Complement A.I.
integration
over
(fK'LIM" (U A X) I V(I'ri i
7
P is first done in
Eq. (A.23), leading to
-ri 1) IlLij I fKLM (Ui
A X)) 3
00
(,yq +'y
ff
de^-
AA/exp
n)Idr
r2 V(r) e-
47r
IBKLBK"L' ( Cr2
di YLm( -)Ypm, (i )-
(27r)N-2C2 detB
d2K+L+2K'+L' d/\2K+Ld/X/2K+L'
[qA2 + q/A/2 + #AA'e. e] i(e x e%,rii(rv) V
X=0'XI=0 e=I,ef=I
(A.162) Here
v
is the
length of the
vector
v
defined
by Eq. (A.122) and
use
is
made of the formula
f 0"(b
x
r),,&
=
47rrii(ar)(b a
x
a),,,
(A.163)
Exercises
294
can
r), ,
-Nf2i [b
=
V.L3'ir [b
rl
x
v2 term e-2c
The
and
proved by using Eqs. (6.53)
be
which
Yj- (i6)]
x
'i, (rv) is expanded
(6.54)
noting (b
x
I,
done in
as was
-
and
Complement
V
A.I: CO
2 V
exp( 2c) -
1
r
-ii(rv)
2
2n+2
(6cr) C
H2n+3
X
the
Eq. (A.158),
of
use
( 6r) C c
+
2(2n + 3)H2n+l
expansion (A.126) and integrating
the
By using
(V2c)
3
(2c) - ' r n=0 (2n + 3)1
v
n
-rj 1) lij
(
V
K+K+L-1
I:
(7771 + 71,q) C
I
H, (X) H2n+3 (X)
F-x(2n + 3)!
V
+ 2 (2n +
E
32
(2 detB
10 V(jX) e-X2
3) H2n+l (x)
C
I dx
min(K,K) X
with
00
n+I -
n=O
X
4
II fKL(u, A, x))
JLLI 'FL(L+,)(2L+,)(2K+L)!(2K'+L)! BKLBK-L 1
- and
over
obtain
we
(fKf L' (u', A, x) I I V(I'ri
X
(A.164)
-
1
Fk-kKI-kL+2k-I (c, q, qr
BkL. L+2k
k=O
(A.165) In the
when
case
V(r)
=
I
to the matrix element. The function
then becomes
L+2k P
we can
H(K
confirm that
A.5. For
a
=
...,
(X1-i
XK) ---7
matrix, calculate
n
=
0 contributes
Fr2k KI-k L+2k-1 (CI q, q, p, -y, -y)
k, q) H(W
-
k, q) H(L
Eq. (A.165) reduces
given function f (X1, X2
f (X1, X2, where:i
-
term with
only the
to
XK)
+
2k, p) Therefore .
Eq. (A-159).
of the form
=1 TcAx + 6,
XK)7
2
(bl,..., bK),
and A is
a
K
x
K
symmetric
Exercises
,)ni
11==02
ef(Xl,X27---,XK
OX,ni Solution. A solution
(A-166)
7XK::--O*
be obtained with the
can
use
of Leibniz's formula
we
show another way to solve this
+
-+AK
as was
done in A.2. Here
Let
operator L denote
an
...
295
problem.
a L
A,
A2
+
(All
Where
e' 'Cef (xI
7
...
7
7XK)
...
*
AK) =
is
an
auxiliary parameter.
ef (x'+6,
ef(XI
7
...
(A-167)
19XK
(9X2
XI
7XK)
'
7
...
Then
we
have
7XK+IMK)
exp1UXV2 (2 +
W + Ax
11)
-
(A-168)
the above equation in power equal to EK i==l ni. Expanding series of 0, collecting the terms of the Nth power in 0, and putting 0 (i =. 1, K) leads to Xi Let N be
=
...,
(f-Nef(xl
7
...
)XK)
[N 21 =
)
Xj=O .... 7XK=O
N!
E k! (N
(1U ) k( b) N-2k. -
-
k=O
2k)!
2
(A.169) problem of finding a solution to Eq. (A. 166) is thus reduced to a simpler problem of expanding the right-hand side of Eq. (A.169) in are expanded as follows: the Ai's. To this end ( b)N-2k and (WA)k 2 The
(N
( b) N-2k
11! 12!
k
2k)! IK.
(Albl)", (AA) 12 k!
(I)k
AA)
2
-
Mll! M12!
2
7nKK!
(XKbK) 1K,
(All A,2)mll
Mij
x
(2Al2AlA2 )M12
Collecting those obtain
a
solution
...
x
(AKKAK 2)MKK.
terms which have powers of
A,nj A2 n2
(A-170) AK nK
we
Exercises
296
K
ani -e
f(XI,X27
....
9X,ni i2L
XK))x
1=07
...
IXK=O
K
k
(')
E 11 ni!H(mij, Aii)H(li, bi)
2
k=O
i=1
Mij K
XH
H(mij, 2Aij),
(A.171)
j>i=l
where
and
mij's
li's
are
all
non-negative integer and
must
satisfy the
following relations K
K
mij
=
k,
mij +
j>i=l
=
(i
ni
K).
1,
(A. 172)
j=1
A.6. Derive
Eq. (A.93).
Solution. First x
E mji + li
(
2 x
we
note that
(Al) '+Ae-2IpX2_,5X.y YLM 41
M)
(,Axe- .j'8X2_jx.Y) X2v+Ay(,Xl) 2
LM
+
e-12 jax 2-,6X.y A.'X 21/+Ay(,\I) LM PC
+2
(Vxe-2.jpX2_jx`.Y) (VXX
M)
2v+,X
-
AI) YL(M P9M
The differentiation in the second term of the
above equation
(AxX
2v+,\
can
be
right-hand side
performed by using Eq. (6.59)
AO
I
The third term
can
be
reduced, by
means
to
(Vxe- .IpX2_jX.y 2
of the
as
AI) YL(M
2v(2z/ + 2A + I)x 2v+,X-2 YL(M PC M (A.42),
(A.173)
VXX
2v+A
AI) YL(M (ic'
of the
(A.174) gradient formula
Exercises
X2v+X-l e
F+' OX
1
oX2 -,6x.,y
(2v + 2A + 1)
1
-A-+I 2A+1
297
X
YX-1 ('' )4
X
Y' Ml LM
2v[[(- x-Jy)XYX+I( 3)11\Xyl( )ILM (A.175)
By using the angular momentum algebra the equation is reduced to
first term in the round
bracket of the above
[[(-OX
-
JY)
,rL3
X
Y)'-' POD'
X
Y' M] LM
OxC(1A-I;A)YL(A')(:t, ) M
+ Sy
A
U(A-1 I L 1; A r,) C(11; r)Y(X-l LM
2A+lt XYL
X1) M
Jy E U(A
r.)
( O,
-
11 L 1; A
K) C(11; r,) YLM
K
(A.176) where
use
Similarly
is made of
Eq. (A.156)
for
C(l A
-
1;
A)
C(A
-
11; A).
the second term becomes
11(- X-*X YA+'(" )IXX Y1MILM XY(,X1) L M PC M I
yE U(A+I ILI; Ais)C(11; r,)YL(A+1r')(1b, M
The contribution of the first term is
(A.177)
easily obtained by using
Exercises
298
2 e- .l#X2_,5X.y
1
+
02X2 + j2y2 + 2,3dx -,y) e-'2,8X
where the last term
involving
the scalar
2
product
-,5x.y
(A.178)
1
x
-
y
can
be reduced
to
X.
Y Y(Ai) LM
=
,
E
-v -3[x
x
y]oo0'1) LM
U(11 ; 0"0 Ry
X
[X
X
Ily
X
YA-1 4A A
X
Y' MI LM
_X
Ry
X
Y +'P
X
YI(MILM*
-V
3
Y
X
X
1WILM
Y
r.=Al
2,X + I
(A.179)
right-hand side of the above equation appear Eqs. (A.176) and (A.177), respectively. Combining these results we obtain the desired equality (A.93) The two terms of the
Ax
(X
2 Ll +,X
l,3X2_,6x.y
e-2
AI) YL(M
30X2 +,32X4 + j2X2Y2 + 2zj(2i/ + 2A +
2 -2Sx 2v+A-l y e-LPx
I
X
x
):+
TA
I
in
1)) 2
X
1 _
-
20(271 + A)X2
2v+,X-2
2
_,SX.y
LpX2_jX.y
e- 2
V/
YL(M
7F
3
(2y + 2A + 1_ OX2)
EU(A-IlLl;Aiz)C(11;K)YL(A-l)(. c, ) M
Is
'A+I -
2A +I
x
(2y
_
OX2)
I:U(A+IILI;Ar,)C(11;r,,)YL K
M
(A.180)
widely used techniques to solve the eigenvalue problems of quantum-mechanical Hamiltonians. Its popularity is due to its simplicity and flexibility. The most crucial point in the variational approach is the choice of a variational trial function. One usually attempts to construct the trial function from some adequate basis functions which contain a number of nonlinear parameters. The direct method of diagonalizing the Hamiltonian matrix on such a basis set may not be feasible, except for simple systems, because of the large number of degrees of freedom involved in specifying the system. One thus faces a problem of selecting the most suitable basis set. It is by the stochastic variational method, that is, by a trial and error procedure with an admittance test that we give an answer to this problem. The stochastic variational method has been developed through the search for precise solutions of nuclear few-body problems. The variational method is
In this method
it enables
us
we
one
of the most
set up the basis element
to test many
parameters
to monitor the energy convergence. The aim of this book is to give
as
after the other because
one
fast
as
possible
and
moreover
unified and
reasonably simple problems with the use few-body recipe of the stochastic variational method and to present its application to various few-body problems which one encounters in atomic, molecular,
nuclear, subnuclear and solid
a
bound-state
for solutions of
state
physics.
quantum systems is in general extremely difficult and challenging, great advances have been made in recent years, especially for systems of a few particles and it has become possible to obtain accurate solutions for the eigenvalue prob-
Though
a
unified
approach
to the diverse
quantum-mechanical Hamiltonians. The main interest the few-body problems lies in, e.g., finding an accurate solution for
lem of various
in
the system
so as
to understand the
dynamics
of its
constituents,
test-
VI
Preface
ing the equation of motion and the conservation laws and symmetries, or looking for unknown interactions governing the system. Quantum mechanics plays a fundamental role in atomic and subatomic physics. It is via quantum mechanics that one can understand the binding mechanism of atoms, molecules and atomic nuclei, that is, the structure of the building blocks of matter. The interaction between the particles depends on the system: For example, the longrange Coulomb interaction dominates in atoms and molecules but the
very different mass ratio of the electrons and the atomic nucleus plays a key role as well. In contrast, the protons and neutrons in nuclei have
equal masses and the interaction between them is short-ranged. The variational foundation for the time-independent Schr5dinger equation provides a solid and arbitrarily improvable framework for the solution of diverse bound-state problems. As mentioned above, the most crucial point in the variational approach is the choice of the trial function. There are two widely applied strategies for this choice: One is to use the most appropriate functional form to describe the short-range as well as the long-range correlations among the particles. Such calculations, however, are fairly complex for systems of more than three particles, and the integration involved is performed by the Monte Carlo method. Another way is to approximate the solution as a combination of a number of simple basis states which facilitate the analytical calculation of matrix elements. We follow the latter course almost
in this book and show that the stochastic variational method selects
important basis set without any bias, keeps the dimension of and, most importantly, provides a very accurate solution. The book is conceptually divided into two parts. The first seven chapters present the basic concepts of the variational method and the formulation using Gaussian basis functions. The latter four chapters of the book cover applications of the formulation to various quantummechanical few-body bound-state problems. In Chap. 2 a general formulation is developed to express the physical operators which are needed to specify the Hamiltonian in terms of an arbitrary set of independent relative coordinates. The linear transformation of the relative coordinates induced by the permutation of identical particles is also established in this chapter. In Chap. 3 we review the basic principles of the variational method with particular emphasis on the case where the variational trial function is given as a linear combination of nonorthogonal basis functions. We introduce in Chap. 4 a key algothe most
the basis low
rithm used in this
book,
the stochastic variational
method,
and show
Preface
that its trial and
error
search
procedure
makes it
possible to
Vii
select the
important basis functions without any bias in the function space spanned by the basis functions. Some other methods to solve few-body most
briefly introduced in Chap. 5. Chapter 6 defines the type of variational trial functions used extensively in the book, the correlated Gaussians and the correlated Gaussian-type geminals. They are chosen because they enable us to evaluate matrix elements analytically and because they provide. us with precise solutions for most problems of real interest. A simple but powerful angular function is introduced to describe orbital motion with nonzero orbital angular momentum. To facilitate the systematic and unified evaluation of matrix elements,
problems
are
it is shown that the above Gaussian basis functions
function. In
7
are
all obtained
show that the
generatChap. deriving the matrix elements of the Gaussian basis functions for an N-body system of essentially any interaction. Explicit formulas axe given in this chapter for the simplest possible Gaussian basis functions, because they are already found to be very useful. The matrix elements for a, general case are detailed in the appendix. We show also in this chapter that the. method can be extended to evaluate the matrix elements of nonlocal potentials and the seniirelativistic kinetic energy as well. Chapters 8-11 present application of the stochastic variational method to various systems: small atoms and molecules (Chap. 8), baryon spectroscopy (Chap. 9), excitonic complexes and quantum dots in solid state physics (Chap. 10), and nuclear few-body problems (Chap. 11). We hope that this book will be found useful by students who want to understand and make use of the variational approach to quantummechanical few-body problems, while it may also be of interest to researchers who axe familiar with the subjects. It will be our pleasure if this book serves to bridge the gap between graduate lectures and the literature in scientific journals, as well as to give impetus to further development in the deeper understanding of quantum-mechanical fewbody systems. We assume that readers have taken courses on quantum mechanics and mathematical physics at an undergraduate level. No special knowledge is assumed of, e.g., atomic physics or nuclear physics. To help readers to understand the text, we have attempted to make the book self-contained, put as much emphasis as possible on clarity, and given several Complements of an explanatory nature. The Complements are intended to further develop or to reinforce the arguments and ideas presented in the text. We have collected the derivation
from
ing
a
generating plays a
function
vital role in
we
VIII
Preface
of formulas that may possibly be difficult for readers solutions at the end of the chapters.
as
exercises with
Depending on their interest, readers may adopt several reading strategies. A thorough-going reader is advised to read all of the text including the Complements. A reader who wants to understand only the basic formulation can omit the Complements. Anyone who is familiar with the variational method may skip Chaps. 2 and 3. Experts, or those readers who are only interested in the performance of the stochastic variational approach or the physical consequences of the results, may skip Chaps. 2-7. The Gaussian basis has long been used in many areas of physics. The correlated Gaussians were first introduced in quantum chemistry by S.F. Boys and K. Singer. The application of the Gaussian basis is one of the key elements of the success of the ab initio calculations in quantum chemistry. The stochastic variational method is actually very similar to the so-called "random tempering", that has been used to find the optimal parameters of the basis in quantum chemistry. In the random tempering pseudo-random parameters axe generated, and the best basis functions are selected by sorting out the states which improve the energy. There exists another method which is also similar to the stochastic variational method, called the stochastic diagonalization. This method, originally developed in solid state physics, attempts to find the lowest eigenvalue of huge eigenvalue problems by randomly testing the contributions of different states. The random selection of the parameters of a Gaussian basis, named the stochastic variational method, was first used by V.I. Kukulin and V.M. Krasnopol'sky in nuclear physics. We are grateful to Prof, V.I. Kukulin for his interest and encouragement during this work. We are indebted to our collaborators, K. Arai, Y. FujiNvara, L.Ya. Glozman, R.G. Lovas, J. Afitroy, C. Nakamoto, H. Nemura, Y. 0hbayasi, Z. Papp, W. Plessas, G.G. Ryzhikh, N. Tanaka, J. Usukura, and R.F. Wagenbrunn for useful discussions and for many of the calculations which press
our
hearty
are
included in this book. We wish to
thanks to Prof. K.T.
Hecht,
ex-
Prof. D.
Baye, Prof. reading of the Despite all these
M. Kamimura and Prof. R.G. Lovas for their careful
manuscript and for much advice and many comments.
efforts mistakes may still remain. Needless to say, these are our own responsibility. Suggestions and criticisms from our readers would be welcomed. We
are
grateful
for the
use
of the computer VPP500 of (RIKEN). One of the
the Institute of Physical and Chemical Research
Preface
for the
authors
(K.V.)
ment of
Niigata University and
is
grateful
would like to thank the
Research of the
hospitality
the Linac
funding agencies,
Iffinistry
of
(Hu gary),
of the
Ix
Physics Depart-
Laboratory
of RIKEN. We
Grants-in-Aid for Scientific
Education, Science and Culture (Japan),
Japan Society for the Promotion of Science (JSPS), the Science and Technology Agency (STA) of Japan, the U. S. Department of Energy, Nucleax Physics Division, and the Hungarian Academy of Sciences (HAS), for their support which has been vital for the completion of the work. Finally, without the support and patience of our families, this book would not have been possible. OTKA Grants
Niigata Argonne September
the
Y. Suzuki K.
1998
Varga
Table of Contents
I..
Introduction
2.
Quantum-mechanical few-body problems
........................................
...........
7
2.1
Hamiltonian ......................................
7
2.2
Relative coordinates
..............................
9
..................................
15
2.3
Symmetrization
2.4
Permutation of the Jacobi coordinates
Complements C2.1 An N-particle Hamiltonian in the heavy-particle center
..............
.........................................
Variational
3.2
The variance of local energy The virial theorem
3.3 4.
principles
......................
19
4.2
21
21
.......................
30
................................
33
.......................
39
................................
39
...............................
43
......................
44
.........................
47
..............................
47
....................................
50
Basis
optimization A practical example 4.2.1 Geometric progression 4.2.2 Random tempering 4.2.3
Random basis
4.2.4
Sorting
4.2.5
Trial and
error
search
.......................
Refining 4.2.7 Comparison of different optimizing strategies Optimization for excited states
4.2.6
4.3
................
.............................
Stochastic variational method 4.1
18
18
Introduction to variational methods 3.1
16
...........
coordinate set
C2.2 Canonical Jacobi coordinates 3.
1
...................................
.................
50 53
.
54
....
56
.
I
Complements
.........................................
61
XII
Table of Contents
C4.1 Minimization of energy 5.
versus
variance
..............
Other methods to solve
5.2
few-body problems Quantum Monte Carlo method: The imaginary-time evolution of a system Hyperspherical harmonics expansion method
5.3
Faddeev method
5.4
The generator coordinate method
5.1
.........
...........
6.
..................................
Variational trial functions 6.1
6.2
6.4
65
67 70
72
..........................
75
Correlated Gaussians
Gaussian-type gerninals Orbital functions with arbitrary angular Generating function The spin function
Complements
..............
.
82
87
................................
94
C6.2 Solid
spherical harmonics C6.3 Angular momentum recoupling C6.4 Separation of the center-of-mass =
as a
basis
.....
96 96 105
.....................
106
motion
.........................
1/2
......................
arbitrary spin arrangement
112
115
......
115
...........................
116
............................................
118
an
C6.7 Six electrons with S
Matrix elements for
=
0
spherical Gaussians, generating function
............
123
..........
123
..............................
125
7.1
Matrix elements of the
7.2
Correlated Gaussians
7.3 7.4
Correlated Gaussians in two-dimensional systems Correlated Gaussian-type gendnals
7.5
Nonlocal
7.6
Semirelativistic kinetic energy
Complements
potentials
.
129
.................
131
.
.
.
.
...............................
......................
.........................................
C7.1 Sherman-Morrison formula Exercises
.
..........................
from correlated Gaussians
C6.5 Three electrons with S
Exercises
momentum
.........................................
C6.6 Four electrons in
75
..............................
C6.1 Nodeless harmonicoscillator functions
7.
65
...................
and correlated
6.3
.........
61
134 137
143
........................
143
............................................
145
Table of Contents
8.
Small atoms and molecules
.........................
8.3
Coulombic systems Coulombic three-body systems Four or more particles
8.4
Small molecules
8.1
8.2
.......
149
.....................
150
...................................
.........................................
C8.1 The cusp condition for the Coulomb potential C8.2 The chemical bond: The H,+ ion C8.4 9.
Stability hydrogen-like Application of global vectors
169
.................
to muonic, molecules
174
................................
177
One-gluon exchange model Meson-exchange model
.....
178 178
............................
181
quark
...........
188
...................................
191
....................
196
method
a
magnetic
204
..........................
213
...
213
.............
216
...............................
223
.........................................
230
11.2 Few-nucleon
systems with central forces
potentials
C11.1 Correlations in few-nucleon systems C11.2 Convergence of partial-wave expansions Pauli effect in s-shell A
C11.4 The "C nucleus
Appendix
202
field. 204
few-body systems Introductory remark on nucleon-nucleon potentials
Quark
....
.........................................
11. Nuclear
C11.3
187
..............................
C10.1 Two-dimensional electron motion in
as a
...............
230
............
233
..........
239
.
242
hypernuclei system of three alpha-particles
...............................................
Matrix elements for
A.1
model
........................
Few-body problems in solid state physics 10.1 Excitonic complexes 10.2 Quantum dots 10.3 Quantum dots in magnetic field 10.4 Quantum dots in the generator coordinate
Complements
171
....
9.2
11.3 Realistic
167
....................
The trial function in the constituent
11. 1
165
167
9.1
Complements
154
........
molecules
Baryon spectroscoPy
9.3
10.
of
149
........................
............................
Complements
C8.3
XIII
general
Correlated Gaussians
A. 1. 1
Overlap
A.1.2
Kinetic energy
Gaussians
..................
.............................
of the basis functions
247 247
247
................
247
.............................
249
Table of Contents
XIV
A.1.3 A.1.4
......................
250
..................
256
Two-body interactions Density multipole operators
A.2
Correlated Gaussians with different coordinate sets
A-3
Correlated
matrix elements
Spin
A.5
Three-body problem and spin-orbit forces
Complements
.................
262 263
with
central,
tensor
..............................
265
.........................................
280
CA. 1 Matrix elements of central potentials CA.2 Matrix elements of density multipoles
CA.3
...............
280
..............
283
matrix elements of the correlated Gaussians
Overlap a three-particle system
for
Exercises
References Index
257
.............................
Gaussian-type geminals
AA
...
.........................
285
............................................
288
..............................................
...................................................
299
307
1. Introduction
There
are a
countless number of examples of quantum-mechanical few-
body systems:
constituent
quarks
in subnuclear
physics, few-nucleon
few-cluster systems in nuclear physics, small atoms and molecules physics or few-electron quantum dots in solid state physics, etc. The intricate feature of the few-body systems is that they develop or
in atomic
depending on the number of constituent particles. The mesons and baryons, the alpha-particle and the 'Li nucleus, or the He atom and the Be atom have very different physical properties. The most important causes of these differences are the correlated motion and the Pauli principle. This individuality requires specific methods for the solution of the few-body Schr6dinger equation. Approximate solutions which assume restricted model spaces, mean field, etc. fail to describe the behavior of the few-body systems. The goal of this book is to show how to find the energy and the wave function of any few-particle system in a simple, unified approach. The system will normally be in the minimum energy quantum state. As forewarned, however, to find this state, the ground state, is a complicated matter. The present stage of the development of computer technology, however, makes a very simple approach possible: Searching for the ground state by "gambling". Without any a priori information on the true ground state, completely random states are generated. Provided that the random states axe general enough, after a series of trials one finds the ground state in a good approximation. The reader individual characters
may find this
a
little
suspicious but there are indeed a number of fine error procedure which makes the whole idea
tricks in the trial and
really practicable. Before bombarding the
reader with
sophisticated details, let us example. Let us try to determine the energy of a Coulombic three-body system: a positron and two electrons, the positronium negative ion, Ps-. The example is simdemonstrate the random search with
Y. Suzuki and K. Varga: LNPm 54, pp. 1 - 6, 1998 © Springer-Verlag Berlin Heidelberg 1998
an
1. Introduction
2
ple enough so t1i at a graphic illustration of the wave function can be given, but its solution is far from being trivial. The ground state of this particular system can be calculated by different methods accurately. In the rest of this chapter we will give just the outline of the gambling method without paying much attention to the details. To look for the wave function of the ground state of the system, we generate random functions. The functions we assume some
continuous a.re
parameters
or
randomly
chosen. For
Tf (a, 0, x,
y)
=
=
are
random in the
depends
on some
sense
that
discrete
or
"quantum numbers" and these paxameters example, we assume a simple Gaussian form
e`2 -
3Y2
(1.1)
7
1r2 -7'31 denotes the distance between the two electrons and I (r2 +r3) /2 r 1 1 denotes the distance between the center-of-mass
where Y
functional form which
x
=
-
of the electrons and the
positron.
randomly chosen generated states are calculated and compared. The one among them giving the lowest value is selected to be a successful paxameter set. Figures 1.1(a)-(d) show examples of four random states and their energy expectation values which are obtained with such successful paxameters. Not surprisingly, the energies of the "configurations" appreciably depend on the shape of the functions, that is, on the random paxameters. The lowest energy, -0.11 in atomic units (a.u.) in Fig. 1.1(d), is higher than that of the exact ground state (-0.262). Here in the atomic units M, (the electron mass), e (the electron charge magnitude), and h axe chosen to be basic quantities of units, so the unit of length is Bohr radius h?/(Mee2) 5.29x1O-11 m, theunit of energyis me4/h? 27.2 ao 2.42 x 10-17 S. (Actually the and the unit of time is h3/(me4) eV, minimum energy calculated analytically with one term of Eq. (1.1) is -0.177, but it may not be known for a general case.) That means that the trial function is not general enough to describe the ground state. To improve the trial function, let us take a linear combination of two of the above functions of Eq. (1.1), where one of the two is fixed as the one already selected and another is newly selected after a number of random trials. Figures 1.2(a)-(d) show examples for random wave functions which are calculated by the two terms. The energy improved because the model space increased but we still miss a substantial amount of binding energy. Increasing the model space further A certain number of parameter sets a and and then the energy expectation values of the
=
axe
=
-
=
by adding one,
we
more
and
more
functions to the linear combination
reach the exact energy and the
wave
function
one
(Fig. 1.3)
by
with
1. Introduction
E--0.09
E--3.15
E=0.18
E--0.11
Examples of the energy expectation value and the (normalized) xyTf (a, 3, x, y) of Ps- for one random basis state. Figure 1(a) is in the upper left, 1(b) in the upper right, 1(c) in the lower left, and 1(d) in the lower right. In each section the x axis denotes the distance
Fig.
wave
1-1.
function
between the two electrons and the y axis denotes the distance between the center-of-mass of the two electrons and the positron. Atomic units are used.
1. Introduction
E-0.14 E=-0.18
0.1
0.0
B-0.12 E=-0.17
0.1 0.0
Fig.
1.2.
Examples
xyjcj-T1(aj, Oj
of the energy expectation value and the
wave
function,
y) +c2Tf(a2, 02 Xi Y) ji of Ps- for combinations of two random. basis states. Figure 2(a) is in the upper left, 2(b) in the upper right, 2(c) in the lower left, and 2(d) in the lower right. See also the caption of -,
Fig.
1. 1.
x,
1
1. Introduction
a
combination of about 150 functions of the form
(1.1).
The
5
conver-
gence of the energy versus the number of the functions in the linear combination is shown in Fig. 1.4. The energy gain is large in the first
few steps and then it
slowly approaches the
exact energy.
E=-0.262
X
Fig. and
1.3. The energy and the wave function of Ps- obtained selection of 150 basis states
by the trial
error
The alert reader may question the importance of the steps that are taken to increase the model space and can ask why not start say, a linear combination of 150 functions and then, by random of all the parameters, one may find the solution. The reason for trials
with,
increasing the number of functions in the combinations is to control "convergence" of the energy. By comparing the energy gain in the successive steps one can guess how far the exact ground-state energy the
is. There is
no
guarantee that the ad hoe 150 functions would be
1. Introduction
6
-0.250
-0.252 M 41
-0.254
-0-256
-0.258 Z
-0.260
-0.262 20
40
60
80
100
Dimension of the basis
Fig.
1.4. The energy convergence of Ps- as a function of dimension of basis are increased one by one with the trial and error selection. The
states that
solid, dashed and dotted
curves
correspond to
sufficient to reach the solution in
cases
three different random
paths.
where the exact energy is not
known. A
skeptic may say that one should, instead of the above gambling method, try a deterministic parameter search such as that furnished by the Newton or conjugate gradient method. While this may be true for small systems, the random trials the
axe more
successful in most of i
in the
I
a. by avoiding being trapped omnipresent local Moreover, one may have discrete paxameters or quantum numbers
cases
where the deterministic seaxch may not be suitable. One may also wonder if the solution we reach is the in
ground state energy. These the succeeding chapters.
and other
really (close to)
questions will be answered
Quantum-mechanical few-body problems 2.
first step toward the goal of giving a unified and reasonably simple recipe for solutions of few-body bound-state problems the basic
As
a
notations and concepts
The motion of the ton
operator
axe
introduced here.
few-body system
(Hamiltonian)
and described
governed by the Hamilby the eigenfunction of the
is
depends on the positions and other degrees of freedom of the particles. See, for example, [1, 21 for textbooks on quantum mechanics. One can define the positions of the particles in several different ways by using single-particle or relative coordinates. The single-particle coordinates are useful if the particles move ah-nost independently of each other. The relative coordinates are advantageous to emphasize correlations between the particles and Hamiltonian. This
to
wave
function
separate the center-of-mass motion. One
can
relate the different
coordinate systems to each other by linear transformations. The definitions and the properties of these transformations are elaborated in this
chapter.
indistinguishable particles. To comply with the Pauli principle the wave function has to be properly symmetrized. In the spatial paxt of the wave function the symmetrization induces permutations of the coordinates. It is shown that these permutations can be imposed by linear transformations in the relative coordinate One often deals with
space.
2.1 Hamiltonian isolated system of N particles with masses ml., and let 7,1,, ,,rN denote the position vectors of the particles. -, mAT All of the particles may be identical or different, or some group(s) of them may be identical. The Hamiltonian of the system, with the
Let
us
consider
an
*
center-of-mass kinetic energy Tcm
being subtracted,
Y. Suzuki and K. Varga: LNPm 54, pp. 7 - 20, 1998 © Springer-Verlag Berlin Heidelberg 1998
reads
as
Quantum-mechanical few-body problems
2.
8
N
N
2
H=E 2mi
-T,.
As the interaction
(2.1)
Vij.
+
unchanged by the displacement of the origin, the above Hamiltonian is translationaUy invariant and depends on the internal degrees of freedom only. One may encounter few-body, problems where the system is subjected to some external field or the particles move in a single-particle potential. The Hamiltonian then reads as IV
H
=
N
2
E2
remains
Vij
+
1: Ui + 1: i=1
i=1
N
(2.2)
Vij.
i>i=l
Even with the presence of the one-body potential Ui it may happen that one wants to remove the contribution from the center-of-mass motion. This will be considered later in
evaluating the matrix elements
of various operators. An extension of the nonrelativistic kinetic energy to the relativistic kinematics will be discussed in Sect. 7.6.
Three-body potentials can be treated in principle but axe suppressed for the sake of simplicity. The two-body potentials are assumed to be local (but nonlocal potentials will also be considered in Sect. 7.5) and can in general be expressed as
Vij
=
1: VP(ri
-
rj) 0?.
P
=
1: f VP (r) J(ri
-
rj
-
r) 0?. dr.
(2.3)
P
Here p is the short-hand notation to specify the component of the potential which is characterized by 0?. (e.g., central, spin-orbit, etc.
and
VP(r)
is the
corresponding form factor.
To
specify
the
particle
motion, several degrees of freedom are in general needed besides the spatial one. For example, the nucleon has spin and isospin degrees
freedom, and the quark has spin, flavor and color degrees of fre&dom. The specific potentials related to nuclear, subnuclear and other of
systems will be discussed in later sections.
2.2 Relative coordinates
9
2.2 RelatiVe coordinates The
separation of the center-of-mass motion
care
of most
is
important to describe intrinsic excitations of the system. The center-of-mass motion is taken mass
conveniently by introducing
coordinates;i
(XIi X2
relative and the center-of-
xN). The symbol- stands for a trans-
pose of a matrix. In particular, x is often used to stand for one-column matrix whose-ith element is xi, whereas ,'c a I row
an
x
N
IV
x
1
one-
matrix. Here XN is chosen to be the center-of-mass coordinate of
the system and the rest of the coordinates
relative coordinates.
independent single-paxtide coordinates
i
-=-
They
(rj,
-'
fXI,
axe
in
-
-
-,
X N-I
general
I
is
a
set of
related to the
TN) by a linear transformation:
N
Uijrj,
xi
ri
=
j=1
E (U-I)ijxj
(i
N).
(2.4)
j=1
We show two
examples
of relative coordinates
is the Jacobi coordinate set, which is defined
by
x.
See
Fig.
2. 1. One
the matrix
C
2
4 3
0
0
(a) Fig.
2.1.
Examples
(b)
of relative coordinates for the
four-particle systein. (a)
the Jacobi coordinate set, xi =rl -r2, X2:-- (MIrI +M2P2)/(MI +M2)'r3 7 X3 = (Tnlrl + M2r2 + M3 r3)/(Ml + M2 + M3) -P47 and (b) the heavyparticle center coordinate set, xi = rl r47 X2 = r2 r47 X3 = r3 r4-
-
-
2.
10
Quantum-mechanical few-body problems 1
-1
MI
M2
M12
M12
M1
M2
0
0
...
0
(2.5)
Ui M12
...
M12
N-I
...
N-I
M1
M2
Tal2---IV
Tnl2---.LV
MN ...
...
M12
...
N
given by
Its inverse matrix is
M2
M3
M12
M123
MIV ...
Tni
M3
M12
M123
M12
---
M
MN ...
(Uj)
M12
...
N
(2.6)
M 12
0
M123
-"M12
0
0
...
N-1
M12 ...N
where M12 i +mi and, especially, 7nl2---N is the total Tnl+Tn2+ the "heavy-particle center" coordinate is Another the of mass system. -
...
set defined
UC
by 1
0
0
0
1
0
-1 -1
...
(2.7)
-
0
0 M12
---
N
...
-1
...
MN
M2
Mj_
...
...
M12
...
given by
and its inverse matrix is
...
N
M12
...
M
N
(UC )-I
M12
...
M12
...
N
M1V-I
M2
MI ...
MN-I
M2
M, M12
M12
M12 ---N
M
N
M12
N
M2
M12
...
N
MN-I
M2
MI
M12
...
N
M12
...
N
M12
...
N
(2.8) The choice of the when the other
mass
particles.
heavy-particle
See
center coordinates may be natural
particle is heavier than the masses of the Complement 2.1. Note, however, that it is always
of the Nth
2.2 Relative coordinates
11
possible to transform from one coordinate set to another. The Jacobian corresponding to the coordinate transformation (2.4) is unity for both U matrices.
Corresponding to the transformation (2.4), the momentum pi is -ih ya conjugate to the expressed in terms of the momentum -xj axj =
coordinate xj: IV =
Pi
N
1: Ujilrp
Wi
=
1:(U-l)jipj
(i
=
1,
...,
N).
(2.9)
j=1
j=1 N
Note that -7rAT
=
Ej=1 p,
is the total momentum. The center-of-mass
kinetic energy is given by T,,n ir2,v/(2MI2 N). The kinetic energy with the center-of-mass kinetic energy subtracted is then operator =
...
expressed
as
P
(27ni 6ij
2mi
i=1
i=1
2m12
...
N)
Pi
Pj
j=1
M-1 N-I
Aij-xi
2 i=1
-
(2.10)
-7rj,
j=1
where N
1: UikUjk Mk
Aij
(i,j
=
11
....
N
-
1).
(2.11)
k=1
To evaluate the potential energy matrix elements for the tions that contain
no
dependence
on
the center-of-mass
wave
func-
coordinate,
it
is convenient to express the single-particle coordinate relative to the center-of-mass, ri xN, the interparticle-distance vector, ri -,rj, and -
difference, pi pp in terms of x and easily done by using Eqs. (2.4) and (2.9):
the momentum This is
-
-x,
respectively.
1V-1
ri
-
XN
=
1: (U-l)ikXk
-W X,
(2.12)
k=1 N-I
'ri
-
r3
1: k=1
((U-l)ik (U-I)jk) -
Xk
=
W(j) X,
(2.13)
Quantum-mechanical few-body problems
2.
12
N-1
2
(Pi
Pj
E (Uki
2
-
(ij )Ir-
Ukj)7rk
(2.14)
k=1
Note that the contribution from the term
momentum,
(Mi
is
when
legitimate
-
Mj)/(2Ml2 N)7rN, ...
we are
proportional to the total omitted in Eq. (2.14). This
is
in the center-of-mass
(,7rlV
system
=
0)
and
interested in the intrinsic motion of the system. The Hamiltonian of type (2.1) is now expressible in terms of independent relative coordinates alone. In what follows
and
x
7r are
relative coordinates unless otherwise
(i.e. (N
introduced to
the notation.
by the
W(i) EN-I k Xkk=1
=
they are (N-1) x 1
stated,
-
1)
1 one-column
x
I)-dimensional vectors) 0), Oj),
simplify
which is formed
-W X
-
represent only the
that
so
Eqs. (2.12)-(2.14), (N
one-column matrices. In matrices
meant to
Wx
E.g.,
is
a
and
(W)
are
Caitesian vector
multiplication of iv(i) and x, i.e., that Oj) and (W) satisfy the following
usual matrix Note
equalities M-1
-((ij)
3 W(ij) (k k
W( ij
=
(-ipw(ij)
k=1 N
W(ij)(1(ij)
W
k
i>i=l
(ij)
F(ij)
i>i=l
kI
N
where the relations UNTj
for k
=7 -
For
N
are
an
(N
-
1) x (N
-
1) symmetric
scalar
products
E xi-(Ax)i with this
i=1
=
1,
EN
Uki
=
0
A,
let the
quadratic
of the Cartesian vectors:
xv)
j=1
convention, that
1V -
matrix
Aijxi-xj. i=1
show,
ai(ri
(2.15)
N-1 N-1
i=1
It is easy to
1),
?ni/?nl2---N, (U-1-)iN
=
N-I =
-
used.
form, bAx, represent
..bAx
N
R, I
Jk, 2
2
i=1
ajw(')J))
x,
(2.16)
2.2 Relative coordinates N
1V
I:
a
(ri _,rj)2
ij
1:
=.i
a
ajjw
-(ij)
(ij)
(2.17)
X.
i>i=l
i>i=l
As
13
special case of the above relations, we have the following equalities
for the moment of inertia around the center-of-mass: N-1
N
N
Mi(ri
-
XN
MzMj
)2
M12
where the reduced is defined
mass
...
N(ri
Tj)
-
corresponding to
ILi
2
(2.18)
the Jacobi coordinate xi
by ?ni+17nl2 M12
...
z
...
(2.19)
1).
N
i+1
We also obtain 2
N
N
(ri
-
T'j )2
N
=
E(ri XN)2
(,ri
_
-
(2.20)
xiv)
i=1
The term are
the
E , (ri
-
XN)
masses
of all the
particles
same.
The total orbital mass
vanishes when the
system
(-xv
=
angular
0)
momentum
be
can
operator L in the center-of-
expressed
in terms of the
operators
relevant to the relative coordinates: N-1
N
hL
(ri
((,ri
-
X
Pj)
Tj)
(XN
X
2
(Pi
X
-
(Xi
7N)
X
7ri),
(2.21)
Pj))
i>i=l N
or i
-
X
'r
Rij)
-X
i>i=l
+
((ri
-
W('j)X
rj) XMi
-
2M12
X
Mj 7rN ...
N
(('j)-7r) =NU. 2
(2.22)
Quantum-mechanical few-body problems
2.
14
Eq. (2.22) use is made of Eq. (2.15) in the last step. Thus the equality of Eq. (2.22) holds in the case one considers the orbital angular 0 or treats a system of paxticles momentum in the system of 7rjV with equal masses. In atomic physics the trial function of Hylleraas type or correlated In
=
exponential type is often used with success. This function contains the exponential form expressed in terms of the interparticle-distance coordinates. T-nstead of the exponential function let us consider its Gaussian analogue IV
T
=
exp
E
2
aij (ri
-
rj )2
(2.23)
i>i=l
aij determine the falloff of the interparticle functions and may be considered variational parameters. Since not all of the N(N 1)/2 interparticle-distance vectors, ri rj, are Here
N(N
1)/2 parameters
-
-
-
(Irl T2) + (T2 -r3), it is convenient to T3 e.g., TI rewrite T in terms of the independent coordinates x. This is needed to independent,
calculate, e.g., the Eq. (2.23) as T
=
exp
aij
are
-1
(
(N
where the
=
-
-
2 -
related
1)
norm
-
of T.
Equation (2.17) enables
one
to rewrite
iAx), x
(N
-
(2.24)
1) symmetric matrix A
and the parameters
by N
N
-
(ij)
AkI
ajjWk
(ij) WI
=
I:
aij
(W
(ij) W(ij)
) kl'
i>i=l
i>i=l N-I N-1
aij
-
1: 1: UkjAkIU1j k=1
=-
-(CTAU)ij
(i
(2.25)
<
1=1
By substituting Akj of the first relation to the second relation one can check the validity of the second relation. The above two forms,
(2.23)
and
(2.24),
are
thus
equivalent.
We introduce both because in
applications the first form or in some others the second form is more advantageous. See Complements 6.4 and 7. 1. The norm (Tf I Tf) becomes finite if and only if A is positive-definite and is then given by some
((270N-1 /detA)
I 2
-with the
use
of
Eq. (6.32)
or
A necessary and sufficient condition of the a
symmetric
Exercise 6.2.
positive-definiteness
matrix A is that the matrix A is
expressible
as
A
of =
2.3
dDG,
where G is
an
(N
1)
-
x
(N
Symmetrization
1) orthogonal matrix containing is a diagonal matrix, Dij diJij,
-
1) (N 2) /2 parameterg and D including (N 1) positive parameters di. (N
-
15
-
=
-
closing this section, we remark that the matrix Uj of Eq. orthogonal in general. In some cases it is convenient to use a (2.5) coordinate system that is obtained from the single-particle coordinates by an orthogonal matrix. We show such an example in Complement Before
is not
2.2.
Symmetrization
2.3
function of the system must have a proper symmetry for the interchange of identical particles. This is achieved by operating on the The
wave
basis function with for bosons
or
the
suitable operator P, which is the symmetrizer antisymmetrizer for fermions. The operator P can in a
general be given by a combination of the permutations P with suitable 1
phases. The permutation transforms the Pr
=
P
=
(PI
2
'V
P2
PN
single-particle coordinates
as
ri
)
-+
of paxticle indices
rp,:
(2.26)
Tpr,
where the matrix Tp is
(Tp) ij E.g.,
=
(i, j
Jj pi
the matrices
Tp
are
1
0
0
0
1
0
T123
1,
=
...'
given as follows
1
(321
T123
(
The
T123
(2.27) in the
case
of three
0
1
0
1
0
0
0
0
1
0
particles:
213
123
T123'
N).
0
0
0
0
1
0
1
0
1
0
0
1
0
0
1
1
0
0
on
the
1
1
0
0
0
1
0
0
1
0
0
0
0
1
1
0
0
0
1
0
T123 132
T123
(3121
23 1
operation of P
wave
it is written in terms of the
function is thus not
(2.28)
a
single-particle coordinates.
problem
when
Quantum-mechanical few-body problems
2.
16
When the
ordinates,
wave
have to note
we
expressed in terms of the relative cothat the permutation P induces a lineaX
function is
transformation of the relative coordinates Px
=
follows:
as
(2.29)
Tpx,
by using Eqs. (2.4), (2.26) given by
where
(2.27)
and
the matrix Tp is
now
N
(TP)ij
1: Uik(U-1)Pki
(i,j
=
'I
...
I
N
-
1).
(2.30)
k=1
Though
we use
matrix is
which is
now
the
same
notation of
(N 1) x (N 1) -
-
Tp,
note that the size of the
because the center-of-mass
coordinate,
unchanged under the permutation, is eliminated. We Will show
Chap. 6 that the properties of Eqs. (2.26)-(2.30) can be used advantage for Gaussian functions. See Eqs. (6.28) and (6.29).
in
to
2.4 Permutation of the Jacobi coordinates
generate different sets of Jacobi coordinates by interchanging the paxticle labels by permutations. Figure 2.2 shows an example of One
can
three different sets,
x('), X(2)
,
and
X(3),
of the Jacobi coordinates for
three-body system. These sets are completely equivalent from the point of view of the dynamical description of the system, but they can be advantageously used in the calculation of the matrix elements. In the Jacobi coordinate set, defined by Eq. (2.5), the first Jacobi coordinate is equal to I'l -r2. By permutations one can create other Jacobi coordinate sets X(k) where the first Jacobi coordinate takes a
rj. The index k refers to the kth permutation and distinguish the permuted Jacobi coordinate set. As the two-
the form of ri serves
to
-
body potential depends on this coordinate, the set created by the permutation will prove to be useful. See Appendix A.5 for an example of using the permuted Jacobi coordinate set. Another peculiarity of the Jacobi coordinates is that if we are in the center-of-mass system (-xiv 0) then by using Eqs. (2.5) and (2.9) --xiv-11 that is, one of the single-paxticle momenta is equal to p1V the relative momenta. By permutations we can express any of one of the single-paxticle momenta in terms of a certain momentum of =
=
the relative momenta. This
simple expression can be exploited in sevevaluating matrix elements of operators
eral ways, because instead of
2.4 Permutation of the Jacobi coordinates
a-
is
*+-3
Mw-
2
3
Ak
a&-
71
IRW
Ah MW
1
2
3
2
2.2. Different sets of the Jacobi coordinates for the
Fig.
tem. From left to
right: x(l), X(2),
containing single-particle an
application
we can
do
so
See Sect. 7.6 for
versa.
three-particle sys-
x(3), respectively.
and
momentum
relative momentum and vice
17
with operators of more details and
to the calculation of matrix elements for the semirela-
tivistic kinetic energy. The permutation of the Jacobi coordinates is defined
by the corresponding cyclic permutation
of the column vectors of Uj and the
permutation exchange of the
(1, 2, the
...,
N)
=
For
masses.
1
(2
2
example, for
N)
3
the
the transformation matrix between
1
singlt-,particle coordinates
r
and the Jacobi coordinates
x(f)
takes
the form 0
1
-1
0
M2
M3
M23
M23
0 ...
0
U(k)
(2.31) M2
-1
MAF
M3 ...
M23
...
M
M23---.LV
N
M12
M2
MI
M23---N
MN
M3 ...
M12---N
M12
...
...
N
M12
---
N
The relative coordinates in the different Jacobi coordinate sets
expressed by
X(k)
=
each other
T(k) X
through
T(k)
=
can
be
the transformation:
UMU-1. i
(2-32)
symmetrization in Sect. 2.3 and the permutation of the Jacobi are formally very similar. The main difference is that while the symmetrization is assumed for identical (and therefore equal mass) particles, in the latter case we allow for particles with different masses. The
coordinates
Complements
18
Complements 2.1 An
N-particle
Hamiltonian in the
heavy-particle
center coordinate
set
that the
Suppose
mass
of
of the
one
particles,
say the Nth
paxticle,
is the heaviest among all the paxticles. In this case the choice of the heavy-paxticle center coordinate may be natural paxticularly when the
position of the Nth paxticle is close to the center-of-mass of the system, like in an atom, where the Nth particle is a nucleus. The Hamiltonian (2.1) can be reduced to the form which contains no explicit dependence the variables of the Nth
on
is
particle.
By using Eqs. (2.7), (2.10) and (2.11) the Idnetic energy operator expressed in the heavy-paxticle center coordinates as N
N-I N-I
2
1
,
P7
E 2mi
-
T.-
=
EE i=1
i=1
( 2mi k 2mjV) +
-
I-ii .7rj
j=1
N-1
1: i=1
N-1
70z
1 +
2jLjjV
-
TaN
E
(2-33)
Iri .7rj,
j>i=l
'where jLjN mimN/(Tni + Talv) is the reduced mass for the ith and Nth particles. The potential energy VjV (i 1, ..., N 1) is a function =
=
of ri -rN on
xi and
=
the ith
can
be considered the one-body potential Uj
On the other hand the
particle. lighter particles can
N-I
H=E i=1
Potential
energy
function of ri -,rj xi thus be reduced to the type of Eq. (2.2)
between the
HamiltoTdan
-
2AjN
=
a
-
xj. The
N-1
N-I
2
^7r'
is
acting Vij acting
+EUi+ 1:
(Vij+
MN
7ri-7rj).
(2-34)
j>i=i
=1
problem is thus reduced to that of an (N-1)-paxticle system with an external field Uj and an additional separable "two-body potential"
The
oo, jLjjv goes to mi and the 7ri .7rj term The effect of the finite nucleax mass in the helium atom
7ri -7rj. In the limit of m1V -
disappears. was
discussed in
-->
[3].
The above formulation
can
also be
applied
to
a
system of identi-
particles when some, or most, of them form an inert core. As an example let us take up the nuclear shell model, where the nucleons
cal
are
divided into two groups, the passive or inactive nucleons and a small number of active nucleons. The passive nucleons form
relatively
C2.2 Canonical Jacobi coordinates
19
by filling the lowest possible single-particle orbits and exert a potential field Ui on each of the active nucleons. The active nucleons may occupy several single-particle orbits belonging to several major shells outside the inert core. In this approximation it is desiran
inert
core
able to describe the motion of the active nucleons without inclusion
arising from the center-of-mass motion. It is clear can be excluded by taking the full Hamiltonian have the form (2.34) with the core as the heavy particle.
of any excitations
that such excitations to
2.2 Canonical Jacobi coordinates
As
was
2.2, the Jacobi coordinates
noted in Sect.
that the transformation matrix Ui of Eq.
is,
neither
following
UjUi-
nor
relations
UjA-'Uj
Uj-Uj
equal
(2.5)
have the property orthogonal, that
to the unit matrix.
Instead, the
fulfilled:
are
L-1,
=
is
x
is not
UiLUi
=
(2.35)
A,
with
Aij
Jij
Lij
(2.36)
Jij, Ai
Mi
N where pi (i 1, 1) is the reduced mass belonging to the ith Jacobi coordinate and is defined in Eq. (2.19), while 1LV is equal to the =
-
the transformation between
the singleUir and p UT-x, of paxticle and the Jacobi coordinate systems, x identities the lead above relations the to and following (2.9), Eqs. (2.4) total
mass
'MI-2
N.
...
By using
=
N
N
E Tnir?
=
z
N
E Nx
Alternatively,
N
pi2
E
-
Mi
one can
introduce
Ir2i
=
-(2.37)
Ai
"canonical" set of Jacobi
a
coor-
dinates: IV
Vijrj
with
V
=
A
2
UiL 2.
(2.38)
j=1
The square root matrices of the diagonal matrices L and A axe simply given by the square root of the elements. This system of Jacobi
coordinates
Vf"
=
belongs
f7v
and therefore
17
to
an
orthogonal
transformation:
(2.39)
20
Complements 1V
1V
EVji j
IV
E 2 Er2
and
(2.40)
=
i
i*
i=1
j=I
i=1
Let the momentum
Simfla,rly
to the
canonically conjugate to i be denoted qi. transformation of coordinates, we obtain the trans-
formation of momenta
follows:
as
N
71i
=
N
E ViiPi,
A
E Viini
=
j=I
(2.41)
j=I
The total Idnetic energy does not take to q and can be expressed as follows:
diagonal
a
form with respect
I?
E Mi
E E (VLfr) ij?7i -77j wLf7rl.
i=1
i=1
j=I
The two systems of the Jacobi coordinates ical Jacobi coordinates X
=
For
ujf '
a
=
=
M1
rT; 1'2
?T 1 M
M2M3
3
M123
12MI23
VMM1232:
1
2
particles with equal I
1
(2.43)
V
vf2-
I
I
v/6-
T6
matrix V reads
0
12
M12
M123
I
'1
73T
disadvantage
masses
mi
axe
M3
mass
2
V"6_
(2.45)
1
73
of the canonical Jacobi coordinates
is that if the
equal then the center-of-mass motion is not sepais not the center-of-mass coordinate).
not
easily ( N
(2.44)
V-Ml'23
-
v/3-
as
0
-
V/2-
rated
=
V(Uj)-Ix.
VMM1123
The
can
-VEM12
L2 M12
and for
Vr
Ujr and the canoneasily be related by x
three-paxticle system the transformation
V
(2.42)
=
3. Introduction to variational methods
The variational method is
popular approaches to tackle quantum-mechanical few-body problems. Though it gives only an approximate solution except for some special cases (the Ritz variational method, for example, gives only an upper bound of the energy), one can get a virtually exact solution with an appropriately chosen function space. The function space is defined by basis states and the wave function of the system is expanded in that, basis. In this chapter we briefly introduce the theorems requisite for obtaining a vaxiational one
of the most
solution.
3.1 Variational Let
principles
physical system whose Hamiltonian H is self-adjoint (Hermitian), bounded from below and time-independent. We are interested in finding the discrete eigenvalues of H and its corresponding (normalized) eigenstates: us
consider
H!P,, The
=
a
En(fi,
energies En
n
axe
=
(3.1)
1, 2,...
real and
are
ordered such that El :! E2<
....
We
that the ground state is non-degenerate. Although H is known, this does not mean that E,, and (fin axe known. In general, it is difficult to solve the eigenvalue equation (3. 1). When we do not know how to diagonalize H. exactly, the vaxiational assume
method becomes useful for any
Theorem 3.1 the state space the is such that E
=
RIHITI) R_ I TI)
type of Hamiltonian.
(Ritz Theorem). For an arbitrary function Tf of expectation value (Mean value) of H in the state Tf
> -
El,
Y. Suzuki and K. Varga: LNPm 54, pp. 21 - 37, 1998 © Springer-Verlag Berlin Heidelberg 1998
(3.2)
3. Introduction to variational methods
22
where the
equality
if and only if Tf
holds
is
an
eigenstate of H
with the
eigenvalue El. Proof.
elementary and can be found in of quantum mechanics. See for example [1, 2]. The ftmetion expanded in terms of the energy eigenstates
The
textbooks T1 may be
proof
of this theorem is
00
(3-3)
ai(fii.
In this
cluded and the
integration
over
(Tf jHn Itp) (TrI Tf )
the
with continuous
eigenvalues are sum must be extended appropriately to include them. Then we can show that for an integer n
expansion
eigenstates
0
Faci=2 A Eln)Jai 12 00, jai 12 Ei= n Y
inan
7
-
En I
-
-
(3-4)
I Clearly the right-hand side of Eq. (3.4) is non-negative for n and if for > if i It vanishes 2. > 0 because Ei 0 for E, only ai i > 2, that is, T, is an edgenstate of H with eigenvalue El. This proves I was used to prove the Ritz the Ritz theorem. Only the case of n theorem but other cases will be needed later to derive Temple's bound. =
=
-
=
approximate deexploited as ground-state energy El, namely the minimization principle of the mean value of H. Suppose that we choose a family of functions Tf (a) which are characterized by a finite number of parameters denoted a. In the case of Eq. (1.1), there axe two parameters a and P. We calculate the mean value E(a) of the Hamiltonian H in this trial function, and minimize E(a) with respect to a. The minimal value obtained in this way is an approximation to tile ground-state energy El of the system. Clearly, whether this variational method produces a satisfactory result or not substantially depends on the choice of the trial functions Tf (a). The Ritz theorem can be generalized to excited states as well: The Ritz theorem is the basis for
a
method of
it is
termination of the
Theorem 3.2
(Generalizecl
of the Hamiltonian H discrete eigenvalues.
value
Proof.
Let
E(TfIT)
=
us
(TfjHjTf)
sumed to be linear term
calculate
an
is
Theorem).
stationary
in the
increment JE of the
The
expectation neighborhood of its
mean
value us' g
+JT-1, where JTf is aschanged of higher order than a terms Neglecting
when Tf is
infinitely small. in JT1, we obtain
Ritz
to Tf
3.1 Variational
(TfITf)JE
=
=
The
(JTf IH
value E is
mean
that
J((TIIHITI)) -
-
23
principIcs
EJ((TIITI))
EITI)
+
(TfIH
stationary if JE
(3.5)
EIJTf).
-
=
0 for any infinitesimal
JT,
if
is,
(,NTtIH
EITI)
-
+
(TIIH
If JTf is chosen to be
the above
number,
-
-(H
EIST)
-
E)Tf,
(3.6)
0.
=
where
-
is
that the
equation implies
an
infinitesimal real
norm
of the function
(H E)Tf is zero, and thus the function (H E)Tf itself must be a null function, namely HTf = ET. Therefore the mean value E is -
-
stationary if and only if the state Tf from which E is calculated is an eigenstate of H, and the stationary value E is the corresponding
eigenvalue
of the Hamiltonian.
generalized Ritz theorem allows an approximate determination of the eigenvalues of the Hamiltonian. If the E(a) has several extrema, they are the approximate values of some of the energies E,,. In most cases for practical applications the. trial function is given as a linear combination of a finite number of independent functions This
!P(a): K
(3.7)
ciTf (ai).
Tf
independence of the functions will be discussed soon later. T' (a K) axe mutually orthogWe do not always assume that Tf (a I) onal because the use of nonorthogonal. functions is in fact quite useful. They can, however, always be made orthogonal if necessary, e.g. by a Gratn-Schmidt orthogonalization procedure. The variational method then reduces to the eigenvalue problem of the Hamiltonian inside the state space VK spanned by the set JTf((YI),...,Tf(aK)Ji that is, the space containing all linear combinations of T (a,), Tf(aK). The mean value E is given by The linear
,
...
i
...
Ctlic E
(3.8)
=
CtBC' where and
-
c
is
a
IC-dimensional column vector whose ith element is
ct is the Hermitian conjugate of c. The K
overlap
matrices W and B
are
defined
by
x
ci
K Hamiltonian and
3. Introduction to variational methods
24
Rij
=
(Tf(ai)JHJT'(cvj)),
Bij
(TI ((Yi) I Tf (aj)).
=
(3.9)
The linear parameter ci can be determined by the generalized Ritz theorem. The condition that E is stationary with respect to an ar-
bitrary, infinitesimal change problem
of ci leads to the
generalized eigenvalue
K
Wc
E&,
=
E(Rij
i.e.,
-
EBij)cj
=
0
(i
=
I,-, K).
(3.10)
j=J
The restriction of the thus
can
eigenvalue problem
of H to the
subspace VK
the solution.
simplify
We discuss the linear Tf (a K). The linear c
except for
c
0
=
can
function. In other
and
only
for
has
E
c
K
if c
independence of the functions Tf(al),..., independence of the functions means that no vector make
words,
0. This is
=
a
linear combination ,
EK i=1 ciTf(ai),
the combination becomes
equivalent
unique solution of c ci!F(ai) 0 is equivalent a
=
=
0.
a
a
null
null function if
that the
equation Bc 0 (To understand this, we show that
to
to Bc
saying =
0. Whe'n
=
EK ci!P(ai)
=
0,
0 for j obviously have Ef I ci (TI(aj) JTf (ai)) I,-, K, which but for 0 nothing j I,-, K. Conversely, when (Bc)j (&)i 0 for j I,-, K, by multiplying cj* (the complex conjugate of cj) and we
=
is
=
=
=
=
=
summing
over
j,
we
obtain ctL3c
=(.EK CiT,(Cv,) I EK I
K
which leads
us
solution
0 for the
Fj-1 ciTf (ai)
1
ciTf (ai))
=
0,
As the existence of
a unique 0.) equation Bc 0 is possible only when detB --A 0, the linear independence of the functions is assured by the condition detB :A 0. Because the overlap matrix B is at least positivectBc > 0 for any vector c, all eigenvalues semidefinite, that is (Tf JTf)
c
=
to
=
=
=
p of B become
Tf (a K) ar e real, positive when the functions Tf (a,), linearly independent. The positiveness of M is understood as follows: For the eigenvector c :A 0 corresponding to the eigenvalue JL, we have the relation pctc As the basis functions Tf are (Tf Ifl. linearly (ai) independent, Tf is not identically zero because otherwise we can make !rf EK ciTf (ai) vanish identically with c =A 0, which contradicts the i= I assumption of the linear independence of the basis functions. Thus (TfITf) is positive and of course ctc is positive, so the eigenvalue y has to be positive. We can also state that if there exists a vanisl-iing eigenvalue of B then the basis set JTf(a1),..-,Tf(aK) is linearly dependent. The solution of the eigenvalue problem, pc(A), gives us an ....
=
=
BcG l
orthonormal set:
=
3.1 Variational
principles
1: C(ii.') Tf (C'j),
01.1
25
(3-11)
2
i=l
where
c(l')
c(") tc(/")
is assumed to be normalized to
=
1. In many
practical problems it may happen that B has one or several very small eigenvalues. Then the eigenstate corresponding to the small eigenvalue has very large expansion coefficients cil") If this occurs then a - Flj,. small error in the matrix elements of R or B can lead to a larger error in the solution of Eq. (110). When the ill condition mentioned above does not occur, the generalized eigenvalue problem (3.10) can be solved safely. The eigenvalues q
(i
est
1,
=
...'
K)
eigenvalue
are
arranged be
El may
a
in
increasing order el :! good approximation to
62
:
the
The low-
...
ground-state
energy El if the state space V_T<- is chosen to include the physically most important configurations. Two functions T and TI' belonging to
the
and ej,
eigenvaluesq
(Tf I Tfl) where
CtBC'
-md c'
c
Rc
=
=
have the
overlap
(3.12)
1
satisfy
EiBc,
respectively,
the
and
following equation
'Hc
=,EjBc'.
(3.13)
I and likewise T', c has to be normalized to ctBc &t&' 1. Using the Hermiticity of R and B and the reality of the 0. Thus eigenvalues Ej in Eq. (3.13) leads one to (Ej cj)ctBc' the two eigenstates belonging to different eigenvalues Ei and ej are orthogonal. In the case of degeneracy, they can be made orthogonal by an appropriate procedure. The relationship between the eigenvaluesEi of the truncated problem and the eigenvalues E,, of the full Hamiltonian is elucidated by
To normalize
=
=
=
-
the Nfini-Max theorem. Theorem 3.3
(Mini-Max Theorem).
operator with discrete eigenvalues El :5 E2
eigenvalues of H restricted to independent set of K functions Tf (a,),
EK be the
the
...
E2:51E2i
El <'Eli
...
i
EK :
I
<
Let H be ....
a
Hermitian
Let El ! e2 <_
subspace Vrc of
Tf(aK).
a
-
<
linearly
Then
16K-
(3.14)
Proof. Let WK be the subspace spanned by the orthonormal eigenstates!Pl,(k I!PK of the operator H. We will first show that there ...
is at least
one
normalized function in Vrf with the property
3. Introduction to vaxiational methods
26
(TfJHJTf) To show this
p
WK;
(3.15)
consider the projected function PTf for any normalized VK, where P is the projection operator that projects
we
function Tf in onto
EK.
>
=,EK I!Pi) (!Pi 1. i= I
There
are
two
(i) there exists a function Tfo in Vr<- such (ii) PTf :A 0 for all functions T, in Vl<-. In
case
(i) Tro
is
a
(ii)
that
PTfo
=
0.
linear combination Of 43K+1,
the normalized Tfo has the case
possibilities:
T)lf+2,..., and hence (Trollll%) : : EK+, ! EK. In fimetions Tf, and Tf2 in Vl<- are
value
mean
any two different normalized
projected to different functions PTfl and PTf2 in _PVK because otherwise P(TV, Tf2) 0, which contradicts the assumption. Namely, any two different normalized functions in V.[<- axe projected by P to different functions in Wl,(. As both VK and )IVI<- have the same dhnension, it follows that there must exist a normalized function T, in V.Tf with the wPrf (a 4_ 0). Then !Tf can be expressed as a!PK + bo, property M =
-
=
where
0 is a PK+1,!PK+2,
(TfJHJTf)
=
normalized flinction
Next
2!
we
;
a
linear combination of
=
(T'IHIT')
largest eigenvalue >
of H restricted to
EK)
V_Tf,
Ell;C.
we
EK.
Vif which
by induction
It is easy to derive the
have
(3.16)
(K- I)-dimensional state space Vr.C-1
of all those functions in to 45K and
value,,.-
mean
_
define the
belonging
as
1. This function has the JaJ2 + JbJ2 Ja 12EK + lb 12(01H10) Erc + lb 12((01H10) ...
Because 16K is the IEK
given
are
can
following
orthogonal
show 'EK-1
to be the set
to the
eigenvector
! EK-1 and
so on.
theorem ft-om. the Nfini-Max the-
orem.
Theorem 3.4. Let E,
of
the lowest
! , E2
<
EK be
a
number K
Hermitian operator H. Let
eigenvalues of be a set Of linearly independent functions. a
Tl(a2)i..., Tf (aK)
Tf(al),
Then
K
1: Ei :! Tr(B`R),
(3-17)
i=1
where the matrices B and R
Proof. to
the
Let el :
162
subspace Vl<-
:5
---
:: -
are
defined by Eq. (3.9).
IEK be the
of H restricted
eigenvalues
of the K functions
Tf(Cfl),
...
i
Tf (Ci K) and let
3.1 Vi-triational
be the orthonormal.
eigenvalueS
6 11
E21
...
16K. Then
i
Oi
is
principles
27
eigenstates corresponding given by
to the
K
E UijT1(aj)
(3.18)
j=1
with U orein
the
satisfying
3.3
orthonormality condition UtBU
K,
K
J:Ej
=
1. From The-
obtain
we
K
EEi=E(OilHloi)=Tr(UtHU)=Tr((BU)-IHU)
:5
Tr(U-'L3-1WU)
=
Tr(B-1R).
(3-19)
right-hand side of Eq. (3-17) is determined only by the subspace spanned by the functions and not by the particular choice of the basis because any non-singular linear transformation of the basis does not change the trace (3.17). Namely, when a new basis It is noted that the
set Xl,...,XK is
K)
with
a
FC
given by a transformation Xi Ei= J Tij Tf ((--ij) (i T matrix non-singular (deff :A 0), we can show that =
Tr(B-'?i),
where
L3i'j
-
21
(XilXi)
and
R'ijtj
=
(XjjHjXj).
formalism where the
This leads to the
density can density-functional calculating the energies of excited states as well as the ground state. See [4, 5] for this interesting idea. The next theorem is fundamental to answer a question of how the eigenvalues obtained in a restricted subspace change as the basis be used
as a
basic vaxiable for
dimension is increased. Theorem 3.5. Let el ! - 162 mitian operator H restricted to the
of independent functions EI
K+1
be the
subspace VK of linear combinations
Tf (a,), If (a2),
...
i
Tf(01K).
of H restricted to the
eigenvalues of independent functions
combinations
Let El <
e2
:5
subspace VTC+l of
Tf (01)i llf (a2),
...
7
...
<
linear
Tf (aK), and
Then
!P(OK+I).
Ell
:! 'EK be Hie eigenvalues of a Her-
< El
:5
E12
:-5
C:2
:5
...
:5 '61K :-5
61K,(+i*
(3.20)
OK be the orthonormal eigenstates correspondProof. Let 011 02 16K- Any function !P in VK+j can be ing to -the eigenvalues 61,162 expressed as 1
...
i
K+1
Tf
CA,
(3.21)
28
1 Introdnetion to variational methods
where the normalized function inake, it.
orthogonal
01-<-+,
is constructed from
to any function in
Tf(aK+l)
Vfc: K
(Tf (a
OIC+I
to
2
1 (0i I Tf (Ct 1,;C+l 12
Tf (a 1,C+ 1))
K
(3.22)
x
The the
eigenvalue problem using simple form 61
0
0
62
-
0
-
0
the basis set
)
h, h2
takes
)
C,
)
C1
C2
C2
=E 0
0
-
h *1
h;2
...
EK
hK
CK
h I*c K
hrc+1
CTC+I
CK
CJf+
(3.23) where
hj
=
%JHJOrc+I) and hj* is the complex conjugate of 11j. The D(E) to determine the eigenvalues reads as
characteristic function
K
D(E)
(hK+1
-
K
E) II(Ei
-
E)
K
II(Ei
-
E)
hIc+i
i=1
(i
Ih-I2 EJ
_
E)
0.
E
(3.24)
I,-, K) axe nonzero. The (K + 1) obtained by finding the roots of the equation K
E-hK+I
E
-
j=I
Here it is assumed that all hi are
-
11 (Ej
Ihi 12
-
K
eigenvalues
K
=
Ihi 12 E
-
(3.25)
ei
The right-hand side of the above
equation becomes negative for E
< C,
and in
positive for E > CK. A graphic solution of Eq. (3.25) is displayed Fig. 3. 1. It is clear that the cross points of curves, namely the new
eigenvalues E , satisfy the relation (3.20). When there are n vanishing hi's, the corresponding ei's become the solutions of Eq. (3.23), i.e., E'i Ei. The remaining (K + I n) new ==
-
3.1 Variational
1N
29
principles
2
'=1E-Cj .................
.......I.. ............V..
.........
....
F-21
611'
.....
...
...........
..................
I 'EK
E-hK+1
Fig.
3.1.
Graphic
solution of
Eq. (3.25)
to obtain the
eigenvalues
of
Eq.
(3.23)
eigenvalues the
same
are
structure
also in this
through the eigenvalue. equation, which has Eq. (3.23). The relation (3.20) trivially holds
obtained as
case.
The significance of this theorem is
that, by including a further term in the basis, the K lowest eigenvalues; cannot become worse. Therefore, this theorem implies the Mini-Max theorem because, in the limiting case that the subspace approaches the full Hilbert space as the basis dimension increases, the K lowest-eigenvaluesEj converge to the exact eigenvalues; of the Hamiltonian. Suppose that one wants to see how the lowest eigenvalue changes when one vaxies only the (K+l)th function-!V(arc+j) while keeping the rest of all. the K functions unchanged. In this case one does not need to solve the generalized eigenvalue problem of type (3. 10) but only needs to find the smallest root of Eq. (3.25). -Of course the latter is much simpler and faster than diagonalizing the matrix. This advantage is used to sample more random trials in selecting a suitable basis function in the stochastic variational method, as explained in Sects. 4.2.5 and 4.2.6.
30
3. Introduction to variational methods
3.2 The variance of local energy It would be nice if there
method to
judge the accuracy of variationally. The expectation value E is the upper bound of the ground-state energy El. If we can calculate a lower boun,d, the. difference between tile, two bounds gives an estimate for how dose E may be to El. This makes sense if and only if the lower bound can be calculated as closely as possible to the ground-state energy. In order to discuss the lower bound, we define the vaxiance of the energy expectation value, o-', by the norm of the residue function were a
the solution obtained
(H
E)TfI(Tf JTf) '21
-
2
(T1J(H-E)2JTf) (TIITf)
-
0'
According
to Weinstein
(TfJH2JTf) R- I Tf
[6, 71
the
following
Theorem 3.6. There ii at least val
[E
theorem
exact
one
(3.26) can
eigenvalue
be
proved.
in the inter-
o-J.
o-, E +
-
E2
-
Proof. By using Eq. (3.3),
the variance
can
be
expressed
as
EMI (Ei E)2 JaiJ2 Fli= 00, JaiJ2 -
2 0'
_
Suppose that Ek
is the
(3.27)
eigenvalue which is
00
i
we
we
have
CO
J:(E Thus
closest to E. Then
2
_E
jai 12
have o-2 >
>
(Ek
(Ek -
-
E)2
E)',
jai 12.
(3.28)
which proves the theorem.
The Weinstein criterion guarantees that there is
an
eigenstate
whose energy is in the interval [E G-, E + o-] but does not indicate which one. In case E is sufficiently dose to the ground-state energy, -
the theorem E-
G-
<
gives
the lower bound
as
El.
(3.29)
Another lower bound called
Temple's
in terms of the variance of the energy
bound
[8]
is also
expressed
as
0'
E
<
-
e-E
-
El.
(3.30)
31
3.2 The variance of local energy
is the energy such that E < e < E2, where E2 conserved quantum energy of the first excited state which has the same numbers as the ground state. The best bound is clearly obtained by Here
-
setting
is
an
e
equal
arbitrary
to
E2. Temple's bound is easily derived
as
follows:
0.2
(E- --E )
E,
1 =
-
e
-
I
E
[
-
(,- + Ej)(E
-
Ej)
+
(TI I H2 I TI) (Tf I Tf)
EZ i=2 (Ej-,-)(Ej-Ej)jajj2 E Ei= 1 jai12
>
00
where
use
Eq. (3.4)
is made of
for E
(TfIH2lTf)l (Tflyf-) _EJ2 (with n 2). The two bounds, (3.29) and (3.30),
-
(3.31)
0,
(with
El
2]
Elf
n
=
1)
and for
=
indicate that the trial function
2
gives the smallest value of (7 is best among various trial functions giving the same expectation value. Temple's bound often gives better that
lower bounds than the Weinstein bound. See
[91
for the extension of
variational bounds. The variance of the
expectation
value
can
also be defined
through
the local energy HTf
Ej., (R) ,
!P7
(3-32)
I
where R stands for the
"configuration?'
of the
particles.
More precisely,
by ri and pi, where pi particle stands for the coordinates other than the position coordinate ri, then R stands for Ir I i P1 ilr2) P2 i 1. The variance defined by Eq. (3.26) is equal to the variance of the local energy if the coordinates of the ith
are
denoted
...
(TfjjEjo,(R)
2 0'
-
E121!Tf)
=
P(R)IEI.,:(R)
-
Ej2dR (3-33)
with
P(R) where as an
E
--
I Tf (R) 12 R I Tf)
(3-34)
energy of H in the state local energy: of the average
E, the
=
mean
(TIjEjo,(R)jT1) 010f)
-
Tf,
I P(R)Ejoc (R)dk
can
also be rewritten
(3.35)
3. Introduction to variational methods
32
Here
P(R),
uration
the
probability density of finding the system at the configpoint R, is non-negative and satisfies the. relation f P(R)dR
1.
H the trial function Tf is the exact
ground state of the Hamiltonian,
the local energy becomes R-independent and equals the ground-state energy El. A similax statement holds for excited states as well. If Tf is
the exact to
eigenstate
E,,. This
with
opens up
energy-&, the. local energy would be equal another possibility of the vaxiational method
instead of the minimization of the
mean
energy: The niinimization of
the variance of the local energy [101. The adjustable parameters of the trial function may be varied to minimize the variance. The calculation of the variance is in general much more difficult than that of the mean energy. An advantage of minimizing the variance of the local energy is, however, that the quantity to be minimized has a known lower bound,
Since in any eigenstate the vaxiance of the local energy is zero, this minimization principle applies to obtain excited states as well as the ground state.
namely
zero.
The
condition for o-' is obtained
stationarity
increment So-' in the
Theorem 3.2.
same.
way
By multiplying
calculating the increment following equation
to
(Tf Jqf-)jO.2 =,6((Tf JH2 Ifl) =
(,5qf- I H2
_
=
(JTf JH2
-
(Tf JH2
+
Here
-
E2
_
2EH + E2
mean
energy in
-
2 _
Eq. (3.26) by (Tflfl and change in JTf, we obtain the
linear
a
_
done for the
as was
an
both sides of
J(E2 (Tf IT,))
0.2 1 Tf) +
2EH + E
by calculating
(Tf I H2
-
_
E2
0.2j((T1JTf))
_
0.2 1 jqf-)
-
2E(Tf JTf) JE
0.2 1 TI)
0.21,5fl.
(3.36)
is made of
Eq. (3.5) in the last step. The stationarity condiequivalent requiring (H2 2EH + E2)(y 0.2TI. When the trial function is given as a sum of a finite number of functions Tf (a) as in Eq. (3.7), this condition reads as the following equation which use
tion is
can
to
=
-
be used to determine the linear parameters ci
(Q
-
2EW +
E2L3)C
=
0.2BC7
Qij
=
(!rf(a,) 1112 1 Tf (Cj)).
(3.37)
Since E defined
by Eq. (3.8) depends on c, the matrix in the round equation also depends on c. Therefore the parameter c must be determined self-consistently: One assumes an initial value for c, calculates the energy-R, and solves Eq. (3.37) to deterbracket of the above
mine
c
and 0-2. As the value
c
obtained in this way may in
general
3.3 The virkd theorem
not be
equal
to the initial
c
value,
one now uses
the solution
33
c as
the
initial value and repeats this cycle until both values of c become the same. The minimization of o-' becomes cumbersome even though the calculation of
Qij
is feasible.
The variational Monte Carlo
(3.35)
to determine the
mean
method of the
(VMC)
calculations
employ Eq.
energy. Tlie VMC method [111, or the of Two-body correlations into Multiple
Amalgamation Scattering (ATMS) [121 similar to it, usually chooses a trial function that attempts to incorporate both the sfi7ort distance behavior and the asymptotic behavior of the correct wave function, leading to complicated functional forms. Thus the analytic evaluation of the matrix elements is in general hopeless. Instead, the VMC requires only the evaluation of the wave function and its Laplacian to determine the local energy. If the interaction between the particles contains the spinorbit force, the first derivatives of tl-ie wave function are required as well. These derivatives are obtained numerically by appropriate difference formulas. The mean energy is then estimated by a Monte Carlo
integration. The sampling of a set of configuration points R is made by, e.g. the Metropolis algoritl-nn [131. The accuracy of the VMC or ATMS calculation is limited by the choice of the trial function as well as by the statistical error inherent in the Monte Carlo integration. To reduce the statistical error one has to sample a great number of points, which is computer time consuming. To optimize the variational parameters of the trial
wave
repeat the mean enof parameters. This is a hard job
function
one
ergy calculation using a different set considering that the VMC or ATMS has
particularly
in the
case one
has
The variational method is
which
can
be
adapted
has to
an
inherent statistical
error
number of parameters. very flexible approximation method
a
a
to diverse -problems. The variational method
physical intuitions lead us to an idea of particularly the qualitative form of the solution. It easily gives good values for the energy. The variational solution may, however, present unpredictable erroneous values for other physical observables. It is unfortunately valuable when
is
very difficult to evaluate their
error.
3.3 The virial theoren, summarize two theorems, the Hellmann-Feyntnan theorem and the virial theorem, which are valid for the exact solution.
In this section
They
can
we
be used to check the
quality
of the
wave
function.
3. Introduction to variational methods
34
Theorem 3.7
(Hellmann-Feynman Theorem).
Let
H(A)
be
a
Hermitian operator which depends on a real parameter A, and POO a normalized eigenstate of H(A) of eigenvalue E(A). Then the theorem states that
d
TA E(A)
=
(3-38)
( P(A)J DA
Proof. Differentiating,
with respect to
A,
the relation
(3.39)
E(A) we
have
a
d
dA
E(A)
=
( P(A)J
(9A
H(APKA)) 19
(9 +
OX The Iast two terms
(fi(A) IHN ON) on
the
+
R(A) W(A) 1
(3.40)
A
right-hand side. of Eq. (3.40)
vanish because
a
(9
PNIH(A)O(A)) E(A)
+
OPNIHNI
J-P(A))
IN
+
CA))
(!P(A) I OA jA-
P(A))]
d =
where
use
E(A) d,X ONON)
=
(3.41)
01
is made of the conditions that
H(A)!P(A)
When the evaluation of the matrix element of the
theorem
E(A)!P(A)
H(A)
and
does not
be used to
Oficulty, Hellmann-Feynman consistency of the vaxiational solution. If H(X) Ho +,XHI, then the matrix elements needed are just the terms of the energy expectation value. It is straightforward to generalize the 11611mannFeynman theorem to the case that the Hamiltonian contains a number of parameters AI A2 We note that the 11ellmann-Feymnan theorem can be stated in an integral form pose any check the
=
7
E(A2)
can
-
E(Al)
=
(!P(A2)JJff(A2) (!P(A2)J!P(A1))
3.3 The viri.-,U theorem
(3-42)
OA
"1
35
equality is trivial for the exact solution and has nothing to do with the I-Iellmann-Feym-nan theorem. The equality holds, however, only approximately for the variational solution and may be used to check the quality of the solution. The following theorem plays a key role in the derivation of the The first
virial theorem. Theorem 3.8. Let (P be
For any operator A
(T)I [H, A] I!P)
=
we
eigenstate of a Hermitian operator
an
obtain
(3.43)
0.
proof is easily obtained by noting
The
H!P
E. To test the accuracy of the vaxiational special operator of the form N
N
.
A=
=
ET, with
solution,
one
eigenvalue
often
uses a
(9 ri
ri-pi
h
H.
(3.44)
-
r. 'ri
i=1
because the commutator
[H, A]
is
easily calculated. We
operator A is a generator of the dilation operator, property, in a single-variable case,
e
aA,
d
exp
(ax dx ) f (x)
=
note that the
which has the
(3.45)
f (eax).
by expanding f (x) in power series and using for any integer value of n. valid nkXn, dX Assume that the Hamiltonian for an N-particle system takes the
This is
easily
(X_A_)kxn
shown
=
form N
H
Whether
or
Pi
E=I2mi
T+ W
=
-
T,,
.
+
W(ri, r2,
can
potential
W is assumed to contain
no
C7 ri-
Ori
P2(-2mi
-Pi'IrN),
'n Ar M12 --N ..
on
(3.46)
the virial theorem.
momentum
then show that
IT,
rN),
not the center-of-mass kinetic energy is subtracted from
the Hamiltonian is irrelevant to the discussion
The
.--,
operators. We
3. Introduction to variational methods
36
1W,
0
ri.
where M12
...
OW
1 c9ri N=
'=
(3.47)
-ri-
ari
EN
mi
is the total
EN
is the total momentum. ..,i= I Pi we obtain
of the system and 7rN
mass
Using Eqs. (3-44), (3.46)
(3.47)
and
N
aw
[H, Al
=
2T
-
with
WA
WA
(3.48) ri
i=1
Substituting
this into
Theorem 3.9 the Hamiltonian
2(!PjTjfl Therefore
77
a
-
(The
(3.46).
virial
Theorem).
following
Let!P be
theorem.
an
eigenstate of
of Eq. (3-44)we
obtain
0.
=
z7 defined
quantity
leads to the
For the operator A
((fijWAj(fi)
(!PIWAI'P) 2(!PjTj!P)
=
Eq. (3.43)
(3.49) by
'
1
(3-50)
vanishes for the exact solution and
be used to check the
can
quality
of the solution.
One has to know WA to make use of the virial theorem. It becomes particularly simple if the potential is a homogeneous function of degree s,
namely
W(Ar I Ar2
ArN)
i
=
i
=
because then
ating, to X
=
A*W(rjr2,
...
I
7 ....
ATNi AM AZN)
rN)7
(3.51)
obtain
with respect to
by the use of Euler's theorem or by differentiX, both sides of Eq. (3-51) and setting A equal
I
WA In this
we
W(AX1 AY1 AZI
=
SW-
special
(3.52) case
Eq. (3.49), together
yields the well-known
with
(!PjTj!P) + ((fijWjfl
=
Ej
relation 2
((!PITI! P) s
Because the
tial
(s
=
+2
E,
PIWI(P)
E
=
s+2
(s :A -2).
(3.53)
homogeneity condition is fulfilled for -the Coulomb potenthe quantity t7 is conveniently used to check the quality
-1),
3.3 The virial theorem
of the solution for
37
system of particles interacting via Coulomb
a
po-
tentials. For
a
general non-homogeneous potential IV
N
W(rwr2,
Ui(ri)
'rIV)
...
must be
E
+
i=t
Eq. (3.52)
Vij(ri
(3.54)
rj),
replaced by
Oui(r)
WA
-
i>i=l
N
IV
function of the form
r-
Or
)
OV,ij (r)
(r
+ r=ri
Or
)r=ri-rj (3-55)
spherically symmetric potential the operator r-ar- reduces simply r-4-. If the calculation of matrix elements of the. operator WA is not dr
For to
a
difficult,
the virial theorem
When the
potential
W
can
be used
depends
as
on a
parameter A,
(9
(!PIWAI(P) Here the
=
((filWA
+A
in the Coulombic we
case.
obtain
d
WI-P)
-
A
dA
(3.56)
E(A).
theorem is used to express the expecW in terms of the energy eigenvalue E(A). In a
Hellmann-Feynman
tation value of
A'&
molecular system X may be a set of parameters which stand for the positions of nuclei and if W consists of only the Coulomb potentials
-W simply reduces WA + A-bX plication
of this relation.
to -W. See
Complement
8.2 for
an
ap-
4. Stochastic variational method
approach to the variational solution of quantumproblems is to diagonalize the Hamiltonian in a state space spanned by some appropriate functions TV (a-,) Tf (a2) Tf (CeK). The applicability of this approach is, however, very limited, because the diagonalization may not be feasible if the dimension K of the state space is very large. This is typical in many-particle problems
The most direct
mechanical bound-state
7 ...I
I
such
as
the Hubbard model
or
the nuclear shell model. This method
approach", because one sets up a basis in a well-defined way, e.g., by using a complete set of states that contains no parameters, and then obtains the energy by a diagonalization. Another possibility is basis optimization, which is actually designed to avoid the problem of the huge basis dimension in the direct approach. In this case one specifically selects the basis states that are really essential to get the energy and the wave function of the syscan
be called
a
"direct
certain accuracy. It is obvious that this selection may be state-dependent: Some functions might be adequate to describe the
tem to
a
others would be
appropriate to approximate an excited state, particularly when the ground state and the excited state have different spatial extensions. The stochastic variational method uses this second route by selecting the most appropriate basis functions in a trial and error procedure. but
ground state,
4.1 Basis
some
more
optimization
It goes without saying that the quality of the variational approximation crucially depends on the choice of the basis functions. Our
primary aim here is-
choose basis functions 1.
They
can
be
IV-particle problems. For this that meet the following requirements:
to solve
easily generalized
for
Y. Suzuki and K. Varga: LNPm 54, pp. 39 - 63, 1998 © Springer-Verlag Berlin Heidelberg 1998
an
N-body system.
we
will
4. Stochastic variational method
40
2. Their matrix elements
3.
They
are
analytically calculable. easily adaptable to the permutational symmetry are
of the
system. 4.
They
are
flexible
enough
to
approximate
even
rapidly changing
functions.
important condition (3) is non-trivial because the permutational symmetry looks very complicated when expressed in terms of the relThe
ative coordinates to be used.
A
possible
choice for the basis functions
tions is the correlated Gaussian of
fillfilling the
above condi-
Eqs. (2.23)-(2.25):
-1 N-1
1
c
exp
Ax)
NE E Aij
exp
i=1
xi
-
xj
j=1
IV
expf -2
E
aij (ri
-
(4.1)
rj
i>i=l
equivalently, aij are nonlinear paxameters are spherically symmetric. To take into account non-spherical states, the basis function has to be multiplied by an appropriate orbital angulax function. In addition to the spatial degree of freedom, the particles may have other degrees of freedom, such as spin and flavor, and thus the function has also to be multiplied by suitable trial functions in these additional spaces. These functions may bring other parameters or sets of quantum numbers. These sets of quantum numbers (e.g., total and intermediate spins, orbital angular The matrix elements
Aij
or,
of the basis. These functions
momenta,
etc.)
be considered
can
often be referred to
as
trial function will be
as
channels in the
given
in
discrete parameters, and will following. (More details on the
Chap. 6.)
The actual form of the basis function is not very important at this stage. The above discussion serves to draw the reader's attention to the fact that the basis function
depends
on
many
and discrete parameters. The parameters define the
linear, nonlinear shape of the basis
function and determine how well the variational function space contains the true eigenfunction. To find the best possible solution, one has to
optimize the paxameters. To have a crude guess of how much optimization amounts to, let us consider an N-particle syswith the simplest basis function (4.1). This correlated Gaussian
work the tem
has ea,r
N(N
-
1)/2 parameters, and, by assuming
combination of K
functions,
-we
face
an
that
we
need
a
lin-
optimization problem of
4.1 Basis
K(N(N 1) /2) -
the number of
41
paxameters. Tb is number increases quadratically with
particles. By taking
N
=
4 and K
end up with 1200 parameters. The main problem of the minimization of
case,
optimization
=
200
as a
typical
we
nipresence of local minima. A local function reaches
a
minimum in
a
There
function is the
om-
point where the
finite interval of variables and the
number of such minima tends to increase
of the
a
minimum is the
exponentially
with the size
are plenty problem. optia function, and these optimizations can be divided into two categories: the deterministic and the stochastic optimizations. A deterministic optimization moves downwards on the slope of the function according to a certain well-defined strategy. There exist many elaborated algorithms (conjugate gradient, Powell (direction set), etc. [141) and they are deterministic in the sense that, starting from a given point, they always reach the same (local or global) minimum. The drawback of these techniques is that they are time consuming and tend to converge to whichever local minimum they first encounter. The solution in these cases may not be the global minimum but a local minimum. These methods are sensitive to the starting point and are
of different methods for the
mization of
unable to search further for
a
better solution after
a
local minimum is
reached.
Stochastic optimizations address the problem of finding the global large number of undesired local minima
minimum in the presence of a by making random steps [15,
161.
One can, for
example, start deterstarting points and then pick up the minimum of these. Numerous strategies have been developed in the last few years, e.g., simulated annealing [171, genetic algorithm [18] etc. The simplest (and actually not very economical) stochastic optimization is a random search where one picks up random points and tries to find the minimum. This may not sound very sophisticated, but the optimization with a random trial and error procedure seems to be the most efficient one. In some cases it would be nearly impossible to apply other strategies than simple random trials. To reduce the load of optimization, an alternative is to shift the burden of minimization from a large number of parameters to a smaller number of more sensitive, "tempering", parameters. Several such possibilities have been explored, e.g., geometric progression [19, 20], random tempering [21, 22, 231, and Chebyshev grid [241. The common property of these methods is that they use a "grW in the parameter space. The grid is given by some ad hoe rule and each point of ministic searches from several random
4. Stochastic variational method
42
grids defines a basis function. The grid can be defined by some simple functions which may depend on some additional- parameters to be optimized. The number of parameters contained in the functions which define the grid is chosen to be much smaller than that of the original function to be optimized. For example, in the case of a three-particle system, each of the basis functions has three parameters: All, A12 A21 and A22 or a12, a13 and C923- Some COnVenient the
=
choices for basis parameters
are
shown in Table 4.1.
Table 4. 1. Choice of parameters. In the random tempering p is an index to a prime number that is used to generate pseudo-random numbers.
denote
See
Eq. (4.8) for the
Geometric
All
=
notation <
i, i
>.
progression -2
(aiqk-i) (a2q2-1) I
A22
=
A12
=
(k
mi)
(k
M2)
(k
I,- mi.)
(k
1:
(k
1
(k
17
mi)
(k
17
M2)
-2
k
0
Random tempering a12
=
OL13
=
a23
=
exp(d, < k,p > +d2 < kp + I > exp(d3 < k,p + 2 > +d4 < k,p + 3 > exp(d5 < k,p +4 > +d6 < k,p + 5 >
,
...
...
M2)
M3)
Chebyshev gTid All
=
A22
=
A12
=
( a2tan ( altan
0
7r
2k-1
2
2 mi
7r
2k-I
2
2M2
4.2 A
distribute
certain number of basis functions to
a
part of the
wave
function and then
the basis elements in
4.2 A
regions
one
43
practical example
might
approximate the bulk
increase the
density of
where finer resolution is needed.
practical example
point of the previous section, let us consider an example for basis optimization. The example will at the same time provide insight into some other aspects of the methods as well. We consider the system of a positron and two electrons, Ps-, used in Chap. 1. The energy of this system has been very accurately calculated by various To illustrate the
approaches and
it has been found to be -0.262005 in atomic units
given in a.u. throughout this chapter unless otherwise mentioned.) The quality of the different optimization strategies used in this section can be judged by comparing their results with this value. Let the positron be labelled I and the electrons labelled 2 and 3. With the electron spins coupled to S 0, the orbital of the trial function that the antisymmetry requires part ought to be symmetric with respect to the interchange of the electron coordinates. The wave function is thus expanded as
(a.u.). (Energy
and
length
are
=
K
1
Tf
=ECk(1+P23) exp (-2-;
,Akx),
(4.2)
k=1
exchange operator P23 is introduced to assure the symr2 and X2 rl metry requirement. The coordinates are x, matrix 2 with three 2 and x Ak is a symmetric (rI + r2)/2 r3, nonlinear parameters All, A12 and A22. The wave function Tf has 3K parameters to be optimized. The linear parameter Ck is determined by solving the generalized eigenvalue problem (3.10). To reach the accuracy required in atomic physics, the minimum basis size of a three-body problem is at least K 100, as will be shown later. That means that even in this simple example we would have 300 parameters. This is already almost beyond the capability of most of the computer codes for optimization and this "full" optimization is out of the question for larger systems. The fact that the optimization of a large number of parameters is not feasible is just a small part of the problem. To have an efficient optimization one needs fast function evaluation as many times as required. There are two steps which are where the
=
-
=
-
-
4. Stochastic variational method
44
necessary to calculate the energy expectation value: The calculation of overlap and Hamiltonian matrix elements and the diagonalization.
the In
a
fall
required
optimization,
one
time increases
(a K3-process)
as
has to recalculate all matrix elements K2) and to
at each calculation of the energy. This is
load
for
(the
the Hamiltonian
rediagonalize
a
considerably
small system. heavy computational Instead, one can try a partial optimization. One can, for example, fix the parameters of all basis states but one. In this partial optimizaeven
a
only one row (column) of the Hamiltonian and the overlap -nn atrix (the required time is of order K). Let -us assume that the Hamiltonian has been diagonalized over the fixed basis states. Then, in the successive step, only one row (column) is changed. As has been shown in Theorem 3.5, after the N x N diagonalization there is no need for an extra diagonalization to solve the eigenvalue problem on the (N + I)-dimensional basis. Consequently, the computational time required by the partial optimization is only a small ftaction of that of the full optimization, and, moreover, only a small number of paxameters (three in the present example) has to be optimized. tion
has to be recalculated
4.2.1 Geometric
First let can
us
try
to
progression
use a
grid defined by
a
define three different sets of Jacobi
which two
particles
are
connected first:
Sects. 2.4 and 7.6 and in these systems to be
given
Fig. by x('), X(2)
and
us
X(3)
(23)1, (31)2,
and
(12)3.
See
denote the Jacobi coordinates The trial function is
IT
assumed
in the form K
3
(P)
qf
Ckl P=1 k=1
with the
2.2. Let
geometric progression. One coordinates, depending on
exp(--l (-P)A(P)x(P))0(11)00(x(P)), 2
(4.3)
1
angular function
0(11)00 (X)
=
[Y1 (XI) X Y1 (X2)100 I
-
(-1)1-M
(4.4)
-: 2= ,+1Y1Ta(X1)Y1-m(X2)-
Here p denotes the arrangement channel and Y1. (,r) is
a
solid spherical
harmonic
YIM M
=
7"YM (P),
(4.5)
4.2 A
where Yl,,, (i
)
polynomial
a
is
a
spherical harmonic.
practical example
The solid
45
spherical harmonic is (see Complement
of degree 1 in the Cartesian coordinate
A() is always chosen to be diagonal. Therefore, the k spherical part of Eq. (4.3) is considered a special case of Eq. (4.1). (See also Complement 8.4.) The trial function of type (4.3) is used in the so-called Coupled Rearrangement Channel Variational Method [20, 25, 26]. The basic idea of this approach is that, by taking into account 6.2).
The matrix
the different Jacobi sets, one can introduce various correlations. This method has been used with great success especially for Coulombic
three-body problems [201. Because of the symmetry of the trial
function,
(2) C
(3) =
k1
The 2
x
cki
we
A (2)
may A
-
requirement imposed
on
the orbital part
that
assume
(3)
(4-6)
"k
k
diagonal matrix A(P) 1, (k K) has two nonlinear k and they are taken as a geometric progression
2
=
...'
rameters
(Ak(')),,
(P)
ai(p) (qi ) In
principle
one can use
k-1
(i
=
1, 2).
pa-
(4.7)
different geometric progression parameters,
ai(P) and qi(p), corresponding to the different arrangement channels and the Jacobi
that
is, they would depend on p and i in order even depend on the angular momentum 1. For simplicity, in this example we use the same parameters, a and q, for all. arrangement channels and all sets of Jacobi coordinates. The main reason for using the geometric progression as parametrization is clear: The number of parameters of the basis is reduced to just 2 (a and q), so the optimization is simple. Another reason for the choice of this specific parametrization is that the overlap integral of the basis functions can be easily controlled by a choice of q and thus the danger of the linear dependence of the basis functions can be avoided. (See Sect. 3 for the danger arising from the linear to
coordinates,
get better convergence. They might
dependence of the basis functions.) The disadvantage of the use of these diagonal matrices is that, without having some polynomials like the scalar product of Eq. (4.4), the energy does not converge to its accurate value. As is shown in Table 4.2, one reproduces only the first two figures (E -0.2618530) =
of the
ground-state energy without the polynomial part, i.e., by taking 0 in the expansion. only I
=
4. Stochastic variational method
46
Table 4.2. The energy of Ps- in different energy is -0.262005. E
arrangement channels. "Exact"
(a.u.)
Channel
Partial
(23)1 (23)1
1
1 =0,2
-0-2392726
(31)2+(12)3 (31)2+(12)3 (31)2+(12)3
1
=
0
-0.2609626
1
=
-0-2619622
1
=
0,1 0, 1,2
(23)1+(31)2+(12)3 (23)1+(31)2+(12)3 (23)1+(31)2+(12)3
At this moment
=
1
=
1
=
1
=
a
wave
-0.2068096
0
-0-2619717
0
-0-2618530
0, 1 0, 1, 2
-0.2619804 -0.2619816
comment is in order. As the
spherical harmonies
angular functions, complete may think that the partial-wave expansion using only one of the Jacobi coordinate sets form
set for
a
one
would be sufficient to represent the wave function. The convergence as a function of I is, however, very slow if only one set of the Jacobi slow convergence in the partial-wave expansion will also be studied in detail in Complement 11.2.) E.g., the energy calculated with the (23)1 channel using the partial waves of
(The
coordinates is used.
up to I
2 is
=
just -0.2392726, which
of the three-channel calculation
(E
higher than the energy -0.2618530) using only I 0.
is much
=
=
Since the lowest threshold of Ps- is the Ps+e- channel at the
en-
ergy of -0.25, this result indicates that this single-channel calculation cannot bind the system. Moreover, the expansion with I has for prac-
tical purposes to be truncated to low values because the calculation of the matrix elements of high I waves generally becomes more time
consuming. Table 4.2 shows that, by combining the expansion Jacobi coordinate sets, the energy converges faster and few terms of the partial-wave expansion are needed. By
in the
the first
only using
10
grid
points for A(') k be
a
600
=
(=
we have found the optimal values of the parameters to 2.6. The variational energy obtained with this 0.06 and q =
2
x
10
x
10
x
with
computational load 180300
alized
(=
x
601/2)
one
repeated
a
calculation is -0.2619816. The
particular basis
set is the calculation of
matrix elements and the solution of
eigenvalue problem
has to be can
600
3)-dimensional
a
gener-
of dimension 600. For the optimization this
few dozen times.
By increasing the basis
get closer and closer to the exact energy.
size
one
4.2 A
4.2.2 Random
practical example
47
tempering
Another
popular tempering method is random tempering [21, 22, 231approach involves the generation of (pseudo-) random numbers with the basis parameters defined by the following prescription: This
I
ak
1: dj < k, j >
exp
(k
=
1,
...,
K),
(4.8)
j=1
where <
k, j
> is a
pseudo-random numer, the fractional part of (k(k + 1)/2)V/P---(j) P(j) being a prime number in the sequence 2, 3, 5, 7,..., that is, P(I) 2, P(2) 3,... By using this tempering number I of one a formula, optimizes parameters, dj, instead of the original 3K parameters. A possible application of the formula for a three-particle case is given in Table 4.1. The origiTi of this formula is the following. One may assume that the ground-state wave function is an integral transform of some known with
=
function with
some
weight
=
function. The known function in
our case
is the correlated Gaussian and its nonlinear parameters are the integration variables. The simplest way to carry out the integral trans-
formation is to
the Monte Carlo method. The
quadrature points integration can be generated by the above formula. As this formula provides "good lattice points" for the integration, these nonlinear parameters can be thought to be adequate to represent the wave function in a variational approach. While the random tempering works in many examples in a superb way, one has to note that it has a serious problem: It often leads to (almost) lineaxly dependent bases. In our actual example, the best energy of Ps- with random tempering is -0.261872, and further hnprovement of this value was difficult due to the linear dependence. The parameters of the random tempering used are, in the notation of Table 4.1, ?nj 7, Tn3 5, di 4, d2 -8, d3 Tn2 4, d4
required
use
in this
=
-7, d5
=
-1, d6
The parameters
are
=
=
1, and the basis dimension nearly optimized.
-11,
di
=
p
=
=
is K
=
245.
4.2.3 Random basis
The above calculations have
suggested
that not all of the
grid points equally important grids give nearly the same energy. The explanation is simple: The basis functions are nonorthogonal to each other, none of them is indispensable; they are dense, that are
but the different
can
4. Stochastic variational method
48
will compensate any of thein can be omitted because some others for the loss. This property of the basis functions and the success of
is,
tempering suggest the idea of a completely random distribution of the parameters. To illustrate this possibility, let us generate K 100 sets of basis parameters in the expansion (4.2). The eleraents of Ak are randomly chosen from a "physical" interval: the random
=
1
1 <
0 <
10,
0 <
10,
< 20.
0 <
-\/FY23
fa13
V/a12 The
<
(4.9)
advantage of using the parametrization by aij instead of Ak 1/.\,ra-jj corresponds to the "distance' between the particles.
is that
e-l"2, the expectation value relevant to the distance is, e.g., (r) V 4/(7a), or A,/-(r-27) VF3/(2a).) The paxameters inside this interval describe the (e+e-e-) system,
(Note that,
for
a
Gaussian function
2
=
=
where the average distance between the positron and the electrons is expected to be less than 10 a.u. and the distance between the electrons is
expected to
be less than 20
a.u.
This limitation
serves
practical pur-
in pose, and it is based on the physical intuition that the paxticles bound system cannot move very fax away from each other.
The difference between the random
a
tempering and the random
distribution of the parameters is that in the former there axe several parameters which generate the parameters of the basis functions and these
optimized, whereas in functions are randomly chosen.
generating parameters
paxameters of the basis
are
the latter the
energies of different random bases are shown in Fig. 4.1. By increasing the size of the basis, the energy goes very close to the exact The
value. See Table 4.3. The energy is better than that obtained with the basis set of geometric progression type, but the comparison is diffi-
geometric progression, the matrix of nonlinear paxameters is diagonal and the partial-wave expansion is introduced to enable the trial function to span the full model space. This paxtialwave expansion increases the basis size significantly. When we use the full A matrix and randomly distribute aij, this partial-wave expansion is not necessary (see Complement 11.2). One may try to choose aij in a geometric progression, but our experience is that both the geometric progression and the random tempering of aij tend to lead to linear dependence. To avoid the linear dependence in the basis (which does not occur cult. In the
case
of the
frequently with the fully random basis), the overlap of the random basis fimctions should be smaller than a prescribed limit. Any random points which do not satisfy this condition are to be omitted and
4.2 A
practical example
49
-0.250
I -0.252-
-0.254-
Ci -0.256-
-0.258
-0.260
-0.269 20
40
60
80
100
Dimension of the basis
Fig.
4.1.
Energies of Ps- for different basis sets. The parameters of the are completely randomly chosen and no preselection is made.
basis states
Table 4.3. The energy of Ps- (in a.u.) for K basis states that are selected randomly. "Exact" energy is -0.262005. The energies Ei are obtained by
starting from different random points. The energy of "Best 100" is the one calculated by sorting the best 100 basis states from 400 basis states, where 50 random trials are probed at each step of the basis selection. Neal is the number of matrix elements evaluated during the optimization. K
=
100
K
=
200
K
=
400
Best 100
El
-0.2617619
-0.2619798
-0.2620032
-0.2619995
E2
-0.2617281
-0.2619793
-0.2620026
-0.2619978
E3
-0.2617918
-0.2619669
-0.2620016
-0.2619956
E4
-0.2618826
-0.2619675
-0.2620008
-0.2619953
E5
-0.2616285
-0.2619824
-0.2620024
-0.2619987
Nevai
5050
20100
80200
80200
4. Stochastic variational method
50
replaced by a new trial. Note that this cure cannot be applied to the procedures of the random tempering and of the geometric progression. In these procedures there is no such prescription. One can just omit some grid points depending on the actual parameters. The energies listed in Table 4.3 are surprisingly good, but the size of the basis seems larger than absolutely necessary. For the solution 400 is excessive. of a three-body problem a basis dimension of K We would like to decrease the basis size by selecting a minimal, indispensable set of basis functions. This will be described in the following =
subsections.
4.2.4
Sorting
One may wonder: Are all the basis states equally important? What happens if we omit a few states from these bases? To answer this ques-
tion,
we
reordered the basis states
by the following random selection
process: 1. A number
n
of basis states
giving the lowest energy restricted basis.
pool,
basis states, were tested in
state.
Again
2. Then
n
were
was
picked
up
randomly and
the
one
selected to be the first element of
a
again randomly chosen from the remaining a
2-dimensional calculation with the first
the state that
produced
the lowest energy
was se-
lected. 3. This
reordering
4.2.5 THal and
is continued until the last basis state.
error
search
randomly chosen nonlinear parameters give good results provided the basis dimension is big enough. For an N-particle problem, howthis ever, the basis dimension might become prohibitively large to use The
4.2 A
practical example
51
simple procedure. One can obviously improve the convergence by selecting the most important states and by not admitting all random trials in the basis set. This increase of basis dimension by searching the best among many random trials is a key of the stochastic variational method (SVM). The original procedure of the SVM, proposed in [271, has recently been developed further [281 and successfully applied to multicluster descriptions of light exotic nuclei [29, 30]. Learning from these applications to nuclear few-body problems, we have generalized and refined the method further to encompass diverse quantumfew-body systems emerging in nuclear and atomic physics
mechanical
[31, 32, 33]. The sorting method has shown that from
large number of random basis states one can select a smaller set without a significant loss of accuracy. Moreover, the accuracy can be improved by increasing the basis size. This experience motivates one to apply the following trial and error procedure, combined with an admittance test, in order to set up the most important basis set: Let Ak be the parameter set defining the kth basis function, and let us assume that the sets A,,- Ak-1 have already been selected, and the (k l)-dimensional eigenvalue problem has already been solved. The next step is the following: a
I
-
Competitive selection A number
s 1.
of different sets of
n
(Ak',
...'
Ank)
are
generated
ran-
domly. By solving the n eigenvalue problems of k-dimension, the corresponding energies (Ek,..., Ek) are determined. The parameter set Ak' that produces the lowest energy from
s2.
s3.
among the set
(El,...,Ek) k
is selected to be the kth parameter
set.
s4. Increase k to k + 1.
The essential different
motivating this strategy is the need to sample parameter sets as fast as possible. The advantage of this proreason
cedure is that it is not necessary to recompute the whole Hamiltonian matrix nor is it necessary to perform a new diagonalization at each time when
a new
parameter Ak is generated. See Theorem 3.5.
The convergence of the energy with the basis selected in this way is shown in Fig. 4.2. The convergence is much faster than in the previous case
in
number
Fig. 4.1, and n
it
of trials. The
dimensional basis sets
can
be made
even
faster
energies found by using K are
by increasing
=
shown in Tables 4.4 and
10- and K
=
the 100-
4.5, respectively.
4. Stochastic variational method
52
-0.250
-0.252
-
-0.254
Ca -0.256
-0.258
-0.260
-0.262 100
80
60
40
2-0
Dimension of the basis
Fig.
4.2.
trial and
Energies of Psprocedure
for different basis sets that
are
selected
by
the
error
Table 4.4.
Energy
of Ps-
(in a.u.) by
different
optimization strategies.
10. The energies "Exact" energy is -0.262005. The basis size is set to K Ei are obtained by staxting from five different random points. The number n denotes the random candidates probed in the trial and error procedure =
of SVM. Ne,,,,l is the number of matrix elements evaluated during the optimization. Tn the case of the full optimization by the Powell algorithm 3900
diagona,lizations were also required. The time in units of seconds is on a Digital Alpha 2100 (250MHz) workstation. The refining cycle is repeated 10 times.
Method
Powell
SVM(n=l)
SVM(n=10)
Refining(n=10)
El
-0.261251
-0.244364
-0.251083
-0.261180
E2
-0.261398
-0.234532
-0.250729
-0.261165
E3
-0.261283
-0.236207
-0.252659
-0.261040
E4
-0.261259
-0.243323
-0.252956
-0.261175
E5
-0.261193
-0.226697
-0.251260
-0.261039
N,-:.v.i
214500
55
550
88055
Time
60
0.8
1.0
11
4.2 A
Table 4.5.
Energy
of Ps-
The basis size is set to K
(in a.u.) by
=
100. In the
practical example
53
different optimization strategies. of the fiffi optimization by the
case
Powell
algorithm 7200 diagonalizations; were required. repeated only once. See also the caption of Table 4.4.
The
refining cycle
Method
Powell
SVM(n
El
-0.26200016
-0.26176191
-0.26199427
-0.26200231
E2
-0.26199947
-0.26172812
-0.26199382
-0.26200351
E3 E4
-0.26200164
-0.26179182
-0.26199733
-0.26200312
-0.26200135
-0.26188261
-0.26199778
-0.26200301
E5
-0.26200193
-0.26162811
-0.26199836
-0.26200271
Neval
35466150
11615
86150
805050
Time
7200
27
43
195
These tables also contain deterministic
procedure.
=
1)
SVM(n
10)
=
=
10)
compaxison with the performance of
a
One
can
conclude that
energy convergence with this simple selection the energy converges to the exact value.
4.2.6
Refining (n
is
a
reach
easily procedure. Moreover, one can
Refining
It is obvious that the basis size cannot be increased forever. Moreover, when the Kth basis state is
the previous states are kept fixed. This means that we try to find the optimal state with respect to previously selected basis states, but actually some of the basis states selected earlier
might
states took
not be
so
important
their role. So
succeeding procedure where the previous the following steps: over
selected,
states
are
one
anymore because the
may include
probed again
as
refining by
a
described
Refinement cycle
(A!,..., Ain)
random paxameter sets are r2. The parameters of the ith basis state ri.
generated. are replaced by
the
new
energies ElK ...Ek axe calculated. r3. If the best of the new energies is better than the original one, then replace the old parameters with the new ones, otherwise keep the candidates and the
r4.
original ones. Cycle this procedure through the
Note
that, again,
orem
3.5.
there is
no
need for
basis states from i
=
I to K.
diagonalization because
of The-
54
4. Stochastic variational method
Table 4.6.
the energy of Ps- in the refining steps. The 100 and 5 random states are probed. Atomic units are
Rnprovement of
basis size is K
=
used.
Refinement
cycle
E
(a.u.)
(Starting value)
-0.261992767
Ist
-0.262002742
2nd
-0.262003677
3rd
-0.262004063
4th
-0.262004178
5th
-0.262004255
6th
-0.262004314
7th
-0.262004339
8th
-0.262004372
10th
-0.262004398
"Exact"
-0.262005
4.2.7
Comparison
of different
optimizing strategies
4.2 A
for
a
longer time,
might get
we
practical example
somewhat better
55
energies, but the
available computer time is limited in any case. Moreover, the other methods reach at least the same energy in a fraction of this time as we
have
seen.
Tables 4.4 and 4.5 show that the full
to converge to different
indicates that the
points depending
procedure
on
optimization seems starting point, which
the
leads to different local minima. This is
by explicit inspections of the parameters. In the third column of Tables 4.4 and 4.5 the energies obtained by using different sets of completely random parameters are given. The
confirmed
next column shows the results of a trial and one
error
selection and the last
10 energy after refinement cycles. We have carried out cycles (each basis state has been cyclically probed 10 times) in
gives the
refining 10-dimensional, and
the
one
refinement
cycle
in the 100-dhnensional
case.
In summary, one may conclude that the stochastic selection of basis parameters leads to energy convergence independently of different random paths and, moreover, to very accurate results in a fraction
computational load than the full optimization. And while the latter is certainly superior in principle (one may need fewer basis states to reach a given accuracy), the former suits more practical applications. Moreover, in more complicated problems of time and
by much
less
where, in addition to the nonlinear parameters, one has to find the most adequate channels (quantum numbers, etc.) to describe the system, the random selection
seems
to be the
only
viable method. Since
the energy convergence attainable is of course not precise in a mathematical sense, one should check the quality of convergence of the wave
function
as
well.
competitive selection the steps is always admitted In the
in
the
relatively
to be
a
best candidate found
basis state
even
though
its
to be very small.
contribution to the energy may sometimes happen newly selected basis has very strong overlap with (linear combinations of) the previous basis states. To avoid this This indicates that the
an
alternative
approach called
a
utility selection [281
select the basis states. The steps go
as
can
be used to
follows:
Utility selection A. Generate
s2. Determine the
value
Ak. energyEl(k) by solving the k-dimensional eigen-
randomly
problem.
a
parameter
set
4. Stochastic variational method
56
s3. Admit the parameter set
gained by including ,El
(k)
< Ej
(k
-
it is
1)
-
Ak to be the kth element if the energy larger than a preset value Je, namely
JE.
Otherwise return to the step (sl) for the next attempt. If no parameter set was found to pass the utility test out of n consecutive
attempts, reduce JE to, say half of its original value, and to
return
(sl).
s4. Increase k to k + 1.
When competitive selection is used, convergence is inferred by the energy curve's flattening out, while when utility test is used, convergence is
signalled by
an
insistent failure to -find further elements that pass
the test. Instead of the random trial strategy described in (sl)-(s2)-(s3)-(s4) and (rl)-(r2)-(r3)-(r4), one may think of more efficient and sophisticated
approaches, like simulated annealing or genetic algorithms. approaches may give faster convergence, although the random selection strategy may have a slight advantage: It is easily implementable in a parallel way because the random trials are absolutely independent of each other. One should note that the basis elements selected by the above admittance tests are in general never optimal, not even locally. A better candidate which will gain more energy could be found in the neighborhood of the basis element admitted. The incorporation of a fine tuning, that is, an additional search for even better parameters I the vicinity of the element that passed the admittance test would clearly The latter
accelerate the energy convergence. When the evaluation of the matrix elements does not require heavy computational loads, this fine tuning is recommended to reach faster convergence. Or even a determini tic selection such as Powell's method [34, 351 might be used to find out
the
locally
4.3
best basis element.
Optimization
for excited states
The Mini-Max theorem the Hamiltonian in not
only for
Sect.
3.1)
shows that
K-dhnensional basis
ground
one
by diagonalizing
gets
an
upper bound
state but also for the excited states
happen predict the energies of the excited
may to
the
a
(see
that the basis set found for the
ground
state is
states with the
as
well. It
fairly good
same
conserved
4.3
quantum numbers
Optimization
for excited states
(angular momentum, parity etc.)
57
the
ground By optimizing only the ground state, however, the energies of the excited states will not necessarily converge (see Fig. 4.3). The energies certainly decrease because we increase the basis size, but except for as
state.
the energy of the second excited state, the upper bounds are not very accurate. (See Chap. 8 for the detail of the basis function used in the
present
section.)
0.4
171
Cd
0.3
CY)
0.2 W
0.1
0.0 0
200
400
600
dimension of the basis Convergence of the energies of the first five 'S states of the Helium atom when only the ground-state wave function is optimized. The energy difference Ej Ei'act is shown in the figure. See Table 4.7 for the exact energies.
Fig.
4.3.
-
As the ith
eigenvalue Ej
obtained
by diagonalization
is the upper
bound of the energy of the ith excited state, one may try to optimize this upper bound to get an accurate estimate for the energy of the excited state. In practice we repeat the same procedure as before, but now the basis selection is governed by the requirement that the ith
eigenvalue, get the ith
ground-state energy El, should be improved. To eigenvalue we need at least an i-dimensional basis to start not the
4. Stochastic variational method
58
with, but practical numerical considerations suggest that it is better to start with a basis in which all the lower eigenvalues (k i 1) axe already "stabld. This means that we need a first guess for the lower eigenvalues, otherwise it may happen that when improving the first excited state, for example, we pick up such components that lower the ground state. As the energies of the excited states of the Helium atom are known to a high accuracy, we will test our strategy in this case. First we 100-dimensional optimize the ground state of the He atom on a K basis. The energy of the ground state is very accurate and even the energy of the first excited state is acceptable. Starting from this K 100-dimensional basis, we increase the basis size one by one, picking up basis states which improve the energy of the first excited state. This procedure quickly improves the energy of the first excited state, while the energy of the ground state also improves a little bit (the important thing to note is that it does not get worse) because the basis size is increased. After reaching the basis size of K 200 we switch to the =
-
=
=
=
second excited state and
so on. The convergence of the result is shown th at the energy quickly converges to the exact value after its turn of optimization started.
in
Fig.
4.4. One
can see
The above
procedure gives us a basis where all the excited states are accurate up to a certain digit. We can of course create bases which give an accurate energy for an individual state, while the energy of the other states might be poor. In that case we simply carry out the stochastic search for a given state staxting from a first guess basis (like the K 100-dimensional basis in the previous case). Tbis optimization may include refinement cycles as well, if necessary. The results we obtained in this way are compared to the "exact" (i.e., the best =
calculation in the
literature)
values
[36, 37]
in Table 4.7. One
for the excited states
for the
can
t1aus
energies ground Examples of caculation of the energies of excited states win be given for the tdA molecule in Complement 8.4, the baryon spectroscopy in Chap. 9 and the four-nucleon system in Sect. 11.2. Before closing this section, we remark that the SVM assumes that the numerical accuracy of the energy calculation in each step ishigh. If the energy is calculated by, say the Monte Carlo method andhas an inherent statistical uncertainty, then the admittance test of the SVM will never work. Even when the energy is calculated by a deterministic get
as
accurate
method,
such
as a
as
state.
diagonalization, the accuracy of the numerical calgood enough to guaxantee the required precision
culations has to be
4.3
Optimization for excited
states
59
0.005
0.004
0.003 Ca >1 CD
0.002
0.001
k
0.000
400
200
0
dimension of the basis Fig.
4.4.
Convergence of the energies of the first five 'S states of the Helium one hundred basis states are selected to optimize the ground
atom. The first
state, the
next
one
hundred
(from
optimize the first excited state, and is shown in the
figure.
101 till 200) basis states are selected to so on. The energy difference Ej Eie"'t -
See Table 4.7 for the exact
energies.
Energies in atomic units of the ground state and the first four 'S excited states of the Helium atom. In column A the basis is optimized successively for all the states as described in the text (K 500). In column B the basis is optimized separately for each state, leading to five different bases (K 600) tailored for the respective states. The "exact" values are Table 4.7.
=
=
taken from
[36, 371.
State
A
B
"Exact"
EiE2 E3 E4 E5
-2.9037243758
-2.9037243769
-2-90372437698
-2.1459737740
-2.1459740452
-2.14597404605
-2-0612718887
-2.0612719880
-2.06127198974
-2-0335865085
-2.0335866779
-2.03358671702
-2.0211312479
-2.0211768312
-2.02117685157
60
4. Stochastic variational method
of the energy. An analytic evaluation of the matrix elements is thus an essential prerequisite in the SVM.
Recently an importance sampling algorithm called the stochastic diagonalization method [381 has been presented to compute the smallest eigenvalue and the corresponding eigenvector of extremely laxge matrices. It is interesting to note that though the algorithm used there has been developed independently of the SVM, it includes procedures analogous to those of SVM. The stochastic diagonalization was applied to matrices of order up to 1035 X 1035. All the above
procedures are based on the minirni a ion of the value. This is, however, not the only possibility expectation energy to find accurate energies and wave functions. As described in Sect. 3.2, the minimization of the variance of the local energy is another
possibility
in
case
will be studied in
the variance
a
can
simple example
be calculated
of the
analytically. This following Complement.
C4.1 Minimization of energy
versus
61
variance
Complements 4.1 Minimization
of energy
versus
variance
by treating a solvable m in the potential of a particle problem, an -Vo exp(-r/a) (Vo > 0). By expressing the wave function Tf V(r) as X/ /Tir, the Schr6dinger equation for this problem takes the form Here
we
compare the two ways of minimization
of mass
S-wave motion of
=
h?
d2X
-Voe'Xp I
2m dr 2
To
(-?')X=EX.
simplifT the notation, 8Tna2 VO
=
(4.10)
a
I
h2
we
and
introduce
V--8Tna
i/
2E,
(4.11)
ji2
exp(-r/2a) reduces Eq. (4.10) change of variable z differential equation of the Bessel functions
The
=
d2X Z-2 The
1 +
dX
--
z
+
dz
1,2)X (1 -Z2 _
=
to the
(4.12)
0.
boundary conditions for bound states, X(r)
=
0 at
r
=
0 and oo,
lead to the solution
X(r)
=
cJ,
( exp (_
r
with
C-2
2a
r
=f'j ( exp(-- ))2
dr,
2a
tj 0
(4.13) where the value of
J-(O
=
is determined
by
the condition
(4.14)
01
and the smallest
(4.11).
v
v
value
In order to have
gives the ground-state energy El through Eq. bound state, the potential strength VO must
a
larger than about 2.405. ground-state energy by the variational method, let us assume a simple basis function for X, rexp(-ar/a), where a is 2for a single basis a variational parameter. There is no minimum of oobtained by variance and the the function. Table 4.8 compares energy using the minimization principle for E or o-2. The parameter values 5.0, a of a are determined by the SVM. For a deeper potential of combination of two basis functions already reproduces the energy rea2.6, however, at least sonably well. For a very shallow potential of
be such that
is
To estimate the
=
=
62
Complements
Table 4.8.
Comparison of variational solutions obtained by E-minimizattion. 2-minimization for an exponential well. Units of E and o-2 are e/(8ma2) and (h2/(8ma2) )2 respectively. The mean values, (T) and and
u
,
(T 2
1/2
,
are
in unit of
with the exact
one
K
5.0
2.6
2
5
a.
is also
The
overlap integral given.
of the variational solution
E
0-2
(r)
(T2) '21
Overlap
E-min.
-3-5840
0.3236
1.4362
1.6169
0.99997
0-2_Min.
-3.5824
0.05437
1.4301
1.6044
0.99983
Exact
-3-5848
0
1.4376
1.6213
1
E-min.
-0.016446
1.275x 10-4
9.4174
12.236
0.99999
U2_min.
-0-016415
3.49Ix 10-6
9.1025
11.619
0.99940
Exact
-0.016447
0
9.4331
12.285
1
C4.1 Minimization of energy
versus
variance
63
(T2-minimization E -minimization
0
............................................... ......... ...............................................................................
i 0
10
20
wave
function-I
30
40
r1a
Fig. exact
4.5. The local energy curve for the shallow potential: wave function is also shown in an arbitrary unit.
2.6. The
5. Other methods to solve
few-body
problems
chapter we briefly show the essential points of other approaches to few-body bound-state problems. There are many different approaches and it is far beyond the scope of tbis book to discuss all of them. Included here are either only those methods which have some sort of connection with our approach or their results are frequently compared with that of SVM, and therefore a short explanation might
In this
be useful.
Quantum Monte Carlo method: The imaginary-time evo lution of a system
5.1
quantum-mechanical system is governed by the time-dependent Schr6dinger equation. When the time t is replaced by an imaginary time, -iha (a > 0), the Schr6dinger equation transforms into a diffusion-like equation The evolution of
Xf (a) aa
a
(5-1)
-HTf (a),
which has the formal solution Tf (a)
=
e-Ha Tf,
(5.2)
th-ne-independent. Here T is an initial wave function Tf (a 0). The imaginary-time evolution of the estimate the ground-state energy. The to used be function can wave
provided
that the Hamiltonian H is =
basis of this idea is Tf lim a-W
that wave
(a)-
a,
(qf- (a) I qf- (a) ) -i
is, the
wave
function at
jail
fimction a
--+
(5.3)
_
oo.
TI(a) approaches To prove
Y. Suzuki and K. Varga: LNPm 54, pp. 65 - 73, 1998 © Springer-Verlag Berlin Heidelberg 1998
this,
the exact
we
only
ground-state
need to
use
the
5. Other methods to solve
66
expansion (3.3) for TI
few-body problems
Eq. (5-2)
that a, :7 0, namely the initial wave function has a non-vanishi -n g overlap with the groundstate wave function. In what follows we assume that this condition is in
and
assume
fulfilled.
If
we
consider
imaginary
T/"(a)
time a, the
bound to the
a
variational trial function value
mean
ground-state
energy
E(a) El.
depending
on
the
of H in Tf (a)
gives an upper In fact it is easy to show t'h at
(Tf (a) I H I Tf (a)) R(a) I Tf (a))
E(a)
-2a(Ei-EI) aj 12 E+E-E ie I I i=2 1+EI i=2 e-2a(Ei-EI)JLi!J2 ai a
E' i=2 (Ei EI+
E1)e-2a(Ei-E1.) I
-
ai e -2a(Ei-El )I 1+1:' i=2 a,
ai
12
2` 12
>
(5.4)
El.
Clearly E(a) goes to E, when a goes to infinity. Another quantity called the asymptotic energy estimator is defined by 1
E(a, -r)
2T
This has the Jim
In
=
(Tf(a) I Tf (a)) (Tf(a + T) I Tf (a + -F))
(5-5)
*
following properties:
E(a, -F)
=
E(a),
(5-6)
-r--+O
+
E(a, -F)
=
E, + -In 2,r
-2a(Ei-El) e EOO I ai 12 i=2 af
(5.7)
'
e -2(a+-r) (Ei -El) I.Es! 12 J:" i=2
I+
a,
Equation (5.6) can be derived by noting that the left-hand side is equal to ( '1/2) -da.Lln(Tf (a) ITI(a)) and that the derivative can be easily calculated by Eq. (5. 1). It is clear that, for T > 0, E(a, 7) ! El and E(a, -r) converges to EI_ as a goes to infinity. In fact, the quantum Monte Carlo (QMC) method or the Green's function Monte Carlo cle to
out the
project point here is not
a
In
general,
method
ground
one
a
[40-44]
uses
state from the initial
construction of
mates the solution but
e-H'.
(GFMC)
a
good
e-Ha
wave
as a
vehi-
function. The
state space which
approxi-
calculation of the imaginary-tiTne evolution of
cannot
compute e-Ha
and H, do not commute, but
by dividing
=
e-(-ffo+H')'
the time
a
into
because rn
Ho
any smaH
Hyperspherical
5.2
steps Aa
=
a/n,
we can
(RIe-H0A'e-HIA'Ik) (RIe
-H,
IRI
=
expansion method
harmonics
approximate G(R, k)
in
=-
(RIe-"0'Ik) by
factorized form. Then the full propagator
a
ff _f G(R, Rn-1) G(Rn-1, Rn-2)
x
-
-
G(RI,k)dPfn-IdRn-2
*
..
be evaluated
use
several time steps Aa and
extrapolate
to
'
-
(5-8)
dRI
the Monte Carlo method. In
can
by
67
practice,
da
=
one
must
0 in order to
eliminate time step errors arising from the non-commuting nature of the kinetic energy and potential energy operators. the propagator (5.8) to calculate both the numerator and denominator of Eq. (5.4)
When
we use
statistical
Eq. (5.4), subject to the
E(a) a're
in
Then the nice feature
of Monte Caflo
integration. ground-state energy may be lost in method. One of the most serious problems in the QMC is the so-called minus-sign problem [43, 451. The propagator G(R, k) is not always positive. This makes it difficult to use importance sampling techniques based on the classical stochastic process such as a Maxkov process or a Molecular Dynamics technique in computing the propagator by the Monte Carlo method. of
having the QMC
an
upper bound to the
Hyperspherical
5.2 We
errors
briefly
introduce
an
harmonics
approach
expansion
which is based
on
method
the
the trial function in terms of hyperspherical harmonic
expansion of
(HH)
functions.
spherical harmonics are useful to expand the angular funca single-particle motion, the basic idea of the HH method is to generalize this simplicity to a system of particles by introducing a global length p called the hyperradius and a set of angles Q. Let us suppose that all of the particles oscillate harmonically with angular frequency w Just
as
the
tion of
N
N
2
Pi
HHO
rniw2Ir2
+
Using
the Jacobi coordinate set,
one can
into the intrinsic and center-of-mass N-I
1 +
2yi
_2
separate the Hamiltonian
parts
N-1
7r2
HHO
(5.9)
2
2mi
/-Ziw2X2i +
7r2N 2rn12
...
2
N+ 2 'rn12---NWXNi (5.10)
5. Other methods to solve
68
few-body problems
where Ai is defined by Eq. (2.19). Among the 3(N 1) degrees of freedom for the intrinsic motion, -
the
p is defined
hyperradius
by
N-I 2
[lix?
P
(5.11)
The
hyperradius contains only intrinsic coordinates, but it can also expressed as E i_=, Mi(Ti XN )21A, which is proportional to the moment of inertia of the system (see Eq. (2.18)). The choice of IL can be axbitrary, The hyperradius is apparently symmetric with respect to the interchange of particle labels. The potential energy of the intrinsic 2P2. Just as the single-particle kinetic part is simply expressed as -Ilzw 2 radial and angular parts, the intrinsic is into separated energy operator be
-
kinetic energy operator is reduced N-1
h2
7r2
I: 21.Lii
2jL
i=1
02
[5;;72
3N
-
as
follows
4 0
+
L
-ap
P
2
(5.12)
p2
where L is called the
grand angular momentum, which is expressible in terms of 3N 4 angles S2. there is no unique way to choose the angles, one possiAlthough bility is to choose 2(N 1) angles &j and 0j, the polar coordinate of N k 1, 1) and the other N 2 angles -yk (0 < -yk E, xi (i 2 2,3,..., N 1) called the hyperangles as follows -
-
=
-
-
...,
-
-VFILIV-1 XN-I V Ak Xk ,//-Ll x,
=
=
=
V/jL P COS7N-1,
- ,FA P Sin7N-1
Vrj-Lpsin-yjv-j
...
...
Then L is related to the usual
(k
Sin'Yk+1COS7k
angular
momentum
LN-D
a2
2
Lk
3(k
-
2
a7k
-
2)cot-yk
Lk-I +
+ COS
'Yk
sin -YA;
...,
N
-
+
(k
4cot2-yk
2,
2), (5.13)
2
=
2,
siny3sint2.
follows
L2
=
la-lk
N
2),
1k
=
"Xk h
X
-irk
as
5.2 2
L1
The
Hyperspherical
harmonics
expansion method
=121.
69
(5-14) of L 2, called the HH
eigenfunctions L 2y/C (S?)
=
K(K + 3N
-
functions,
5) Yjc (Q)
(5.15)
by the quantum number )C which comprises K (K 5 eigenvalues of the angular operators. 0, 1, ...) Expressing the intrinsic wave function as a sum, over k, of products are
chaxacterized and 3N
-
of radial functions
RK(p)
and the HH functions
Y)c(Q),
3N-4
Ep-
-V
(5-16)
RIC (P) Yk
2
/C
N-body Schr8dinger equation, after eliminating the center-of-mass motion, is reduced to the matrix equation for the radial functions the
d2
'C(C + 1) P2
(_W [ dp2 21L
E)
(p)
N
+
1: (YK 11:
Vii I yic) RIC, (P)
=
0,
(5-17)
with
L=K+
3(N 2
-
(5.18)
2).
The volume element is
given by
N-i
P3N 4dpdQj
fj dxi
MIL2,
AN-I
N-1 x
fj df2kSin3k-4,YkCoS2,YkdYk,
(5-19)
k=2
df?k Sin?ykd'OkdOk is the usual surface element. Because C is positive for N > 3-particle systems, it turns out that the "centrifugal baxrier" is always present in N > 3-particle systems even for zero total
where
orbital
=
angular
For the HH ation of the over one
momentum states.
expansion method to be practical, of course, the evalu-
potential
has to have
energy matrix elements must be feasible. Moreguide for the truncation of the HH basis, even
a
5. Other methods to solve
70
few-body problems
the
expansion converges with a few K values, because the deof the HH functions is high. It is also necessaxy to construct generacy the HH functions of proper symmetry for a system of identical par-
though
ticles. This is
a
difficult
HH
expansion method
(N
<
5).
A
sophisticated
a
to
with FT. In
it limits the
application of the systems of rather small numbers of particles
version of the HH
correlation factor F
(5.16)
problem, and
expansion method [461 employs
N
=
[36,
113'>i=l hi
ri
-
rj) [471, replacing
TV in
481 the correlation factor is chosen to
Eq. satisf ,
special boundary condition like the cusp condition for the Coulombic system. (See Complement 8.1 for the cusp condition.) The use of the correlation factor accelerates the convergence and leads to precise a
solutions. The HH expansion method was introduced in 1935 by Zernike and Brinkman and reintroduced 25 years later by Delves and Smith. See
for
example [49, 50]
for details and
[511
for recent
developments.
5.3 Faddeev method
T
=
'01
+
V)2
+
(5.20)
?P31
where for the bound state each component is related to Tf
*j
GoViTf,
=
where the free
or
(E -110)0i
three-body propagator
=
is
by
(5.21)
ViTf, given by
I
Go
-
E Here
Ho
-
is the
netic energy
-
Ho' three-body kinetic
being
(5.22)
energy with the center-of-mass -kisubtracted. The potential Vi is expressed in terms,
5.3 Faddeev method
V12 It V23 V2 V31 V3 notation, that is, V1 is easy to see that Eqs. (5.20)-(5.22) is equivalent to the Schrbdi-nger HO + VI + V2 + V3 equation (E H) Tl'= 0 with H We show two forms of the Faddeev equations which are widely of "odd
man
out"
71
: --
:::--
i
-
i
=
-
used for numerical solutions. One is the differential form which results
by rewriting the
(E
(E (E
-
Ho
-
-
Ho
-
-
Ho
-
Another is the
Oi
=
second form in
Eq. (5.21):
VI),01
=
V1 (02
+
03)
V2)02
=
V2 (03
+
01)
V3)7P3
=
V3 (01
+
02)
integral
(5.23)
form of the above
(j 7 ij
GiVi(Oj + Ok)
k
equation
=7 i, j :7 k),
(5.24)
with
Gi The
(5.25)
=
E
integral
-
Ho
form
-
Vi'
can
G,
02 03
be
simply expressed 0
0
0
G2
0
0
0
G3
in
matrix form
a
0
V,
Vi
V2 V3
0
V,2
V3
0
'01 2
03
(5.26) satisfy the boundary condition that Oi large distances. This condition is, a particular value of energy E, which is notbing but the energy eigenvalue of the three-body system. The Faddeev method is thus a direct method to solve the eigenvalue problem. It does not rely on the variational principle. There are three possible choices for the Jacobi coordinates X(1), X(2)7 X(3). See The bound-state solution has to all of the components however, met only for
Fig.
vanish at
2.2. It is natural to express the Faddeev
of x(l), and likewise
02 by X(2) and 03 by X(3)
component 01 in .
The most difficult part
in numerical works of the Faddeev method then
comes
uation of matrix elements between the functions
of the different Jacobi coordinate sets. Readers for
some
details for nuclear and Coulombic
problems. In Chap.
11
we
will show that
related Gaussians with the
use
a
are
from the eval-
expressed
in terms
referred to
three-body
[55, 561
bound-state
optimization of the corSVM gives solutions that are
careful
of the
terms
5. Other methods to solve
72
few-body problems
in excellent agreement with the results of the Faddeev method for
nuclear
three-body problems.
A real power of the Faddeev method is
probably not only in boundstate problems but in applications to scattering and continuum problems. See [541 for the recent developments on this subject. It is possible to extend the Faddeev method to four-particle system. The Faddeev equations are then called Faddeev-Yakubovsky equations which consist of seven components, four for I + 3-partition of four particles and three for 2 + 2-partition. Numerical solutions become increasingly difficult.
5.4 The
generator coordinate method
There is
a
which is
a
called the generator coordinate method (GCM), natural extension of Eq. (3.7) to the case of continuous
theory
superpositions. The GCM is a powerful method and has a variety of applications such as the restoration of symmetries or good quantum numbers (e.g., angular momentum and particle number) In many-body wave functions, the microscopic description of reaction dynamics between two composite particles and collective motions of many-body systems. The GCM
assumes
that the trial function is
a
continuous superpo-
sition of the basis function Tf (a), which is often called the
generating
function: Tf
=
f f (a)
Tf (a) da.
(5.27)
f (a) is called a weight function. A real or complex parameter a specifies the generating function. It is called the generator coordinate to distinguish it from a physical coordinate. The symbol The function
a
may
tegral
represent in
more
Eq. (5.27)
than
is then
one a
parameter:
the
equation known
principle as
=
multidimensional
the parameters. The choice of the important in the GCM. The variational
a
generating
a2l
...
).
integration
is used to determine
f H(a, a')f(a)da' El B(a, a')f (d)da', with
,
functions
the Hill-Wheeler equation
=
(al
f (a)
Tf(a)
The inover
all
is most
and leads to
[571 (5.28)
5.4 The generator coordinate method
(a, a) Here
=
li(a, a)
(Tf (a) I H I Tf (a)), and
13(a, a')
B (a,
are
the Hamiltonian kernel and the
a)
=
(Tf (a) I Tf (a)).
73
(5.29)
linear integral operators and called
overlap (or norm) kernel, respectively.
the GCM looks very similar to the diagonalization of the Hamiltonian in the basis states TV (a). This is true if the parameter a
Formally
is discretized and if
only a finite number
of functions Tf (al) ,
...
i
Tf(aK)
are used. Then the HiU-Wheeler equation reduces to the generalized eigenvalue problem (3.10). For further details see [57-601. Examples
of
using discretized generator
10.4.
coordinates
are
given
in Sects. 8.4 and
6. Variational trial functions
Due
care
of the correlation between the
particles
is
important
for
a
precise variational solution. The variational trial functions used in this text, correlated Gaussians and correlated Gaussian-type geminals, are formulated
by using a generating function g. Orbital functions with arbitrary angular momenta or Cartesian polynomials around arbitrary centers are constructed from g by using a simple, well-defined prescription. It is also shown that the generating function can be related to the product of single-particle Gaussian wave-packets through an integral transformation. This 'uncorrelated' form is useful to extend to a many-body system of identical particles. The spin function for an N-fermion system is briefly discussed.
6.1 Correlated Gaussians and correlated
Gaussian-type geminals approach is the choice of the trial function. To solve an N-particle problem, it is of prime importance to describe the correlation between the particles properly. The correlation between the particles can be described by functions of appropriate relative coordinates. The correlation is then conveniently represented by a correlation factor, F rl3',,i=, fij (ri rj). There are two widely applied strategies: (1) One is to use this form of F directly by selecting the most appropriate functional form to describe the short-range as well as long-range correlations. Such calculations are, however, fairly involved for systems of more than three particles and the integrations involved require the Monte Carlo technique. (2) An alternative way to incorporate the correlation is to approximate fij by a number, possibly a large number, of simple terms which facilitate the analytical calculation of the matrix elements. We follow the second course by using an expansion over a correlated Gaussian "basis". Most crucial for the variational
=
Y. Suzuki and K. Varga: LNPm 54, pp. 75 - 122, 1998 © Springer-Verlag Berlin Heidelberg 1998
-
6. Variational trial functions
76
important that two basic requirements axe satisfied in the second course. First, in the limit of large dimensions the basis functions should become complete, so that the results obtained by means of some systematic increase of the number of basis functions could converge to the exact eigenvalue. Second, for the approach to be practicable, computational effectiveness is required for the basis functions. The It is
matrix elements have to be evaluated with ease; otherwise calculations witli combinations of a great number of basis functions would be
extremely difficult. In our opinion only the correlated Gaussian basis meets these requirements. The usefulness of Gaussians was already suggested in 1960 independently by Boys [611 and Singer [621, and since then exploited by many authors [63, 22, 64, 65, 661. Though a mathematically sound proof may not be available for the completeness of the Gaussian functions, a heuristic argument for it is possible as follows: As is discussed in Complement 6.1, any square-integrable, well-behaved function with angular momentum 1m can be approximated, to any desired accuracy, by a linear combination of nodeless harmonic-oscillator functions eter
a
(Gaussians)
(Eq. (6.37)). By generalizing
N-particle
continuous size paramthis to the N-particle system, the
basis function then contains
N
rj)21
F=
(r113" ,,j=jexpf--!a-2
This
simple correlated Gaussian
U
calculations. For
a
specific
for
cases,
see
-
?,
discussion
is
on
of
=
a
a
product
eXpj
-.1 2
of these Gaussians:
EN )2j. j>i=l ozij(ri -,rj
actually widely used in variational the completeness of Gaussians for
example [67, 351.
Of course, we have to keep in mind that the Gaussian is not economical in describing the asymptotic behavior of the wave function at
laxge
distances
(see Fig. 6-2). Moreover,
it does not
predict
a
correct
specific quantities such as the cusp ratio. See Comasymptotics and the cusp ratio could be well plement described with exponential functions. For example, welinow that the Rylleraas-type functions give very accurate results for Coulombic fewbody system (see, e.g. [68, 691). The correlated exponential functions are not, however, amenable to analytic evaluation of matrix elements for a system of more than three paxticles. This makes it difficult to use the exponential functions as a vahational trial function for a general N-particle system. value for
some
8.1. Both the
We extend the above argument further to define the correlated we consider separately two cases for two types
Gaussians. For this
of Hamiltonians. First in be
expressed
hn
terms of
case a
of
set of
Eq. (2.1) the Hamiltonian can independent relative coordinates
6.1 Correlated Gaussians and correlated
,7
x
(xi,
=
...
,
xN-1).
As
was
shown in
Gaussian-type geminals
Eqs. (2.23)-(2.25),
77
it is then
con-
venient to express F in terms of x, instead of N(IV- 1) /2
interparticlefunction, a so-called
distance vectors, ri rj. An N-particle basis correlated Gaussian, then looks like -
I:
Type where A is
TI
(N
an
exp
=
-
1)
(_2 :TbAx)
x
(N
0 (x),
(6-1)
1) positive-definite, symmetric
-
matrix
of nonlinear parameters, specific to each basis element. As mentioned in Sect. 2.2, the matrix A with these properties can in general be written
6DG with the use of an orthogonal matrix G and a diagonal
as
matrix D with all
positive diagonal elements. The function O(x) is a generalization spherical harmonics Y of Eq. (6.56) to the many-particle case. More details will be given below. of the solid
The second each
is
is suited to the Hamiltonian
case
governed by both
(2.2).
The motion of
the
single-particle potential and the two-body Eq. (6.1) to include an independent motion of the ith paxticle around some point R, Here PI, is not a dynamical coordinate but just a parameter vector. In this particle
interaction. It is thus useful to extend
case we
but
do not need to
can use
the
use
the relative and center-of-mass coordinates
single-particle coordinates. Therefore we are led to
the
following type of correlated Gaussians which are often called correlated Gaussian-type geminals N
expf
XeXpf Here
r
-
aij (,ri
-
2
-
2
Y
R stands for
i(ri a
N
E j>i=, aj(,r, _,rj)2 A defined N
3=i+l
can
through Aij as
Type
11
:
Tf
=
x
Ri
)210(,r
be written
=
exp
-
T3)2
set of vectors
aij, the correlated
expressed
_
Aji
=
firl
R). RI,
-
compactly
-aij
(i
<
as
-
R),
Ar
-
2
(r
-
---,,rN
-
Ar with
j), Aii
Gaussian-type geminals
1 -2
O(r
_
=
can
R)B(r
-
RNJ_ a
matrix
E''-=, aji
in
R)
As
general
+
be
I (6.2)
78
6. Variational trial functions
6.1 Correlated Gaussians and correlated
Gaussian-type geminals
79
optimization of nonlinear parameters allows us to obtain high-quality solutions with expansions containing not too many terms. The Gaussian of type I is a special case of type IL To unify the description for both cases, we change the notation in what follows throughout this chapter according to the following conventions: As it is cumbersome to use x or r depending on the type of problems, we in the
of the presence of an external field. To have N as the number of xi coordinates in both cases, we consider (N + 1) particles for the basis of type I and N particles for the basis of type II,
will
use x even
respectively.
case
Table 6.1 summarizes
our
conventions for the notations.
Table 6.1. Convention of the coordinates jo
`:
(XII X21
---I
XN)
in the
cor-
related Gaussians and the correlated are
abbreviations of
Number of
Meaning
Gaussian-type geminals. SP and CM 'single-particle' and 'center-of-mass', respectively.
particles
of
x
T ansformation matrix
from SP coordinates
SP coordinates relative -
ri
Gaussian-type geminals (Type II)
N+I
N
Relative coordinates U: Ur X =
SP coordinates U=1:
X=r
IV
I
XN+1
Ei==1 mixi
7nl2---N
(U-1X)i
Xi
-
XCM
XCM
Relative distance vector -
Correlated
(Type I)
r
CM coordinate: xcm
to CM: ri
Correlated Gaussians
(U-I-X)i
-
(U-'x)j
Xi
-
Xj
rj
Before
discussing the
function
O(x)
in
Eq. (6.1)
we
distinguish two
types of variational calculations: One is a type of calculation called a vaTiation before projection. Here the trial function used is not neces-
sarily an eigenstate of some of the conserved quantities belonging to the symmetries of the Hamiltonian such as paxity and angular momentum. After the variational calculation, the conserved quantities are proje-ted out from the trial function. For example, we know that an eigenstate of a rotationally invariant Hamiltonian should have a good angular momentum. Yet one often uses a variational trial function that does not have the proper rotational symmetry but, after the
6. Vaxiational trial functions
80
variational calculation, the symmetry is restored by angular momentum projection. The restoration of good angular momentum is ensured
superposition of the rotated variational solutions and the weight function involved in the superposition can be determined, e.g., by the
by
a
generator coordinate method of Sect. 5.4. Another type of calculation
after projection, where the trial function
is what is called variation
constructed
so as
is
quantum numbers of the conthe variation after projection is superior to
not to mix different
served quantities. Clearly the variation before projection because the variation in the former
case
only in the state space which has the same symmetry as solution, while in the latter the variational solution tends to reach a minimum in the space including the basis states with different quantum numbers. Even a calculation of variation before projection type, if thoroughly done, would reach a solution that has the proper symmetry in the limit that the state space is complete. See Sect. 10.4. Of course calculations of the variation after projection type become more challenging because it is in general not easy to calculate matrix elements with trial functions that are eigenstates of the conserved observables. Our objective is, however, to use trial functions with the
is carried out
the exact
proper symmetry. The function O(x) in
Eq. (6.1) describes non-spherical motion. To angular momentum L and its projection M, a direct generalization of solid spherical harmonics Y (4.5) (see Complement 6.2) to a many-particle case is a vector-coupled product of the solid spherical harmonics of the relative coordinates: describe the orbital motion with the orbital
[[[Yi, (XI)
OLM*
X
)(
...
Y12 (X2)IL12
X
Y1' *V)
I
X
Y13 (X3)IL123
LM
N
C.
r-=JM1.IM27 where c,, is
a
...
(6.3)
Ylimi (xi),
7MNI
product, (IIM112M2 IL12 M1+M2) (L12 M1+M2 13M3 IL123
M1 +Tn2 +M3)
-
-
-
(L12 N-IMI+?n2+-+TnN-1INmNJLM),
of the
...
Clebsch-Gordan coefficients needed to couple the orbital angular mospecified quantum numbers. Here each relative motion
inenta tG0 the
has
a
angular momentum. See, e.g., [2, 70, 71, 721 for the deangular momentum algebra. Angular momentum recoupling
definite
tails of
coefficients used in t'his book will be defined in
Complement
6.3.
6.1 Correlated Gaussians and correlated
Gaussian-type geminals
81
Since the
angular momentum of the relative motion is not a conquantity, it may be important for a realistic description to include several sets of angular momenta (117 12 IN; L12, L123 ) This is the case especially in nuclear few-body problems [32, 251 because the non-central components of the nuclear potential necessitate higher partial waves. For example the tensor force couples S waves to D waves. It is also noted that a faster convergence is in general obtained by allowing the use of different sets of relative coordinates together with suitable sets of angular momenta [30, 20, 251, because a parserved
7 ....
i
...
-
ticula,r type of correlations can be best described in the coordinate set conforming to the type of correlation. From the fact that OLM(X) can
be
expressed by
different
paxtial-wave decompositions
in different
relative coordinate systems, one can conclude that the use of partial waves may not be so important after all. Besides, the various possible
partial wave contributions
increase the basis dimension.
calculation of matrix elements for this choice Of OLm (x) becomes too
Moreover, sooner or
This choice is therefore
complicated. especially as the number
the
later
obviously particles increases and/or different sets of relative coordinates are employed. This difficulty can be avoided by adopting a different generalization of Y for OLM(x) [33, 311: nient
OLM(X)
inconve-
of
V2KYLM(V) N V 2K+LyLob)
Only
the total orbital
number in most
cases
pression. The real
with
v
Uixi
which is
angular momentum,
(at
vector ii
a
=
fix.
(6.4)
good quantum
least =
approximately), appears in this ex(ul,..., uAr) defines a global vector, v, a
linear combination of the relative
coordinates, and the wave function angle b. The vector u may be
of the system is expanded in terms of its considered a variational parameter and
one
may
try
to minimize the
energy functional with respect to it. The energy minimization then amounts to finding the most suitable angle or a linear combination
of
angles.
The
of the parameter u can be more advantavariational calculation than the discrete nature of the set of
continuity
geous in a the angular momenta
(111 12
IN; L12, L123 -) because the change of the energy functional can be continuously seen in the former case. The factor of v2K+L plays an important role in improving the short-range 7 ...,
i
.
behavior of the basis function. A remarkable
advantage
of this form
6. Variational trial fimetions
82
of
is that the calculation of matrix elements becomes much
OLm(x)
simpler
than in the former
momenta is
completely
case
coupling of N angular appendix for details.
because the
avoided. See the
Eq. (6.2) the construction of the trial function with good orbital angular momentum may not be an For the function
0(,r
-
R)
in
immediate concern, because the vector R is intended to determine a specific shape of the system. The function 0(r R) is chosen in the -
spherical
basis
as
N
O(x or
-
R)
_nz.12kiy1 imi (X,
IX,
in the Cartesian basis N
0 (x
-
i=1
(6-5)
IZZ
as
3
fl 11 (xip
R)
_
-
(6-6)
Rip)nip,
P=I
where the index p = 1, 2, 3 stands for x, y, z components of the vectors xi and R, and nip is a non-negative integer for the p-direction of the
simple transformation for the polynomial parts between the spherical basis and the Cartesian basis, the above two representations are actually equivalent. See Complement 6.2. In the following we assume that 0 is given by Eqs. (6.3) or (6.4) for the correlated Gaussian of type I and by Eq. (6.5) or Eq. (6-6) for the correlated Gaussian-type geminal of type II. The Gaussians of type I have definite parity of either ith
or
particle.
Because there is
a
(_l)L, depending on the choice Of OLM(X),
type II
parity
are
not
while the Gaussians of
eigenstates of the parity operator. To project out good function, one has to take a combination of two
from the trial
Gaussians with the centers at R and -R.
6.2 Orbital functions with
arbitrary angular
momentum The construction of
a
trial function with
good angular
momentum
is very important for obtaining solutions of a rotationally invariant Hamiltonian. In the previous section we introduced the forms of
OLM(X) Since
to describe the orbital fimction with
Eq. (6.3)
good angular momentlan. Eq. (6.4) takes
appears to be better established but
6.2 Orbital functions with
arbitrary angular
momentum
83
particularly simple form and makes the calculation of matrix elements very simple, it is useful to understand the relationship between the two angular functions. a
Any functions of type
Theorem 6. 1.
N
v2KyLM (V) can
be
with
expressed
2ki
2k2
-
-
all
*
*
X
*
X
Y12 (X2)] L12
YIN
of terms
Y13 (X3)] L123
X
(XN)l LM7
non-negative and satisfy 2k,
+
11 + 2k2 +12 +
-+2kv+IN =2K+L.
Proof. We V
are
linear combination
a
[[[Yl (XI) X
where ki and li
uixi
of
in terms
x2k'
X2
XI
v
=
prove the theorem
by induction.
For
U1XI + U2X2, the assertion is true because
equality [73, 747 75] (see Exercise
6.1 for
v2KYLM(V)
D
a
a
we
two-variable vector know the
following
simple proof):
KL
kj.Ijk212
U2k3-+llU2k2+12 2 1
2kl.+11+2k2+12=2K+L
XX
where
DKL
kjIjk212
DKL kjIjk212
-
-
is
-
[Y11 (X1)
same
Y12 (X2
LM7
Bkj.,,Bk2l2 (2K + L)! Q1112; L). BKL (2k, + I,)! (2k2 +12)1
(6.7)
(6.8)
by
47r(2k + 1)! 2kk!(2k + 21 + 1)!!'
and C is the coefficient needed to
the
X
given by
Here Bkj is defined
BkI
2klX2k2 2 1
(6.9) couple two spherical haxmonics with
argument [70]:
1YI 00
X
and reads
as
YI'001 LM
=
C(Il; L) YLm (i),
(6.10)
6. Variational trial fimetions
84
1 +::1: (21+1)(211+1) L (101'OILO). 4 4v(2L 47(2 v(2 + 1)
F2
Qll'; L)
The coefficient C vanishes unless I + I' + L is For
general
a
of the vector
case
VN-1 +UNXN With VN-1
vector-coupled products and
x
U1X1 +
=
V2KyLM(V)
formula to show that
2kN
v *
*
even.
-,LulvxN , we put v +UN-lXN-1 and use the above ulxl +
=
*
(6.11)
can
be
*
-
expressed 2K12
of the terms of
v
...
N-1
1
in terms of the
YL12
...
(VN-1)
Y1N (XIV) with 2K12 m-1 + L12 N-I 2K + L 2kN 1Ndecomposing VN-1 to VN-2 + UN-IXN-I, one can apply =
N
N-I
...
By
further
the
same
2KI2 N-1 to vN-1 YL12 Then it is clear that we ...
argument
repeatedly.
-
-
...
...
N-1
can
(VN-1)
th is process show that the statement is and
use
true.
Theorem 6.2. A
two solid
vector-coupled function of
spherical
harmonies
[Y11 (Xl) be
can
X
(X2)]
Y12
expressed
(-1)11+12
with
=
(_I)L
LM
in terms
of a
linear combination
of terms
X2ki.x2k2V2qyLM(V), 2 1 where
2k, + 2k2 the- form Of V =
Proof.
For
a
2q
+
=
(6.12) 11
+
UIXI + U2X2
L
given
L and the vector
of each term has with appropriate coefficients ul and U2
12
value, 11
-
v
-
+
12
-
L is
even
and non-negative, and
thus 11 + 12 may be set equal to 2k + L with a non-negative integer k. We use induction with respect to k. First we show that the statement is true for k 0 (11 + 12 L). As Eq. (6.7) shows, for the K = 0 =
case
there
are
=
with
(see also Eq. (6. 108)), and (L + 1) terms of [YI(xi) x By taIdng (L + L), each multiplied by UI1 UL-1. 2 L with + + X2 (i aix, 1) ai =7 0, we
only terms
k,
=
k2
=
0
V
=
UIXI + U2X2 consists of
YL-I (X2)] LM (I
=
0,
YLM(V)
With
...,
1) different vectors, vi obtain
=
L
YLm(vi)
=
-21+
71 + 1)! 4-Ax(2L 1. 21 + I)! 1 1+ 1) (2L
E(aj)' 1=0
X
One
can
[YI(XI)
view the above
equations for [Yl(Xl)
X
X
-
YL-1(X2)]LM-
equation as
a
(6.13)
system of simultaneous linear
YL-I(X2)]LM (multiplied by
the square root
6.2 Orbital functions with
arbitrary angular
factor).
The determinant of
nothing
but Vandermonde's determinant and it becomes
(L + 1)
(L + 1)
x
85
momentum
(ail)
coefficient matrix
nonzero
is
for
different
ai's. Therefore the solution of the linear equation exists, and hence it is possible to express [YI (XI) X YL-1 (X2)] Lm as a combination of terms of YLm(vi). Thus our assertion has been proved. Next we assume that the statement holds for k < K 1, and will show that it also holds for k K. To prove this, we note that a general [YII (XI) X Y12 (X2) I LM With 11 + 12 2k + L 2K + L which satisfies the triangular relation Ill -121 :! L < 11 +12 takes the form [YK+l (xi) x YK+L-1 (X2)] LM (1 0, L). What we have to show is that the term -
=
=
=
=
...,
[YK+1(Xl)
YK+L-I(X2)]LM is expressible as a combination of terms of the form of Eq. (6.12). Now, looking at Eq. (6.7) in its full generality, the expansion of V2KyLM(V) contains all of these (L + 1) functions, each
X
multiplied by
u
K+I u
K+L-1
2kt+li
and
further terms. These have
some
the form const. x u 1
2+12X 2kjX2k2 lyll (XI) 1 2
at least
nonzero
one
U2 of k, and k2 is
+ L with k
k2)
=
K
-
ki
k2
-
< K
Y12(X2)ILM
in these terms
appropriate
v
are
k,
-
1.
-
Equation (6.7)
vectors.
where -
Thus, by the assumption that 1, all factors [YI, (xi) x expressible in the form of Eq. (6.12) -
statement of the theorem holds for k < K
with
Y12 (X2)]LM7 equal to 2 (K
X
and 11 + 12 is
can now
be rewritten
as
L
E
V2KyLM(V)
u
K+L-1
K+1
U2
1
CKLI [YK+l
(X1) X YK+L-1 (X2)ILM
1=0
+
(6.14)
...'
where CKLI is a suitable constant factor and the symbol indicates the terms that axe already expressed in the form of Eq. (6.12) as stated above. in
exactly
By taking (L
the
+
different vectors vi in V2KyLM(V) before, Eq. (6.14) can be viewed as
1)
same manner as
system of lineax equations for [YK+I(XI) X YK+L-1(X2))LM, which is also solvable. The solution yields [YK+I(XI) X YK+L-I(X2)lLm as a
combinations of terms of is clear from the
uniquely
proof,
Eq. (6.12).
note that the
completes the proof. As vector v in Eq. (6.12) is not
This
determined.
It is easy to see that the following result.
repeated application
Theorern 6.3. A vector-coup led product
of Theorem 6.2 leads to
of solid spherical haT7non-
ics
[[[Yll-(Xl)
X
Yl,2(X2)IL12
X
Y13 (X3)IL123
X
X
YIN (XAr)lrm
6. Variational trial functions
86
(-1)11+12, (_j)L123 (_l)L12 (_I)Ll2 N-I+IN can be expressed in
With
=
...
(_I)L12+13,
=
...,
(_ 1) L
and
of a linear combination of
terms
terms N
I
(Xi Xj) kjj V2qyLM(V) -
with 2
of each Uj's.
li i= Ei> j=1 kij + 2q X:N is given by EN 1 UiXi With
kii
j=1
term
+2
=
v
Since the factors roles in lish the
-
1
xj2
=
(xi xj)
and
-
are
L, where the
pprop,riate.
a
vector
co
scalar, they play
v
j cient8
no
active
the rotational motion. Theorems 6.1 and 6.3 estab-
desci ibing equivalence
between the
angular
Eqs. (6.3) and angular momenta a-Te the parity oL the basis
functions of
(6.4)
under the condition that the intermediate
restricted
stated in Theorem 6.3 and that
as
L
given by (_ 1) through the angular momentum L. The basis function whose parity is given by (_l)L is called tohave a natural parityThe construction of a general angular function with unna ural parity must be based on the vector-coupled form of Eq. (6.3). Unfortunately there is no simple function analogous to Eq. (6.4) for the unnatural parity case. One way to construct the angulax function with unnatural parity is for L > 1 function is
OLM (X)
V2K [YL-1(V)
.
X
W]LM
with v
=
iix
and
Sij (xi
w
x
(6.16)
xj),
skew-symmetric matrix which satisfies Sij 0-, YL-I(v) must be replaced -Sji. For the special case of LI with YI(v). A slightly simpler angular function would be possible by introducing another vector v' as follows:
where S is
an
N
x
N
=
OLM(X)
=
V2K [YL(V)
X
VILM)
Vr
=
I?X-
(6-17)
Eq. (6.7) that in the case of K 0 both ki and k-2 are limited to zero and only the stretched coupling, namely 11 + 12 L, is allowed. See Exercise 6.1. With an increasing K value the possible values of partial waves 11 and 12 increase including the case of nonWe note in
=
=
stretched
coupling.
This
applies
to the
case
of many variables
as
well.
6.3
To increase K is thus
one
way to include
Generating ftmction
higher partial
waves
87
in the
calculation. The matrix A of the correlated Gaussian is often assumed to be
diagonal in order
In this
case
to reduce the number of nonlinear
parameters.
has to increase K when the contribution of
one
high
important. However, in the case where expected partial A is not diagonal, additional and important partial-wave contributions come from the cross terms of the exponential part of the correlated 0. E.g., the term exp(-Aijxi xj), when Gaussian even with K expanded into power series, contains many terms of the form (X,.X,)n, which can describe high partial waves associated with the coordinates relation xi and xj. This is easily understood by noting the following to be
is
waves
-
=
for
arbitrary
vectors
(a .,r)n
a
and b:
Bkja
2k
r2k
2k+l=n
E
Yj,,,(a)*Yj,,,(r) M=-1
Bkja 2kr 2k(_1)1,,
F21-+l[Yl(a)xYl(r)loo, (6.18)
2k+l=n
Eq. (6.45) and the addition theorem (6.54) for spherical harmonics. This implies that even the basis function with 0 is expected to be useful if a general matrix A is used in the K variational calculation. The calculation for the dtl-L molecule given in Complement 8.4 will clarify the point discussed in this paragraph.
which results from
=
6.3
Generating function
The calculation of the matrix elements becomes
generating
function for the correlated Gaussian.
simpler if one uses a In fact, the following
function g, which contains an auxiliary "vector" A (8 1, ..., SV), is found to generate the correlated Gaussians of both type I and type 11 =
conveniently:
g(s; A, x)
=
exp
To relate the use
the
following
(-2 ;Mx 9x).
(6.19)
+
generating
function g to the correlated Gaussian
formula
B kja2k+l r 2ky1M (,r)
=
fYjm (ol) (a
.
r
)2k+lda
we
6. Variational trial functions
88
f Y,.(
=
,
(
92k+l
Aa-r
dal
e
a)
OA2k+l
(6.20)
A=O
which is
easily proved by using Eq. (6.18). Then the vector-coupled product (6.3) can be generated from the factor egx of the function g as follows: By choosing si aiti with a unit vector ti we obtain =
N
-I bAx)
exp
YI,., (xi)
2
dii- Y ,,., (Fi)
B01i
Oai
g(a It; A, x)
(6.21) By a symbol alt we mean a "vector" such that given by aiti, where & (a, aiv) with ai =
I
(ti,
...,
tN)
is
I
a
The
x
N
-
- -
one-row
7
each component is a real number and
matrix of Cartesian vectors ti.
in the above
key point equation is Eq. (6.20) which relates the solid spherical harinonics to e ". Since Eq. (6.20) yields a more general term r2kyl., (,r) than just the solid spherical harmonics, it may also be useful to use a simpler relation, which generates just the solid spherical haxmonics. The relation (6.58) serves for this purpose. By choosing tj (1 T,j 2, i(I +,ri 2), -27-j) (j 1, N), we obtain =
=
...,
IV
exp
2
TbAx)
Y1,m, (xi)
N
a Ii-Mi
01i
I
H=1 Climi Oaili O-Fili-mi x
g(alt; A ,X)
1
t =(,-, 2,i(l+, j j j
U=11
---
2),-2-rj)
(6.22)
IN)
where
47r(I M) 1 (21 + 1) (1 + m)! -
Cim
=
(-2)111
-
(6.23)
Note that the vector ti satisfies ti-tj -2(-Fi-'F,j)2, particularly ti 2 0. This formula requires only differentiation in contrast to Eq. (6.21), =
6.3
where both differentiation and
integration
Generating
function
needed. To
are
89
couple
the
spherical harmonics to the desired function in Eq. (6.3), one has multiply Eq. (6.21) or (6.22) by q, and sum over x. To construct the correlated Gaussian-type geminals from g we note
solid to
2RBR
exp
=
exp
(6.5)
f
-2
1
iAx
-
2
BR; A + B, x)
---
(x
to
serves
+
-
R)B(x
-
Eq. (6.6).
f
-2 (x
Jc- Ax
-
2
-
R)
+
(x
-
R)j-
generate the function O(x
It is easy to show that
or
exp
9R) g(s
1 -
0('-R)
The factor
-
R)B(x
-
R)
-
(6.24)
R)
of
Eq.
I
IV X
IIIX,_Ri12ki y1irni (X,
-
Ri)
N
02ki+li
fj BkjIj f dtiYjj.j (ti) Oaj 2kj+1j
i=1
x
x
exp
(
-
-1 kBR
-
2
-J-tR)
g(alt + BR; A + B, x)
(6.25)
1
aj.=O,...,CXJV=O ItAr 1=1 it, 1=1-, ...,
or
fn (A, B, R, x) I
I
expf -2 j Ax 2(x -
-
R)B(x
-
R)j
xip i=1
N
3
anip
fj fj atipnip
i=1
x
-
Rip )nip
P=j
exp( 2kBR iR) -
-
P=I
g(t + BR; A + B,
x))
1
(6.26)
6. Variational trial fimetions
90
where
n
stands for the set of
In,,, n12,n13,...,nNj,nN2,nN31-
Tn the
Cartesian representation the parameter a plays no active role but the x,y, and z components of each vector ti, (tillti2lti.), serve to construct the function
O(x
-
R).
To construct the
vector-coupled product OLM(X) of Eq. (6.3), one has to sum over mi's with appropriate Clebsch-Gordan coefficients in Eqs. (6.21) or (6.22). Apparently this is a very tedious task particularly when the number of particles is large. The choice Of OLM(X) of Eq. (6.4) leads us to the following very simple equation which relates the correlated Gaussian to g. By choosAe with a unit vector e, we obtain for t2 tN ing t1 =
V
=
iiX
fKLM (u, A, x)
' =
BKL
f
=_
exp
YL M (' )
(_2I FcAx (
v
2KYLM(v)
d2K+L
g(Aeu; L WA-2_K+
A, x)
dL
(6.27) We
from
Eqs. (6.21), (6.22), and (6.25)-(6.27) that the are explicitly constructed from the generating function g. Depending on the choice of the vector s, g leads to different forms of the correlated Gaussians, when followed by suitable operations acting on s. These are surn-ma J ed in Table 6.2. The construction of fKLM (u, A, x) is simplest among others and it has a wide range of applications as will be shown in later chapters. The correlated Gaussian of Eq. (6.27) contains only the relative coordinates and the center-of-mass motion is dropped from the outset. Thus there is no problem arising from the coupling between the intrinsic motion and the center-of-mass motion. If one uses the singleparticle coordinatesr instead of x in Eq. (6.27), the coupling between them occurs in general and has to be taken caxe of appropriately in can see
correlated Gaussians
order to calculate the energy of the intrinsic motion. A suitable choice of A and u will, however, lead to the result that the center-of-mass motion in
separated from the intrinsic motion. This will be discussed Complement 6.4. Chap. 2 we discussed the linear transformation of the coordi-
can
more
In nates
x
be
detail in
induced
by tJae permutation
P. It is
important
to Imow the
6.3
Generating
function
91
Relationship of the two types of the correlated Gaussians to generating function g of Eq. (6.19). The symbol alt indicates a set of vectors f aiti amtN I. Bkj and Cl,,, are defined in Eqs. (6.9) and (6.23), respectively. Table 6.2.
the
Correlated Gaussians
-I.;v2
exp
Ax) rIN
i=1
(I rIN
1
P01-i f
-
i= 1.
2
i= I
i=1
x
Ylimi(Xi) ali-Mi
ali
I
rIN
jg(alt; A, x))
diiYiini(Z)
-!.,'cAx) IIN
exp
Y1 irni (Xi)
climi aaili ajli-mi
g(alt; A, x)
I
t =(I-, 2,i(l+, j j j
2),-2-rj)
i=O"ri=O (i=l,.. N) .,
ld Ax) IV12Ky f dgYLm( ,) (
exp 1 =
Correlated
Uixi)
i=1
,
x)
).X=O,e=je-j=3-
Gaussian-type geminals
-UMx
-
(In,iv exp
.1 2
2
x
EN
=
d2K+L j,-X2K+L g(Aeu; A,
F3 K-L
exp
(V
J(V)
L IV
2
I
-
,
Bkjjj 2
(x
-
R)B (x
f dii-Yi
RBR
-
I fT7
R)
=,
Ixi _RZ12ki Y1 imi(Xi_ 14)
a2ki+li iM
JiR) g(a It
-
+
BR; A + B, x)
expf .1.7cAx I(x--RR)B(x R) I IIN JJ'=j(Xip =ffrff, lip-=, t-,77ri- jexp(-1:YZBR-!R) g(t+BR; A+B, x))
-j=O,jtjj=j
3
-
-
-
-
, i=-
2
2
3
a7"P
i=
2
p
P
x
ti=o
p
-
Rip )ni,
6. Vahational trial functions
92
effect of P
the correlated Gaussians. Since the correlated Gaus-
on
generated from the generating function, it suffices to examproperty of g due to P. By using the relation Tpx (see Eqs. (2.26) and (2.29)), we obtain a very simple result
sian is
ine the transformation
Px
=
Pg(s; A, x)
=
g(Tp_.9; Tp-ATp, x).
(6.28)
One
only needs to change the matrix A and the vector s appropriately. An important fact is that the generating function preserves its functional form under P. This is also true for a more general linear transformation T of the coordinates x, e.g., a transformation from one set of coordinates to another. Combining this fact with Eq. (6.27), we
obtain
a
very useful
property of the correlated Gaussian fKLM-
Namely for the transformation of Tx
TfKLM(u, A, x)
7--
=
Tx,
we
have
fKLM(TU TAT, x).
(6.29)
Thanks to this nice property one only needs to redefine the parameters A and u of the basis function to construct the transformed wave function. This
property plays
important role
an
in
evaluating
the matrix
elements. The
generating function (6.19) plays a key role in generating the and, moreover, facilitates the evaluation of the
correlated Gaussians matrix elements of
physical operators.
the formulation based
It is therefore desirable that
the
generating function In a many-body system as well. In extending the results to laxger systems of identical paxticles we need to cope with the symmetry adaptation one can use
of the
the
wave
function. Tn such
generating
function
were
a case
on
it would sometimes be useful if
expressed
in
an
"uncorrelated7 form of
the coordinates x, because the symmetry adaptation can then be shnplified by using the technique of Slater determinants or permanents. In fact to the
we can show that the generating function can be related product of the Gaussian wave-packets centered around -4 =
(R,,..., RN) through an integral transformation. Using the definition Eq. (6.51) of Complement 6.1, we can express the product of the
of
Gaussian
wave-packets
as
N
det-V
_'i
ORj (Xi)
)
4
1 exp
:iFx + 2
k_Vx
1 -
2
kFR) (6-30)
with
an
N
x
N
diagonal
matrix
6.3
-YJ
0
0
72
Generating ftmetion
93
0
...
(6.31)
0
'YN
A direct calculation using
1:Mx +
exp
2
2
x)
dx
-9A-18
exp
detA
following equation which relates product of the Gaussians of Eq. (6.30): proves the
2
g of
Eq. (6.19)
(6.32) to the
g(s; A, x)
(detr)3
4
expf 'g(r, Arisl -
-
(47r-) N (det(rT
2
N
g(_V(.V-A)-1s;A(F-A)-'FR)
X
'Y'
ORi (xi)
dR.
(6.33) Note that
A(r
From this
we see
A)-' r
r(r
A)
-1
r
F is
symmetric matrix. The function g in the right-hand side of the above equation depends on the integration variable R and serves as just a weight which is needed to convert the product of the Gaussian wave-packets to the generating function. Equation (6.32) is proved in Exercise 6.2. We have seen that the correlated Gaussians are all generated from the same generating function as summa ized in Table 6.2. Furthermore, the latter can be obtained by the integral transformation of Eq. (6.33) involving the product of single-particle Gaussian wave-packets. -
=
-
that the calculations
-
can
a
be reduced to the matrix
N-particle wave functions involving the product of the wave-packets. The width parameters -yi of the wave packets can be chosen arbitrarily. To choose uniform width parameter for all of them is most convenient if xi indicates the single-particle coordinate of the identical Particles, because then the permutational symmetry of the wave function is simply reduced to that of the "generator coordinates" R. Even when x denotes the set of relative coordinates, this elements of
Gaussian
6. Variational trial fimcdons
94
nice property
be used
by including the center-of-mass coordinate as [311. integral representation of the generating function has been successfully employed in accurate solutions of few-body problems [311 as well as in microscopic descriptions of nuclear systems can
shown in
The
in multicluster models
6.4 The
X.L M
[29, 30].
spin fanction
=
1
cosO -1 .1 (0),
2
Alternatively,
22
.1; 2
-IM) 2
.1 + sin# .1 22 (1),
may set up the
we
.1; 2
.1 2
M).
spin function with
(6.34) a
continuous
parameter such as that of Eq. (6.34) by an elementary method instead of using the successive coupling. Its merit is that the construction of the
spin function is simple, that the evaluation of spin matrix elements
is easy, and
V in
moreover
Eq. (6.34),
that
one can use
continuous parameters such
The construction is done
as
follows. The
Young diagram
spin function with spin S for the N-fermion system
[(N12)
+S
as
in the variational calculation.
(N12)
-
S].
The maximum
weight function
is
for the
[n+ n-I
with M
=
=
S,
XSS, must have n+ spin-up functions and n- spin-down functions. The number of terms distributing n+ spin-up functions among the N
particles terras:
is
NS
=
(,+). Therefore xss is expressed n+
as a sum over
these
6.4 The
spin fimcdon
95
NS
XSS (A)
=
E Ai ii)
(6.35)
-
i=1
characterizing the spin function, satisfy the condition, are not independent 0, where S+ is the spin-raising operator. Acting with S+Xss(A) S+ on XSS (A) yields n+ + 1 spin-up functions and n- 1 spin-down N functions in each term and thus leads to (n++,) independent terms.
ANs)
The coefficients
of. each other but must
=
-
Since the coefficient of each term must N
I
V (n+ 1) (nl+) +
1
=
vanish,
one
(2S + 1)N! + S + 1)! (IN (IN 2 2
has
-
S)!
(6-36)
independent parameters to specify the vector A completely. Here the last minus one of Eq. (6.36) comes from the normalization of the spin function. XSM is easily obtained from Xss(/\) with the use of the spin-lowering operator and thus denoted as XSM(A). The overlap of two spin functions is independent of M and sh-nply given by iV. The independent pa(XSS(A)IXSS(/X')) (Xsm(/\)IXSM(/\')) varied be /X can continuously in the variarameters needed to specify =
=
tional calculation. The ner.
isospin
exactly the same manthe spin and isospin parts in Eq. (6.35), leading to a
function is also constructed in
permutation P on reordering of the terms
The action of the
simply produces
a
linear transformation of the vector /X to another vector denoted
A(P).
96
Complements
Complements 6. 1 Nodeless harmonic-oscillator functions
as a
basis
In variational calculations for bound states it is very important to have a set of basis functions which can approximate square-integrable
functions to any desired accuracy. For a single-variable function one may use the well-known complete, orthonormal eigenfunctions of, e.g., the harmonic-oscillator
(HO)
Hamiltonian. Or
one may try to use functions which may not be complete mathematically able to cope with many problems flexibly and, from a practical
nonorthogonal but
are
point of view, accurately. The amine the
possibility
functions. We will
by studying
purpose of this
of nodeless HO functions the
see
performance
Complement
is to
ex-
set of such "basis"
as a
of the nodeless HO functions
how well
they approximate a given function f (r). The nodeless HO functions with angular momentum Im have
continuous
parameter
-Palm (T)
=
Nal
( V3)
14 I/r
exp
2
T
with
a
solid spherical harmonic
the normalization constant
(2 1+2 a"
a
a:
2)y7n(,VF
(6.37)
I/r
1
(see the following Complement). where
Nj
is
given by
2
2
Nal
Here
(6-38)
=
(21 + 1)!! is introduced to scale the
length. The Gaussians employed in Complement 8.1 to solve the ground state of the hydrogen atom are 0. The overlap of two nodeless HO functions is special cases with I simply given by v
=
(-Vaiml-Va-'im)
aa1
Na I IVa'I
2
2
(6.39)
_
=
.
+.,1)2
(a+a)2 2
2
We attempt to approximate a normalized function momentum ITn in terms of combinations of
fl,,, (r)
with
angular
K
Am (T) The ance
error
cirailm (T) of the
approximation
(6.40) is estimated
by calculating
the vaxl-
C6.1 Nodeless harmonic-oscillator functions
0-
2(f)
=
fI
cillil,,-,
,,
97
a
set of
Or)Idr.
parameters
(P.41)
jal, a2,..., aKI by
the trial and
may set them up
by SVM. Once they are selected, minimizing 0-2 to determine the ci's priate way
basis
2
fl (r)
One may choose or
as a
error
leads to
some
appro-
procedure a
linear
of the
equation
K
E(Failml-Vajim)cj =' (Faiiralfim)
(i
=
1,
...,
(6.42)
K),
j=1
andthena 2 is As
a
I
given by
test function
K
-Y:ij=iCi*Cj(-Vai1mIrajIm)-
fim(r)
we
first take up the HO
wave
functions
angular frequency w hy/m, where n is the number *,,Im (r) of radial nodes. We employ a phase convention such that the radial HO wave functions are all positive for r greater than the outermost nodal is needed to point. The overlap of the functions Fal (r) and calculate o-2. It can be obtained by using a generating function of the with the
=
..
HO functions
Ik
4
A(k,,r)
exp
2
+
Nf2-vk-r
2
1 VT
-
2
21 (6.43)
0,,jm(r)*P,,jm(k), n1m
polynomial of the complex Bargmann space variable k, is the Bargmann transform [76, 771 of the HO wave functions in a spherical basis and its explicit form is given in terms of the solid spherical harmonics of (4.5) [781
where
Pnjm(k),
a
:(2
P,,lm(k) where
Bn1 n -
1) n + 1)! (2n
(6-44)
(k k) ny1m (k), -
Bnj, defined in Eq. (6.9), is the coefficient needed to Legendre polynomials Pl(x):
express xm
in terms of the
xm
21+1
E
=
4r
(6.45)
BnIP1(
2n+l=m
Expanding
(6.44),
we
the
overlap (F ,lm (r) I A(k, r))
obtain
in terms of the
polynomials
Complements
98
(-PaimlOnlm) Obviously,
(2n + 21 + 1)!! (2n)!! (21 + 1)11.1
the
1
(1+ )n (
2n)+!!
-
approximation
in
-
a
Eq. (6.40)
2
2Va
a
1 +
(6.46)
a
becomes less trivial
the number of radial nodes of
as
1 f increases. For Onlm (r) wit"h n 2, the combination of merely a few (::5: 3) terms leads to very good 5 it is hard to make u 2 less than approximations, whereas for n 10-10 with a double precision calculation. This is so because all ai's tend to take a value of unity, reflecting the fact that such a HO function can be expressed by combinations of higher order derivatives of Falm(r) with respect to a at a 1, and the optimization procedure tends to construct such derivatives out of -Tal,,, of almost equal parameters. One thus has to deal with an almost singular overlap matrix ((Faiiml-Vajim)) to obtain the solution of ci's. In stead of approximating each of the 0,,,. (r) with different sets of =
or
=
=
1),
(j:IV n=O 0r2(0nIM))/(N-.L
minimized the average value, (U2 )N with a single set of ai's. The values of the
ai's,
we
follow
a
=
geometric progression and
cessive terms
were
determined
of ten nodeless HO functions The next test
example
k
ai's
were
assumed 11-o
its first term and the ratio of
by Powell's method [141.
yielded (a 2)8
=
0.6
x
suc-
A combination
10-10 for I
is the shifted Gaussian defined
=
0.
by
I
( 3)4 (_IV(,r2+S2))jI(I/'Sr)y V
(r)
=
Fj
exp
-
where the function
ii(x)
(6.47)
7 IM
2
7F
is the modified
spherical
Bessel function of
the first kind
ij(X)
=
F '7-x'I,+.! (X) 2
X2k+1
1: (2k)!! (2k + 21 + 1)!!'
(6.48)
k=O
and where the normalization constant Fj is 4e
2
'us
with
F,j
2
(6.49)
2
To get the normalization constant, 00
fo
x
e-ax21, (bx) 1, (ex) dx
which is valid for Rea
>
Gaussian
wave-packet
I =
2a
0, Rev
shifted Gaussian because it is
we
>
have used the formula 2
exp(b +c2),,'(bc)'
-
4a
1. The function
closely
centered around
2a
(6.47)
related to the s
(6.50) is named
single-particle
C6.1 Nodeless harmonic-oscillator functions 3
,ps' (r)
99
(6.51)
11
-
2
T
through
basis
1 exp,
-
as a
the relation
4V'7-r Os
(6.52)
1M
Fj 1ra
Here
use
eVr_5
is made of the
=
evrscos'o
=
equation
E(21 + 1)ij(vrs)Pj(cosO)
(6.53)
1
with the rem
angleO between r and spherical harmonics
s, and the well-known addition theo-
for the
I
4w-
Pi (CosO)
1:
=
21+1
yjm(i )Yjm( )*
ra=-l
=
4-x-(-I)' /2_71-+I
lyl(p)
X
(6.54)
YIM100.
8 The radial paxt of the shifted Gaussian is hence peaked at r shifted the be to therefore would It approximate challenging -
/_2, Iv.
Note that large s in terms of the combinations of A e-211 A of of in terms as Eq. (6.43) expressible (NF2'91 T). 0,v(r) The overlap between the functions F,,la(r) and 7p,jn(r) is
Gaussian with
1
2
is
(r,im 1,0SW
v/-2-
N,, 1 Fs
1(1 + a)'+' 2
P(_ 1+a
ex
V I/
(6.55)
Assuming again that the ai's follow a geometric progression, 0-2 was minimized by Powell's method for a given value of ;. For the case 2 0 it is possible to make o- < 10`0 with ten nodeless HO of I =
functions for the shifted Gaussians of up to :! 10. All ai then tend to take values close to each other for large,;, and this requires a very
equation (6.42). Increasing the number 2 of nodeless HO functions gives us even smaller o- values and widens precise solution of the
the range of maximum approximated well.
linear
(;
value in which the shifted Gaussian
can
be
examples strongly suggest that the nodeless HO functions can approximate square-integrable functions to any desired accuracy, though the number of nodeless HO functions needed depends The above
Complements
100
on
the
shape of the
test function. A remarkable
shows that the linear function
approximated to high
r
itself, defined in
accuracy in terms of a
in
example given
[66]
finite range, can be combination of Gaussians a
e-ar2 ) We stress two remarkable points of the nodeless HO functions. One a combination of a few nodeless HO functions can approximate
is that
the shifted Gaussian is easy to make
2 a
even
with
less than
nodeless HO functions. In
an
a
large
value of
10-1-0 with expansion
a
;.
For
example, it only 15 HO functions,
combination of
in terms of
would need many more terms to obtain the same accuracy. This indicates the flexibility of a nonorthogonal Gaussian "basis" compared one
to the
orthogonal basis such as the HO functions. Another point is that there are many, possibly an infinite number, of sets f a,,..., aKI which approximate a given function equally well, even though the number K is fixed. This is what we have experienced in the above examples. We display graphic illustrations of Gaussian expansions for different functions with I 0. The most appropriate nonlinear 0, m paxameters ai are determined by optimization, while the linear ones ci are given by the solution of the least square equation (6.42). To point out the dependence on the number of Gaussians, we use different numbers of terms in the expansion (K 5, 10 and 20). In the first example we approximate the HO function 05oo(r) by =
=
=
Gaussians. This function is smooth but oscillates and
asymptotically falls
off like
a
(it
has five
nodes)
Gaussian function. As shown in
10 and K 20 Gaussians give a perfect fit to the 6.1, K function, so these curves are practically indistinguishable. =
In
=
the second
to fit
Fig.
exact
an exponential function e-', the wave function of the ground state of the hydrogen atom. The asymptotics of this function is quite different from that of a Gaussian. To approximate the asymptotic part of this function one needs many terms of Gaussians as is illustrated in Fig. 6.2. By increasing the number of Gaussians one has better and better agreement in the asymptotics. After a certain distance, the Gaussians fall off much more rapidly than the exponential function. In many practical applications, especially for bound states, however, one can always use enough Gaussians to reach the required accuracy. We note that the Gaussian fit gives a poor value (zero) for the derivative at the origin (the exact value is -1) as will be discussed in Complement 8.1. In the next example we try to approximate the absolute value fimction f (r) 12.5-rl. The Gaussian expansion, again, does a pretty
f (r)
=
=
case
we
attempt
C6.1 Nodeless harmonic-osefflator functions
as a
basis
101
10
5
0
-5
-10 0
4
2
6
8
r
0, m 5, 1 0) and its Fig. 6.1. The harmonic-oscillator function (n approximations by Gaussian expansions. The solid curve is the (UM3.ormalized) harmonic-oscillator function f (r) V 500, and the dotted, dashed and 5-, 10- and 20-term Gaussian expansions. long-dashed curves are the K The 10- and 20-term. expansions are practically indistinguishable from the -
=
exact
curve.
100
IT
-C
20
10-40
--60
10
40
10
0
10
30
20
40
50
r
Fig. 6.2. The exponential function and its approximations by'Gaussian exp(-r), expansions. The solid curve is the exponential function f (r) and the dotted, dashed and long-dashed curves are the K 5-, 10- and 20-term Gaussian expansions. =
=
102
Complements
good job (Fig. 6.3), would expect
of further Gaussians
by the inclusion
and
an even
one
better fit.
3
2
V
1
0
0
1
2
I
I
.
3
4
5
r
Fig. 6.3. The absolute value function and its approximations by Gaussian expansions. The solid line is the absolute value function f (r) 12.5 -,rl, and the dotted, dashed and long-dashed curves are the K 5-, 10- and 20-term Gaussian expansions. =
=
The approximation for the step function, f (r) 1 if 7- < 2.5, 2 if r > 2.5, is less impressive (Fig. 6.4), but it goes without f (r) that it is not trivial to fit that function. One can improve the saying =
=
fit
by increasing
the number of
Gaussians,
be pretty slow. The last example shows that
adequate
in
some cases.
Let
us
with Gaussians. This function is it simulates the behavior of
a
try
but the convergence
might
Gaussian expansion may be into
approximate f (,r)
practically
a wave
zero near
function of
a
the
12 =
r
2
e-T
origin, and
system -with very
core. The Lenard-Jones or other hard-core potentials such a function. Figure 6.5 shows that the Gaussians produce may generally give a good fit to this function. By scrutinizing the inner
strong repulsive
part of the approximation
reality
(see Fig. 6.6)
the Gaussians do not
produce
one can
exact
see,
however, that
zero near
the
in
origin, but
C6.1 Nodeless harmonic-oscillator functions
as a
basis
103
3
2
C, -
0 0
Fig.
6.4. The
1
2
3
4
5
step function and its approximations by Gaussian expansions.
The solid line is the step function f (r) = I if r < 2.5, f (r) and the dotted, dashed and long-dashed curves are the K 20-term. Gaussian
2 if
r > 2.5, 5-, 10- and
expansions.
the
approximate function oscillates around the exact one even for 20 Gaussians. This oscillation may be very unpleasent: To tame the hard core potential, we need a wave function which is effectively zero near the origin. The oscillating function leads to numerical problems, which makes it rather difficult to obtain the solution of interaction has
a
very
It is
problems
where the
strong hard-core part.
to know the
important completeness of the basis functions in L (square integrable functions), H1 and H2 (first and second Sobolev) spaces because the bound state in quantum mechanics is traditionally 2
formulated in L 2, and the mathematical solution of the
equation is formulated in H' and H2
spaces.
(The
Schr8dinger
space HI is
a
set of
functions whose derivatives of up to the mth order are all square integrable. Note that the kinetic energy operator requires the second order derivative of
a
basis
variational method
ple
in
[791,
and the
function.) are
The convergence properties of the Ritz demonstrated in H1 and H2 spaces, for exam-
completeness
in L 2 is not sufficient to
guarantee
the convergence of the Ritz method. The proof of the completeness in the space of L 2 and in the spaces of H' and H2 is presented in
[801.
This work proves that any function
can
be
approximated
to any
Complements
104
200
150
100
50
0
-50
-j
-100 0
1
2
3
4
5
r
Fig.
expansions. dashed
The solid
curves are
10 and 20 term
curve
the K
=,r
and its
is the ftmction and the
=
expansions
12e_,r2
approximations by Gaussian dotted, dashed and long5-, 10- and 20-term Gaussian expansions. The
f (r)
6.5. The fimction
are
practically indistinguishable from
the exact
curve.
0.002
.
.
.
.
.
i
.
.
I
I
0.001
0.000
-0.001
-0.002 0.0
0.1
0.2
0.4
0.3
0.5
0.6
r
in Fig. 6.5, but the inner paxt is 5- and 10-term Gaussian expan i n scope and not drawn here.
Fig.
6.6. The
curves
same as
with the K
=
magnified. axe
The
out of the
C6.2 Solid
prescribed
accuracy with
spherical
linear combination of
a
that the number of Gaussians in the the parameters of the Gaussians
harmonics
105
Gaussians, provided
expansion is sufficiently large and
are
appropriately
chosen.
6.2 Solid
spherical harmonics The solid spherical harmonic Yl,(r) r1YI,(i6) of Eq. (4.5) is a 0. 0, of the Laplace equation, V2f (,r) solution, regulax at r The irregulax solution is given by r-1-'YI (f). The solid spherical harmonic is a homogeneous polynomial of degree I in the Caxtesian =
=
==
..
coordinate:
21+1 +
rlyl.(,P)
(1 + Tn)! (I
47r
-
m)!
p! q!
p
-
q)!
pq
(_X+i )p (X _iy)q Y
X
2
where p and q p + q <
1,
p
-
=
(6.56)
7
positive integers which satisfy the conditions The complex conjugation leads to Yl,,(,P)*
are
q
ZI-p-q
2
m.
(-I)MY1-09It is easy to show that with 7-21i(l + 7-2) -2-r) we obtain
(t-T)i
=
(X
+
iY
2 -
-F
(X
_
iY)
I
M)! where C1. is it
(
given
in
vector t
(I
-
2TZ)I
-F'-'YI,,, (r),
C1
=
(6.57)
Eq. (6-23).
the condition 0 < 1
-
m
<
al
aal This
_
special complex
For a given 1 the power of -F must 21, which guarantees the range of m should. Equation (6.57) leads to the equality
satisfy as
a
eat-r
=
CIMYIM(r)-
(6-58)
a=O
simple relation was used
in Sect. 6.3 to
generate the angular part
of variational trial ftmctions. 2
By expressing the Laplacian as V2 the solutions of the
I =
a
T2- Yr-
a (r 2 -a-,
we see
that
equation
[V2_(n-1)(n+1+1) I f(r)=0 r2
(6-59)
Complements
106
expressed
be
can
When
n
1
=
in
or n
polar coordinates as r'Y,,,,(r) or -1 1, the equation reduces to
=
-
the
Laplace
equation. The inverse relation of
Eq. (6.56)
where 1 takes the values n,
given
it is
as
follows:
n
-
2,
...,
0
or
(6.60)
npq 1. A hint to calculate BIM
in Exercise 6.3.
jjIM1j2TtI2j3Ta3)
IT
expressed
BPq Yl,,, (i6),
XPY qZn-p-q =,rn
is
is
(6.61)
j
=
U1 +j2) +j3
=
il
U2 +j3)
+
possible
J12 +j3
to choose
ii
a
+
(6.62)
J231
basis in which
j212, j2, J, j2, 2 j2, 3 1 j2,
are
diagg-
onal
(6.63)
ljlj2 (J12) h; jM) 7
or a
basis in which
2 j2, 1 j2, 2 j2, 3 123, J2i Jz
jil j2h (J23); JM) i
expressed
in terms of
ljlj2 (J12), h; JIVI)
diagonal
(6.64)
-
The transformation from the it is
axe
=
a
one
basis to the other is unitary and
U-coefficient
as
E U(jlj2 Jh; J12 J23) jil, j2h (J23); JM) J23
C6.3
Angular
momentum
recoupling
107
(6-65) The U-coefficient is often called
by
the
overlap
U(jlj2 Jh; J12 J23) and is in fact
a
unitary Racah coefficient.
It is
given
of the two basis states
(il j2h (J23); JM ljlj2 V12) h; JM) (6-66)
:::::
independent
1
i
i
of the
magnetic quantum number
M.
By using Clebsch-Gordan coefficients, the unitary Racah coefficient can be expressed as
(jITn1j2?n2jjI2MI2
U(jlj2 Jh; J12 J23) MIM2M3MI2M23 X
(JI.2Ml2j3M31JM)(j2M2j3M3lJ23M23)UIMIJ23M231jm)(6-67)
Note that in the above
equation M
is fixed to
certain value in the
a
range of -J < M < J, so that actually only two m values are independent in the summa i n. As the Racah coefficient is real and
unitary, the (U-1 Ut =
inverse transformation of
Eq. (6.65)
is
simply given by
CT)
=
U(j1j2 Jh; J12 43) ljlj2 (42) h; JM)
jil j2h (J23); JM)
1
7
J12
(6-68) If the basis of the it is
Eq. (6.63)
is to be transformed to
and
angular momenta j, coupled with i 2 to
transformation coefficient is
ljlj2 (42) h; JM) 1
=
=
j3
axe
the total
first
angular given by
coupled
a
to
basis in which
J13 and then J, then the
momentum
(-l)jl+j2-J12 ji2il (J12)
(-l)jl+j2-J12 1: U(j2jI Jh; J12 J13) jj2
i
i
h; JM)
j1h (J13); JM)
J13
=
E(-J)jl+J-J12-JI3 U (j2jl Jh; J12 J13) ljlj3 (J1-3)
7
j2; JM)
J13
(6-69) unitary Racah coefficients are transformation coefficients becomplete sets of states, so that they obey orthonormality and
The tween
Complements
108
completeness relations. The orthonormality condition (jl,j2j3(J231); JMjjI j2h (J23); JM) Jj ,3 j,,,,, reads, with the use of Eq. (6.68), as 7
E U(jlj2 Jh; J12 J23) U(jlj2 Jh; J12 J231)
:_
(6-70)
JJ23 J23'-
J12.
The
relation may be
completeness
E (jlj3 (J13)
7
expressed
as
j2; JM ljlj2 (J12) h; JM) i
J12
(jlj2 V12) h; JIVII-ljlij2j3(j23); JM)
X
7
-
i
i
in terms of the Racah coefficients to
equation can be transcribed following relation
This the
(6.71)
Ulh V13) j2; JMIj1 j2h (J23); JM)
E (-I)jl +J-J12 -JI3 U(j2jI jj3; J12J13) U(jlj2Jj3; J12J23) J12
(-1)32+j3-J23 U(jlj3Jj2; J13J23)There
are
symmetry relations with respect
among the six seen
curly
(6.72) to the
interchange
momentum labels. These relations
angular by introducing the
so-called
6j symbol [70, 72, 751
can
be best
written
by
brackets
U(jlj2Jj3;j]-2J23)=(-I)jl +j2 +j+j3 Af(2JI2+1)(2J23+1) X
1
il h
j2 J
because the symmetry relation of the one can show that
U(jlj2Jj3; J12J23)
=
(-l)j2+J-JI2-J23
=
is rather
+ 1)(2J, J1 223+1)
(2j2
+
simple. E.g.,
U(Jj3jlj2; J12J23)
U(jlJj2j3; J2342)
=
rL2
U(Jl2jlj3 J23; j2 J)
(6.73)
6j symbol
U(j2jlj3J; J12J23)
U(j3Jj2jl; J12J23)
X
J12 J23
1) (2J + 1)
C6.3
If
one
U(j2 J12 J23 J; j1h)
of the
(6.74)
-
efficient has the value +1. If J12 =
=
109
recoupfing
2
labels, ji, j2, j3,
U(jlj2jlj2; J120)
momentuin
(2JI2,+ J,2+ 1)(2J23 + 1) + 1) (2j3 + 1) (2j, +
(_l)jl+j3-J12-J23 X
Angular
or or
J, is zero, the unitary Racah J23 is zero, then we have
co-
U(jljlj2j2; OJ12)
(-l)ll+j2-J12
p2
2J12 +1
+ 1) (2j2 2j,+ (2j,
The above discussion is extended to the
case
+
(6.75)
1)
involving
four
com-
muting angular momentum operators, jlj2,j3,j4. A basis with a good total angular momentum J and its z component J, is constructed as, for example,
jj1j2(JI2)J3j4(J34);JM)
or
ljlj3(J13)ij2j4(J24);JM)- (6-76)
The transformation ftom. the basis in which J12 and
quantum numbers numbers is
to the basis in which
J13 and J24
are
J34 are good good quantum
again real and unitary:
ljlj2 (J12) j3j4 (J34); JM) i
J13 J24
31
32
h J13
j4 J24
J12 J34
ljlj3 (43) j2j4 (J24); JM) 1
-
J
(6-77) The transformation coefficient is
recoupling coefficient between four angular momenta called a 9j symbol in unitary form. It is independent of M and given by (jIj3(JI3)J2j4(j24); JMjjIj2(j12)J3j4(J34); JM)i a
that is
il h J13
j2 j4 J24
J12 J3 4 J
M I M 2 M3 M4
IVI 2 M3 4 MI 3 M2 4
(jlMlj2M2ljl2MI2)(j3M3j4M4lJ34M34)(Jl2Ml2J34M341JM) (il7nlj3Tn3lJl3Ml3)(j2M2j4M4lJ24M24)(Jl3Ml3J24M241JM)(6-78)
Complements
110
again that M is fixed to a certain value in the range of -J < M < J, so that only three m values axe independent iTi the above summation. The inverse transformation is given by
Note
ljlj3 (J13) j2j4(J24); JM) i
A
j2
h
j4
J12 J34
J24
J
J13
J12J34
ljlj2 V12) j3j4 (44); JM) i
(6-79) The
orthonormality and completeness relations for the unitary 9j
coefficients
can
il h J13
J13 J24
be derived
as
before:
J12 J34
j2 j4 J24
il h J13
j2 j4 J2 4
A
J2
h J13
j4 J24
J12 J3 4
J
(-l)j3+j4-J34 J12J34
L
j3-j4-J23+J24
There
are
many
il j3 J13
JJ12JI2"JJ34JS4"*
J
J
J
A
j3
j4 J14
j2 J23
L
il j4 J14
j2 h J2 3
J1-2 J3 4 J
J13 J2 4
j
(6.80)
J
symmetry properties of the unitaxy 9j coefficient.
These will have their
[70, 72, 751
J121 J341
simplest form in curly bracket
terms of the familiaz
9j symbol
written in
j2 j4 J24
J12 A4
-\/(2JI2+1)(2J34+1)(2Jl3+1)(2J24+1)
J
X
1
il h J13
j2 j4 J24
J12 J3 4
Using the symmetry properties of the 9j symbol, il h
j2 j4
J12 A4
j2
j4
J13 J24
J13
J24
J
J12
J34
J
J3
(6-81)
J we
have for
example
C6.3
Angular
momentuin
j + 1) (2J34 + 1) (243 + 1) (2J12 (231 + 1) (2h + 1) (2J + 1) (2jj -
-
(243 + 1) (2J24 + 1) (2J34 + 1) =V, 3
(2j3 (233(2. 3.3 + 1) (2j4 + 1) (2J + 1)
(-l)jl+j2+j3+j4+JI2+J34+JI3+J24+J
Ill
recoupling
j2 j4 J24
J12 J3 4 J
j3 J13
J13 il j3
J24
i
j2 j4
J12 J34
33
34
il J13
j2 J24
31
J34 J12 J
(6.82) For
One may consider other basis states than those given in Eq. (6.76). example, the basis in which J12 and J34 are diagonal can be trans-
formed to
pled
as
basis in which the
a
angular
momentum is
successively cou-
follows:
ljlj2 (42) j3j4 (J34); JM) 7
E U(J12j3 Jj4; J123 J34) jj1j2 (J12)
7
j3 (J123) j4; JM) 7
-
J1,23
(6-83) The
9j coefficient
rewrite the state in
coupling,
as
be
expressed in terms of products of unitary Eqs. (6.69) and (6.68) enables one to Eq. (6.83), which is obtained by the successive can
Racah coefficients. The
use
of
follows:
jj1j2 (J12) h (J123); j4; JM) i
(-J)jl+J123-JI2-JI3 U(j2jlJl23j3; J12J13) J13 X
jj1j3 (43), j2 (J123), j4; JM
(-j)j1+J123-J12-J13 U(j2jlJl23j3; J12J13) J13 J24 X
U(J13j2 Jj4; J123 J24) jj1j3 (43), j2j4 (J24); JM) (6.84)
Complements
112
Substituting this result leads to
useful
a
il h J13
j2 j4 J24
into
Eq. (6.83) and comparing with Eq. (6.77)
identity J12 J34 J
E(_j)jI+JI23-JI2-JI3 U(Jl2j3Jj4; J123J34) J123
U(j2jljl23j3; J12JI3) U(Jl3j2Jj4; J123J24)-
X
(6.85)
application of the above equation, we note that if one of angular momenta is zero the 9j coefficient is expressible i-n terms of Racah coefficients. E.g., we have As
an
nine
ilh J13
0
31
j4 j4
A4
U(jlj3Jj4; J13J34)i
J 0
h J13 0
h h
6.4
j4 J24 j2 j4 J24
:
6
J
2 243 + 1
2j, (2j,
J
+
1) (2j3
+
1)
U(J13jI Jj4; h J24)
i
32
(-l)j4+J-J24-J34 U(j2j4Jj3; J24J34)- (6-86)
J34 J
Separation of the center-of-mass
motion from correlated Gaussians
The correlated Gaussian
(6.27) in the global vector representation is, among the several possibilities of the correlated Gaussians, most easily constructed from the generating function. Thanks to this simit has
wide range of applications. The relative coordinate (xi, ___j xiv-1) is used to represent the correlated Gaussian, so that the center-of-mass motion is dropped from the beginning. The
plicity ;
a
=
aim of this
Gaussian is
Complement is to expressed in terms
fKLM (u, A, r)
=
exp,
show
that, even when the correlated of the single-particle coordinates r as
(_2'fAr) V2Ky M(V) L
C6.4 Separation of the center-of-mass motion
113
N V
(6-87)
wri,
possible to separate the intrinsic motion, expressed in terms of the relative coordinates, from the center-of-mass motion by requiring some special conditions on the parameters A and u. Here A is an N x N symmetric matrix and u is an N x I one-column matrix. Therefore, in it is
the correlated Gaussian basis the
always absolutely necessary used
use
of the relative coordinates is not
but the
single-particle coordinates can be following. correlated Gaussian of Eq. (6.87) can be
well. We shall show the conditions in the
as
By using Eq. (2.4), the expressed by the relative coordinates x and the center-of-mass dinate xN, which are again denoted as x:
fKLM (Ui Ai 7)
=
fKLM (U A! X) i
coor-
(6.88)
7
with
A'
=
U---IAU-'
The matrix elements
U,
and
A!Nj
=
=
A!jN (i
U--IU. 1, give
=
...'
and center-of-mass coordinates and
(6.89) N
-
1)
rise to
pendence of the function. The element A!Nlv
is
a
connect the relative a
center-of-mass de-
parameter describing
the center-of-mass motion which has Gaussian form. Thus the separation between the intrinsic motion and the center-of-mass motion is made
possible by requiring that
XNi where
c
=
is
0
an
(i
=
I,-, N
1)
and
XNjV
=
c,
arbitrary positive constant. As (U-I)iN=l (i
the above conditions N
-
are
rewritten
as
=
11.... N),
follows:
N
I:I:Ajk(U-I)ki=O
and
j=1 k=1 N
N
E 1: Aik
=
(6-90)
C-
j=1 k=1
The center-of-mass contamination in the
YLm(v)
can
be removed
by requiring that u1V
angular function V2K =
0, that is,
IV
Euj i=1
=
0.
(6.91)
Complements
114
and
(6.90)
The conditions
(6.91)
ensure
that the correlated Gaus-
is free from any contaminations due to the center-ofmotion. The intrinsic motion is described by the correlated
(6.87)
sian mass
Gaussian of type (6.27) and the center-of-mass motion is given by exp(-cx'N /2). The number of parameters contained in the function
N(N + 1)/2 + N, whereas since there axe (N + 1) condi(N 1)(N + 2)/2 free parameters. This number is of course equal to the one which the correlated Gaussian (6.27) has: (N 1)!V/2 + (N 1) (N 1) (N + 2)/2. It is instructive to see how Eq. (6.90) leads to the separation of the (6.87)
tions
is
we
have
-
=
-
-
-
center-of-mass motion. For this purpose part of Eq. (6.87) can be rewritten as
note that the
exponential
1
I
(- 2 FAr)
exp
we
exp
a
-2
j(,r, _,rj)2
2
(6.92) where aij
and,3i
are
related to the matrix A
by
N
aij
-Aij (i 7 j),
=
gi
=
1: Aki-
(6-93)
k=1
The
coupling
between the intrinsic motion and the center-of-mass
tion arises from the second term in the exponent. As ri is
mo-
expressed
as
IV-1
7'i
E (U-l)ijXj + XN7
(6.94)
j=1 we
have N-I
N
X2N
Ep,r? i
i=1
N
(
Pi(U-l)ij
+2 j=1
xj
*
xN +
i=1
(6-95) Here the symbol
...
denotes the terms that
coordinates and have
quadratic in the relative xN. Substituting 3i of Eq. are
dependence (6.93) and using the condition (6.90), we obtain that 0. EN EN j= 1_ Aki(U-l)ij k= I EN j= 1)3i ((,T-l)ij =
no
on
=
EiN=1,3i
=
c
and
C6.5 Three electrons with S
6.5 Three electrons with S Let
M
construct the
us
=
1/2
11)
1/2
115
1/2
=
function with S
spin
=
=
The basic terms for
1/2.
are
I ITI),
=
12)
=
13)
1111),
=
(6.96)
I ITT),
assumed to be in increasing
order, e.g. in the state 11) the first and second electrons are in a spin-up state while the third electron is in a spin-down state. The requirement that S+X(A) must vanish leads to the condition Of /XI + -X2 + A3 0. By taking account of the normalization the spin function can be parametrized by a single variable &(-,xl2 < 6 < 7r/2) as follows:
particle indices
where the
are
=
1
(V/jcos,0 VilsinO) I TIT) ( F, coso 4 sin i ) I IT). 2
X.L
i 22
sinO
(A)
+
-
The,O value is chosen such that,&
=
with the intermediate Spin S12
0 whereas V
S12
(6.97)
I
+
=
0
corresponds =
to the
7r/2
spin function
to the
one
with
I-
=
By acting with S-
on
Eq. (6.97),
we
obtain
2
X.1
_.L
2
2
(A)
=
+
These
baryon
:3sin,& 1111) (61 41sin,&) (Vj cos,0 4 sin,&) 1111). -
_
-
+
spin functions
wave
c Os'&
can
functions. See
6.6 Four electrons in
be used for the
Chap.
spin part of three-quaxk
9.
arbitrary spin arrangement Complement 6.5 gives us the following of the four-electron system. For function general spin an
Arithmetic similar to that of result for the S
=
0:
Xoo (A)
(2
(6.98)
in,&
+
COO + 2
sinO)
(2Icos,&
I -
inO)
cosO + 2
ril-isind)
Complements
116
I
+
(2
COO
-
FWIi.6)I lilt) F3 sinO I JITT),
(6-99)
+
where the paxameter 6 satisfies -T/2 < V < -Ft/2 and is chosen such that,O 0 corresponds to the spin function with S12 07 S123 1/2; =
=
=
1i S123 corresponds to the one with S12 and two of electrons molecule two consisting positronium 1/2. positrons, the most important spin function is such that the spins of 0. the identical paxticles are coupled to zero, which corresponds toO
whereas V
=
-r,/2
=
=
In the
See Sect. 8.3. For S
=
1:
ezzsindsin o
X11 (A)
2 +
3
F1!is 4's 4S
sin,& cos o
( ilc -(61
+
osv
-
cos'O +
-
in,&
sin o)
in,& cos p
-
r!2is r!is
in ? sin
p)
in 6 sin
W)
1
inO cos W +
7
(6-100) where the parameters V and W satisfy --F,/2 < 0 < -F,/2 and W < 7r/2. The three independent spin functions with definite
-Ti-/2
<
S12 and 0 S123 values correspond to the V and (p values as follows: for S12 and S123 O for I and 6 and S12 S123 0; 1/2, 1/2, T-/2 I and for and 6 and S12 S123 0; 3/2, z/2 Ti-/2. W o =
=
=
=
=
=
6.7 Six electrons with S
=
=
0
spin functions may be constructed from those of a smaller group of electrons through angular momentum couplings. As an example, let The
spin function of the six-electron system with S 0, Xoo (A), from two groups of three electrons. Clearly X00 (A) is given as us
a
construct the
=
combination of two terms with xoo (A)
As two
tion
=
122 00)
cosO .1 .1;
S123(= S456)
+ sin V
=
1/2
or
3/2:
1!!; 00).
already mentioned, the spin function with S123 0 or 1, and similaxly independent states with SI-2 with S456 has two independent states with S45 1/2 was
=
=
(6.101)
22
=
1/2
has
the fimc=
0
or
1.
C6.7 Six electrons with S
Therefore the first term of the
parametrized I I
00)
as
=
right-hand
side of
=
0
Eq. (6.101)
117
can
be
follows:
COS
sin 61
611 (S12
COS
62
=
1 (S12
0)"17 (S45 2 =
00 0) '1; 2
=
0) 2 (S45
sinVI sin V2 COS
V31 (S12
sin 01 sin #2 sin
631 (S12
=
=
=
00 1) '1; 2
1) 2 (S45 1) "2 (S45
=
=
I
00 0) *'; 2
1) '1; 2
00)
(6-102)
hand, the spin function with S123 3/2 or S456 3/2 has just one independent state and needs no angle parameters. Each term of the above equation and the second term of the right-hand side of Eq. (6.101) can be easily written down in the form (6.35), e.g., On the other
(S12
=
=
=
0) 211 (S45 I -
2v,f2-
00 0) '1; 2
=
(I
-1 ITIJIT)
+
+
(6.103)
+
The four parameters of the five
6, #1-, 02,,03 determine the relative weight independent spin states for the six-electron system with
S=0It is of
course
possible
to start with other
groups of four electrons and two electrons trons.
or
groupings such
as
two
three groups of two elec-
Exercises
118
Exercises Eq. (6.7).
6.1. Prove
Solution. Let
IT
Of
= -
f
course
us
consider the
V)2K+Ldb
YLm (&) (a
I is
equal
E
=
With
U
=
2 1 + -C2-
BKLa 2K+L V2KyLM(V) from Eq.
to
calculate I in another way,
(a.V)2K+L
quantity
(6-104) (6.18).
To
the relation
we use
(2K+L)! p!q!
(a-xj)P(a-X2)q.
(6.105)
p+q=2K+L
[YI, (a)
x
x
Y1, (xl)loo [Yi, (a) 11 12
0
A
A
0
x
Y1,,(a)],\ x1YI1 (XI-)
2A+I
(211+1)(212 X
IYX(a)
X
Y12 (X2)] 00
0
11 12
[[YI, (a)
X
X
Y12 (X2)]AIOO
+,)a11+12C(l, 12; A)
1Y1J-(XI)
X
Y12(X2)1,\100-
(6.106)
coupled form [Y,,_ (a) X Y12(a)],x, is reduced to a 11+12C(II 12; A) Y by using Eqs. (6.10) and (6.11). When using Eq. (6.106) in Eq. (6.104) and integrating over b., we see that A is restricted to L, because otherwise the integral vanishes due to the orthogonality of the spherical harmonics. We thus obtain The
f YLM01)[YA(a)
X
1YII(XI)
X
Y12(X2)],X]00d&
119
Exercises
[YI XI)
JAL
X
Y12 (X2)1 LM
(6-107)
-
Combining these results leads to Eq. (6-7). 0. Then p + q is Consider Eq. (6.105) for the special case of K equal to L. Since 11 and 12 have to satisfy the condition L < 11 + 12 ":: L. Equation (6.7) then simplifies p + q, 11 + 12 must also be equal to =
to the well-known formula
YLM(XI
X2)
+
=
JX1
+ X2
I'F'YLM( XI +X 2)
L
EDOL
=
010 L-I
1XI (Xl)
YL-1 (X2)] LM
X
1=0 L
1+41')
4-) 47r(2L + 1)! [Yl(Xl) 7) 1)! (2L 21 + 1)! (2 + 1) (21
E
=
1=0
X
Yr,-I(X2)]LM-
-
(6-108) 6.2. Prove
Eq. (6-32).
positive-definite, symmetric matrix A can be diagonalized by a suitable orthogonal matrix T, namely tAT becomes a diagonal di being positive. With the matrix D with its diagonal element Dii use of a transformation x --> y (x Ty), the integral.T of Eq. (6.32) Solution. A
=
=
is evaluated
I
where
as
follows:
exp
(deff)
Though
3
Dy
2
is the Jacoblan
detT is 1 in
Y) (detT)
+ Ts
general,
3
(6-109)
dy
(functional determinant) det(ax/Oy). we
may
assume
dy. preserve the volume element, dx the Since D is diagonal, integration
it to be +1 in order to
=
rately
j
in each
component of y, which reads 1
exp
Therefore
2 we
can now
djyj2 + (ts)j -yj dy
be
performed
sepa-
as
(27r)
3 2
e 2di
(6.110)
di
obtain
N
(27r) d-
3 2
exp( 2di
(6-111)
Exercises
120
N
n,'=, di
It is easy to note that
and
E Jtsfi/dj j=
to detD
equal
expressed
=
det(TAT)
=
detA,
E ,(D-1)jj(!fs)j ( fs)j
as
-
=
j=
-
TsD-lts
9T(TA 'T)ts
9A-1s. Using these results
-
in
Eq. proof. It is useful to generalize Eq. (6.32) for the case where all the vectors and si are d-dimensional. Following the above derivation we get
(6.111) xi
be
can
Z
-Z-
is
=
-
-
leads to the
d
1.,r-,Ax +
exp
where the scalar
(a,, a2,
--.,
x) dx
2
ad)
detA
I
)
exp
2
M-1s),
(6.112)
(inner) product (a b) for d-dimensional vectors d (bl, b21 bd) is to be understood as (a b) -
and
-
=
=
....
i:d Some useful formulas related to this
integral axe collected below. By differentiating both sides of the above equation with respect to the mth component of the vector si,
(xi)m exp
(si)m,
(-2I: Mx 9x) +
we
obtain
dx
d
((A-'s)j)m
(
detA
)
exp
(2 9A-1s)
Further differentiation with respect to
(xi),a (xj),, exp
(
I
TcAx +
-
2
(sj),,
leads
(6.113) us
to
9x) dx
f(A-l)ijJm,, ((A-1S)iW(A-18Wnj +
d 2
exp(19A-1s).
X
detA
Setting
m
=
n
and
f (xi xj) ( -
exp
(6.114)
2
summing
over m
I
7cAx +
-
.
2
leads to
9x) dx
d(A-')jj ((A-1s)i (A-1s)j) +
-
d
X
detA
exp
2
s).
(6.115)
Exercises
121
bl,,,,, with ln 1, 0, -1) a,,,b,, (&,) (M /-4-x/3aYj"' a,,, I rIL) (Im -L: etc. for three-dimensional vectors a and b that is, a, (a,; +iay), \/2 --!-(a,, iay). Here n 0 or I or 2 and -n < [t < r,. ao a, a-, v/2Tn the
case
of d
=
3, let
define
us
product, [a
tensor
a
x
=
=
=
=
-
,
--
=
The scalar
(a b) -
=
=
-
product
-vf3-[a
is
special
a
of the tensor
case
I
bloo
x
that
product,
whereas
=
r,
=
is,
I and
give a vector (outer) product and a second rank tensor. See also Eq. (11.2). The use of Eq. (6.114) leads to the following result r,
=
2
1
[xi
x
xjl,,,,exp
9x) dx
:IAx +
-
2
v/3- &,OJ ,O (A-1)ij
[(A-1s)i (A-1s)j]
+
x
3
2exp 2 gA`s).
X
detA
6.3. Derive
B,,,Pq
a
Eq. (6.60).
npq explicit formula for Bj'.
Solution. The
give
in
hint for its derivation. First
x=
(6-116)
we
r V:r(Yj--j(P)-Yjj(P)), V
is
bit
a
lengthy,
and here
we
recall that
F -r(Yj-j(P)+Yjj(P)), V V
y=i
3
3
VL37'y 7F
Z=
(6.117)
10
Thus the power
XpyqZn-p-q
can
be
expressed
q
P
(7 )n r
XPY qzn-p-q
=
rn
p+q
2
2n-
2
iq
3
E t1=0
X
(yi, (i6))
The power of the k
"+"
(YI -1 (,P))
(k,m)
(Y1jr (j )) =ED L a
(p) E (q) (-1? A
p+q-g-v
spherical harmonics
as
I/
V=O
(y10 (,,))
can
be
n
-p-q.
expressed
(6-118) as
YL km (f)
(6.119)
L
(k,m) where D L
can
be derived
calculated ftom the equation
by using Eq. -(6.10) successively
and it is
Exercises
122
(k,m)
DL
V/47r
=
E LIL2
...
Lk-I
k
xll(C(Li-ll;Li)(Li-l(i-1),ralmlLiim)),
(6.120)
i=1
where
LO
0 and
=
determined
as
Li
=
L. In the case ofTn Lk 1, Li is uniquely 0 there are several possibilities for i, but for rn =
=
=
product of the spherical the following:
the value of Li. Thus the
(6.118)
can
be reduced to
(Y11 (i6))
A+-
(Y,
-1
(f))
p+q-A-v
(Y10(f))
harmonies I
Eq.
n-p-q
(p+q-,p-t/,-I) D(A+"")Dp+q-IL-il E D L(n-p-q,O) 11+V
L
X
Y[t+v li+v(P) Yp+q-A-v
-p-q+IL+v
(f) YLO (i )
product of the three spherical harmonics spherical harmonic by using Eq. (6.10):
The
YIL+z,
IL+v
(i6) Yp+q-tt-71
-p-q+IL+zl
is
(6.121)
-
coupled
to
a
single
(i ) YLO (i6)
E C(IL+v p+q-1-t-7,1; L) C(L'L; 1) L11
X(A+v1-t+vp+q-tz-v -p-q+A+YIL'-p-q+2A+2z;) x
(L' -p-q+21j,+2v L 0 11 -p-q+2A+2v) Y, (6.122)
The above formulas
B npq I.
provide
us
with the
ingredients
needed to derive
7. Matrix elements for
spherical
Gaussians
chapter we show that the generating function introduced in the previous chapter can be used to advantage to derive the Hamiltonian matrix elements of an N-particle system for both the correlated Gaussians and the correlated Gaussian-type geminals. Matrix elements of various physical operators between the generating functions will be tabulated in a compact form. Examples of matrix elements for spherical or primitive Gaussians are given in this chapter and more general matrix cases will be treated in the appendix. We can thus calculate for interaction a many-body system. We elements of essentially any also show that the matrix elements involving the correlated Gaussians become particularly simple in two-dimensional problems. An extenIn this
sion to the matrix element for nonlocal potentials is also presented. We discuss the kinetic energy operator for the relativistic kinematics and calculate its matrix element for the correlated Gaussians as an
example.
7.1 Matrix elements of the
generating
As the correlated Gaussian basis function
can
function
be constructed from
function g (6.19), it is convenient to derive the matrix element between the basis functions from that between the functions This is true even though we use more than one set of coordinates
the
generating
g.
because the transformation of the coordinates
simply leads to
a
change
of the parameters A and s, as shown in Eq. (6.28). It is convenient to use g not only because thereby all types of correlated Gaussians can be constructed in a simplified manner but also because the matrix elements between functions g can be evaluated with ease. Table 7.1 lists formulas for the matrix elements of some basic operators 0
Y. Suzuki and K. Varga: LNPm 54, pp. 123 - 148, 1998 © Springer-Verlag Berlin Heidelberg 1998
7. Matrix elements for
124
spherical
elements, M (g(s; X, x) 101g(s; A, x)), o-Foperators generating functions g of Eq. (6.19). Here we take all vectors d-
Table 7.1. Matrix 0 between
dimensional.
(a
b)
x
and
x is
=
a
N
E,=1 wi xi.
short-hand notation for
tensor
a
product [a
x
b]2,,
A vector
product
defined for three-dimensional
are
XB-1s AB--s. P is A + X, v a and b. B s + s' and y Tpx. permutation operator and the matrix Tp is defined by Px
vectors a
Gaussians
=
=
=
-
=
0
eXP(2!' B-1V)
detB
fvB-1-vMo
X
IdTr(B-'Q) +i B-1QB-1vjMO
aiQx (Q:
symmetric matrix)
a
( X [fV-X
X)
(iv-B-1-v x B-lv) Mo
CX121L
[fvB-lv x (B-1Vj2ttMO -ih&Mo
X X
(7rj
=
-ih
axj
h2
-iA7r
(A:
d 2
(2,X)
MO
a
-
VAy
JMO
symmetric matrix)
L.
(rzL
IdTr(AB-XA)
-i =
Ei(Xi
X
E,, (B-')ij(s'i
x
sj),Mo
7ri)) 2
L2
(fVX
(d X
J(iv-x
J(i7vx P
-
-
-
I)WB-1-s
-
( Ej(B-1)jjZz Sj) I.M0 X
r)
-ih
(fvB-1-v &) Mo
r)
Mi.
=-
(27riv-B-1w)-2
x
exp
r)
(iv-x
x
r)
x
.4
-ihf (r
X
f
1
2iv-B-lw
&) +
(r
AB-lw ivB--Iw
-
(r
I MO ibB-lv) IM,
bB -lV)2 x
(g(s ; A, x) Ig(Fp-s; TpATp, x))
125
7.2 Correlated Gaussians
A4
=
(7.1)
(g(s; W, x) 10 Ig(s; A, x) ,
positive-definite, symmetric matrices. The fxj,..., XNJ, can be integration over a number N of vectors, x carried out by using Eqs. (6.112)-(6.116). where A and X
are
N
x
N
=
leaving this section, we note that the formulation with the generating function, though quite useful and powerful, is still severely restricted in its application to a many-body system by the number of paxticles because of the following properties of the method: Before
1. The number of nonlinear
of
one
basis function is
in the matrix A
parameters contained
N(N + 1) /2. By increasing
particles the optimization hibitively time consuming. of
the number
of these parameters becomes pro-
2. To calculate the matrix elements between the basis
functions,
one
needs to calculate the determinant and the inverse of the N x N matrix B A+ A. The number of operations needed to calculate =
the determinant and the inverse increases
as
N3. The symmetrizaof this calculation N!
the
repetition postulate would require overlap and the kinetic energy and (N(N 1)12)N! times for the potential energy matrix elements, respectively. As the system gets more complex, the number of basis functions needed to represent the wave function increases considerably and
tion
times for the
3.
more
As
was
-
matrix elements have to be calculated.
shown in
Chap. 4,
a
number of random trials
are
probed in the
SVM to select a basis element. To calculate the energy for each trial, the determinant and the inverse of the corresponding matrix B have
computational load mentioned in the second item would become very heavy even for systems containing a relatively small number of particles. Fortunately, when one repeats the calculation of detB and B-1 by changing only one element or a few elements of B, as is often the case in the SVM selection procedure, to be calculated many
one can use
times,
so
the
the Sherman-Morrison formula
involved in the calculations. This will be
[141
to reduce the load
explained
in
Complement
7.1.
7.2 Correlated Gaussians We
assume
form of
that the Hamiltonian of
Eq. (2.1).
an
N-particle system takes
The set of coordinates :Z
=
(xl,..., xN-I)
the
stands
126
7. Matrix elements for
spherical Gaussians
for the relative coordinates chosen to describe the basis function. As a
example of correlated Gaussians we basis function is given by Eq. (6.27):
concrete
that the
fKLM (u, A, x)
exp
(-2I bAx
assume
in this section
IEX12K YLM(ijX).
(fKILIMI (U 7 X7 X-)IOIfKLM(U; A7 X)) I
BKLBK"L'ff X
de--d
f
YL m (e^-) YL, lw,, (, f)
(g(A'eu; X, x) 10 Ig(Aeu; A, x))
*(
d2K+L+2K'+L"
dA2K+LdAj2KI'+L-' (7.2)
X=O,,N,=O' e=jej=1
e"=jeIj=I
As the matrix element
(g(Aeu'; X, x) 10 Ig(Aeu; A, x))
between the
generating functions is given as a function of A, A', e, and e' for various operators by using Table 7.1, one only needs to perform the operations prescribed in the above equation. As will be shown in the appendix, the operations can be done very systematically for many cases of practical interest for arbitrary L. The correlated Gaussiajas with L 0 what we call spherical Gaussians take the particularly simple form in the special 0. As case, K they give a fairly good solution for a variety of few-body problems, we list the explicit for-m of the matrix elements for this case in the following. Because the parameter u of Eq. (6.27) is redundant in this approximation, we denote v'r4-7rfooo (u, A, x) simply as =
=
FA(x)
=
exp,
(-2 Jc'Ax).
(7-3)
127
7.2 Correlated Gaussians
Since this function is
equal
to
g(s
0; A, x),
=
special
a
generating function (6.19), the needed matrix elements given in Table 7.1. The overlap of the spherical Gaussians is given by
(FA' IFA )
(
=
)
A)
in fact
(7.4)
.
The matrix element of the kinetic energy is obtained
(2.10)
are
3 2
(2T)N- I det (A +
of the
case
through Eqs.
(2.11) by
and
2
P
(FAI
-
Tm IFA)
2mi
3OTr (A(A + A) -'A!A) (FA, I FA).
(7-5)
2
The matrix element of the
two-body potential
(FAIIV(ri-rj)IFA)=(FAIIFA)(
is
given by
j)3j V(r)
Ci
2
.1
e-2 C,,r2 dr
(7.6)
27
where I
-
=
w(ij) (A + W) -lw('j)
(7.7)
Cii
w('j) defined by Eq. (2.13). The matrix element of the total potential energy is easily obtained by summing Eq. (7.6) over i, j. As a by-product of Eq. (7.6), we show that the matrix element of Iri rj I' can be easily obtained because then the integral in Eq. (7.6) can be reduced to is
given through the column
vector
-
a
r
-
e
LCT2
2
dr
47r
f
r
a+2
LCr2dr
e- 2
0
a+3
2-x
(2)
x
e -xdx
2
----
C
0 "+3
27r
(2 ) C
2
(a+
where F is the Gamma function. Thus
1
(7-8)
2
we
obtain
7. Matrix elements for
128
(FA'117'i
rjl'IFA)
-
spherical Gaussians
(FA, IFA)
=
2
2
T
Cij
X
where cjj is given by Eq. (7.7). For a spherically symmetric
potential
the
(7.9)
2
integral
W
I(k, c) can
is
a
>
-1
e-
!
CT2
(7.10)
dr
analytically for
a
certain class of
potentials.
If
V(r)
in the form
V(r) then
k
V(r)r
be calculated
given
n(n
fo
=
re-"
=
2-br
(7.11)
expression for I(k, c) becomes possible for -k) by using the following formula closed
an
integer
00
1
ee-ar2-brdr
0
n+I 1
-
T
7F(- I)n
ni
n
Ek=O (n-k')! k! fk gn-k
2 ,fa-
n!
for
for
bn+l
a
> 0
a=O
b>0
(7-12) with
V
fo
go
=
exp
b
(4a) erfc ( 2V/a-
and for k > I
[k121
E V (k
A
k! -
i=O
b
2i)!
k-2i
)
2
=(-I) kH'k-I
A
Here
the
Hn(x)
is the Hermite
complementary
erf (x)
(7.12).
=
(2/V9T f
In
(7.13)
case
error
polynomial and erfc(x) function, where erf(x) is the
e-t2dt.
where
an
1
-
error
erf(x)
fis
function:
See Exercise 7.1 for the derivation of
analytic
evaluation of the
integral
is
Eq. impossi-
ble, one has to rely on a numerical integration. Examples of application of FA(x) to various quantum-mechanical few-body problems and 11.3.
are
presented
in
[811.
See also
Complements
11.1
7.3 Correlated Gaussians in two-dimensional systems
129
7.3 Correlated Gaussians in two-dimensional
systems In this section ments in the
we
case
review the trial
wave
of two-dimensional
function and its matrix ele-
(2D) problems. A real application
of the results in this section will be made to the in
problems In 2D
on
few-body
quantum dots in Chap. 10.
problems
the motion of the
The vector
xy-plane.
study
r
particles
is constrained to the
has therefore two components
r
=
(x, y),
and
for relative coordinates and other vectors introduced
all
expressions 0. The scalar previous sections remain valid by setting z will include variables and over only the spatial integrations products in the
xy
=
components. The main difference between the 2D and the 3D momentum. In 3D the basis function
bital angular eigenfunction
2
of the operators L
,
case
was
is the
taken
L ,:
rlyi-00In 2D L
2 =
as
or-
the
(7.14) L
2
so
Z ,
that
one
of these operators becomes redundant
the fact that the system may be invariant under rotations only. The corresponding orbital angular functions
reflecting
around the z-axis take the form
r'e"mw
=
(X
W,
(M
>_
0),
(7-15)
is, the states are characterized by the quantum number "m". This function, apart from normalization, can be derived from its 3D coun0 (or 6 m and z r/2 in polar coordinate terpaA by setting I that
=
=
=
system). The matrix elements derived for the 3D
case
used because they already include an integration
cannot be
over
directly
the z-component.
One can, however, derive the matrix elements in a very similax way by using generating functions and exploiting the simplifications present in the 2D
function.
the
The first step toward this is to construct a generating Similarly to the 3D case, we assume a trial function, by using
global
case.
vector
fm(u7 A7 x) where the
x
x
representation,
=
(V1
+
in the form:
iV2)' exp
I -Je' A x
(_2
,
(7.16)
is the vector of relative coordinates in 2D and v, and V2
and y components of the vector
are
7. Matrix elements for
130
spherical Gaussians
N-1 V
(7.17)
Uixi.
Note that the
complex conjugate of fm (u, A, x) is an eigenfunction of L, with eigenvalue -mTo derive matrix elements, one can use a particularly simple generating function for f,,, (u, A, x): dr"
f (u, A, x)
--
..
dtra
g(s, A, x)
(7.18) t=O
with
g(s, A, x)
=
exp
1-2
- CAX +
where the 2D vector sj (j vector (tuj, ituj). Note that
=
*1
+
I,-, N
iV2)
-
1)
(7.19) is defined
by
a
*
complex 2t-eui Uill
they satisfy si sj 0, si sj'3 generating function remain valid as listed in Table 7.1 by substituting d 2 and assuming 2D vectors. By using the matrix elements of the generating function all the necessary matrix elements can be easily calculated by carrying out the differentiation prescribed by Eq. (7.18). The matrix elements for some basic operators -
=
-
=
The matrix elements of the
=
are
collected in Table 7.2.
Table 7.2. Matrix elements A4 ...... B is equal to A+ X. A is defined in same as
be
defined in
changed
0 If,,) for two-dimensional
Eq. (2. 11). Q, R,
Eqs. (A.2), (A.9), and (A.18), integral I is defined in Eq. (7.10).
to 2. The
0
Mram' M,,,J IV
P2
Ei=l 2mi IiA7r
-
T,,.,n
..
.,
with
Mm
(27r)N-1 detB
1(2p)m
+ mQ)MmJmm, -lh2(R 2 P
=
L;,
V(J ri
MMmJnm' -
ri 1)
(27r)'V-'-c(7nI)2I5mM, J:M detB
case.
p, p, -y, -y, c are the but the factor 3 in R must
n=O
(2#)m-'(-y-y')' (m-n)!(nl)2 1(2n+1,
c)
7.4 Correlated
7.4 Correlated We
the form of
(Irl,
...
Gaussian-type geminals N-particle system now takes single-particle coordinates j
that the Hamiltonian of the
assume
Eq. (2.2). We
the
use
=
in this section. The transformation matrix U defined in
rN)
Eq. (2.4)
131
Gaussian-type geminals
is
now
redundant
or can
be set
equal
to the unit matrix.
The basis function is chosen to be the correlated
Gaussian-type geminal of Eq. (6.26). The function is characterized by three parameters. The matrix A describes the correlation among the particles, the diagonal matrix B determines the spread of the single-particle wave(RI, RN) give the centers of the packets. This packets, and k type of basis function is employed in molecular physics and nuclear cluster models. The permutation P of particle indices transforms the coordinates as defined by Eqs. (2.26) and (2.27). This transformation changes the basis function to =
...,
(TpATp, TpBTp, TpR, r),
Pf,,(A, B, R, r) where
use
stands for
p-I Thus the
is made of the relation
fnq,,, nq,-2 PI
P2
1
2
(7.20)
'n
,
nq13 ...
...
, ....
N
permutation acting
=
nqN, ,
PIV
Tp-i (Tp)-' for nq,3) nq,2, 1
) (q, on
2
and
Tp-n
N)
...
q2
Tp-
=
qN
...
the basis function
just leads
to the
simple transformation of the parameters, A, B7 R, and n and still keeps the form of the basis function invariant. It is not nice, however, that the value of n changes under the permutation. The integral needed to evaluate the matrix element with the correlated Gaussian-type geminal is 3N dimensional and of multi-center type. Such integrals are extremely hard to calculate and to handle. See, e.g., [82, 831 for some developments made for those multi-center integrals which appear in molecular physics. To avoid the complication, a few simplifications are introduced in practice. For example, only one pair of particles is allowed to be correlated in each basis function. This is equivalent to assuming that the only non-vanishing elements of the matrix A are, Aii = Ajj correlation between the ith and
expf-aij(ri _rj)2 /21.
Another
=
aij and
Aij
=
Aji
=
-aij, if the
is taken into account
jth particles simplification
is to
use
by only primitive
0. Primitive Gaussians Gaussians, that is, the basis function with n yield solutions of good quality for few-electron systems. In this section we calculate the matrix element in the primitive Gaussian basis =
7. Matrix elements for
132
spherical Gaussians I
fo (A, B, R, r)
=
=
allowing the
most
exp
-2
exp
2
I
FAr
-
2(r
-
R)B(T
-hBR) g(BR; A
general type
R)
-
B, r),
+
(7.21)
of correlations. The formulation pre-
sented here will be found to be very simple and lead to compact results for various matrix elements. The matrix elements with higher angular
momenta, when needed,
be obtained from the matrix elements
can
generating function by straightforward but tedious differentiaalgorithm for a systematic evaluation of the differentiation is presented in Appendix A.3. The overlap matrix element is readily available from Table 7.1. It in the
tion. An
reads
as
MG
(fo (A!, B', k, r) Ifo (A, B, R, r))
=
3 2
exp
detC
2'DC-1v 2RBR 2kB -
(7.22)
-
where
C=A+B+X+B',
v
=
BR+ B'k.
Note that formulas derived in this section metric matrix B
or
are
(7.23)
valid for
a
B' provided that A + B and A! + B'
general are
sym-
positive-
definite. The matrix element of the Idnetic energy is also available from Table 7.1: N
(fo (A! B'; k 7
h2 =
2
JKI ((A
I? P'
2mi
+
I fo (A, B, R,
B)C-'(X + B)A)
-
VAyjMG
(7.24)
with y
=
(X + B)C-'BR
where A is
a
-
(A + B)C-'B'k,
diagonal matrix with Aii
=
I/mi.
(7.25)
The matrix element of
the center-of-mass Idnetic energy is calculated very easily as -well. As Eq. (2-10) shows, it is calculated by the above formula with a shnple
7.4 Correlated
modification: The matrix A is all i and
now
to be taken
we
make
Aij
:--
1/?n12
...
for
N
one-
and
two-body poten-
following relation
of the
use
(fo (W, B/I k, r) I V (I iv- r =
as
j.
To calculate the matrix element of both tials
133
Gaussian-type geminals
B, R, r))
S 1) 1 fo (A,
-
f V(x) (fo (A!, B1, k, r) IJ(,Cvr
-
S
-
x) I fo (A, B, R, r)) dx, (7.26)
where iv-
(WIi
:--
...
i
WN)
is
an
auxiliaxy parameter
and S is
a
3-
dimensional vector. Substituting the matrix element of the J-function leads to
Eq. (7.26)
in Table 7.1 into
(fo(W, Bf, k,,r)IV(Ifvr
27r)
=MG
-
2
Sl)lfo(A, B, R, r))
expf
V(X)
C
C _
2
(X + S
_
VC-IV)2ldX
17
0,0
FTCVS C
1CS e_2
2f
-I
V(x) x e-!i
CX2
(ecx
-
e-
csx)dx,
(I 0
(7.27) where
c-1 =,W-1w were
and
s
=
jW-1v
-
(7.28)
S1
introduced.
two-body matrix elements are calculated in a unified way with an appropriate choice of w. The matrix element of the one-body potential, V(Iri SI), can be obtained by N). If V(x) takes the form of Eq. (7.11), choosing wj Jij (j then the above integral is obtained analytically for n (n > -1) As the most important application of this formulation, we give below the matrix element for the Coulomb potential: We show that both
one-body
and
-
=
=
-
1
(fo (X, BI k' r) 1 I
Iri
-
S1
1 fo (A, B, R,
( F?)' c
erf
MG S
where c-' is
now
defined
by
(C-1)jj
(7.29) and
s
is
given by I (C-lv)i
-
S1
-
7. Matrix elements for
134
The
the
matrix element
two-body
by
same manner
the
the matrix element of as
Wk
=
-
Coulomb
the substitution of c-1 s
J(C-'v)j
=
tor e-P'
can
also be obtained in
of
exactly
Eqs. (7.27) (7.28). needs to choose w one only SI) rj the of the matrix element 1, E.g., rj 1, can be calculated from Eq. (7.29) With use
V(Iri
Jik Jjk (k potential, 1/ Iri
:--
spherical Gaussians
-
and
To derive
-
-
(C-1)jj + (C-l)jj (C-1)jj (C-1)jj (C-lv)jl. When V(x) takes the Gaussian form
-
=
-
-
and fac-
2, the matrix elements for both the central and the spin-orbit
potentials
can
be
expressed
particularly simple
in
forms:
(fo(X,Bf,k,r)je-P(' '_-r)2 Ifo(A,B,R,r)) -9
c
c+2p
)
2
CP
expf
-
c
(fo (A!, Bf, k, r) I e-P(fVr) 2(for
+
x
2p
(iV-C-1V)2j.
(7.30)
p) I fo (A, B, R, r)) Ii
=
-ih( W-lv
X
&)-MG (
C
c
+
CP
2
2p) eXpj
c
+
2p
(CVC-1V)2j. (7-31)
It is clear that the above formulation
N-body
matrix elements
as
applies
to the calculation of even
well.
7.5 Nonlocal
potentials
In this section
show
simple example of the extension of the calculational technique to nonlocal potentials. The application of nonlocal potentials will be presented for positronic atoms in Sect. 8.3 and for the nucleus 12C iTj Complement 11.4. The potential is assumed to be of separable form we
a
N
E ki
V
(Ti
-
rj)) (W. (ri
-
Tj)
(7.32)
a
with
a
Gaussian form factor
O a Or)
-
=
This form is to other
e
-21
ar
2.
(7-33)
quite useful in many practical cases, but generalization is straightforward by using the same trick as for the
cases
7.5 Nonlocal
that is
potential functions, function
J(ri
-
rj
-
135
potentials
by substituting the function
o,,
by
a
delta
r).
The first step of the calculation of the matrix elements of the above potential is to substitute the relative coordinates x by a new set of
Jacobi coordinates Y fY1; --- YN-11. The Jacobi coordinates y are the from obtained original set of coordinates x in such a way that yj =
=
,ri
The matrix of the
(see Sect. 2.4).
-rj
corresponding transformation
y=T (k)X,
(7.34) 1, is defined in Sect. 2.4. The kth (k) T is defined in such a way that the first
det(Ox/Oy)
whose Jacobian is in
permutation
deriving equal
Jacobi coordinate is
to ri
=
-
rj.
The next step is to show the matrix elements for the generating function (6.19). The matrix element of one term of the potential reads as
(g(s'; X, x) I
o,, (ri
-
rj)) (W,, (ri
-
rj) Ig (s; A, x))
(g(ts; tXT, y) IW,,,(yj) (W,,(yjjg(ts; tAT, y)),
(7.35)
integration has to be carried out first over yj in both sides and then over the remaining coordinates, yj's. We use the same notation y to denote these remaining (N 2) variables, as well. To carry out the integration implied in Eq. (7.35), we introduce the following where the
-
notations: -
B:
an
row -
b:
(N
-
2)
x
(N
-
2)
matrix obtained
and the first column of the matrix
an
(N
-
by suppressing tAT.
2)-dimensional vector I (TAT) 12,
the first
(TAT) I N-1
-
bi: the first diagonal element of the matrix (TAT), i.e., (TAT),,.
-
t:
-
tj: the first vector
a
set of
(N
-
2)
vectors
j(t8)2,
---I
(tS)N-11-
(Ts) 1.
corresponding to the bra side are introduced in exactly the same way and distinguished with a prime symbol from the quantities of the ket side. By separating the yl-dependence explicitly from the generating function g as The notations
g(Ts; TAT, y) g(t; B, y) exp and
by using
the formula
(-2 bjy2 (6.32)
1
by. yj
one
obtains
_
+ tI. yj
(7-36)
7. Matrix elements for
136
Gaussians
spherical
(w,, (yJ lg(i s; i AT, y)) i 2
(2T
(2d
exp
d
t2
g
(t
-
d
bti; D,
y),
(7.37)
where 1
D
d=a+bl,
B
=
-
bb.
(7.38)
d
As this function is again a generating function, the integration over the remaining variables y is just the overlap of the generating functions, and from Table 7.1
one
(9(8'; A!, X) kPa (Ti
obtains the final result
-
Ti)) ((Pa (Ti
ddI det (D -+D 1)
exp
rj) Ig(s; A, x))
3 2
(27r)N
X
-
as
-
)
I 1 Iti2 + 2d' t/ 2+ iTo (D + D) -1vO 2
(2d
(7.39)
1
with vo =t-
Ibt, + t'
I -
d
&
b't'l.
The above result becomes Gaussian with L
(FAd Wa (ri
-
=
(7.40)
paxticularly simple for
0 used in Sect. 7.2 and reads
TA) (Wa (?'i
-
ddIdet(D + DI)
Ii) IFA)
)
(7.41)
To calculate the matrix element for nonlocal
factor,
(J(y,
-
we
as
32
(27r)N
form
the correlated
note from
Eq. (7.36)
potentials
of an
arbitraxy
that
r) lg(i s; i AT, y))
g(t
-
br; B, y) exp
which leads to the
general
(g(s'; A!, x) 1,5(,ri
-
rj
(-2 blr2
+tl-r
result
-
r)) (J(ri
-
rj
-
r) Ig(s; A, x))
(7-42)
7.6 Semirelativistic kinetic energy 3 2
(2r)N-2 det (B +
x
B)
(_2Ibir2+tl-r
exp
137
-
Ib' rt2 + t/ rf+Iij (B + B') -1v .
1
2
1
2
(7.43) with v
=t- br
+t'- b'r".
By introducing the
(7-44)
short-hand notations
ti .Z
W
C
-
b(B + B)
-1
(t + iV)
=
I
ti b,
-
6(B + B')-b
-
(B + B)-'(t + t)
-6(B + B")-lb'
=
1
41(B + B)-lb Eq. (7.43)
can
be
expressed
(g(s'; X, x) I J(ri
-
rj
-
V,
-
(7.45)
61 (B + B') -Ib'
in terms of g in
compact form:
r')) (J(ri
r) Ig(s; A, x))
-
rj
-
3
N
2 (2 det(B + i ')
X
2
exp
-
2
(t + t) (B + B)
-1
(t + t)
g(W; C' Z).
(7.46)
7.6 Semirelativistic kinetic energy This section shows the calculation of the matrix element of the semirelativistic Idnetic energy. This will be needed in the
nuclear systems among others. See Chap. 9. We assumed the nonrelativistic Idnematics in
application
Chap.
2. The
to sub-
single-
Idnetic energy T is then related to the momentum p by p 2/(2m). In the relativistic Idnematics it is replaced by T
particle T
=
V4c__2p2+,rn2c4
=
mc2. The speed of light c is set to unity and the term, -mc2, is dropped unless otherwise stated as it just shifts the -
138
7. Matrix elements for
energy
expectation value. The kinetic
spherical Gaussians energy
we
consider takes the
form IV
T.r
I"
=
rIzP12 +Ira?
(7.47)
z
The evaluation of the matrix element of the operator (7.47) is never trivial, because the separation of the center-of-mass motion requires
special care.
The contribution of the center-of-mass motion is removed
for the nonrelativistic
case
in Sect. 2.1
by subtracting
the center-of-
kinetic energy from the Hamiltonian. This prescription led us to the expression (2.10) for the intrinsic kinetic energy, which, as it mass
should, does not contain any dependence on the total momentum -r, IV. However, this method no longer applies to the present case and must be replaced by a more general procedure. We assert that the sought procedure is to evaluate the matrix element in the center-of-mass system, that is, in the system of IV :=
,7r,v
E pi
(7.48)
0.
i=1
Note new
that, when applied to the nonrelativistic kinetic energy, this, procedure reduces to the previous prescription of subtracting
the center-of-mass kinetic energy from the total kinetic energy as expected. This is easily seen as follows: By substituting Eq. (2.9) into N
(1/2) ENJ j= E3=,Aij-7ri--rj with Aij being defined by Eq. (2.11). (The center-of-mass kinetic energy was subtracted from the beginning in Eq. (2.10), so that the suffices i and j go up to N 1 and Aij was defined for i, j :! , N 1. In Tnr
=
i:N p2/ (2mi), j= I
i
we
obtain Tn,
=
-
-
the
case
where the center-of-mass kinetic energy is included, it is not see that i and j take I to IV and Eq. (2.11) is still valid for
difficult to
requirement (7.48) then restricts the sum over practice instead of up to N. Thus the matrix j element of the operator T,,, evaluated in the center-of-mass system is equal to that of the intrinsic kinetic energy of Eq. (2.10). Now we will show a method of calculating the -matrix element of T,
(i, j
=
i and
11
....
N).)
up to N
The
-
I in
in the center-of-mass
system in
two
steps. The first step is
to express
the operator in terms of the operators defined in the relative coordinates. As Eq. (2.9) shows, the single-particle momentum operator pi can use
be
expressed
in terms of the N momenta 7rj (j x defined by the tran
the Jacobi coordinate set
N).
=
r-ma
i
-n
If
we
matrix
7.6 Semirelativistic kinetic energy
139
(2.5), the momentum PN becomes just --7rN-1 thanks to the condition (7.48). This peculiarity was already noted in Sect. 2.4. Note that none of the other momenta takes such than
one
-7r's. Let
denote this
us
a simple form; they contain more particular Jacobi coordinate set x, the
and the momentum 7rIv_1 defined in this co(N) Ov) I and -7r N_ respectively. Obviously we can
corresponding matrix Qj, as
X(N),
1)
other sets of Jacobi coordinates
ordinate set
(N
define
-
each of which
J
x(k) (k
1,
=
1
acting with the cyclic permutation (1, 21 k-times
on
2.4 and
Fig.
tion
be obtained to the
original Corresponding
2.2.
froM
permutation (1, 2, of the
U(1) U(3) J 1 7
to each set
Eq. (2.4) defines TT(". i
to
IV)
=
...'
masses
U(2) i
from
-
a
N)
and
so on.
same
Likewise,
1), by
2
...
N
3
...
1
x(M. See
also Sect.
linear transforma-
It is easy to
and at the
mi,...' miv..
,
X(k)
(2 see
the N column vectors of
by rearranging
permutation
...,
pattern of the Jacobi set
analogous
N
to the relative coordinate set obtained
corresponds
the
...'
U(1) i
can
N)
U(i
time
U(2)
that
according by cyclic
can
the
be obtained
As the transformation between
and relative coordinates is
the
single-particle always given in the form of Eq. (2.4) for any Jacobi set X(k), the corresponding transformation of the single-particle and relative momenta is also given as in the form of Eq. (2.9): N
E (U(
Pi
i
))
7r,(k)
i
=
3
ji
1,
...'
N).
(7.49)
i=1
we
Using the special form of the matrix U(k) and the condition (7.48), following useful relation
obtain the
Pk
=
(k) -7r.-i
(k
=
1,
...,
N).
(7.50)
That is, the kth single-particle momentum is equal to the negative of 1-th relative momentum defined in the kth Jacobi coordinate the N -
set.
Therefore,
energy
(7.47)
with the
can
be
help of Eq. (7.50) the semirelativistic kinetic effectively replaced in the center-of-mass system
by N
T.r
)2 +M2 _(7rN-02 J2 Mi' MI (')
+
Note that this operator has
be paid is that
one
(7-51)
no cross terms such as -7ri--7rj. The price to has to transform the coordinate systems conforming
7. Matrix elements for
140
W
to 7rN-,. There is
work with
can
spherical Gaussians terms when
difficulty in evaluating the cross
no
one
definite set of the relative coordinates. That is the
a
followed that route for the nonrelativistic kinetic energy reason why in Sect. 2.2. In the case of the sen-lirelativistic kinetic energy the cross we
term has to be avoided
will be shown below.
as
The second step is to show how to evaluate the -matrix element of the operator (7.51). One has to calculate the matrix elements term by show
method
term. For
brevity
Gaussian
FA(x)
set of the
Jacobi coordinates X(N)
we
of
Eq. (7.3).
Let
N-
only
=
simplest correlated
for the
x
denotes the last
jX1'...' x1v-11.
To calculate the
us assume
(7r(i) J2 +,rn?,
matrix element of nate
a
we
that
have to transform the coordi-
Z
system conforming to the ith Jacobi
set. This
can
be achieved
by
using the relation X
where last we
=
V(i)X(i)
V(')
row
is
(7.52)
1
an
(N
-
1)
and coWmn of
x
(N
-
1)
matrix obtained
U(N) (U('))-l. By using
by omitting
the
this transformation
have
(FA, (x) I
(7r(')
N-I
)2
+
Ta2i IF ,(X))
(detV(') )3 (FA,(i)(x('))IV(,7r(') N- 1)2+Ta,21FA (,)(X(i))), where
(detV('))'
is
Jacobian
a
corresponding
to the
(7-53)
change of
inte-
variables and
gration
A(')
=
A!(')
00AV('),
=
00WO).
The kinetic energy operator and the relative coordinates of the lated Gaussians are now expressed in the same coordinate set.
To cope with the square root,
we
(7.54) corre-
go from the coordinate space to
the momentum space:
(FAf(,)(X(i))l
/(7r(i)
N-I
)2+Ta3jIFAW (X(i)))
ff (FA,(i)jk')(k'j (-7r'(') 1)2+mj2jk)(kjFA(i))dkdk, IV-
where k as
(k" I k)
(7.55)
(27t-)-2R-'V-1-)6i 0 is normalized kiv-11 and (xlk) Iki of the correlated Gaussian The Fourier transform J(k -k). =
. ....
7.6 Sen-Arelativistic kinetic energy
and the matrix element of the square root operator a very simple form:
( kIFA)
(xlk)* exp
=
=
(detA)
-,)2 fixj(v'
(k'I
+
-
2
(-2 FcAx)
V/ (w
=
(7.56)
Vh 2 k,2v-
=
+
into
j
I
+
;j J(k
Eq. (7.55),
-
k').
(7.57)
one can
reduce the
M2 FA (x))
(detV(') )3 (detA(')detX('))-32 x
=
dx
,
mj2 I k)
_1)2
be written in
FA- (k),
By substituting these expressions integration over k as follows:
(FA, (x)
can
(FA,(j)
(k) I
-i
Vr(7r,(i 1)2)2
+
M3 mj2i IFAM
(detV('))3(detA(')detA!('))--:32i x
(FA,(j)
(k) IJ(klv-l
-i
-
(k))
f vh2q2
q) IFA(j)
-,
+
Ta2i
(k))dq.
(7.58)
To calculate the matrix element of J(kN-I-q), werewriteit
q)
with
make
141
an
use
I x (N
auxiliary
-
1)
one-row
matrix Cv
of the formula in Table 7. L Then
(FA, (x) I
V (7 (Vi r
1
)2
+
M
if 2
27r
e-2Iciq2
(0,
as
J(iv-k-
0, 1) and
obtain
i2 I FA (x) )
(detV(') )3 (detA(') detA! x
we
=
(FA, (j)
vh2q2
+
-1
(k) I FA(j)
-1
(k))
Mi2 dq 3
(FA, I FA) f (ci, mi)
(2-x)
(det(A
IV-
+
A)
f (Ci, 7ni),
(7.59)
with
Ci
(f,7(i)A(A+A!)-'A'V('))
where the function
f
is defined
by
N-1
N-1,
(7.60)
142
7. Matrix elements for
spherical Gaussians
(2v) X je-2-Ixq2 2
Ax M)
X
47r
( ) 27-1
and
use
a2
j,)O
h2 -+M2 dq I
(FAIIFA) =
-=
\//-h2q2 + M2 q2 dq,
(7-61)
0
is made of the fact that the
with respect to the
2
e-!! xq
change
overlap has the following property
of the Jacobi coordinate set:
(FAI(x)IFA(x))
(detV(') )3 (FA,(i) (x(')) IFA(i) (x('))) i
(detV(i))3 (detA(')detA'())-i (FA,(j)-i- (k)IFA(i)-i (k)). 2
(7.62) The
overlap (FA, I FA)
is
explicitly given by Eq. (7.4).
143
Complements 7. 1 Sherman-MorTison
fonnula
To show that the Sherman-Morrison formula
vantage let
in
us use
selecting
a
[14]
can
be used to ad-
basis element from among many random trials, (4. 1) or Eq. (7.3) as the basis
the correlated Gaussians of Eq.
functions. The matrix elements
Aij or, equivalently, aij are nonlinear and of the they are related to each other through Eq. basis, parameters (2.25). As is shown in Sect. 7.2, the calculation of the matrix elements, (FA, I FA)
and
(FA, I HI FA), requires
and inverse of the matrix B Let
that
=
the evaluation of the determinant
A + X.
optimize the symmetric matrix A of nonlinear paxameters by changing only one element, say Ak1 Alk, or aij (j > i), but by keeping all other parameters unchanged. Instead of changing all of the nonlinear parameters randomly at once, we change one particular element randomly and then proceed to other elements step by step. The latter type of optimization of A is certainly us assume
we
attempt
to
=
a
very restricted way, but in this
the computer time
case
required for tremendously reduced, as will useful in selecting a successful
the evaluation of the matrix elements is be
seen
below. And it is
actually
very
candidate from among a number of random trials. We see from Eq. (2.25) that changing only one element aij to aij + A produces a change in A as follows: A
--+
A+
where the
(2.13). k,
Alk
--+
-
is just
OPOP
a
(and 1, k) Alk + A) can e(')
as
e(')
-
1)
x
(N
-
1) matrix,
Eq.
whereas
AN
,
=
follows:
A
where
(N
is defined in
-
e(k)
A+ 1 +
an
Oi)
hand, changing directly only Alk of A as AN AN + X (and be achieved by introducing (N I)-dimensiOnal
element
unit vectors
is
column vector
number. On the other
I
A
(7.63)
I)-dimensional
Note that
Oi) 0A the
(N
Aw(0w(ii),
+
e(l) eo(k)
(7-64)
Jkl
is defined
ui and vi stand for
by
(N
-
given by Eqs. (7-63) and
(e('))i
1). By letting I)-dhnensional vectors, both of the changes (7-64) are in general expressible as =
Ji, (i
N
-
P
A
--->
A+
Aiuigi,
(7.65)
Complements
144
where p is either I or 2. As B is equal to A + leads to the
following
X,
the
change B,
of A
as
given by Eq. (7.65)
modification of
P
B
---+
B+
(7.66)
Aiuigi.
Therefore the calculation of the matrix element for the above
change
of A results in the calculation of the inverse and determinant of the
special form of matrix, B + a, Aiuigi. When the modification is given by just one term of the form AuO, the Sherm an-Morrison formula can
be used to obtain
Aub)-'
(B
+
det
(B + Aub)
B-1
-
-AB-1u,6B-1,
1 + AbB-lu
(7.67)
and
(I + A,5B-1u)detB.
(7-68)
See Exercise 7.2 for the derivation of the Sherman-Morrison formula.
advantage of these formulas is appaxent: By knowing B-' and detB one can easily calculate the right-hand side of the equations, and the A dependence is given in a very simple form. To change A, therefore The
there is
no
need for the evaluation of inverses and determinants of the
modified matrix B
(which would require (N
get the desired results by
_
1)3 operations),
but
we
simple multiplication and division. When the modification of B is given by Eq. (7.66) in fiL11 generality, then the inverse and determinant can be calculated by using the Woodbury formula [141, which is the block-matrix version of the Sherman-Morrison formula. If one wants to change a few of the aij's or one-column (and one-row) elements of A at the same time, the summa ion in Eq. (7.66) has to be further extended appropriately. a
Exercises
145
Exercises
Eq. (7.12).
7. 1. Derive
Solution.The the
integral
case
as
of
a
=
0 is
1 00rnCar2-brdr
dn 1
(_I)n x
case
of
a
>
0,
we
may
get
dn
(-I)n
=
0
By putting
In the
simple.
follows:
=
dbn 2
bl(2,Va-),
1000e-ar2-brdr I-exp ( ) erfc( \,Fa
dbn
V
b
4a
a
the above
2
(7-69)
-
equation becomes
00
fo
rne-ar2-brdr
=
V,-(_I)n
(
1
2
- fa- )
n+1
n
nt
k=O
(n
-
k)! k!
A (X)
gn-k (X) ,
(7-70)
where
fk (X)
=
A (X)
=
(_X2) (eXp (X2))
eXp
(k)
eXP(X2) (erfC(X))(k).
(7.71)
fk (x) defined Eq. (7.13). Remembering that the
It is easy to show that
above is
in
Hermite
not
(n)
(-I)n eX2 (exp(_X2)) difficult to check that gk(x)
.(X)
=
7.2. Derive
and
can
(erfc(x))
be
equal to the one given polynomial is given by (1)
expressed
_X21 V/,-X-,it is -
=
as
-2e
in
Eq. (7.13).
Eqs. (7.67) and (7.68).
Solution. Let X represent u,&. Then the inverse as follows:
(B + AX)-1
may be
calculated
(B + AX)
-1 =
(I + I\B-'-X) -'B-1 CO
=
j:(-A)n(B-IX)nB-1.
(7-72)
n=O
special form of X, XB-'X reduces to cX, where the constant factor c. is given by OB-'u. Therefore repeated use of this relation leads to the follwoing result: Because of the
Exercises
146
(B -I.X)n B-1
B-'(XB-'XB-'X
=
d'-'B-'XB-1
=
By substituting
this result into
(B + AX)-' which is
nothing
=
......
(n
Eq. (7.72),
>
we
B-'XB-17
-
(7.74)
Eq. (7-67).
but
given by Bij
P(A)
obtain
I+Ac
+ Auivj, let
us
(B + AX)
=
-
=
ao +
aj.X +
-
-
+
-
(i, j)
with its
suppose that the determinant is a The function P(A) is a polynomial
P(A) det(B+AX). degree (N 1) and can be expanded
function of A:
of at most
(7.73)
1).
A
B-1
To calculate the determ In ant of the matrix
element
B-'X)B-1
as
follows:
aN-O:I F-1,
(7-75)
where the coefficient ak is calculated by k!p(k) (0). The rule of dif0 ferentiating determinants leads us to the conclusion that ak =
for k ao
=
2 because of the
>
P(O)
=
special
form of the matrix X. We have
detB. The coefficient a, is obtained
as
B1,
B12
B, N-1
V1
V2
VN-1
Biv-, I
BN-1 2
BN-1N-1
follows:
N-1
Ui
a,
N-1 N-1
IV-1 N-1
E Y uivj,6ij
E T ujvjdetB(B-')jj
i=1
i=1
j=1
j=1
(,DB-1u) detB.
(7.76)
Aij is the (i, j) cofactor of the matrix B and use is made of the detB (B-1)jj. Thus P(A) is eq ial to (I+ADB-'u) detB, relation Aij Here
=
which is what
we
7.3. Calculate the
want to derive.
one-body density matrix for the generating function
g: Pi W,
r)
(g(sf; X, x) IJ(,ri
-
xN
-,r'f)) (,5(,ri
-
xN
-
r) lg(,s; A, x)).
(7.77)
147
Exercises
(2.12) shows,
Solution. As Eq. W The
ri
-
can
xN
be
expressed
as
EN-I 1=1
argument made for the nonlocal potential in Sect. 7.5 in exactly the same way. The can, therefore, be applied to this case only necessary change is to replace w('j) with 0). The density matrix When the wave function pi (r, r) takes the same form as Eq. (7.46). has the proper symmetry for a system of identical particles, the onebody density matrix does not depend on the suffix i.
w, xj.
Reproduce the nonrelativistic kinetic energy formula (7-5) by using the formulation presented in Sect. 7.6. 7.4.
Solution.
By using Eq. (7.50), the matrix element of the nonrelativistic
kinetic energy operator in the center-of-mass system becomes IV
N
2
(FA'(-X)JE
IFA(X))
2Tnj
=
(FA, (x) 11:
(i)
(7rN-l)
IF",(X))
2m,
3h2
(7.78)
(FAI IFA)
2mici
3h2 / (2mx) for the nonrelativistic kinetic energy is f (x, m) obtained by replacing Vlfh-2q2 + M2 in Eq. (7.61) with h2q 2/ (2m). To perform the summation over i in Eq. (7.78), we need to know the special matrix element M9 N-1: k
where
=
N
VM
(U(N) )kI (TT(i)
-
us
vectors of
Uj(N)
TnI , M2 ,
....
(1, 2,
N)
can
0) i
recall that
...,
mv
can
Uj and
EE
(7-79)
i
kN-i
Let
-I
IN-I
be obtained same
time
with the
i times. Rom this construction it is easy to
be obtained from
constructed from
N)
Uj('
(N)
in
Uj'
but
the column vectors. Then
the N column
by rearranging the masses the operation cyclic permutation
at the
according to
by rearranging
exactly
the
see
that
same manner as
by rearranging the row obtain (see Eq. (2.6))
U(') i U
M is
vectors instead of
we Mi
for
1: k i
for
I
M12---N
UW
(7-80) I N-1
-(I
'i
-
) M12---N
=
i.
Exercises
148
Using
this result in
0 for k
Eq. (7.79) and noting the relation
N enables
VW siimma
IV
i
n
3h2 -
2mici
over
3eT 2
which
was
(
Eq. (7.78)
(A(A
+
X)
can
easily
j(kv)
U
as
(7.81)
(UJ)ki
be done to obtain
-'A!A)
being
(7.82)
defined
kinetic energy from the total kinetic energy. This exercise serves as an indirect evidence for the
formulation
-
by Eq. (2.11). This agrees with Eq. obtained by explicitly subtracting the center-of-mass
with the matrix A
(7.5),
i in
I
to obtain the desired matrix element
(U'(V))ki
-
kIV-1
The
us
E'V I=
given
in Sect. 7.6.
validity of the
8. Small atoms and molecules
examples for the application of the method to atomic and molecular systems. The interaction between the charged particles is the Coulomb force. The long-range character of this force makes the solution of the few-body problems difficult, especially in the case of scattering. We restrict our attention to bound states, where This
chapter
contains
many different methods have been elaborated in the
These calculations
provide an
excellent
past decades.
possibility to test the
of our method. Relativistic effects in atoms
perimental precision of today, which calls
are
efficiency
withi-n the reach of ex-
for very accurate theoretical
calculations.
8.1 Coulombic
systems
The systems of charged by the masses (MI jn2 ,
particles
can
be characterized and classified
mN) and the charges (qj,q2,...,qN) of distinguish systems formed by either equal (unit) or unequal charges, and depending on the masses of the particles one can classify the systems as adiabatic and nonadiabatic ones. ITI the unit charge systems of more than two particles the constituents form a molecule and the binding energy depends only on the mass ratio(s) of the particles. Atoms are good examples for systems with unequal charges. The distinction between the adiabatic and nonadiabatic cases is dictated by the possibility of a simplified treatment in the former case. In the adiabatic case the masses of a group of particles are considerably heavier than those of the rest. The classical examples are the molecular ion (see Complement 8.2). In these H2 molecule and the H+ 2 cases the electrons move faster, while the protonic frame may rotate and vibrate by moving considerably slower. This physical picture is expressed in mathematical form as the Born-Oppenheimer approximation, where the electronic motion is first calculated by assiimin the constituents. One
....
can
Y. Suzuki and K. Varga: LNPm 54, pp. 149 - 176, 1998 © Springer-Verlag Berlin Heidelberg 1998
8. Small atoms and molecules
150
form
CkA
JeXP
2
i AkX) jVk 12K+L YL
M
(f,-I") X Slus
k
(8.1)
UkiXi
Vk
The operator A is introduced to impose proper symmetries on identical particles (antisymmetry for fermions and symmetry for bosons) of the system. In most calculations the index K is assumed to be zero or at most zero or one. In the case of the PS2 molecule of Sect. 8.3, K is set
equal
to
zero.
The details of the calculations
can
be found in
some of the system presented in this chapter. The vari[33, 84, 851 ational parameters included in each basis function are the elements of
for
the -matrix
Ak and the coefficients
Uki, which define the
vk. Since the Hamiltonian used in this
chapter
global
vector
commutes with the
spin operator, the variational trial function can be chosen to 'have a definite spin value S. The value of S influences the symmetry of the orbital part of identical particles. Therefore, possible S values 'have to be tested in general to obtain the ground state. For treating the adiabatic system of small molecules in Sect. 8.4, we use combinations of the generating function g itself as the variational trial function.
8.2 Coulombic
three-body systems
8.2 Coulombic
13-16
figures, that the solution of
lem is
one
151
three-body systems
the Coulombic
three-body prob-
of the most useful benchmark tests to compare different
methods. The accuracy pursued in Coulombic cases is not of purely academic interest, but highly motivated by the high precision of the For
experiments.
example,
the fine-structure
splitting of the
state of the Li atom has been measured with
(parts
per
million).
include relativistic
including
an
the
18 22p2pj
accuracy of 20 ppm numbers one has to
experimental and quantum electrodynamics (QED) corrections, To
explain
terms of the second- and third-order in the fine-structure
require very high accuracy. In addition to the accurate reproduction of the energies, another motivation is that in the variational calculations, even though the energy is good, other physical observables might be less accurately determined. The increased accuracy of the energy, as we will demonstrate, eventually will lead to constant, which
very accurate values of other observables
as
well.
three-body general (m+M+m-)-type stability C A B system with unit charges has been thoroughly explored [861, and the like requirement for stability can be phrased as an empirical rule: In masses" ac[861. charges have to be borne by equal or nearly equal cordance with this prediction, systems such as H- (pe-e-), H2+ (ppe-), Ps-(e+e-e-), HD+(pde-), HT+(pte-), or tdg- are all bound, while (ppe-) or (pe+e-) are most probably not. Here p, p, d, and t are proton, antiproton, deuteron (2 H) and triton (3H) respectively. In the latter case, the particles with opposite charges form an atom, which does not bind the third particle. Some systems, e.g., the muonic molecules such as (ttft-) or (tdft-), remain bound even for L =-;,k 0 orbital angular domain of
The
a
"
,
momenta.
by
A second group of the Coulombic three-body problems is formed systems where not all the particles carry unit charges. The rep-
resentatives of this category are the helium atom (ae-e-) and the helium-like ions, where a stands for the 'He nucleus. These systems often form bound states with L atom
(ape-)
angular
has been observed
momentum states
We have
challenged
(L
our
=
=,4
0
as
[871 30
-
well. The
antiprotonic helium
and studied in very
bigh orbital
40) [88, 891.
method to calculate the
energies of
some
of these systems. In the calculations to be presented the number of basis functions superposed is mostly limited to be modest because our
primary
purpose is to demonstrate the overall
performance of the
cor-
related Gaussians and not to compete with well-established methods sharpened for these systems. We will increase the basis dimension to
8. Small atoms and molecules
152
reach
high
accuracy
only
in
cases
where such calculations
sidered to be important. The results of the calculations
are con-
axe
listed in
Table 8.1. Table 8.2 presents the parameters used in the calculations. The results are compared with those of other (mainly Hylleraas-type or
correlated
hyperspherical haxmonics basis)
calculations. Our results
reasonable agreement with other calculations. In most cases a basis size of K 200 was used in the SVM. The precision of the axe
in
=
results
be
improved by increasing the basis size as can be seen on the selected examples of Table 8.1. In these cases we reach almost the same precision as the other methods. The calculation extends to nonzero orbital angulax momentum states as well, including an e,,c 31 state of the antiprotonic helium atom or the a,mple for the L 3pe bound of the H- ion. These nonzero orbital angustates slightly can
=
lar momentum states the
as
well
as
L
=
0 states have been
investigated
by using global representation. The recovery of the results of other calculations (which are based on several different represenvector
tations of the orbital part of the
usefulness of the
wave
function)
convinces
us
of the
global vector representation. See Complement 8.4 for tdl-t molecule with the global vector
calculation of the
compaxative representation. a
To illustrate the convergence of the energy and the expectation values of average separation distances, the results at different basis sizes
are
tabulated in Table 8.3 for Ps-.
increasing the itself. One
Actually
accuracy is the conventional
can
the limit of further
precision of the computer
notice that at the basis size of K
100 the energy the first four figures of =
is accurate up to six decimal digits, but only the separation distances can be precisely determined.
the basis size, the virial coefficient falls below
high
accuracy of the calculation that all the
calculation
are
By increasing
10-'0, showing by the digits
of the reference
recovered.
Quite a few very accurate methods have been developed to solve the three-body Coulomb problem. It is very difficult to go beyond their precision. This is especially true for the methods which have been elaborated for a given system only, incorporating as much physical intuition as possible into the trial function or into the solution. In contrast with these methods, we use the same trial function, which is of Gaussian nature and therefore it is not tailored to Coulomb problems at all. Still, as the examples prove, one can get a sufficiently good solution in a unified and automatic way knowledge about the systems to cope with.
in
without
a
priori built-
The real power of the
8.2 Coulombic
three-body systems
153
Energies of different Coulombic three-body systems in atomic basis dhnension. See Table 8.2 for the constants which are the K is units. used in the calculations denoted superscripts a, b and c. Table 8.1.
System
State
K
SVM
Other method
K
Ref.
Ps-
Ise
600
-0.262005070226
-0.2620050702328
1488
COI-i-
600
-0.527710163
-0.527751016523
850
200
-0.1252865
-0.1252865
90
200
-112.97300a
-112.9730179a
200
-110.26210a
-110.2621165
a
ttA
Ise 3pe Ise 1PO
200
-105.98292
b
UIL
1D'
-105.982930b
2250
ttA
IF'
200
-101.43131'
-101.43'
200
MIL MIL td[t
IS'
200
-111.36444a
-111.364511474a
[691 [90] [911 [23] [231 [921 [931 [231 [20] [20] [941 [951 [681 [68] [681 [68] [68] [681 [681 [681 [681 [891 [96]
H-
UP
b
b
500 500
1400
IP,
200
-108.17923
1D'
200
-103.40849a
-103-408481a
1566
'He
Ise
600
-2.9037243769
-2.903724376984
700
He
Ise 3se
200
-2.9037242
-2.903724372437
100
He He He
He 'He
-108.179385
2662
200
-2.1752291
-2.175293782367
700
IP0 3po
200
-2.1238423
-2.123843086498
700
200
-2.1331635
-2.133164190779
700
1-D' 3D'
200
-2.0556201
-2.055620732852
700
700
-2.055338993068
-2.055338993337
700
IF' 3F'
200
-2.03125504
-2.031255144382
700
He
200
-2.03125506
-2.031255168403
700
He
IGe
200
-2.02000069058
-2.020000710898
700
He
3Ge
200
-2.02000069062
-2.020000710925
700
VHe+
L=31
300
-3.50760
-3.50763486
1728
'Li+
Ise
300
-7.279913
-7.279913
He
Table 8.2. The constants used in the calculations. The of the electron
mass.
energy in eV.
Set
a
masses are
m,,=7294.2618241, mp=1836.1515. R.
Set b
Set
c
Mt
5496.918
5496.92158
5496.918
Md
3670.481
3670.483014
3670.481
MI,
206.7686
206-768262
206.769
2R,,
27.2113961
27.2113961
27.2116
is the
in units
Rydberg
8. Small atoms and molecules
154
Energy and different separation distances for the (e+e-e-) three-body system as a function of the basis dimension K. The virial ratio 71 is defined by q 11 + (V)/(2(T))I. See Eq. (3.50). Atornic Table 8.3. Coulomble
=
units
are
-E
(,r2+_) (7-2
1 2
used. SVM
SVM
SVM
Hylleraas
(K=100)
(K=200)
(K=600)
[691
0.26200465
0.2620050648 0.262005070226 0.2620050702328
5.489
5.48962
5.489633252
5.489633252
8.548
8.54856
8.548580655
8.548580655
6.958
6.95832
6.95837
6.95837
9.65284
9.65291
9.65291
1
9.652
0.46
77
approach will
x
10-4
be
0.34
more
number of particles is most
as
The
or more
expedition
nium molecule
to
10-6
0.54
x
10-10
x
10-10
where the
than three and the method still works al-
while the other methods need tedious efforts.
particles
larger Coulombic systems
(PS2).
0.23
highlighted in the following sections, more
easily as before,
8.3 Four
x
starts with the
positro-
This exotic molecule consists of two electrons
and two
positrons. The possibility that the PS2 molecule or in general electron-positron system consisting of p positrons and q electrons form a bound system was originally suggested by Wheeler [97], and this question has been extensively studied since then. The existence of the positronium. negative ion Ps- (p 1, q 2) has experimentally been observed [981. The binding energy Of PS2 was first calculated by Hylleraas and Ore [991. To date, it has not been observed yet due to the dffficult experimental circum tances, and this fact has intensified the theoretical interest in solving- this Coulombic four-body problem [65, 100, 69, 101-1041. Actually the positron-electron annibil i n limits the
=
=
the lifetime of Ps2 to few nanoseconds. I-n
obtaining the
solution for the
PS2 molecule, it is useful to note PS2 is invariant with respect to the charge permutation, that is, the exchange of positive and negative charges. that the Hamiltonian for
The trial function should therefore either remain
unchanged or change
sign under the charge permutation operation. The ground PS2 turns out to be even under the charge permutation. its
state of
8.3 Four
or more
particles
155
The convergence of the energy Of PS2 against the increase of the basis dimension is shown in Table 8.4. The fact that the best vari-
ational calculation
[103]
in the correlated Gaussian basis is
already
at the basis size of 400 illustrates the power of the random
surpassed trials.
Table 8.4. The total energies (in a.u.) of the ground state and the bound excited-state of the PS2 molecule in atomic units. K is the basis dimension.
PS2
Method
(K (K SVM (K SVM (K SVM (K SVM (K CG [1031
SVM SVM
=
=
=
=
=
=
100) 200) 400) 800) 1200) 1600)
(L
0)
=
PS2
(L
=
1)
-0.516000069
-0-334376975
-0.516003119
-0.334405047
-0.516003666
-0.334407561
-0.516003778
-0.334408177
-0.5160037869
-0.334408234
-0-516003789058
-0-3344082658
-0-5160024 -0.51601+-0.00001
QMC [1041
possible existence of bound excited-states of the PS2 molecule [84, 85] by taldng all possible combinations of states with L 0, 1, 2 spins. By 0, 11 2,3 orbital angular momenta and S a bound excited state we mean such a state that cannot decay to any dissociation channels. The results of the calculation were negative in I (with negative parity) and all but one case. In the case of L of a second bound-state the existence S 0, the calculation predicts We have examined the
=
=
=
=
of the PS2 molecule. This unique bound-state has been found to be odd under the charge permutation operation. The convergence of the excited-state energy is shown in Table 8.4. Figure 8.1 summa izes the energy spectra of the bound states made up of two positrons and two electrons
together with the
relevant thresholds.
One may ask the question of why the second bound-state cannot decay to two Ps atoms in spite of the fact that it is located above the threshold of Ps (1S) +Ps(IS). Since the total spin of the state is to zero, it
that
they
dissociate into two Ps
can
have
equal spins
(ground state)
and the relative orbital
atoms
coupled provided
angular momentum
1. (Recall that the Hamiltonian preserves spin, between them is L orbital angular momentum and parity.) However, this is apparently =
impossible
equal spins are bosons and their L. Consequently, the PS2 molecule
because two Ps atoms of
relative motion
can
only have
even
156
8. Small atoms and molecules
0
Ps(2P)+e++e-0.1
-0.2
-
Ps(lS)+e++e-'
Ps-+e+
Cd -0.3
IP0
-
Ps(IS)+Ps(2P)
>1 0)
-0.4
-
W
-0.5
-Ps(IS)+PS(IS)
-0.6
Se
PS2
-0.7
Fig. 8.1. The energy spectrum are given in atomic units.
with L
=
I and
of electron and positron systems.
Energies
negative paxity cannot decay into the ground states of
two Ps
atoms, that is, the lowest threshold of Ps(IS)+Ps(IS). Since the energy of this L 1 state is calculated to be E -0-3344 a.u. (see =
Table
=
8.4), (-0.3125 au.) of Ps(IS) + Ps(2P), this state is stable against autodissociation into this which is lower than the next threshold
channel. The
binding energy of this state is 0.5961 e-V from this second threshold, about 40% more tightly bound than that of the ground state Of PS2 whose binding energy is 0.4355 eV from the lowest threshold. are
The expectation values of vaxious listed in Table 8.5. In the
quantities for the PS2 molecule
PS2 molecule we deal with antiparticles, so the electronpositron pair can annihilate. The second bound-state may decay either by annihilation or by an electric dipole transition to the ground state [851. The most dominant annihilation is accompanied by the emission of two photons with energy of about 0.5 MeV each. To have an estimate for the decay width due to the annihilation we have substituted
8.3 Four
or more
particles
157
Properties of the ground and excited states of the PS2 molecule. positrons; are labelled I and 3 and the electrons are 2 and 4. Because (r14) of charge permutation symmetry, some equalities hold, e.g. (r12) Table 8.5. The
=
(r32)
=
(r,34).
Atomic units
PS2
(L
are
used.
0)
=
PS2
(L
=
1)
(r12) (r13)
4.4871530
7.56881891
6.0332070
8.8575844
(r212
29.112633
80.173836
46.374735
96.085514
2
r13 3
( r12) 3 (r 13 )
253.04611
1041.3251
443.85244
1226.7955
(412)
2807.2718
15612.112
4 13
5202.0371
17939-574
(r 21) (r.3' -2) r12 -2) r13
0.36839693
0.24082648
0.22079007
0.147244820
0.30310361
0.16081514
0.073444303
0.032230158
(1'12'TI3) ('r12 'T14) (5(rl2)) (5(rl3))
23.187368
48.042757
(r ) 12
5.9252651
32.131079
0.0221151
0.0112091
0.0006259
0.00014591
(V2) 1
-0.258001894
-0.16720401
(V1 V2)
0.1307732538
0.091656853
(VI'V3) 11 + (V) /(2(T))
-0.0035446132
-0-016109693
*
the
0.3
10-9
x
probability density
(J(r12))
,
of
into the formula
rannihi
=
47
x
10-6
electron at the
position of
a
positron,
[1021
(MC622 ) 2hc(TIjJ(rj-T2)jTf) 62
=
an
0.36
4ir-(hc)
4
hcaOI(5(rI2))i
(8.2)
equal to J0 (Tf 15(rl r2) ITf) with ao Roughly speaking, the Metime is inversely probeing portional to the probability. The Metime due to the annihilation is estimated to be 0.44 ns. This is twice that of the ground state (0.22
where
(,S(r12)), given
in a.u., is
-
the Bohr radius.
ns). dipole transition from the excited state emits one photon with energy of 4.94 eV. The decay width Idjp&,, for this transition The electric
8. Small atoms and molecules
158
is calculated
through the reduced transition probability B(EI) dipole operator D.:
electric
16v
Tdipole
':--
(E) 3B(EI;
I-
0+),
-->-
he
9
for tile
(8-3)
where I
B(EI; 1-
--+
0+)
=
)7 I (Oind I DI, ,
_
M) 12
(8.4)
1
with 4
qi 1ri
Djz
X4
-
I Ylp (ri
-
X4)
IIIere X4 is the center-of-mass of the
(4.94 eV)
tation energy
B(El)
dipole
P-92 molecule and E is the exci-
of the second bound-state. We calculated the
value and obtained
electric
(8-5)
-
B(EI)
=
0.87e2a 2. The lifetime due 0
transition has been found to be 2.1
ns.
The
to the
branching
dipole transition is thus about 17 % of the total decay rate. Therefore, both branches contribute to the decay of the excited state of the PS2 molecule. Its lifetime is finally estimated to be about of the electric
The excitation energy of 4.94 eV found for PS2 is different by 0.16 eV from the corresponding excitation energy (5.10 eV) of a Ps 0.37
ns.
atom. This difference
to be
large enough to detect its existence, e.g. in the photon absorption spectrum of the positronium gas. Before discussing the spatial distribution of the PS2 molecule, let us recall that the average distance (r+-) between the positron and the electron is 3 a.u. in the ground state of the Ps atom, while it is 10
a.u.
first excited state. The root-mean-square radius
in its
(ri
-
X'j
)2),
culated to be 3.61
surprising a
system of
of the
a.u.
5.66 a.u., 1.5 times
if
seems
larger
one assumes a
ground
The
rms
PS2 molecule
is cal-
radius of the second bound-state is
than that of the
ground
state. This is not
that the second bound-state is
Ps atom in its
(spatially extended)
state of the
(rms),
ground
state and
a
excited state. To check the
essentially
Ps atom in its first
validity
of this
as-
sumption, we have restricted the model space to include only this type of configurations. This can be achieved by a special choice of the uki parameters of that is, the
Eq. (8-1). The energy converged to -0.323 a.u., Ps(1S) +Ps(2P) system with zero relative orbital angular
momentum forms state of the
a
bound state with energy close to that of L
PS2 molecule, therefore this configuration
is
lik-ely
=
to
1
be
8.3 Four
the dominant
configuration in this molecule.
uration, the Ps- + e+
or
Ps+ + e- with L
=
or more
There is
particles
159
second
config-
a
I relative orbital
motion,
intuitively may look important because two oppositely charged particles attract each other, but it is barely bound (E -0.315 a.u).
which
=
1 state The average distances in Table 8.5 show that in the L the two atoms are well separated. In fact we can estimate the root=
mean-square distance d between the two atoms
by
2
d2
I'l +7'2
7'3 +7'4
2
2
(2(r12 ) 2
4
(,r213)
+
-
) (8-6)
2(rl2*rl4)
The symmetry properties of the PS2 wave function are used to obtain 6.93 a.u. the second equality. Using the values of Table 8.5 yields d =
for the L
=
I excited state and d
state. One cannot
give
a
direct
Arij
The correlation function defined
gives
(TfIJ(ri
=
more
-
-
(r,2
of the
(,rj)2
ground ground or
is
0
laxge.
by
(8.7) on a
system than just various average
quantity can be calculated for the correlated Gaussians
by using Eq. (A.30) =
=
=
r) ITf)
detailed information
distances. This
Tf with L
rj
for the L
a.u.
geometrical picture
excited state because the variance
C(r)
4.82
=
or
(A.136).
0, C(r) becomes
monopole density.
a
ground-state wave function of only r, which is called the
For the
function
For the excited-state
wave
function with L
=
1,
monopole and quadrupole densities. and the electron-positron electron-electron the 8.2 displays Figure correlation functions r 2C(r) for the ground-state of the PS2 molecule. The peak position of the electron-electron correlation function is shifted to a larger distance than that of the electron-positron correla-
C(r)
consists of the two terms of
tion function. The latter has much broader distribution and reaches
farther in distances
compared
to the
corresponding function
of
a
Ps
atom.
Figure
8.3
displays
the electron-electron and
electron-positron
cor-
relation functions for the second bowid-state of the PS2 molecule. As I state consists above, the correlation function for the L of the monopole and quadrupole densities and their shapes depend on the magnetic quantum number M of the wave function. Of course the M-dependence of the shapes is not independent of each other
mentioned
=
8. Small atoms and molecules
160
0.020
0.015
Cd 0.010
0.005
0.000 0
Fig.
4
8.2. The correlation functions
molecule. The solid dashed
the
2
curve
r
6
8
r
(a.u.)
2C(r)
for the
12
ground
14
state of the
PS2
denotes the electron-electron correlation and the the electron-positron correlation. For the sake of comparison, curve
electron-positron correlation function for
dotted
10
a
Ps atom is drawn
by
the
curve.
but is related
by the Clebsch-Gordan coefficient. See Eq. (A.136). quadrupole density is contributed from only the P wave of the electron-positron relative motion, while the monopole density is contributed by both S and P waves. Figure 8.3 plots the correlation funcThe
tions for both
(a)
M
=
0 and
(b)
M
=
1. As the correlation function
axiaUy symmetric around the z axis and has a reflection symmetry with respect to the xy plane, the correlation function sliced on the xz plane is drawn as a function of x (x > 0), z (z > 0). The electronelectron correlation function has a peak at the point corresponding to the average distance of 7.57 a.u. The electron-positron correlation function has two peaks reflecting the fact that the basic structure of the second bound-state is a weakly coupled system of a Ps atom in the L 0 state and another Ps atom in the L 1 spatially extended state. The peak located at a larger distance from the origin is due to the P-wave component of the PS2 molecule. The hydrogen and positronium molecules can be considered as members of the same family as both are quantum-mechanical fermio-nic is
=
=
four-body systems of two positively and two negatively charged identical particles. But they are at the opposite ends of the (M+M+m-m-)-
8.3 Four
x
x
(a.u.)
(a.u.)
x
z
(a.u.)
z
(a.u.)
0 x
or more
(a.u.)
161
particles
z
(a.u.) z
(a.u.)
(a.u.)
Fig. 8.3. The correlation fimctions rC(r) in atomic units, raultiplied by one thousand, for the second bound-state of the PS2 molecule. The z comI for 0 for (a) and M ponent of the orbital angular momentum is M (b). Drawn on the xz plane are the corresponding contour maps. =
=
8. Small atoms and molecules
162
type Coulombic systems called biexcitons biexciton and two
(or
biexciton
holes,
molecule),
is observed in
a
a
or
bound
variety
excitonic molecules. The
complex of
two electrons
of semiconductors
[105, 106].
See also Sect. 10.1. The biexciton molecule is characterized mass
ratio
o-
=
m/M. Apart
by the connection, however, their
from this
properties are radically different, e.g., H2 is an adiabatic but PS2 is a highly nonadiabatic system. Moreover, while in the case of the H2 molecule many bound excited-states have been observed experimentally and later studied theoretically, in the case of the PS2 molecule only the ground state and the unique excited state discussed above have so far been predicted theoretically. See also Complement 8.3 for the stability of the biexciton molecule. Figure 8.4 displays the dependence of the binding energy of the biexciton molecule on the mass ratio a m/M. The changes of the binding energies in the ground state (L 0) and the excited statues with L 1 and negative parity is si-milar. Both the ground and excited states become less bound by changing the mass ratio from H2 to PS21 though the binding of the excited state decreases to a somewhat lesser =
=
=
extent. The energy of the transition from the excited state to the
ground state is also shown in this figure. This transition may take place in an external field, for example. By increasing the mass M of the positively charged paxticles tbowa,rd infinity, one arrives at the energy of the C I H,, 2p-x state of the H2 molecule. This state is formed by an excited H-atom and a groundstate HI-atom. Consequently, a statement similar to the case of the PS2 molecule is valid for the biexcitonic molecule: The second bound-state of the biexciton molecule is of
a
ground-state
dominantly formed by
exciton and
an
L
=
an
interacting
pair
1 excited-state exciton.
The rule that the Pauli
principle forbids odd partial waves between identical bosons also applies to the biexciton with L I and negative The second bound-state of the biexciton molecule cannot decay parity. to two ground-state excitons. A somewhat similar situation exists in the 3p, state of the H- ion as well, where its second bound state cannot decay due to the parity conservation. By changing the mass ratio in that (M+m-m-) system, however, this kind of state disappears for oI and the Ps- ion is known to have only one bound state. =
=
Tables 8.6-8.8 show tem
our
results for various other Coulombic sys-
.
The tivated
investigation of the stability of positronic atoms has been nioby the use of positrons as a tool for spectroscopy (positron
8.3 Four
or more
163
particles
0.15
0.10 CU
0.05
0.00 0.0
0.4
0.2
0.6
0.8
to
M/M
Fig.
8.4. The
mass
ratio
o-
binding
=
m/M.
state, and the solid
energy of the biexciton molecule as a function of the curve is the binding energy of the ground
The dotted
curve
is that of the first excited state with L
Table 8.6.
curve
Energies of different Coulombic four-body systems
I and
=
is the energy difference, multiplied negative parity third, between the first excited state and the ground state.
The dashed
by one
in atomic
units. K is the basis dimension.
System
State
K
SVM
Other method
K
Ref.
PS2
Ise IP0
800
-0-516003778
-0.516002
400
[102] [107] [1071 [1071 [1031
800
-0.334408112
600
-7.478058
-7.47806032
1589
Li
Ise 1PO
1000
-7.410151
-7.410156521
1715
Li
'D'
1000
-7-335520
-7.335523540
1673
'HPS
Ise
1200
-0.7891964
-0.7891794
PS2 Li
8. Small atoms and molecules
164
Table 8.7.
Energies of different Coulombic five-body systems
in atomic
units. K is the basis dimension. The Li- energy with K = oo is the extrapolated one [1091, where E = -7.500577 is given by multiconfiguration
Hartree-Fock calculations with K
=
2997.
System
State
K
SVM
Other method
K
Ref.
Be
Ise Ise
500
-14.6673
-14.667355
1200
600
-7-50012
-7-50076
00
[1081 [1091
Li-
Ise Ise
(27r+, 3,-1-) Li +
e+
Table 8.8.
Energies
200
-0-5493
1000
-7.53218
of different Coulombic
six-body systems
in atomic
units. K is the basis dimension.
System
State
K
SVM
(37,-+,37r-)
IS' Ise Ise
300
-0.820
600
-7.73855
1000
-14.692
Li + Ps
Be+e+
annihilation
spectroscopy) is whether
in condensed matter
physics.
An
intrigu-
or a chemically stable system containing a ing question positron or a positronium could be formed in the various targets. This
not
be answered
only by a sophisticated calculation or experiment because the mechanism responsible for binding the positron to the neutral atom is the polarization potential present in the atom+e+ system. The boundness of the hydrogen positride (positronium hydride) HPs was predicted theoretically by Ore [99] in 1951 and it has recently been created and observed in collisions between positrons and methane [1101. The properties of HPs is discussed in [851. The use of the SVM proved for the first time that the positronic lithium (Li+e-r [1111 and the positronic beryllium (Be+e+) [1121 are stable. We see from the tables that the positron separation energy of the positronic lithbun is 0.054 a.u. Below the Li+e+ channel the Li++Ps channel is question
can
open and the
energy of the
positronic lithium is only 0.0022 a.u. against the dissociation into the Li++Ps channel. A calculation has to be accurate at least to 10-3 a.u. to answer the stability of the
binding
positronic litbium. Likewise, the positron separation energy of the
Be+e+ system
typically
is
only
about 0.01
0.025
a.u.
a.u.
one
Due to the
tiny binding energies of
has to. be able to reach
high
accuracy
8.4 Small molecules
165
6-particle systems. A naive picture of these systems is that the positron orbiting around the neutral atom slightly polarizes the negative electron cloud, and the positron is bound by the resulting in these
attraction.
(as a fall N-body solution) for the investigation of the stability of much larger systems (e.g. Sodium plus positron) To extend the method
question. One can, however, try to use a "frozen core approximation?'. In this approximation the positively charged core is considered to be passive (its polarization is neglected) and the problem is is out of
model space where the single-paxticle orbitals are orthogonal to the core orbitals. One has to solve the modified Schr6dinger
solved in
a
equation of the form
(H + AP)Tf
=
with
ETI-
P
(8-8)
Oi) (Oi iGoccupied
produces wave functions that are orthogonal to the core orbitals provided the positive constant A is large enough. The projection operator P is an example for the nonlocal potentials discussed in Sect. 7.5. See also Complement 11.4. One can validate this approximation by comparing it to the "exact" fuU N-body calculation for Li+e+. This approximation turns out to be very accurate, reproducing the first six digits of the result of the full calculation [1121. Assuming that the accuracy holds for larger systems, one seems to find the stability of positronic sodium (Na+e+) [1121. which
8.4 Small molecules As it case
was
mentioned at the
demands
a
special
beginning
of this
assumed to take combinations of the form 1
g(s; A, x) where
Aij
and
functions. The x.
exp
=
Note that
s
(_2
=
is
a
Mx +
f8li S21
"generator x
chapter, the molecular
treatment. The variational trial function is
-7
(6.19)
9x),
SN-11
coordinates"
(8.9) are s
parameters of the basis
are
chosen
conforming
to
set of relative coordinates. Our aim here is
to calculate the energy of the
system which
can
be
directly
com-
pared to experiment without recourse to the adiabatic treatment like the Born-Oppenheimer approximation. Each basis function includes !V(IV 1) /2 + 3 (IV 1) parameters to be optimized. These parameters -
-
8. Small atoms and molecules
166
describe various correlations. The matrix A describes the electronic
correlations and motions, while the generator coordinate s makes the function flexible and allows us to represent several "peaks" of
wave
the
distribution
density
the holes
when, for example
in the
well separated and the electrons them. around 0 the function By choosing s are
=
hydrogenic limit, "atomic orbits"
axe on
(8.9)
at the
'has its
maxhnum
origin and this limit is suitable around a1, when m/M the paxticles with nearly equal masses are moving equally fast. At the hydrogenic limit, when the motion of the heavy particles are very slow compared to the light ones, the density distribution has several peaks =
=
axound the attractive centers, and to represent these configurations need to shift the maximum of the trial functions out of the origin
we
by choosings appropriately. The usefulness of the generator coordinates in the basis function (8.6) can be understood by the following example. Let us try to calculate the energy of the IH+ this basis with and without 2 by using (that is by setting them to zero) the generator coordinates. The latter
form
corresponds
to the correlated Gaussians for L
of that system is -0.6026
basis of K
=
a.u..
300 Gaussians
0. The energy Without the generator coordinates a
give
-0.5999
a.u.
=
for this molecule. The
inclusion of the generator coordinate immediately h-nproves the con10 basis states vergence and one can get -0.6024 a.u. by using K =
only! Table 8.9 shows ions
consisting
examples
of calculations for the molecules and the
of protons and electrons..
Table 8.9. Ground state
energies of small molecules
in atomic units. K is
the basis dimension.
System
K
"OH+ 2 0OH2 "OH+ 3
SVM
Other method
K
Ref.
50
-0.602634429
-0.602634214
160
100
-1-17445
-1-174475714
1200
100
-1.34351
-1.343835624
600
[1131 [1131 [1141
167
Complements 8.1 The cusp condition for the Coulomb potential It is desirable that the trial function satisfies the proper asymptotic behavior or the special boundary condition as demanded by a given
special boundary condition, the cusp condition [115] known for the Coulomb potential, by using the hydrogen atom as an example. The local energy for the hydrogen atom is given by Hamiltonian. We discuss
h2 1
Ej"'c
a
192 Tf
-
-r-2
2m Tf
where hl is the
angular
h2
2 (9Tf + r
-5r- )
momentum
1
e2
2
(8-10)
1 IIf
+
.M r2 Tf
r
operator. As
was
discussed in
Sect. 3.2, the local energy for the exact wave function turns out to be a constant. The Coulomb potential in the local energy gives a singular 0. For the local energy to be a constant, this singular behavior at r =
behavior must be
for
compensated
by
the kinetic energy term. For
an S wave, where the wave function Tf has no angular dependence, 12Tf 0 and the constancy of El.,r requires that the second term in =
the bracket in
Eq. (8.10) cancel
the
singular
behavior of the Coulomb
potential: I
2
aTf)
Tf 9r
-me h2
r=o
ao
where ao is the Bohr radius. It is easy to see that an exponenexp(-r/a) for the radial part of the ground-state wave
tial form of
function leads to sen
a
constant local
density
if and
only
if
to be ao. The constant of the local energy is then
-0/(2ma2) drogen
=
_Me4/(2h2)
=
-El
as
expected,
atom ionization energy without the
a
is cho-
equal
where E, is the
to
hy-
proton recoil effect, that
is, the well-known energy of 13.6 eV.
angular dependence, it has to take care I/r singularities. Then Tf may be expressed I/r as a product of radial and angular parts: Tf r'R(r)Y(S?), where s 0. is a positive constant and R(r) is assumed to be nonzero at r Substituting Tf into Eq. (8.10), we obtain For
a
general
case
when Tf has 2
and
of both the
=
=
h?
Eloc 2m
+
1
a2 R
( R -5r-2
h2 1 2 __1 Y 2Tar2 Y
2(s + 1)
8(8+1)
1 M
+
+ r
R 9r
r2
2 -
_.
r
(8-12)
Complements
168
As R does not vanish at
r
0, 1IR gives
=
origin. The condition that the local following result: 1
I
( OR) 9r R
We know that the second
12y equation
no
singularity
at the
energy is constant leads to the
=
(s + 1)ao'
=o
rise to
(8.13)
S(S + 1)y
is satisfied if and
only
if
s
is
a
positive integer 1, and then the first equation determines the correct behavior of R near the origin as R(r) oc exp(-r/(l + I)ao). Equations known to be the cusp conditions. Let us attempt to solve the S-wave hydrogen atom variationally with Gaussian basis functions, exp[-(a/2)(r/a0)2J' where a is a vari-
(8.11)
and
(8.13)
are
ational parameter. We functions that lead to
use a
Gaussians
as
an
example of such basis
rather accurate solution but
are
poor in
satisfying the cusp condition. When a single basis function is used, the optimal value of a is 16/(9z), giving the minimum energy of -8/(37t-)EI -0.849EI. A combination of a few terms approximates the ground-state energy quite well. The parameter values of a are determined by the SVM. Table 8.10 shows sample results of such calculations obtained with the code given in [81]. The calculation with five Gaussians already reproduces the energy up to three digits. The wave function obtained with ten Gaussians can reproduce both the energy and the mean values of r and I/r fairly accurately. The over=
lap of the wave function with the exact wave function is very close to unity. We may conclude that the Gaussian basis can predict physical quantities to high accuracy. Of course, the solution does not satisfy the proper asymptotic behavior at large distances and, moreover, always gives zero for the cusp value of Eq. (8.11). The local energy displayed in Fig. 8.5 for the variational wave function indicates that with increasing K it tends to show smaller and smaller deviations from the exact wave function except for the singular points mentioned above. The local energy at large r deviates from the correct value because the Gaussian basis has the wrong asymptotic behavior. It is possible to generalize the above arguments for the cusp value in a system of particles interacting via Coulomb potentials. Evaluating the cusp value for a pair of particles with charges qj and qj, we obtain
(If 16(ri
-
2'j) alriarj I Of)
(T/IJ(ri
-
rj)ITI')
I-tijqiqj h2
(8-14)
where tzij is the reduced mass of the two particles. The left-hand side of Eq. (8.14) is expressed with the matrix elements involving
C8.2 The chemical bond: The H+ ion 2
J(r)
=
J(r)/(47rr2)
used to test the
The cusp values for a pair of paxticles quality of the variational solution at the .
are
169
often
particles'
coalescence.
Table 8.10. Variational solution for the
hydrogen
of Gaussian basis functions. The last
shows the exact values. Ei is the is the Bohr radius.
hydrogen K
atom ionization energy and ao
E
E,
((_L_)-2) ao
atom with
a
number K
row
ao
((_E_)2) ao
ao
Overlap
1
-0.8488264
1.131774
0.8488284
1.499996
2.650706
0.9568351
3
-0.9939585
1.903352
0.9939409
1.491519
2.922759
0.9987560
5
-0.9996191
1.986219
0.9995692
1.499147
2.991284
0.9999446
10
-0.9999998
1.999700
0.9999958
1.500004
3.000014
0.9999999
-1
2
1
1.5
3
1
K= 1
0-, IN.
-4-
K=3
K=10
1 %
0
5
10
15
20
rlao Fig. 8.5. hydrogen
The local energy
curve
plotted for the variational solution
of the
atom. K denotes the number of Gaussian basis functions.
8.2 The chemical bond. The
H+ 2
ion
Quantum mechanics enables us to understand the chemical bond, which is responsible for the formation of molecules from isolated atoms. The chemical bonding phenomenon involves the delocalization of electrons in an atom to gain attraction from the other nuclei when
Complements
170
the atoms
close to each other. We take up the simplest possiH2+ ion, to understand what an important role the
come
ble molecule, the
Hellmann-Feynman
and virial theorems
[1]
of the chemical bond. See
play for clarifying the origin
for detail.
ffilly quantuin-mechanical description of a molecule is a complex problem. This problem is usually simplified by using the BornOppenheimer approximation, where the electronic motion is separated from the nuclear motion, considering the fact that the electron mass is much smaller than that of the nuclei. One starts with determining the motion of the electrons for a fixed configuration R of the nuclei and The
ground state, of energy U(R), of the electronic system. Then one assumes that,when R varies, the electronic system always remains in the ground state corresponding to R, that is the electrons follow adiabatically the motion of the nuclei. The chemical bond is then determined by studying the nuclear motion in a potential energy V(R) which comprises the Coulomb repulsion between the nuclei and obtains the
U(R)Ht 2
Let R be the distance vector between the two protons of the ion and
v
be the
vector of the electron with
position
respect
to the
center-of-mass of the protons. The electron motion is determined the Hamiltonian
2[t
Note that R is
equal
is
the
m
to
e2
e2
p2
H,e( R)
IT
just
a
-
by
stage. The reduced mass jL where M is the mass of the proton and
parameter
2Mm/(2M + m),
(8-15)
IT + RI 2
Al 2
at this
of the electron. The electronic energy U(R) is the lowest of the Hamiltonian (8.15). It is clear that U(R) becomes
mass
eigenvalue
only. The Schr8dinger equation for the Hamiltonian completely separated in elliptical coordinates with respect (8.15) to the foci, R/2 and -R/2. We do not need its exact solution in the following discussion, but note that it is well approximated by the variational calculation using a trial function of the form a
function of R is
Tf
-01"
where at
s.
(Z' R) +'OL' (Z' R),
01, (Z, s) is the charge Z is
The
(8.16)
_
2
2
ls a
hydrogenic orbital
of radius
variational parameter and its
ao/Z
centered
optimal
value
is deternUned to miniraize the energy for each R. The optimal value 1 for R ---* co. At 0 to Z 2 for R of Z decreases froin Z --+
cc
the system will switch
=
=
=
R
over
to
a
configuration
of the
hydrogen
C8.3
atom and the
Stability
proton. Between these
of
hydrogen-like
molecules
extremes, Z is
two
function of R. The energy U(R) is -4E, (2M/(2M + and approaches -E, (MI (M + m)) for R oo.
a
m))
171
decreasing for R
=
0
By using the virial theorem (3-49) and Eq. (3.56) with A R, Ry7 R, we can show that the expectation values, (T) and (W), of the kinetic energy and potential energy of H,..(R) satisfy the relation d
2(T)
+
(W)
+R
dR
U(R)
=
(8.17)
0. '9
Here we have used the fact that WA+R. aR W R d dR to
=
-W and
Rr-aR -U(R)
U(R), enables us equation, together with (T) + (W) U(R). the protons, between of the potential express (T) and (W in terms This
V(R)
=
U(R)
+
=
(e 2IR),
as
follows:
d
d
(T)
=
-U(R)
-
R
dR
U(R)
=
U(R)
=
-V(R)
-
R dR
=
2U(R)
+R
dR
For the chemical bond to must have
(T)
,
clude from
occur
2V(R)
in the
+R
dR
V(R)
-(8.18)
-
R
H+2
system, the potential V(R) --+ oc) -Er at some point
V(R V(Ro:) < -E-r have to be met. Since oc (see Eq. (3.53)), we can con-2E, at R at equilibrium (R Eq. (8.18) that, RO), the electronic
a
deeper
minimum
that is, V(Ro) Er and (W)
Ro,
e2
d
d
(W)
V(R),
=
than
-
0 and
-- -
-
=
kinetic energy is increased and the electronic potential energy is decreased. The lowering of the electronic potential energy is large enough to cancel the
repulsion
for the chemical bond.
and
V(Ro)
-
V(R
-+
between the protons, and that is
According to
oo)
=
an
exact
responsible 2.00ao calculation, Ro -
-2.79 eV.
challenging to perform a nonadiabatic calculation in which no separation of the electron and nuclear motion is made. The validity of the Born-Oppenheimer approximation can be tested in such a calculation. Furthermore, the development of the nonadiabatic treatment for a smaU molecule is of importance in its own right because the adiabaticity may be questionable when the electron is replaced with heavier particles like the muon or the pion. The excellent results obIt is
tained in Sects. 8.2-8.4 indicate that realistic nonadiabatic, calculations are
in fact
possible
in the correlated Gaussian basis.
172
8.3
Complements molecules
Stability of hydrogen-like
The existence of bound states of systems composed of particles with unit charge attracts considerable attention. We discuss this problem here
by applying
of the
some
principles discussed
in
Chap. 3. always bound
The system of the hydrogen-like atom, (M+m-), is and its binding energy is equal to jL/2 in units of e
1, where p stability of a
Mm/(M+m)
=
is the reduced
-mass.
=
I and h
=
What about the
hydrogen-like molecule (M+M+m-m-)? This system is characterized by the mass ratio om/M. Two well-known examples include the hydrogen molecule (a < 1) and the positronium molecule PS2 (o1). Another example is the biexciton molecule [105, 1061. See =
=
Sect. 8.3. The value of 0 <
Emits,
< I.
c-
o-
of the biexcitons
A molecule is bound
can
vary between the two
provided that
the threshold
of any dissociation channel is higher than the lowest energy of the system. The lowest dissociation channel is (M+m-) + (M+m-) for this system, and its threshold energy is Eth the hydrogen-like molecule has been studied
(see Fig. 8.4). However,
-ft. The
stability of
numerically
in Sect. 8.3
=
theoretical argument [1011 makes it possible to prove that the system is bound for arbitrary values of o-, that is, the ground-state energy E of the system is lower than Eth. The proof relies
on
stability
the
a
scaling property of the Coulombic Hamiltonian and the point of the proof given in [101] will be
Of PS2. The basic
shown below. The Hamiltonian of the system -ff
=
-ffS
+
1
_as
=
4[t +
(P21 +P22+P32 +P42)
(
2 e
1
1 +
the
mass
nothing
I -
r34
7'12
1
4-M
is
4m
1 -
r13
) ( 2+ Pi
r14
-
T'24
r23
),
a
-+
=
A H(PS2), M,
(8.20)
2
P44:) P22_P2_ 3
but the Hamiltonian of
mX of
I
I -
a
(8.21)
-
system
(X+X+X-X-)
particle X being equal to 2p. By applying transformation ri Ari (,X m,/(21L)), we obtain I-Is
as
(8-19)
-
HS
is written
HA,
I
HA
(M+M+m-m-)
a
with
scaling
=
(8.22)
C8.3
where the
m,.
is the electron
Stability and
mass
molecule. This
positronium HS, ES,
state energy of
by Es
=
(2A/Tne)E(PS2).
from the the
can
of
hydrogen-like molecules
H(PS2)
is the Hamiltonian for
indicates that the
equation
be obtained from that Of
Note that this relation
groundPS2, E(PS2),
also be obtained
can
Helh-nann-Feynman theorem and the virial theorem. In fact Eqs. (3.38) and (3.53) enables one to obtain (mx 2/-t)
of
use
173
=
d
a
ES
=
dTax
1
( fisjTax-Hsjfis)
=
-- !PSjTSjPS) Tax
I
(8.23)
ES,
-
Tax
which leads to the solution that is the
(Ps
ground-state
wave
E,,/mx
independent of mX. Here function of HS and TS is the kinetic is
energy of HS.
According to the Ritz theorem following relation
the
ground-state
energy
El of H
satisfies the
El :! ((fiS IHS + HA PS)
(8.24)
-
The term
HA is odd under the interchange P of the masses M and -HAP. Then by using that (PS is invariant under i.e., PHA
m,
=
(0SjpHApj(pS) _((pSjHAp2j0S) ((PsIHA10s) The side of 0. Eq. (8.24) is thus equal right-hand -(0sjHAj(Ps) to ((fisjHsj0s) ES, and we obtain P
we
have
El
-
=
Eth :! ES
-
(-A)
=
2A =
M,
(E(PS2)
-
(-
=
M.
2
))
(8.25)
-
The energy of -m,/2 is the threshold for P,92 to decay into two Ps atoms. If the stability Of PS2 is established, that is, E(PS2) +
(m,/2)
0, then
<
we can
the
hydrogen-like
[991
calculated the energy
showed its
immediately from Eq. (8.25) that always stable. In fact Hylleraas and Ore Of PS2 by using a simple trial function and
conclude
molecule is
stability.
It is of interest to examine the
stability of a hydrogen-antihydrogen
like molecular system, (M+m+M-m-) [1041. The lowest threshold of this system is now Eth -(M/4) (m/4) corresponding to the dis-
sociation
(M+M-)
+
(m+m-)
and gets
deeper
and
deeper
as
M in-
creases, which is in contrast to the case of the hydrogen-like molecule. It is un I ikely that the hydrogen-antihydrogen system gets lower energy
Complements
174
than the threshold. It is thus
expected
that there is
a
critical
mass ra
M/m beyond which the system becomes unstable. According to the calculations hn [104, 1161, the stability limit is M/m < 2.1.
tio,
Application of global vectors to muonic molecules The aim of this Complement is to show the utility of the global vector v of Eq. (6.4) for describing the angular part of the variational trial function. As a test example we take up a Coulombic, three-body system, the tdIL molecule, which has attracted much attention in relation to muon catalyzed fusion [1171. The excited P state especially plays a key role in the fusion since it lies close to the threshold for the decay 8.4
to the
ttt
atom and the deuteron.
The basis function
we use
is
fKLM of Eq. (6.27).
matrix elements for this function
Chap.
7 and the
appendix.
can
be calculated with the method of
Without loss of
parameters ul and U2 which define U +U22 1. Each basis function for =
The Hamiltonian
v
generality
can
be normalized to
given
a
the variational
satisfy
set of KLM values thus
contains at most four nonlinear parameters, three of which
come
from
positive-definite to assure the 6DG, general be expressed as A
the matrix A. The matrix A has to be finite
where G is one
Of
norm
a
fKLM,
2x2
and
can
in
=
orthogonal matrix
(
cosO
-sinO
sin,& cos
?
specified by just
parameter V and D is a diagonal matrix including positive d, and d2 values.
two
diagonal
elements of
The accuracy of the variational solution depends on how the parameters ul, V, di, and d2 are given [331. The most naive choice would
be to take G
ing only
a
as a
single
unit matrix
(6
=
0)
,
which is
equivalent
set of relative coordinates x, and then to
to
us-
try
to
im-
by including successively higher partial appropriate choice of K and ul values. Many examples show [20, 29, 30, 311, however, that this type of single-channel calculation does not work well, especially in the case where the adiabatic approximation is questionable as in the present example. Gx correspond to other The matrix G may be chosen to let y particular coordinate sets such as the so-called rearrangement cha n nel [201. (For Gx to correspond to the rearrangement channel, the length scales of x and y in general have to be modified appropriately.) Three possible rearrangement channels expressed in the Jacobi reach convergence
plied by
waves
an
=
particles d, t, and tL are labelled particle 1, 2, and 3, the three patterns in Fig. 2.2 correspond to (tlL)d, (/-td)t, (dt)IL arrangements, respectively. If x is understood coordinate set
are
drawn in
Fig.
2.2. If the
C8.4
Application
of
global
vectors to muonic molecules
175
to stand for the coordinate set of the
(tl-t)d arrangement,
ordinate set
arrangement is obtained
to the other
the
co-
corresponding by choosing such an appropriate,& value that is uniquely determined by the change of the coordinates. With this choice of ?Y, we can write ,'N DGx i Ax dj y', + d2y22 Dy The following three simple types of bases were chosen: =
=
=
*
(i)
K
0 and A
=
dDG,
=
trices
connecting only explained above.
K
(ii) (iii)
=
K
0
or
1 and the choice of A is the
0 and A
=
where G is restricted to the
=
special
ma-
the three sets of rearrangement channels
dDG,
where G is
same as
now an
in
case
as
(i).
arbitrary orthogonal
matrix.
As
mentioned in Sect. 6.2, the angular part with K 0 describes only the stretched configuration, 11 + 12 L, and therefore was
=
-
the basis of type (i) allows rather limited angular correlations between the particles. In fact, the possible (Ili 12) values are given by
(1, L
-
1), (1
=
0,
...,
L).
(ii)
Basis
is
an
clude the non-stretched
extension of basis
(i) to in1, possible (11,12)
coupling. With K 0,..., 1), (1 L) are also allowed. Though K is set to in basis zero (iii), the angular correlation can be taken into account through the cross term of the exponential part of the basis function. In bases (i) and (ii) the factors exp(- FbAx/2) are always "Correlationfree" in a particular channel, that is, it contains no cross term in the exponential part. In these bases it is through the inclusion of all rear-
(I
+
1, L
-
I +
rangement channels that type ment
for
(iii), (that
=
=
=
one
the contrary, is, coordinate
on
through the general
takes
one
care
of the correlation. In
set) explicitly;
case
(i)
contained in basis
form of A.
expected, the
re-
to other bases. The correlation
poor compared is too restricted to obtain
are
(i)
basis of
just any one arrangethe correlations are allowed
Table 8.11 shows the results of calculations. As sults with
a
needs to consider
a
realistic
description
of the system. The angular correlation, which is taken into account in basis (ii), certainly improves the energy over case (i). The basis functions
first six for L
(iii) give
figures 0
even
better
of the most
energies. In fact they reproduce the precise variational calculations [20, 23, 921
2 states. Even with K
0, the use of the full A matrix incorporate important correlations between the particles. A Gaussian basis of type (ii) is employed in the Coupled=
enables
--
one
to
=
the
Rearrangement-Channel Gaussian (CRCG) basis variational calculation of [201, where the angular part is, however, represented by the
176
Complements
coupling of type (6.3). The fact that the D state energy with the type (iii) calculation of dimension 200 becomes slightly lower than that of [20] with 1566 basis functions confirms the importance of successive
optimization of the nonlinear parameters. Further increase of the dimension can improve the accuracy rather easily in the angulax function of Eq. (6.4) as seen in the case of S and P states. The basis function used in [23, 921 is correlated exponential (CE) which a
caxeful
takes the form of
exp(-air,
-
T21
-
JT2
-
T31
-
7IT3 -'r1j)
for the orbital motion with
Multiorbital
nonzero plied by some polynomials determined The are momentum. by the , parameters a, -y angular calculations These 4.2.2. in Sect. random tempering explained using the exponential function give very precise results.
Table 8.11. The total
energies of the tdIL molecule. Parameter set a of by superscript b, where set
Table 8.2 is used except for the case indicated b is used. Atomic units are used. Dimension
E
200
-111.29346
200
-111.36398
200
-111.36444
600
-111.3645077
CRCG[20] CE[231
1442
-111.364507
1400
-111.364511474
(i) SVM (ii) SVM (iii) CE[231
200
-108.13122
200
-108-17914
200
-108-17940
1800
-108.1795424
800
-108-179361
CRCG[20]b CE[921b
2662
-108.179385
1900
-108.1793881
(iii)b CRCG[201b CE[921b
800
-99-660367
2662
-99.660548
1900
-99.6605507
SVM
(i) (ii) SVM (iii) CRCG[201
200
-103.37067
SVM
200
-103.40824
200
-103.40849
1566
-103.408481
L
Method
S
SVM SVM SVM
P
SVM
SW
P*
D
(i) (ii) (iii)
(ifl)
b
SVM
9.
Baryon spectroscopy
This
chapter is devoted to applications of the SVM to the constituent quark model of baryons. There are several different models of baxyons and this is a very hot topic nowadays. The discussion of the different models and their relative merits are far beyond the scope of this book, and our only aim here is to show the applicability of the SVM to solving tbree-body dynamics in constituent quark potential models. The constituent quark model assumes that the baryons comprise three quarks with dynamical quark masses and the quarks interact via phenomelogical potentials. These potentials contain a confining term and other terms which determine the fine structure of the spectra. are no free quarks observed in nature, the confining inter-
As there
action hinders the "ionization7 of the the
baryons in the constituent usually taken as a power
models. This interaction is
quark potential potential (V(r) rP). The second part of the interaction is traditionally chosen as the one-gluon exchange potential (OGEP) which is a color analogue of the Breit interaction of quantum electrodynamics (QED). There is an alternative possibility, the meson exchange interaction, which is motivated by the fact that, despite its quite successful reproduction of the ground-state spectra of the mesons and baryons, the OGEP may not be adequate for describing the excited states as mesonic effects or quark-antiquark excitations become important. This chapter includes examples for both choices to illustrate law
-
how the SVM
can
be used for these systems with different interac(nonrelavistic and semirelativistic) forms of
tions and with different
the kinetic energy. These simple quaxk models have obvious limitations
(e.g., large relativistic corrections can be anticipated for light quarks), but their success
indicates that relativistic and field theoretical effects
incorporated
in the paxameters. In
so
Y. Suzuki and K. Varga: LNPm 54, pp. 177 - 186, 1998 © Springer-Verlag Berlin Heidelberg 1998
far
as
are
somehow
sophisticated approaches
9.
178
Baryon spectroscopy
quantum chromodynamics (QCD) do not provide results for these systems, the application of constituent quark models is justified. based
on
9.1 The trial function in the constituent
quark
model carries space-, spin-, flavor-, and color degrees of freedoTn. The flavor is an extension of the isosphn to include up(u),
quark (q)
The
down(d), strange (s), charm(c), bottom(b),
and
top(t) quarks.
baxyons are colorless: The three quarks in a baryon form a color-singlet state, that is, they are totally antisymmetric with respect to the interchange of the color coordinates. For example, the eight baryons of spin 1/2, N (p, n), A, Z (Z+, V), Z-), and EF (EE,", belonging to a family of octet baryons, consist of three quarks of u, d and s flavor varieties. The product of the spin-, flavor-, and spacepart of their wave functions has to be symmetric to comply with the generalized Pauli principle. The spin paxt is represented by the spin functions X(S,,)Sm, (see Complement 6.5): The
TIT),
xmi a
X(O).1.1 22
2
r6 ( -V
11 X(1) 'TT
The flavor
IR
=
I (
S exp
where X and metrizer.
9.2
same
wave
-
I III)
functions
wave
baryons with the The trial
21111)
chaxge
function for
-1 -
are
2
are
-v/2
(I TIT)
-
I
ITT)), (9-1)
I ITT)
shown in Table 9.1, where the
arranged baxyons is a
are
in each column.
combination of the terms
6Ax) X(S12)SM (T12)TMT
the
(9.2)
spin and flavor functions, and S is
a
sym-
One-gluon exchange model
one-gluon exchange potential (OGEP) model has proved to be quite successful in describing the spectra of the ground states and
The
various
properties of
the
baryons [118, 119, 1201.
In this model the
9.2
Table 9. 1. Flavor to
wave
model
One-gluon exchange
ftmctions of baryons
179
arranged columnwise according
charge +2
Baryon N A
UUU
+1
0
uud
ddu
uud
ddu
-1
ddd
71,= (ud du) d + du) 71, 72 (u
A
-
s
2 '
UUS
dds
s
ssd
SSU
S?
SSS
7L (ud
Ac
-
2 ,
El-
72= (ud
UUC
+
du) c du) c
ddc
72 (ud
Ab
1
uub
Eb
vr2-
-
du)b
(ud + du) b
ddb
quarks exchange massless but color-charged gluons and the qq potenas the color analogue of the Breit interaction [3, 1211. Consequently, the simplest version of the OGEP includes a Coulomblike I/r piece and a color-magnetic ("hyperfine") piece:
tial is taken
7r -
i<j
6mimj
(AI7. AC j)(ffi-0j)j(rj-rj)'
where the Gell-Mann matrix
generator for the ith uct,
1:8
=1
introduce masses
dence,
(9-3)
S
quark
AF (a) AC (a). (In
A9 (a) (a
and
(A'9 Ajc) -
a more
tensor and other
spin-orbit,
=
z
is the color
SU(3)
scalar
prod-
stands for
a
elaborated framework order
higher
essentially
terms.)
one can
The
quark
very weak flavor depenflavor-blind. In a nonpertur-
mi in the denominator introduce
but the interaction is
8)
1,
a
bative treatment the contact interaction leads to
a
Because
singularity.
of this and because of the finite spatial extent of the constituent
quarks
the delta function should be smeared out, and it is to be substituted by a Gaussian or a Yukawa form factor.
Among the several different parametrizations of the OGEP we have potential [1221 as an example:
chosen the AM
3(A9 8
Vij H
-
A(7) 2rr.'
x
--+Ar-A+ r
3mpaj
exp
Tij ij 1.
7F 2
a
3
Pij
-
-
O'j
,
(9.4)
180
9.
Baryon spectroscopy
where the
length r is given in units of inverse energy (I fin 1/197.327 MeV-'), and the smearing parameter pij is given iTi the forioa Pij
=
-B
(
A
_
"
2mimj
mj + Taj
)
(9-5)
The
315 MeV, quark masses used in the calculation are Tn,, Tnd 577 MeV, m, 1836 MeV and ?nb 5227 MeV. The parameters m , were determined by a fit on a large sample of meson states in every ffavor sector [1221: They are n 0.1653 0.5069, rs' 1.8609, A GeV2 A 0.8321 GeV, B 0.2204 and A 1.6553 GeVB-'. The matrix element of the color part can be easily obtained because the baryons are in a color singlet state. By noting the relationship between the color-exchange operator P53 and (AF Ajc), =
=
=
=
=
=
=
=
=
=
=
7
-
PC== zy
2
C (A9-Af)+-, 3 '
(9.6)
J
the matrix element of
(A9.A'7) 3
between the
S
simply
to
states reduces
-8/3.
The SVM results
[1221
color-singlet
[81, 1231
are
compared
with the calculations of
in Table 9.2. The
agreement between the two results is good. data [1241 are included for guidance. One has to
The
experimental compaxe the energies relative to the nucleon ground state and one has to bear in mind that this potential gives good results for the mesons and no free parameter has been introduced to fit the baryons. Table 9.2. The
tial
[1221.
masses
of baryons
The numbers in
(in MeV)
parentheses
axe
as
the
predicted by the AU poten-
masses
relative to the nucleon
mass.
Baryon
Ref.
N
[1221
SVM
Experiment [1241
998
995
939
A, Z. Ab Zb
1154(156) 1231(233) 1343(345) 1674(676) 2296(1298) 2466(1468) 5642(4644) 5849(4851)
1306(311) 1149(154) 1228(233) 1339(344-) 1674(679) 2290(1295) 2467(1472) 5635(4640) 5849(4854)
1232(293) 1116.(177) 1192(253) 1318(379) 1672(733) 2285(1346) 2455(1516) 564150 (470250)
-Fb
5808(4810)
5803(4808)
A
A E
0
9.3
9.3
Meson-exchange
model
181
model
The OGEP model is successful in and
Meson-exchange
describing ground-state energies
properties of baryons, but for the excited
states
a
number of
delicate problems remain unsolved. No model has been able to
explain, orderings in light- and strange-baryon spectra in a simple three-quark description of baryons. The source of the problem can be easily understood by the following argument: The ground state of the nucleon N(939) has positive parity followed by the positive paxity 1/2+ Roper resonance N(1440) and the negative-parity 1/2- and 3/2- states N(1535) and N(1520), respectively. The parity of the states follows as (+, +, -, -). In the case of the A, the order of states follows a (+, -1 +1 -) pattern. The order of parity of the spectra in a simple harmonic-oscillator quark model is (+, -, +, In the case of the linear confinement the level spacing changes, but the order for
example,
the correct level
of states remains the
The
same.
wave
function of N and A differs
only in flavor, but the OGEP is flavor independent (though it slightly depends on flavor through the mass difference of light and strange quarks in Eq. (9.4)). Therefore it predicts the same level order for both N and A in contradiction to the experiments. To resolve this problem an explicitly flavor-dependent mesonexchange interaction has been introduced [1251. The essential difference between the two approaches is that while the OGEP of type (ai o-j) V(rij) acts on the color and spin space, the Ei<j (AF AjC) 3 -
-
%
meson-exchange potential Ei<j AFA- 37(aj-o-j)V(rjj) acts on the flavor and the spin degrees of freedom. The meson-exchange potential combined with a semirelativistic form of the kinetic energy is the second example of the application of the SVM for baryon spectroscopy. In this example no gluon-exchange mechanism is explicitly introduced to describe the qq interaction but only a meson-exchange potential is employed to generate the baryon '1
spectra. The
three-quark Hamiltonian is a sum of the semirelativistic Idmeson-exchange potential V and the confinement potential VO + Cr: netic energy, the
3
3
H=E /--2+ ?
Mi2 P71i +M?
where the
+
(Vii
meson-exchange potential
+
V"
+
Clri
V reads
as
-
ri
1)
(9.7)
9.
182
Baryon spectroscopy 7
3
Vij (r)
V7r (r)
=
ij
1: AF (a),XjF (a) + VI
a=4
+ VI7 (r) AF i (8) ij
The form factor of the
g2
f
1
47-,
2
AjF (8) + 3 Vj'j7' (r)
Oj
-
aj
meson-exchange potential
V
ij
AF i (a) AjF (a)
(r)
3
a=1
12mimj
2
I
6 -Ay
e-A-yr
A 'I
A2'Y
r
r
I
paxametrized
is
as
(9-9)
1
coupling constant gy is taken to be g8 for the pseudoscalar octet mesons (7r, K, n) or go for the pseudoscalar singlet meson (n,), respectively and the cut-off mass Ay is assumed to be given by
where the
Ay
=
Ao
(9.10)
+ riLy
q, 77. For the constituent quark masses we take the 500 MeV. Parameters typical vdues, Mu, Md 340 MeV and m, of the Hamiltonian are listed in Table 9.3.
for each -/
=
K,
-Ft,
=
=
=
Table 9.3. Parameters of the semirelativistic constituent
quark model with
meson-exchange potential Fixed parameters
Quark
masses
Meson
[MeVI
masses
[MeVI 928
Mui Md
M,
Ar
AK
An
A'a
340
500
139
494
547
958
41r
0.67
Free parameters
(golg8)2 1.34
Ao
[fin-]
K
Vf0 [Me,V
fln-21 2.33
-416
0.81
2.87
C
The -matrix elements of the semirelativistic kinetic energy in Eq. (9.7) can be calculated with the method presented In Sect. 7.6. The matrix elements of
culated
by-using
AF(a)AjF(a) 71
in the flavor space
Table 9.1 and the
explicit
can
be
easily
calm-
form of the GeH-Ma=
Meson-exchange model
9.3
183
matrices. It is useful to know the action of the
AF(a)A-3 (a)
ator
the flavor
on
wave
ffavor-changing operfunctions. As is summarized in
Z
9.4, for example the ffavor function uidj, when acted AF (a) M'(a), transforms to 2djuj uidj. I
Table
E
3
upon
by
-
a=
Z
3
Table 9.4. Effect of
Operator 3
flavor-changing operators
on
quark pairs
UU
dd
SS
ud
us
ds
Ar (a) X3 (a)
UU
dd
0
2du-ud
0
0
1
AF (a) AF j (a)
0
0
0
0
2su
2 sd
4
-1 UU
-1 dd
4UU
lud
-!us
-ids
3
3
Z
/\F(8),XjF(8)
3
3
3
3
_
predictions of this model axe shown for all light- and strange-baryons with mass up to M < 1850 MeV; the nucleon mass is normalized to its mass of 939 MeV, which determines the value of the confinement potential parameter V0. All masses corresponding to three- and four-star resonances in the most recent compilation of the Particle Data Group [124] are included. The result is good, reproducing the spectra of all low-lying light and strange baryons [123, 126]. In particular, the level ordering of the lowest positive- and negative-parity states in the nucleon spectrum is reproduced correctly, with the 1/2+ Roper resonance N(1440) falling well below the negative parity 1/2- and 3/2- states N(1535) and N(1520), respectively. Likewise, in the A and Z spectra the positive-parity 1/2+ excited baryons, A(1600) and E(1660), fall below the negative parity 1/2-In
3/2-
9.1 the SVM
Fig.
states,
A(1670)-A(1690),
and the
1/2-
state
E(1750),
respec-
tively. In the A spectrum, at the same time, the negative-parity 1/2-3/2- states A(1405)-A(1520) remain the lowest excitations above the A
ground
state.
remark is in order about the necessity of employing a semirelativistic kinetic energy operator in the three-quark Hamiltonian (9.7). Certainly, this is only an intermediate step toward a fully At this
place,
a
covariant treatment but it
already
allows
us
to include the kinemat-
ical relativistic effects. In any nonrelativistic approach these effects get compensated by the potential parameters, but there one faces a
disturbing
consequence of
constituent
quark
and
c
v1c > 1,
is the
where
velocity
of
v
is the
light.
mean
velocity of the
184
9.
Baryon spectroscopy
18001700-
El
E----1
L-J
=
16001500M
[MeV]
14001300-
12001100-
N
1000-
A
900-
1
1+
1-
3+
3
2
2
Y
Y
5+ 5 Y Y
1-
3+ 3i -2
Fig. 9.1. Comparison of low-lying baryon spectra predicted by the mesonexchange potential model with experiment. The solid lines denote the calculated spectra, and the shaded boxes show the error bars.
experimental
masses
with
their
(Continued
on
the next
page.)
One may think that the idea of the meson-exchange interaction acting between quarks is unfamiliar though it has produced the good results. To
judge
its merits the present calculations have to be
tended into several directions
ex-
(prediction physical observables, e.g. factors, decay widths, etc.). The complexity of the NN interaction is basically due to the compositeness of the nucleon. As N, A or Z belong to the same octet family, it is natural to attempt a unified description of the NN, AN or EN interaction from the underlying qq potentials. This goal is at present too difficult to achieve directly on the basis of QCD. In so of
form
far
as
QCD as yet presents no results, the effective theory formua microscopic quark-cluster model [1271 seems to be justified.
lated in
9.3
(Continued
from
Fig
Meson-exchange
model
185
9. 1.)
E] 1800
1700 1600
-
-
F-7
17.71
t
H
r--.7.
-
r__,_T
p G=
r_.j
0
p"
-
-
1500M
[MeVj
1400-
13001200-
1100-
A
1000900
1
1
_2
1
2
_2
_2
1+
1-
3+
1
_2
1800
1700 1600
1500 M
[MeVJ
1400
1300 1200 1100
1000
900 3-
1
11
3+
_2
_2
1
_2
186
The
9.
Baryon spectroscopy
intermediate-range
attraction of the
cannot be accounted for
by
tribution between the colorless the
baryon-baryon interactions produces no con-
the OGEP because it
It is not clear yet whether introduced above leads to a realistic de-
baryons.
meson-exchange potential scription of the NN interaction or not. It is probable that the OGEP combined with the meson-exchange potential is a useful, effective vehicle to obtain a reasonable description of both baxyon spectra and baryon-baxyon interactions at the same time. See, for example, [128] for the attempt along this line.
in solid state
10.
Few-body problems physics
This
chapter is a collection of examples of the application of the SVM to few-body problems in the field of solid state physics. The examples include excitonic complexes, biexcitons, and quantum dots with and without magnetic field. The biexciton and two
(or excitonic molecules), the bound state of two holes
electrons, provides
an
interesting few-body problem,
since it
may be thought as a positronium. or a hydrogen molecule with variable electron and hole masses realized in semiconductors. The biexcitons
also often confined and
are
can
be modelled
as
two-dimensional
(2D)
systems. In recent yeaxs there has been much excitement in the possible applications of ultrasmall. systems with a length scale of 10-100
A,("nanostructures"). The technological motivation (in electronics and optoelectronics)
is that the smaller components
to manufacture very small
contain
very few electrons
only a
are
faster. It is
possible
nanostructures, often called "dots", which
(N
<
10).
The
quantum-mechanical
effects in these small systems axe very important and their properties are strongly N-dependent. The energy spectra of few-particle quantum
dots call for theoretical interpretation. In confined in
are
a
box of sizes
L, L.,
a
nanostructure the electrons
and L,. If
L,
-
Ly
>>
L, the
quantum dot is quasi 2D. The confinement is usually modelled by a harmonic-oscillator potential. This system is somewhat similar to an atom in
nature, where the "confining" potential is the Coulomb
traction of the electrons to
quantum dots
are
Throughout this chapter
we use an
tron. When the dielectric constant of
length =
a
effective
mass
me*
for
material is denoted
an
by
elec-
r., the
and the energy will be measured in "atomic units" with Bohr h2n/ (M*e2 ) and haxtree (two times the Rydberg energy) e-
radius a*
2R*
at-
nucleus in the atom, and therefore the often referred to as "artificial atoms". a
=
e2/(Ka*), respectively.
Y. Suzuki and K. Varga: LNPm 54, pp. 187 - 211, 1998 © Springer-Verlag Berlin Heidelberg 1998
188
10.
Few-body problems
10.1 Excitonic
in solid state
physics
complexes
The excitonic
complexes have considerable importance in the development of semiconductor physics and spectroscopy. The bound state of a positively charged hole with an effective mass m and an electron with an effective mass m,* is called an exciton in semiconductor physics. The mass ratio ocan between 0 (hydrogenic limit cchange m*/m*h and oI (positronium limit) depending on the material and other factors. Like in the case of hydrogen, where not only the hydrogen atom but the hydrogen molecule (H2), or the H2+, H-, H,+ ions are bound, the system of N, electrons and Nh holes can also be bound. The latter system is called an excitonic complex. These excitonic complexes, including the charged excitons, have been subject of intensive experimental [105, 1291 and theoretical [99, 130-1351 investigation. The properties and structure of these systems strongly change with the mass ratio, and, by approaching the two limiting cases, one arrives at two completely different worlds. The interest in these systems has =
=
e
=
been intensified when the advance in semiconductor
technology has possible the fabrication of artificial nanostructures with diameters comparable to atomic distances. This restricted geometry has a prominent effect on the dynamics of the excitonic species. In the following we present the ground-state energy and some other properties of 2D excitonic complexes. The general Hamiltonian made
be written
can
IV.
H
as
N
pi,
i=1
2m,
i=IV'+I
N
j>i=N' +l x
TC.
r"Iri
-
e2
+
2Mh
3>z=l N,
e2
E where
Ne
pi,
+
=
N
e2
E E
rj I
i=1
j=N,+1
is the dielectric constant of the
material,
and the position
2D vectors in the xy plane. In atomic units introduced in this chapter, the Hamiltonian becomes dimensionless and the envectors
axe
depends only
(Note
that
do not need to
specify what dependence is hidden in the atomic units.) This is easily understood by introducing dimensionless coordinates and momenta by ri* ri/a* and pi* pil(hla*). Then the ergy
value of
K
on o-.
is used because the
r,
=
Hamiltonian
(10-1)
we
is reduced to
=
10.1 Excitonic N,
H
2
2P'
2
P
2(o-N,
i=N +I
+N
N
+ Z
i>i=l
3
-
N'-
E
+
j>i=N,+l
2
0-
*2
E
+
189
N
/
1
2R*
complexes
Jr
-
'r
7,
3
P'
N,) N
E E
I
i=1
j=N,+l
J'r j*
r 3
I'
At this point, a comment is due. What we consider here is a welldefined quantum-mechanical problem to be solved. This is, however,
experimental situation. In the realistic case there considered, for example the effect of other electronic bands, the possibility that the interaction is different from the pure Coulomb force due to the confinement in the z direction, etc. The discussion of the importance of these effects is beyond the scope only are
model of the
a
many other factors to be
of this book.
As the
problem here is practically the same as that discussed in the chapter of small atoms and molecules (Chap. 8), the same trial function is used (see Eq. (8.1)), except that it is tailored to the 2D case, that is, the position vectors have only x and y components. The results for case
o-
=
0 and
o-
=
1
are
and in Table 10.2 for the 2D
shown in Table 10.1 for the 3D
case.
between the 3D and 2D results is the
The most
large
striking difference binding
increase in the
0) in energy in the 2D case. E.g., the binding energy of R2 (with o2D is about 8 times of that in 3D. The increase of binding has been =
found
experimentally
Table 10. 1.
as
well.
Energies and binding energies (in a.u.) of 3D
of electrons and holes for two orbital
angular
cases
momentum and
of the
mass
ratio
o-.
exciton complexes
L and S
are
The asterisk refers to the states which
are
found to be unbound. The binding
energy is understood with respect to the nearest threshold.
E(o-i)
B(o-=O)
B(o-=I)
-0.500
-0.250
0.500
0.250
-0.528
-0.262
0.028
0.012
-0.602
-0.262
0.102
0.012
-1.174
-OM6
0.174
0.016
System
(L, S)
E(o-=O)
eh
(0,0) (0,1/2) (0,1/2) (0,0) (0,1/2) (0,1/2)
eeh ehh
eehh eeehh eehhh
*
-1.344
the total
spin of the exciton complex, respectively.
0.169
190
10.
Few-body problems
in solid state
physics
Table 10.2. Energies and binding energies (in a.u.) of 2D excito-n complexes of electrons and holes for two cases of the mass ratio o-. M is the z component of the total orbital angular momentum and S is the total spin. See also the caption of Table 10.1.
System
(M, S)
E
eh
(0,0) (0,1/2) (0,1/2) (0,0) (0,1/2) (0,1/2)
-2.00
-1.00
2.00
1.00
-2.24
-1.12
0.24
0.12
-2.82
-1.12
0.82
0.12
-5.33
-2.19
1.33
0.19
eeh ehh eehh eeehh
eehhh
(o-
=
0)
E
(o-
=
1)
B
(o-
*
-6-82
1.50
=
0)
B
(o-
=
1)
10.2
equality 2(r+-)
distances have to fullfil the
=
Quantum
dots
V(r--)2 + (r++)2.
191
Ta-
ble 10.3 shows that this is not satisfied. Moreover, the fact that the uncertainties
Arij
(,ri2j
-
(rij 2
are
large
means
that
no
such in-
terpretation is possible. There is only one case where the uncertainty is in very small, that is the distance between the (heavy) positive charges with accordance in that In can limit. case one the hydrogenic assume, the adiabatic approximation, that the positive charges are fixed at the distance of 0.37 a.u. The equilibrium distance in the H2 molecule in 3D is 1.4
a.u.
Table 10.3. ratio
mass
Average distances in 2D biexcitons (eehh) as a function of the E.g., (r--) is the mean distance between the two negative
o-.
charges. Atomic 0-
(r++) (,r+-) 2 (r _) (r 2+) 2 (r _)
=
0
units o-
are
0.4
=
used. a
=
0.7
0-
=
0.67
1.26
1.55
0.37
1.14
1.49
1.80
0.47
0.93
1.16
1.38
0.59
2.09
3.19
4.33
0.14
1.69
2.95
4.33
0.31
1.29
2.05
2.88
1.80
dots
10.2
Quantum
Rapid
advances in semiconductor
rication of nanostructures called
[136, 137, 1381. electrons,
are
In
I
quanturn dots
technology
have led to the fab-
quantum dots a
few
or
artificial atoms
electrons, typically
2 to 200
bound at semiconductor interfaces. These few-electron
(2D) electron gases (MOS) structures are
systems arise when homogeneous two-dimensional metal-oxide-semiconductor
of heterojunctions
or
laterally confined
to diameters
comparable
to the effective Bohr
ra-
dius of the host semiconductor. The interest in quantum dots arises not only from the prospect of new technological applications but also from the desire to understand the in
physics of a few interacting electrons
external field. Needless to say, we concentrate on the few-electron where the effect of electron-electron interaction and their corre-
an
case
lation
seems
to be very
important.
192
10.
Few-body problems
The axtificial atoms
in solid state
consider here
we
of N electrons confined in 2D
physics are
modelled
by
a
system
by harmonic-oscillator potential. Before starting with artificial atoms in 2D we present a calculation for the energies of 2D "natural" atoms where the "co-ofi-ohn ' potential is a
Coulombic. The results for few-electron atoms and ions Table 10.4. The much
binding energies
are
again, due
are
to the 2D
given
in
geometry,
than in the 3D
case. Otherwise the properties of these laxger systems axe rather similax to those of their 3D counterparts. For ex-
ample,
akin to
3D,
the
H, Li
or
Be atom
can
bind
an
electron,
extra
while the He atom cannot.
Table 10.4.
Energies
of atoms and ions confined in 2D. M is the
nent of the total orbital
units
are
(M, S)
H
(0,0) (0,1/2) (0,0) (0,1/2) (0,0)
He Li
Be
momentum and S is the total
z
compo-
spin. Atomic
used.
System
H-
angular
Energy -2.00000000 -2.24
-11-89 -29-87 -56.77
The
ex6mple of quantum dots considered here is a system of 2D electrons in a quadratic confiniD potential. The Hamiltonian reads as N
N
9
Pi
H
+
_M
2m*
(V__
+
ivy)
M
where v., and v.
by
N
I 2
e2
*WO2,r? +
(10.2)
exp( IiUr),
(10.3)
-
2
are
the
x
and y components of the 2D vector
v
defined
10.2
Quantum dots
193
N V
(10.4)
Uiri.
The parameters ui and A of the basis function SVMDue to the external field
are
determined
(the single-particle potential),
by the
the Hamil-
tonian (10.2) of the system is not translationally invariant. Correspondingly, the basis function (10.3) is written in terms of single-
particle coordinates instead of relative ones. This basis function depends on the center-of-mass coordinate, and the energy obtained will be the total energy of the system, unlike the previous cases where the energy of the center-of-mass motion was always subtracted. The case of the quadratic confinement is very special, because in that case the energy of the center-of-mass can be separated. In general, for an arbitrary single-particle potential, however, this cannot be done. 1 electron is trivial and its solution is a simple The case of N 2 an analytical solution exists for cerharmonic oscillator. For N I a.u., the ground state tain frequencies [139]. For example, for hwo which is 3 is precisely reproduced by the numerical cala.u., energy culation. Not surprisingly, as is shown in Table 10.5, if the quadratic =
=
=
confinement is strong, the spectrum is close to that of a harmonic oscillator. In the realistic cases (hwo < I a.u.), however, the eigen-
strongly deviate from those of the harmonic oscillator. This example illustrates the importance of these systems: by changing the confinement one can "tune" the properties of these artificial atoms.
values
Energies of N-electron quantum dots with the z component of angular momentum M and the total spin S. The numbers parentheses are the eigenvalues without Coulomb interaction. Atomic
Table 10.5.
the total orbital in
units
are
used.
N
(M, S)
hwo
2
(0,0) (0,1/2) (0,0)
3.000
212.198
7.220
525.12
3 4
Figure
=
I
hwo
(2.000) (5.000) 10.600 (6-000)
10.1 shows the
=
654.45
100
(200.00) (500.00) (600.00)
energies of the ground and excited
states
of the energy of a harinonic-oscillator quantum) of two electrons as a function of the inverse square root of the oscillator constant, -1/2 If there would be no Coulomb interaction between the elec-
(in units
(hwo)
.
10.
194
Few-body problems
in solid state
physics
trons, the energy divided by hwo would be given by lines parallel to the horizontal axis., The deviation from the
straight lines is entirely interesting feature is that states but the ground state
due to the Coulomb interaction. Another level
crossings
always
occur
remains M
between the excited
0.
12
10
8
6
4
2 0
2
4
6
8
10
(hcoo )-1/2 10.1.
Fig. potential
Energy
with
an
levels of two electrons confined
by
a
oscillator constant wo. Atomic units
harmonic-oscillator
are
-used.
Quantum dots
10.2
195
3.5 3.0 2.5
2.0 ;-4
t__I (14
1.5 1.0 0.5 0.0
1
Fig.
3
2
4
(a.u.)
r
10.2. Electron densities of two-electron
quantum dots
as a
function
origin. The solid, the dotted and the dashed 5, respectively. Atomic 0.2, hwo 2, and hwo
of the radial distance from the curves
units
correspond to hwo
are
used.
1.2
1.0
0.8
A
0.6
0.4
0.2
0.0 0
4
2
6
r
Fig.
8
10
(a.u.)
(dashed curve), four- (dotted curve), (solid curve) quantum dots. The oscillator frequency of the
10.3. Electron densities of two-
and six-electron
quadratic confining potential
is hwo
=
0.2. Atomic units
are
used.
196
Few-body problems
10.
in
solid state
physics
peak density inreases with the number of electrons, that is, the equilibrium configuration is realized on rings of expanding diameter. In the case of N=4 electrons, for example, the paxticles may move along a ring, while they are situated at the vertices of a square to TniniTni e the Coulomb repulsion.
10.3
dots in
Quantum
magnetic field
A dramatic feature axises when the quantum dots are subjected to magnetic fields: The ground states are stabilized into magic-number states. When the
state
Tn
from
jumps owing
to the combined effect of the electron correlation and
appear
the Pauli
agnitude of the magnetic field is vaxied, the ground magic number state to another. Magic numbers
one
principle [140].
The Hamiltonian of the system is N
N
I
I
H
=
E 2m* (pi + e-A,)2 + -
N
2
us assume
ward in the
z
2
Wd 'ri
e2
E
1' Iri
i>i=l
Let
2
,,
-M
-
C
+
given by
that
a
(10.5)
-,rj I*
homogeneous magnetic
field B is
direction. The above Hamiltonian
can
applied
down-
then be rewritten
in the form
H
P'
2
+
-m w
2
2m*
N
2-
r?-
1 -
2
hw,
Ii,
e2
rdri
-
rj I
where u),
eB/(m*c) is the cyclotron frequency and w
We
the
=
V/W--22. rl4+wo
ignored magnetic interaction energy due to the electron spin. See Complement 10.1 for details. It is shown there that the part of
the Hamiltonian
corresponding to the single-particle motion can easily
be diagonalized by the use of associated Laguerre polynomials. Depending on the strength of the magnetic field, both the single-particle
10.3
Quantum dots
in
motion and the correlation due to the Coulomb
sive roles in
determining
function is the
same as
197
repulsion play deci-
the structure of the quantum dots. The trial
in the
For the two-electron with the
magnetic field
previous
case our
solution
section.
calculation is in
perfect agreement
[1411. The spectrum of two elec0) is shown in Fig. 10.4. See also
in
given analytical spin-singlet state (S Complement 10.1. The harmonic-oscillator frequency wo is taken as 0.01 a.u. The most intriguing phenomenon seen is the level hwo crossing. Due to these level crossings, the ground state changes with the strength of the magnetic field. This is shown in a magnified scale on the right-hand side of Fig. 10.4. If there is no Coulomb interaction between the electrons, no level crossing occurs. trons in the
=
=
10.0 18 9.5
16
9.0 14 +It
+Z
8.5
12 8.0
10 7.5 8
7.0 0
1
2
3
4
0.0
0.5
1.0
1.5
2.0
2.5
(,)C/ ("0
(J)C/(00
10.4. Energy levels of a two-electron quantum dot in magnetic field function of the cyclotron frequency w,. The figure on the right-hand side magnifies the level crossing. The oscillator frequency of the quadratic 0. Atomic units are 0.01. The total spin is S confining potential is hwo
Fig. as a
=
=
used.
Figures
plot the energy levels of a three-electron total spin being S 3/2) against the orbital
10.5 and 10.6
quantum dot
(with
the
=
198
10.
in solid state
Few-body problems
physics
25
20
15
10
5 0
2
4
6
8
10
12
14
COC/ coo 1.0.5.
Fig. z
Energy
levels of
a
three-electron quantum dot
component of the total orbital angular
IVI
=
6
(dotted curve),
and M
=
9
momentum M
(dashed curve)
frequency of the quadratic confining potential S
=
3/2.
Atomic units
are
used,
is hwc)
belonging
=
3
to the
(solid c=e),
orbital motion. The =
1. The total
spin
is
10.3
Quantum
3
4
dots in
magnetic
field
199
15
14--1
13
12
44
10-
9
8
7
6 0
2
1
5
6
7
9
8
10
M
Energy levels as a function of the z component of the total angular momentum, M, of a three-electron quantum dot in a low 3/2. Atomic 1). The total spin is S 0.2, hwo magnetic field (hw,
Fig.
10.6.
orbital
=
units
are
used.
=
=
200
10.
Few-body problems
of these orbits
in solid state
physics
larger (see Eq. (10.16)). The energy of the dots interplay of the single-paxticle energies and the interaction energy. By increasing the strength of the magnetic field, similarly to the case of low magnetic field, the ground state does -not take all possible values of M. By changing the magnetic field the states with M 3,6,9,12,... "magic numbers' become ground states. is determined
axe
by
the
=
40-
35-
30-
25
-
201
10
0
1
2
3
4
5
6
7
8
9
M 1.0.7. The
Fig. hwo
=
1).
same as
Atomic units
in
are
Fig.
10.6 but for
high magnetic
field
(hw,-
=
6,
used.
change of the ground-state angular momentum quantum numbers as a function of the magnetic-field strength is illustrated In Fig 10.5 for a three-electron quantum dot with spin S 3/2. This figure shows the energies of the states with M 3 (solid curve), M 6 M and 9 orbital The calmomentum. angular (dotted) (dashed) culation shows that by changing the strength of the magnetic field the orbital angulax momentum of the ground state of this system is M 3 for w,lw < 5, it is M 6 for 5 < w,lw < 11, and then it The
=
=
=
=
=
switches
=
over
to M
=
9, reproducing the "magic number"
sequence
10.3
(M
3,6,9,..).
Quantum dots
in
magnetic field
201
States with other orbital
angular momentum never ground state and are not included in the figure. The previous examples dealt with spin polaxized electrons (S 3/2). Actually, spin polarized (S 3/2) and spin unpolarized (S 1/2) states compete to be the ground state as the magnetic field changes. This very interesting phenomenon is illustrated in Fig. 10.8. As the magnetic field increases, the ground state changes as (M, S) (1, 1/2) -+ (2,1/2) --+ (3,3/2) and so on. At low magnetic field the lowest energy state is spin unpolarized (S 1/2) and as the magnetic field increases the spin becomes polaxized. The orbital angular momentum (M) changes continuously (in steps of unity). =
become the
=
=
=
=
=
5
4
3
C5
2
0 0.0
0.5
1.0
1.5
2.0
2.5
(j)C/ (1)0 1-0.8. Spin (S) and z component of the total orbital angular momentum (M) quantum numbers of the ground state of a three-electron quantum dot
Fig. as
a
function of
denote M and
magnetic field strength. hwo S, respectively. Atomic units are
=
1. Solid and dotted lines
used.
202
10.
10.4
Few-body problems
Quantum
in solid state
physics
dots in the generator coordinate
method In Sects. 10.2 and 10.3 the correlated Gaussian basis of
Eq. (10.3)
is
to obtain the solution of the
employed not the only
quantum dots. However, th is is Gaussian basis function for the quantum dots. In fact, the
quantum dots offer
(8.9)
that
was
application of another basis function study of small molecules. That basis func-
very nice
a
used for the
by the matrix A which describes the correlated particles and also by the generator coordinates s which represent several peaks of the density distribution of the
tion is characterized motion of the
allow
us
to
system. In this section
consider the quantum dots in 3D and
we
field. The
state of the
quantum dot
apply belong
can ground 3angular momentum (for example, in the N electron case). The application of the basis function (8.9) gives a very simple way to obtain accurate energies of the dots even with nonzero orbital angular momentum, without the hassle of partial-wave expansion or angular momentum algebra. As this basis function has no definite orbital angular momentum, in principle one has to project out the component with good angular momentum quantum numbers. In practice, however, the converged wave function already belongs to the correct ground-state quantum numbers and the energy gain due to an orbital angular momentum projection is negligible. As the Hamil2 tonian and the L operator commute, the optimization of the wave 2 function filters out the eigenstate of L as well. As was mentioned in Sect. 6.1, this type of calculation is called a variation before projection. In the limit that the variational trial function is chosen to be flexible enough, the solution of this type of calculation would reach the 2 eigenstate of the conserved quantity like L to a good approximation.
no
to
magnetic
nonzero
orbital
=
In Table 10.6 the results of the SVM with the basis function of
Eq. (8.9)
is
compared
to that of
The shell model works
a
extremely
large
scale shell-model calculation.
well in this
case
because the
con-
fining potential is a harmonic-oscillator well. By choosing the singleparticle wave functions to be the eigenfunctions of this single-paxticle potential, the diagonalization of the Hamiltonian with the relatively weak Coulomb interaction between the electrons is simple and gives an accurate solution. The results of the two methods are in perfect agreement. This agreement shows that the optimization of the basis of
Eq. (8.9) finds the
correct
ground
state
automatically
even
for the
10.4
case
with
sis size is
Quantum
nonzero
dots in the generator coordinate method
orbital
angular
sufficiently large. An
momentum
orbital
angular
provided
203
that the ba-
momentum
projection
may lead to a faster convergence, but as it can only be carried out by a three-dimensional numerical integration, it would prohibitively slow
down the calculation.
Energies of few-electron quantum dots in 3D calculated by SVM and by a large scale shell-model (SM) basis'. L is the total orbital angular momentum, S the total spin, and -x is the parity. K is the basis 0.5. Atomic units are used. dimension. hwo Table 10.6.
=
N
(L, S, -ri)
(1,1/2,-) 4(0,0,+) 5 (1,1/2,-) 3
a
P.
(SM)
K
E
4.01324
100
4.01324
6.35025
300
6.3506
39 hwo 14 hwo
9.00331
500
9.0032
6 hwo
E
(SVM)
Navratil, private
communication.
SM space
(in hwo)
Complements
204
Complements 10.1 Two-dimensional electron motion in
a
magnetic field
vaxiational solution to the
have
presented a previous section we few-body problems of quantum dots. In this approach a correlated Gaussian basis has been used and the problem has been formulated In a relative coordinate framework. The examples from atomic physics (Chap. 8) convincingly prove that this is a very powerful -way to cope In the
with the correlation between the electrons in the field of
an
attractive
only possible approach. One can, alternatively, use a basis set made up from products of single-particle states to diagonalize the Hamiltonian. The properties of these singleparticle states axe discussed in this Complement. The Hamiltonian of interest is given by center. This is,
H
however,
-2m*
not the
(P eAi)2 +
+
9*ABB
c
-
2
aj
I
+
vori)
2
(10.7)
+ X
i 'i
1ri
-,rj I
where m* is the effective electron mass, AB eh/(2m,c) the Bohr magneton of the electron, g* the effective g-factor of the electron, =
and
r.
is the dielectric constant of the bulk material. The
magnetic
interaction energy due to the electron spin is included in the Hamilltonian. The magnetic field is directed downward in the z direction, i.e. B
B
-
with B
> 0. For high magnetic field of, e.g. T, the value of ILBB becomes 0.5788 meV. The vectors ri
10
and pi
(0, 0, -B)
=
axe
2D vectors, which have
x
and y components. The
one-
body potential V(r) is to confine the quantum dots and is chosen to be quadratic, m* W02T 2/2. A typical value of the confinement en5 meV. By taking a symmetric gauge vector potential, ergy is hwo -
A
-r x
=
(P
B/2 2
e
)
_Ji2' j +
A
+
where A
B(y/2, -x/2, 0),
c
_2B 2
we
e. hB
2
(10.8)
4C2 +
_ey2-
and I-, is the
obtain
c
z
component of the angular
mo-
mentum.
To express the Hamiltonian in terms of dimensionless introduce
vaxiables,
-we
C10.1 Two-dimensional electron motion in
e,B W,
W
=
WC2
=
M
a
magnetic field
205
hI + W02
and
p
.
=
(10.9)
W U)
Here wc is the cyclotron frequency and p is the that the magnetic length is expressed by p
magnetic length. (Note using Bohr and hartree 2R*.) Then the Hamiltonian is expressed by the =
radius a*
- -2R*-1hwa*
dimensionless variable ri: 1
H
=
hw
A,
+
2
1
lr,2 2
hw, 2
E lj ,
-
2
g*AB B
aj ,
i
e2
+XE 1ri
with -
j>i
Here the is
hwc
=
length is
rjI
2R*a*
(10.10)
-
=
P
rIP
in unit of the
(2ra,/m*)ABB.
X
magnetic length. The cyclotron energy
The ratio of m,/rn* is of order 15 for the 2D
electron system in GaAs-Ga,,All-,,As heterojunctions. The strength of the Coulomb potential is of the order of X, while the level spacing of the single-particle energy is determined by hw. Therefore the ratio
Xlhw
-
VI'2--R*-Ihw is
the
quantity which
charac-
terizes the electronic motion. If the ratio is much smaller than
the
independent
tion. As it
the
motion of the electrons becomes
a
unity, good approxima-
increases, the correlated motion of the electrons overwhelms
independent
motion.
Recall that the second and third terms of the above Hamiltonian are
consistent with the well-known fact that
mass
M and
charge
qe in the
magnetic
a
spin 1/2 particle with magnetic
ffied B has the
interaction energy -IPB, where the magnetic moment tL of the particle is in general expressed by tL g1l + gs in units of ehl(2Mc). Here g, and g, are the respective gyromagnetic ratios for the orbital and =
spin angular momenta. The gi value is equal to q, while the Dirac 2q for a point charged paxticle like the equation would give g, 2.0023 ILB for the The measurement electron. gives I g, I precision electron and the deviation from 2 can be very accurately computed by the higher order corrections of quantum electrodynamics. On the other hand, the experimental values for the nucleons are far from the 5.586 for the proton Dirac value expected for point paxticles: g, =
=
=
and g, AN
=
=
-3.826 for the neutron in units of the nuclear
eh/(2mpc).
nucleon.
magneton
This fact is ascribed to the internal structure of the
206
Complements
The eigenvalue of the single-particle Hamiltonian, Ho hw (-,A/2 +r2 /2) -(hw,12)1., -(g*p.BB12)o-,,, is obtained easily. Rewriting the spatial coordinate as a complex variable z (x iy)/-\/-2, we defme =
-
I
at
72= (
=
0),
*
Z
-
'0
I
--.,f2- (z- OZ* ),
bt=
(z
a=
OZ
=
[at, b]
Hamiltonian
Ho
=
=
[at, bt]
=
OZ*
(z*
b
0 +
OZ
[a, at]
with the commutation relations
[a, bt]
+
0. We
can
[b, bt]
=
=
[a, b]
1 and
then express the
single-particle
as
hw(ata + btb + 1)
Ihw,(ata
-
-
2
btb)
Ig*ttBBc-z-
-
2
(10.12)
The operator at creates a right circular quantum of counterclockwise rotation about z axis, and bt creates a left one. Let a non-negative
eigenvalue of the total oscillator quanta ata+btb at a b b. and m denote an eigenvalue of the angular momentum 1,, Then the eigenvalues of A and btb are given by (N + m)/2 and (N m)/2, respectively. Since they should be positive integers or zero, the possible values of m for a given N are m -N+ N, N-2, N-4, 2, -N. The single-particle energy becomes integer N denote
an
=
-
-
=
...,
I
E,,,,,,
=
hw(N+1)
hwm
-
-
2
g*ABBS,
I =
with Sz
quanta
hw(2n + Iml
being
either
N is set
1/2
For
of
hw(N + 1),
1) or
to N
equal given N, possible
n.
a
+
hwm
-
-
2
-1/2, =
2n +
values of
(10.13)
g*ABB$,
and where the number of total
Iml
with
n are
0,
1
a
non-negative integer [N12]. The energies
....
with N + I-fold
degeneracy, corresponding to the 2D are independent of m conforming to N. The singleparticle energy is shown in Fig. 10.9. In the case of no magnetic field or very weak magnetic field (w,lwo < 1) the single-particle level shows a shell structure with degeneracy of N + 1. The single-particle level is then in the order of haxmonic-oscillator
(n, m)
=
(0, 0); (0, 1), (0, -1); (0, 2), (1, 0), (0, -2);...
The level
cross-
V/-I/-2
ing of the low-lying single-particle levels occurs at W,/WO 0.71 and the level with (n, m) (0, 2) becomes lower than the =
=
with
(0, -1).
Z-'
one
CIO. 1 Two-dimensional electron motion in
a
magnetic field
207
If wo 0, that is, there is no confining potential present or the magnetic field is very high (wc.Iwo > 1), w is (nearly) equal to (112)w, and the energy becomes E,,,,, hw,(n+(Iml -m)/2+1/2) -9*ABBS,, de=
=
(Iml
pending only
on n +
dent of m for
positive
-
m.
m) /2. Clearly the energy becomes indepen-
This
gives
rise to
infinitely degenerate levels
called the Landau levels with the Landau level index The fact that the
levels with
an
single-electron degeneracy
infinite
motion is
quantized
number in the
+
n
case
(IM I
-
m) /2.
to the Landau
of
no
confining
potential reflects the fact that the energy is independent of the center of the cyclotron motion. In the presence of the confining potential, the
single-particle energy increases with m as seen from Fig. 10.9. By using the polar coordinates r and 6 the eigenfimetions single-particle Hamiltonian are expressed as
rp2
ni n!
(n-t TP 2(n+lml)!
(r)
1-1
L (I ' 1) n
P
r2
(p2
-
r
ey-P
of the
2
2p2
(10.14) where
Ln(c) (x)
is
an
associated
Laguerre polynomial defined by
d L,(,o I (X) =ex-' (e-XX n+a n! dXn n
E k! (n
'k=O
F(n+a+l) (-X),. k)! -V(k + a + 1)
(10.15)
-
parity operation changes V to V + 7, so the parity of the singleparticle eigenfunction is given by (-l)'. For the lowest Landau states (n 0, m > 0) the larger angular momentum state corresponds to a wave function which is more extended from the origin:
The
=
2
(0o,jr loom
(Iral
The Han-d1tonian
+
1)P2.
(10.10)
(10.16)
contains the Coulomb interaction be-
electrons, which leads to interesting effects on quantum interplay between the magnetic field and the Coulomb interaction is an essential ingredient for studying the dots. The relative importance of various pieces of the Hamiltonian depends on m*, K, g*,
tween the
dots. The
and B. For two electrons the Hamiltonian
can
be
separated in the relative By introducing
and center-of-mass coordinates of the two electrons. the coordinates
7 C)
Is 6
5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
10
O)C /0)0
Fig. 10.9. The single-particle energy E,,,,, of a quantum dot as a function of magnetic field strength. The Zeeman spin splitting of the energy is not included.
C10.1 Two-dimensional electron motion in
I
R=
-\f2-
('rl +r2),
Ir
a
magnetic field
I-(Irl -r2),
=
209
(10.17)
A/2
the Hamiltonian is reduced to H
=
hwf
I
,AR
-
+
2
1R21
'&V,, IR ,
-
"A'r + 21,r2 2
+ hW
hWcl
,T Z
2
-
C
.
2
2
g*[tBB(Sj,,
+
S2 ,)
X 1 +
(10.18)
-
vF2 r
by exactly the same Hamiltonian as the single-particle Hamiltonian and its energy and eigenfunction are given by Eqs. (10.13) and (10.14). The eigenfunction b(T) of the relative motion Hamiltonian may be obtained in polar coordinates. By separating the angular part as b(r) u(r)e " (m and the the function radial eigenvalue E, are obu(r) 0, 1, 2, ...), tained from the following equation The center-of-mass motion is determined
=
d2u
1 du +
-
.-
r
where
u(r)
r2
+
hW
dr
has to
M2
V2-X 1 +
satisfy
=
2Er
W, 7n
U
-
=
0,
(10.19)
r
the condition that its norm,
fo' drru(r)2, is
finite.
analytical solutions are available provided Xlhw belongs to a certain enumarably infinite set of values [1411. For a general case we can obtain the eigenvalue by numerical integrations. The eigenvalue 0, 1, 2,... for each m. We may be labelled by a quantum number n note the symmetry property of the eigenfunctions. The interchange of the position coordinates ri and V2 leads to r -r, which is # + 7r. Therefore the eigenfunction equivalent to the change of 6 and labelled by n m receives a phase change e"' (-I)m, so that it is symmetric for even m and antisymmetric, for odd m. Figure 10.10 displays the energy E, of the relative motion as a function of the cyclotron frequency. The most striking feature is the level crossing in the lowest level (ground state). With increasing magnetic field the so (0, 0) to (0, 1), (0, 2), (0, 3), ground state changes from (n, m) that the angular momentum m of the ground state increases one by one. (This figure is basically the same as Fig. 10.4 which is obtained by the variational calculation, but included here to show the significant Paxticular
=
---).
--*
=
=
role of the Coulomb
Finally
we
electrons in
a
..
-,
interaction.)
importance of the correlated motion of strong magnetic field has already been recognized in
note that the
the quantum Hall effect
[1421.
The
broadening of the Landau levels
210
Complements 12-
11
10I f I I
9-
8 It
7
*11
IA 41/
*, -
3-
2
1
0
-
1
0
1
2
3
4
5 0) C
Fig. dot
6
7
8
9
10
/ (00
10.10. The energy E, of the relative motion of a two-electron quantum function of magnetic field strength. The Zeeman spin splitting of
as a
the energy is not included. The solid curves are for states with even m and the dashed curves for states with odd m. hwo = 1. Atomic units are used.
CIOA Two-dimensional electron motion in is caused
by
scatterers such
as
a
magnetic field
211
impurities and lattice defects. The repulsion between many electrons
correlations due to the Coulomb
lead to the Anderson localization and the fractional quantum Hall effect. Laughlin [1431 successfully used a correlated basis of Jastrow
type
to describe the electron motion.
11. Nuclear
few-body systems
originally developed to solve physics. In this chapter we collect a
The stochastic variational method
was
few-body problems in nuclear few possible applications of the SVM for nuclear systems. The nuclear force, due to the effect of the underlying quark structure and relativistic motion, is a very complicated interaction. h addition to the spinisospin dependent central part, the two-nucleon interaction contains spin-orbit and tensor forces, and L2- and (LS)2_dependent terms. The two-body interaction designed to fit the nucleon-nucleon phase shift and the properties of the deuteron fails to reproduce the binding energies of the three- and four-nucleon nuclei, and a three-body interaction has to be introduced to get the correct binding energy. As the nucleons are not structureless point-like entities, the spatial part of the nuclear force contains a strong repulsion at short distances. These circumstances altogether make the solution of the nuclear few-body problem a
formidable task.
application is restricted to central forces. These schematic forces are used in simple model calculations and serve for benchmark tests. Examples of more realistic problems are shown J-n the last secThe first
tion.
Introductory remark
11.1
on
nucleon-nucleon
potentials The nucleon
(N)
has
spin and isospin degrees of freedom
in addition
of Eq. (2.3) is a product of these spatial one. The operator 0?. V factors, one acting on each coordinate of the three types:
to the
0?. 73
=
(0,,ij (space) 0,.ij (spin) -
Oij (isospin),
Y. Suzuki and K. Varga: LNPm 54, pp. 213 - 246, 1998 © Springer-Verlag Berlin Heidelberg 1998
214
11. Nuclear
where,
e.g.,
few-body systems
0,,,,j (space) (IL
=
n,
r.
-
-K)
1,
is
a
spherical tensor a scalar product
spatial part, and product (-I)"A/2__K+ I[T,,, x U,,Ioo, which (T,,,.U,,) is defined with Clebsch-Gordan coefficients by operator of rank
acting
K
in the
stands for the tensor
[Tr-1
X
Ur-2]r-393
=
E
(11.2)
rv1jL1K2A2jK3A3 >TK,jL1 Ur-2/L2'
<
IIIA2
e.g., [2, 70, 71, 72] for angular momentum algebra. The NIV interaction is a typical example of the strong interaction
See,
and is still not known in fiiU detail. There are, however, several versions of the nuclear potential whose parameters have been determined fairly
accurately so
to
as
reproduce two-nucleon
bound state and
scattering
data. The nuclear force reaches at most up to 2 fla (I fin = 10-13 cm). An NN potential which is consistent with the scattering data is called
"realistic". The most important ingredients of the realistic include central, ,tensor, and spin-orbit components. The central part of the nents:
0-'12
is thus
==
Ii
TTT27 0'1*(7'2,
expressed
V12
=
potential
is characterized
and(71'72) (o-j-a2).
potential
by four compopotential
The central
as
+ V' (,r) -rl'72 + VCO_ (r) 01 0'2
V(,r)
*
-
+ VcTO
(r) (Irl 72) (Ol U2) *
*
expressed by the space- (Pj'2), spin1+'rr72 and isospin-exchange (P172) operators. (P12 2 2 The generalized Pauli principle requires the relation These
components
can
also be
=
Pi2P12'P12_
=
Therefore the central force
VI 2
=
V12
=
(11.4)
_1* can
also be written
VB H Pl' 2
VW H
+
VM (r) PJ'2
(V-t (r)
+
V` (r) 7, '72) S12)
+
+
as
VH (r) PI'2 PI 2,
(11.5)
(11.6)
11.1
remark
Introductory
nucleon-nucleon
on
potentials
215
with the tensor operator 3 (o-1
S12
*
f6) (0`2 f6) *
U1 Cr2 *
/E4T (Y2
"
V
The deuteron
X
5
consisting of
bound through the force. Here
-
a
0212) (p)
proton
and
coupling of the 3SI and 3D,
2s+'Lj
neutron
a
waves
(n)
becomes
due to the tensor
denotes the two-nucleon channel with the
spin S
and the relative orbital
angular momentum L being coupled to the However, no such coupling occurs in the case of two protons or two neutrons because the Pauli principle (11.4) forbids two like nucleons with L 0 or 2 to be in a spin-singlet state. The interaction between like nucleons is just too weak to produce a total
angular
momentum J.
=
bound state. The
spin-orbit force has non-vanishing matrix elements only spin-triplet states, too. It is parametrized in the form V12
=
in
(Vb (r) + Vb-r (r) -ri '72) (L S) 12 -
with
(L-S)12
r X
2h
h
where p
=
-ih-L, ar
(r
X
hl
(PI
-
P2))
P)' (Ol + 2
=
(r
x
p),
2
(Ol
+
6'2))
and
(
Cr2)
2 x
)
(Ol
472))
+
indicates
a
product. The form factors of the various components pressed in terms of Y(x) e-xlx and its derivatives.
vector
are
(outer)
usually
ex-
=
As is evident from the above
discussion, one of the characteristic features of the realistic NN potential is its strong state-dependence, namely it differs in the four states of singlet-even, triplet-even, singletodd, and triplet-odd character, where even and odd refer to the parity of the orbital motion.
The tensor force makes it necessary to introduce at least D-waves in the calculations. Another important feature of the most commonly used
family of NN potentials is the strong repulsion at short distances, say within 0.5 fi-n. This requires due care in so far as short-range correlations must be properly taken into account in theoretical models. All of these aspects of the NN interaction really make a variational calculation for a nuclear system very challenging. In these circumstances
11. Nuclear
216
few-body systems
frequent to use some "effective" potentials which axe not necessarily designed to reproduce the scattering data but determined to fit some bulk properties of nuclei such as the binding energy and the size. Although relativistic effects should be fairly large in nuclei ((V/C)2 0.08, which is estimated from the kinetic energy TF 40 MeV belonging to the Fermi level), the ambiguity of the nuclear force makes it rather difficult to extract these clearly from experimental it is rather
-
-
data. The
complexity
of the NN interaction is due to the
substructure of the nucleon
as was
compositeness of mesons exchanged
11.2 Few-nucleon
discussed in
Chapt.
quark-gluon 9 and to the
between the nucleons.
systems with central forces
performance of the SVM by performing model calpotentials, e.g., the spin-averaged Malffiet-Tjon (MT-V) potential [1441, the Volkov no. I (Vi) "supersoft" core potential [1451, the Afnan-Tang S3 (ATS3) potential [1461, the Afinnesota potential [1471 and the Brink-Boeker (BI) potential [1481. Parameters of these potentials are given in Table 11.1. Some of these model problems have already been solved to high accuracy by various methods and therefore we can directly compare the solutions. The results presented in this section do not include the contribution from the Coulomb potential. Each calculation has been repeated several times staxting from different random points to check the convergence. The energy as a function of the number of basis states is shown in Fig. 11.1 for the case of 6Li with the V1 potential. The energies on different random paths approach each other after a few initial steps, and converge to the final solution. The energy difference between two random paths as well as the tangent of the curves give us some information on the accuracy of the method for a given size of the basis. The root-meansquare (rms) radius is calculated in each step and found to be rapidly convergent to its final value. By increasing the basis size the results can be arbitraxily improved when needed. The number of basis states required to reach energy convergence increases with the number of particles, but it depends on the form of We have tested the
culations with different NN central
well. This latter property is illustrated in of the alpha-paxticle. The soft-core Volkov (Vi)
the interaction for the
case
as
Fig. 11.2 potential
11.2 Few-nucleon
systems with central forces
217
Table 11J. List of the parameters of the nucleon-nucleon central po-
tentials. The potential consists of a few terms; each is expressed as Vf (IL, r)(W + MP' + BP' + HP'P'). See Eq. (11.5). For the Gaussian
potentials (VI, ATS3, Minnesota, Bj), the form factor f ([L, r) is eXp(_/tr2)' the potential strength V is in units of MeV, and the potential range p is in units Of fM -2, while for the Yukawa potential (MT-V) f (A, r) is defined by exp(-[tr)/r, V is given in units of MeVfin, and IL is in units of fin-, respectively. The Majorana, mixtures m and m' are set equal to zero in the calculation. For realistic values of m and n see the original papers. The parameter u of the Minnesota potential is set equal to unity. V
A
W
M
B
H
MT-V
1458.05
3.11
1
0
0
0
[1441
-578.09
1.55
1
0
0
0
VI
144.86
0.82
[1451
-83-34
1.60
ATS3
1000.0
[1461
-326.7
Potential
-2
M
M
0
0
1-M
M
0
0
3.0
1
0
0
0
1.05
0.5
0
0.5
0
-166.0
0.80
0.5
0
-0.5
0
-43.0
0.60
0.5
0
0.5
0
-23.0
0.40
0.5
0
-0.5
0
u/2 u/4 u/4
(2 (2 (2
-2
Minnesota
200.0
1.487
[1471
-178.0
0.639
-91.85
0.465
Bi
389.5
0.7
[1481
-140.6
1.4-2
shows
-2
rapid convergence, requires more basis states
1
-
I-M 1
-
n
while the to
get
an
-
-
-
u)/2 0 u)/4 u/4 u)/4 -u/4
0
(2 u)/4 (-2 + u) /4 -
M
0
0
MI
0
0
repulsive-core ATS3
interaction
accurate solution. The
relatively
fast convergence for the MT-V potential of a strong repulsion can be explained by the simplicity of the spin-independence of this interaction.
results, together with results of others, potential to N=2-7-nucleon systems. application 41.47 MeV fi-n2. In the table the ground-state enerWe chose h2/m E and matter rms radii (r 2)1/2 are given. The basis dimenpoint gies sion K of the SVM listed in the table is such that, beyond it, the energy and the radius do not change in the digits shown. For three-body systems, the solution of the Faddeev equation is known to be the method of choice (see Sect. 5.3), but the SVM can easily yield energy of the Table 11.2 shows the SVM
of the MT-V
for the
=
11. Nuclear
218
few-body systems
-65.5 -65.6 -65.7 -65.8
5
-65.9
0)
-: W
-66.0
-66.1 -66.2 -66.3
-66.4 -66.E 0
50
100
150
200
250
300
350
40(
dimension of the basis
Fig.
11.1.
Volkov
Convergence
of the
(VI) potential [145]
6Li
energy
on
different random
paths.
The
is used.
-29.0
-29.5-
Minnesota
................................................................. -30.0-
ATS3 2
Vo-lkov
-30.5-
W
-31.0
MTV -31.5
-32.0
0
20
40
60
80
100
120
140
16(
dimension of the basis
Fig. 11.2. Convergence potentials
of the
alpha-particle
energy for different central
11.2 Few-nucleon
systems with central forces
219
Energies (in MeV) and root-mean-square (rms) radii (in fin) of interacting via the Malfliet-Tjon (MT-V) potential [14 . The value of K denotes the basis dimension beyond which the energies and the radii of the SVM calculation do not change in the digits shown. Table 11.2.
N-nucleon systems
N
(L, S) J'
Method
E
(T 2)1/2
2
(0,1)1+
Numerical
-0.4107
3.743
SVM
-0.4107
3.743
3
4
(0,1/2)1/2+ Faddeev[150, 1511
(0,0)0+1
(0,0)0+ 2 5
6
7
(1,1/2)3/2-
(0,0)0+ (1,1/2)3/2-
5
-8.25273
ATMS[121 CHH[1521 GFMC[411 VMC[1531a
-8.26
SVM
-8.2527
FY[1491 ATMS[121 CRCG[261 GFMC[411 VMC[1531a
-31.36
SVM
-31.360
CRCG[261
-8.50
2000
SVM
-8.49
150
ZLO-01
1.682
-8.26
0.01
1.682
-8.27
0.03
1.68
-8.240
1.682
80
1.40
-31.36 -31.357
1000
-31.3
0.2
-31.3
0.05
1.36 1.39 1.4087
150
VMC[153]a
-42.98
SVM
-43.48
VMC[1531a
-66-34
SVM
-66.68
1.52
800
SVM
-83.4
1.68
1300
'Calculated with Coulomb tracted
K
perturbatively.
The
0.16
1.51 1.51
0.29
500
1.50
potential, the Coulomb contribution then subpotential strength used in the VMC [153] cal-
-578.17 MeV, which is culation is Vi=1458.25 and V2 from that used in the calculation. See Table 11.1. =
slightly different
220
11. Nuclear
few-body systems
accuracy. As the MT-V potential is a preferred benchmark test of the few-body calculations, there are numerous solutions available. same
Table 11.2 includes
a
few of the most accurate results. The nice agree-
ment for the four-nucleon case corroborates the fact that the SVM is as
accurate
as
the direct solution of the
Faddeev-Yakubovsky (FY) Amalgamation of Two-body corre-
equations [1491, Multiple Scattering (ATMS) [121 the method of the
lations into
Carlo
[411
(VMC) [153]
methods. The basis used in the
Gaussian-basis the
SVM,
or
the variational Monte
and the Green's function Monte Carlo
(CRCG)
(GFMC)
Coupled-Rearrangement-Channel.
variational method
[26]
but the Gaussian parameters follow
is similar to that of
geometric progressions. ground
The fact that the basis size needed in the SVM for both the
and first excited states of the four-nucleon system is much smaller proves the efficiency of the selection procedure. See Chap. 4. The re-
sults of the VMC calculation for the five- and six-nucleon systems are also in good agreement with the results of the SVM- The MT-V potential has no exchange term; therefore, unlike nature, it renders the
five-nucleon system
bound,
and the nucleus tends to
collapse
the
as
binding energy increases with the number of particles. Neither the MT-V nor the V, potential contains the full set of change components of the central potential, rather simple. The Minnesota potential, spin-,
and
isospin-exchange
realistic. In fact it has been
so
ex-
the calculation becomes
however,
contains space-, and is considered to be more operators,
successfully
used in
microscopic cluster-
model calculations tential
was
[28, 29, 301. An extensive calculation with this pocarried out for N=2-6 nucleon systems in [31], where all
possible spin and isospin configurations were allowed for and all angula.r functions that give non-negligible contributions were included. The agreement with experiment was surprisingly good. The energies and the'radii of the triton and the alpha-particle converge to realistic values. The Minnesota potential, correctly, does not bind the N=5 system, but it binds 6He and slightly overbinds ILL The radius of 6He is found to be much larger than that of 4He, consistent with the neutron-halo structure of 6He
[29, 301.
The SVM calculation of
[311
has
made it
possible to test for the first time the Minnesota force without assuming any cluster structure or restricting the model space by any
other bias.
The last
example
is the
application
to Efimov states
how well the correlated Gaussian basis describes the totics. A
three-body system
with
short-range
[1541
few-body
to show asymp-
forces may have several
11.2 Few-nucleon systems with central forces
221
bound states. If the
two-body subsystem has just one bound state (close to) zero, the three particles can interact at long
whose energy is distances and an infinite number of bound states may appear in the three-body system. These "Efimov states" are extremely interesting
experimental and theoretical points of view because of their distinctive properties. These states are very loosely bound and their wave functions extend far beyond those of normal states. By increasing the strength of the two-body interactions these states disappear. from both
A system of three identical bosons of nucleonic mass is considered for simplicity. The potential between the particles is taken as the Poeschl-Teller interaction because this
can
be
easily
V(r) Here
r
potential
is
analytically
solv-
body case, and therefore the two-body binding energy
able for the two-
set to zero;
625.972
1251-943
sinh(l.586 r)2
cosh(I.586 r)2
(11.10)
in MeV.
=
is in units of fi-n. A
spatially
very extended basis is needed to
represent the
bound states. In this
basis. The
basis is created
predefined diagonal elements of the matrix A. These diagonal elements are taken as geometric progression:
wealdy predefined
case we use a
by using only
the
1
A,-,
(i
-
('
-
(aoq6
1))2
=
1,
...,
n),
1
A22
(j
-
( ao q(j-'))2 10
This construction defines
(i, A 0.1,
(I, I),-, (n, n).
=
q0
2.4. One
=
an n
can
=
reveal
2-dimensional basis corresponding are chosen as n 20, ao =
to =
find the first three Efimov states with this
can
expJ-2vJ
more
(11.11)
The parameters
basis size. The ratios of the
Ei+IlEi
n).
rule
energies of the bound states follow the [1541. By increasing the basis size one
bound excited states. Note that the value of
roughly corresponds to the spatial extension basis in this example goes out up to aoqon-I
aoqo'-'
of the basis. The present =
0.1
x
2.419
=
1674990
This extension is necessary to get the third bound state. For a comparison, to calculate the ground-state energy of -4.81 MeV it 7 and the basis covers only the [0, 101 (fi-n) is enough to choose n
(fin).
=
interval. The
rms
radii of the
ground, first and
second excited states
are about 1.5, 25 and 6000 fin, respectively. As shown in Fig. 11.3, these facts illustrate the tremendous spatial extension of the Efimov
222
11. Nuclear
few-body systems
Ground state
.0 .0 0 10.0
Efimov state
0.1
0.0 M
x
Fig. IXM,
Wave fimetion
amplitudes
of the
ground
state and the first
Efimov'stAte. Note the different scale (in fin) of the lengths of the Jacobi coordinates
x
and y.
11.3 Realistic
223
potentials
states. These results indicate that the extended correlated Gaussian
basis
gives
a
practical
to describe the
means
tics of the Efimov states. See
Complement
asymptotics characterispossibility of
11.4 for the
Efimov states in the 12C nucleus.
11.3 Realistic
potentials
The trial function is
given
as a
combination of the correlated Gaus-
sians:
'O(LS)JMTMT(-'C 14) 1
Ajexp(- 1.+,Ax) [OL (X)
=
XSJ jM
X
2
where
x
=
jX1i
...
XN-11
is
a
TITMTli
set of relative
(11.12)
(Jacobi) coordinates,
the operator A is an antisymmetrizer,, XsMs is the spin nTmTis the isospin function. The matrix A is an (N
-
symmetric, positive-definite
function, and
1)
x
(N
-
1)
matrix of nonlinear variational parame-
angular part of the wave function. It is taken as a vector coupled product of partial waves i3)]L123 I LML This is re[[[Yli 41) X Y12 (ZAL12 X Yls (X3 OLML (X) ferred to as the method of partial-wave expansion. The spin and isospin functions are also successively coupled, for example XSM,
OLML (x) represents
ters. The function
the
:`
...
.
=
X(E12SI23 )SMs ...
we
L
used the
=
0, 1,
satisf3 g
2.
=
[[[XI/2
X
XI/21SI2
partial-wave channels For the alpha-particle,
the condition 11 +
12
+
X
X1/21S1,23 ...J,sms
with
(11, 12)L,
For the
triton,
where 1i < 2 and
partial waves ((11, 12) L 12 13) L 13 :! 4, 1i :! _ 2, 111 121 <
all
1
-
0, 1, 2 have been tried. As the triton has J 1/2, there are three spin and two isospin chan1/2 and T 0, 1. For the alphanels; (S12)S (0)1/2,(1)1/2,(1)3/2 and T12 0, T 0) there are six spin and two isospin chanparticle (J nels unless very small isospin mixing is taken into account; (S12 S123) L12 !
11
+
12 and L
-
=
=
=
=
7
(0, 1/2)0,(0,1/2)1,(1,1/2)0,(1,1/2)1,(1,3/2)1,(1,3/2)2 T123) (0, 1/2), (1, 1/2). =
and
(T12,
=
dependence of realistic nucleon-nucleon interaction is extremely important. To extend the application of the SVM calculations to nuclear systems interacting with realistic potentials, in addition to the previously applied random selection of the nonlinear parameters of the correlated Gaussian, a random selection of the spin and orbital components of the wave function The
spin, isospin and orbital angular
momentum
11. Nuclear
224
few-body system
is introduced. The main motivation for the random selection is that
the numbers of channels and nonlinear parameters are prohibitively large, therefore the calculation of all of the matrix elements and di-
potentially important basis states is out alpha-particle, for example, there partial-wave channels even in our truncated-partial wave expansion. Just to show up an alternative, it is worthwhile assessing what dimensions a deterrni-ni tic optimization would require. The simplest choice of the nonlineax parameters of the Gaussian basis is to use a diagonal matrix A. One of the best deterministic choices for the diagonal elements is a geometric progression [251. To reach good accuracy, at least five values for each of AJ-1 A22 3 125 functions in a given and A33 have to be used, requiring 5 channel. The spin-isospin and space paxt, therefore, would result in a all
agonalization including
of the question. In the case of the are 12 spin-isospin channels and 32
7
=
basis size of about a
one
hundred thousand
"direct" calculation for the
(2
125
x
x
12
x
in such
32)
alpha-particle.
The literature is very rich in papers devoted to few-nucleon prob3 nuclei it is possible lems [155, 156, 157, 12, 25, 158-1611. For N =
to include
enough
performed, N
4 nuclei
=
channels and
but other methods some
results of the
an
exact Faddeev calculation
the
give essentially
of these methods become too calculations agree
existing
only
same
complicated,
within
a
can
be
results. For
and the
few hundred
keV.
To test the SVM
used two, from the
we
form,
rather different interactions. The
(AV6
and
ond,
AV8) [1621,
the Reid
is
a
the
first,
view of
spatial Argonne potential
well-behaved smooth function. The
potential (M) [1631
linear combination of
point of
is
more
singulax
exp(_/tr2)/,rk-type terms.
sec-
and includes
a
The latter
potential AV6 The includes can certainly cause some problems. potential only central and tensor components and serves as a good test case for the inclusion of non-central forces. The AV6 and AV8 potentials axe used without Coulomb potential, while in the case of the RV8 potential the Coulomb interaction is added. To
keep
the number of nonlinear parameters
matrix A to be
diagonal
in
one
of the
possible
low,
we
restricted the
Jacobi coordinate sets.
of the three-nucleon system we have only one possibility of the Jacobi coordinate, (NIV) + N. For the alpha-particle we use
In the
case
possible coordinate sets, (3N) + N ("K") and (2N) + (2N)type ("IF) Jacobi coordinates. Note that the basis function is fuHy both
antisymmetrized.
This choice is also dictated
by physical
intuition
as
11.3 Realistic
it is natural to consider
potentials
225
3H+P 3He+n and d+d-type partitions in the ,
Test calculations show that tbis choice
alpha-paxticle. satisfactory results.
already gives
As is shown in Table
11.3, the convergence is relatively fast both for the triton and the alpha-particle with the AV6 and AV8 potentials, but in the case of the RV8 potential, due to its singular nature and stronger repulsive core, the convergence is slower. About 50 basis states for the triton and about 200 basis states for the alpha-particle give fairly good binding energies. The energy of the alpha-paxticle with the basis size of 200 is already within 0.5 MeV of its final value. It is very satisfactory that the SVM has succeeded in reproducing a reasonably good energy with just 400 basis states.
Table 11.3. The convergence of energies alpha-particle for different potentials, AV6
(in MeV)
[1621,
for the triton and the
AV8
[1621,
and RV8
[163].
K is the basis dhnension.
'He
311 K
AV6
AV8
RV8
K
AV6
AV8
RV8
25
-6.63
-7.36
50
-7.04
-7.69
-6-53
100
-24-37
-25.15
-23-35
-7.41
200
-25.05
-25.50
75
-7.11
-24.15
-7.74
-7.54
300
-25.28
-25.60
100
-7.15
-24.35
-7-79
-7.59
400
-25.40
-25.62
-24.49
The results of the SVM and other calculations Tables
11.4 and
11-5. We show and compare
compared in the results not only are
for the energy but for the rms radius and average kinetic and potential energies as well. The results remain the same by repeating the calculations several times
staxting from different random values.
comparison for the triton with RV8 demonstrates that the SVM are in very good agreement with the Faddeev results. Since Faddeev calculations are considered exact for three-body systems (see The
results
Sect.
this agreement confirms that the basis selection of the SVM performed efficiently even for the singular potential of RV8.
5.3),
has been
The correlated
hyperspherical
harmonics
(CHH) expansion
method
(see 5.2) also agrees with the SVM for the triton. The results of the SVM and GFMC are very close to each other. Except for the case Sect.
of the
alpha-particle
the statistical
errors
AV6, the energies of the SVM axe within of the GFMC, though the expectation values of with
11. Nuclear
226
few-body systems
Table 11.4. Energies (in MeV) and radii (in fin) of the triton calculated by different methods and with different interactions. (T), (V), and (V6) are the expectation values of the kinetic energy, the total potential energy, and the of the central and tensor components, respectively. (, 2) 1/2 denotes the root-mean-square radius. The value of PL shows the percentage of the component with the total orbital angular momentum L.
potential energy
SVM
Faddeev
GFMC
CHH
VMC
AV6
(T) (V6) (r 2 1/2
44.8
PS PP PD
91.2
E
-7.15
-51.9 1.76
-43-7(1-0) 1.95(0.03)
-52.0(3.00) 1.75(0.10)
< 0.1
8.7
-7.22(0.12)a
-7.15'
-6.33(0.05)'
AV8
(T) (V6) (VLS) (T 2)1/2
46.3
PS PP PD
91.1
E
-7.79
-52.9 -1.2
1.75 < 0.1
8.9
_7.79d
-7.79'
RV8
(T (V) (T 2)1/2
52.2
PS PP PD
90.3
E
-7-59
aRef.
-59.8
1.75
54.0(0.20) -62.0(0.20) 1.68(0.07)
52.2
-7.54(0. 10)b
-7. 59b
-59.8
1.76
< 0.1
9.7
[1641 bRef. [1651. .
'Ref.
-7.44(0.03)'
[1571. dRef. [1591.
'Ref.
[1611.
-7-60'
11.3 Realistic
potentials
227
Table 11.5. Energies (in MeV) and radii (in fin) of the alpha-particle by different methods and with different interactions. The isospin mixing due to the Coulomb potential is not taken into account. See also the caption of
Table 11.4. SVM
CHH
vMC
FY
GFMC
AV6
(T) (W) (r 2)1/2
100.1
-125.4 1.49
PS PP PD
84.3
E
-25.40
-122.0(3.0) 1.50(0.04)
-122.0(1.0) 1.50(0.01)
-25.50(0.20)a
-22.75(0.01)a
0.5 15.1
AV8
(T) M (W) (VLS) (r 2)1/2
98.8
PS PP PD
85.5
0.3
E
-25.62
-25.75(0.05)
111.7
109.2(0.20) -137.5(0.20) 2.45(0.23) 0.71(0.02) 1.53(0.02)
-124.4
-124.20(l.0)
-121.5
-2.9 1.50
1.51(0.01)
14.2 ,d
-25.60'
-25.31-
RV8
(T) (V6) (VLS) (KOUI) (T 2)1/2
-139.1 2.1
0.75 1.51
84.1
PS PP PD
0.4 15.5
15.5(0.20)
E
-24.49
-24.55(0.13)b
aRef.
[1641. bRef. [165].
cRef.
-23.79f
[1571 dRef. [1591 .
-24.01e
.
eRef.
-23-9c
[1611. fRef. [1551.
11. Nuclear
228
the kinetic and the
few-body systems
potential energies
VMC and GFMC have
a
integration. Though the of the trial function
statistical error
well,
as
are
error
somewhat different. Both
bar of VMC is influenced
the
error
by the
choice
of GFMC is considered
purely
statistical in the limit of infinitesimal time step (see Sect. 5.1). Variational calculations were considered to be inappropriate
realistic
not be able to
for
the
reproduce large potential energies [1581. Our calculation shows that, with the careful optimization of the nonlinear potentials,
as
they might
of
involved in the Monte Carlo
cancellation between the kinetic and the
variational parameters, this is not the case. We energies even in the case of the RV8 potential.
can
obtain accurate
experimental energies of the triton and the alplia-particle are MeV, respectively. We see that the calculation using the realistic potentials underbinds; about 0.9 MeV for the triton and about 3.8 MeV for the alpha-particle. This indicates that the binding energies of few-nucleon systems cannot be accounted for in two-nucleon potential models but call for other mechanism to get more attraction. One possibility for the mechanism is a three-body potential which can be produced by two-pion exchanges among three nucleons. Once we have a basis for a given potential, one naturally expects that the same basis may give fair ground-state energy for other potentials of similar nature as well. We have checked if it is really the case. The basis optimized for the RV8 potential gives excellent results (within 100 keV) for AV6 and AV8 as well. Due to the singular nature and stronger repulsive core of RV8, however, the basis optimized for AV8 gives about 500 keV less binding for the alpha-particle with RV8 than the basis optimized for RV8 itself. This result is still not so bad and can be easily improved by refining the nonlinear parameters as explained in Sect. 4.2. Therefore, one does not have to look for an entirely new basis set for another interaction, but can use the same basis for a given system, with some "fine tuning" if necessary. It is obviously important to calculate accurate binding energies of light nuclei with realistic forces. For example, the two-neutron separation energy of 6He is about I MeV, so that a few hundred keV less binding is thought to change its neutron "halo" structure significantly. One of the advantages of the SVM is that it is relatively easy to extend it to N 5-, 6-, 7-nucleon systems [31]. The low dimension of the bases required to solve the N 3- and 4-nucleon problems The
-8.48 and -28.295
=
=
confirms that the SVM is suitable for
treating larger
nuclei with
alistic forces. The formalism and the computer code itself is
re-
general
11.3 Realistic
[311,
so
potentials
229
applicability is mostly limited by the memory and speed of no dffficulty in including three-body
the
the available computer. There is forces if necessary.
utility of the global vector representation of Eq. (6.4) in describing the angular part of the system interacting via realistic potentials. We calculated the ground-state energies of the triton and the alpha-particle by using the AV6 interaction [162]. The results by the global vector representation and by the partial-wave expansion are compared to each other in Table 11.6. The model space allows for natural parity states only. The partial-wave expansion is restricted for partial waves with orbital angular momentum 0, 1 and 2. These partial waves give a good description of the ground states of these We show the
nuclei the
even
The table shows that
[321.
global
vector
in the
more
different orbital
one can
reach the
same
results with
representation as with the partial-wave expansion complicated case when the tensor force couples the
angular
momenta. The
nice illustration of the main
case
of the
alpha-particle
is
a
of the
global vector representaadvantage partial-wave expansion: Even in the expansion truncated to the natural parity states, one needs 24 partial-wave combinations due to the large number of combinations of the individual partial waves, while in the global vector representation only 4 sets of (K, L) ((K, L) (0, 0), (1, 0), (0, 2), (1, 2)) were used to reproduce the same result [1661. By increasing the number of particles the partial-wave expansion would contain a prohibitively large number of combinations. The number of sets (K, L), on the contrary, does not depend on the number of particles. Only the number of parameters ui increases (linearly) by adding particles to the system. tion
over
the
=
energies of the ground states of the triton and the alphaparticle [1621. Only natural parity states have been taken into account for the orbital motion. Two types of angular functions are used; the global vector representation (GVR) and the paxtial-wave expansion (PVVE). K is the basis dimension. Table 3-1.6. Total
with the AV6 interaction
3
411
11 Method
K
GVR
100
PWE
100
Method
K
E
-6.80
GVR
300
-23.29
-6.80
PWE
300
-23.27
E
(MeV)
(MeV)
230
Complements
Complements 11.1 Correlations in
few-nucleon systems significance of the correlation we calculate the binding energies of the triton and the alpha-particle by using the correlated Gaussians and the correlated Gaussian-type geminals. The spinisospin parts of the triton and the alpha-particle wave functions are considered totally antisymmetric and, as the nucleon is a fermion, their spatial parts must be totally symmetric. The spatial motion in the ground state of the N 3- or 4-nucleon system may thus be To elucidate the
=
treated
as
motion of bosons. We obtain variational solutions for the
triton and the
alpha-partile interacting via a spin-isospi-n independent two-nucleon potential. The orbital wave function is assumed to be given by combinations, of the correlated Gaussians with L 0, see Sect. 7.2: =
Tf
CkS eXP
(_2
,7c A k X
k=1
Here
x
is the Jacobi coordinate set and S is
positive-definite symmetric
matrices
Ak
a
symmetrizer. Each of the
contains nonlinear variational
parameters, three for the triton and six for the alpha-axticle. The
paxameters
can
be determined
with the code of chosen
so as
[811.
was
done
The number K of the basis functions has been
1V =
function,
The function
Ffj (-v)'Iexp( P,
a
(Gaussians),
one
may
try the following
corre-
3
-
product
1Vri2).
2
IF
tions
The calculation
to reach convergence.
As another type of trial lated wave function
Tl'= F P
by the SVM.
of the nodeless harmonic-oscillator func-
describes the
independent single-particle motion,
whereas the function F takes into account the correlations. The
Jastrow-type correlation factor [471
is
taken,
as
IV
F=
(I+aexpj-b(rj
fj
_
ri)21
i>i=l KN
E ak exp (__2 i;BkT') k=1
in
a
single Gaussian form,
CILI Correlations in few-nudeon systems
with KIv
231
1, where ak is given by an integer power of a, and Bk is an N x N symmetric matrix of type, 2b Ei uiiii, where ui is an N-dimensional vector which has only two nonzero elements. =
2
2
Therefore the trial function TJ- becomes lated
Gaussian-type geminals (6.26)
needed formulas for matrix elements
a
with
combination of the
n
=
0 and R
=
corre-
0, and the
involving such special geminals
in that section. Three parameters v, a, and b can be varied to minimize the energy. Note that Tf is totally symmetric and that were
given
the center-of-mass coordinate x1v can be factored out from the intrinsic motion. This makes it possible to calculate the
its
dependence
on
intrinsic energy. We also consider contains
only
the
a special type of the correlation factor which pair correlation
K
F2
=
I +
N
1: ak
exp
bk (ri
_,rj)2)
k=1
Here the
pair correlation
can
be better described with combinations
of K Gaussian functions than with
Jastrow-type
correlation becomes
correlations of ear
more
than two
parameters bk and
v
Eq. (11.15). more
It is
useful in
particles play
a
where various
major role. The nonlin-
and linear parameters ak
minimize the energy. We use three potentials, the MT-V
expected that the
cases
are
[1441, V, [1451
determined to
and B,
[1481*
potentials listed in Table 11.1. The space exchange operator PI of the potential can be set equal to unity because of the bosonic symmetry of the
spatial wave function. The Majorana mixture parameters.
energy is thus
independent
Table 11.7 summarizes the results. The result for N
[311
=
of the
3 and 4
calculated with the correlated Gaussian basis is known to be in
good agreement with other accurate calculations available in the literature. It is interesting to note that, except for the alpha-article with the MT-V potential, the pair correlation with K 2 terms-already gives lower energy than the Jastrow-type correlation described with a single Gaussian function. (The alpha-paxticle has more compact structure than the triton. Because of this property, the strong repulsion of the MT-IT potential has to be fully taken caxe of in order to obtain the ground-state energy particularly for the alpha-particle. The correlation factor of Jastrow type appears to be more suited than just the pair correlation of Eq. (11.16) in this regard.) The small difference =
Complements
232
between K tion
can
=
2 and K
pair correladescription of
3 calculations indicates that the
=
well be described with two Gaussians. A better
pair correlation appears to be important. The trial function with K 0 is nothing but a product of the single-particle wave functions the
=
and contaiins
no
lot of energy. This potentials such a.-, the MT-
correlation. This function misses
evident in
a
singular significant energy difference between potential. MT-V potential. This suggests that with the results the CG asid CGG at least a triple particle correlation, e.g.,
becomes
more
more
Note that there is
V
a
IV
E
exp(
-
b(Tj
-
Tj)'
-
b(Tj
-
Tk
)2
-
b(Tk
_
T,)2),
(11.17)
k>j>i=l
has to be taken into account in the correlation factor for this potential to obtain accurate energy. This problem was investigated in [1671, which takes into account the two-, three- and even more-particle correlations and shows the importance of triplet correlations in producing accurate energy.
Table 11.7.
Energies
thealpha-particla in
of the triton and
the correlated
(CG) of Eq. (11-13) and the correlated Gaussian-type geminals (CGG) of Eq. (11.14), which axe chosen to represent correlations in one of two forms. The correlation factor F of CGG is defined by Eq. (11.15) for Jastrow and by Eq. (11.16) for the number K of pair correlations. See Table Gaussians
11.1 for the nucleon-nucleon
potential.
The
am
in units of MeV.
CGG
CG
MT-V
energies
Jastrow
K
=
0
K=I
K=2
K=3
[1441
triton
-8.25
-6.00
unbound
-5.78
-7.11
-7.27
alpha
-31.36
-29.04
-6.41
-27-03
-28.47
-28.93
triton
-8.46
-6.99
-6.66
-6-99
-8-11
-8-11
alpha
-30.42
-29.14
-27.92
-29.13
-30-06
-30-13
triton
-11.64
-9.95
-7-00
-9.87
-11.20
-11.20
,alpha
-38.34
-36.73
-28.19
-36.02
-37.60
-37.60
V1
B1
[1451
[1481
spin-isospin dependence, the correlation state-dependent. The evaluation of matrix
When the interaction has factor is also
expected
to be
CII.2
of
Convergence
partial-wave expansions
elements with such correlation factors would then be
Refer to related
11.2
[1681
for
a
recent
development
on
the
more
application
233
involved.
of the
cor-
functions to heavier nuclei.
wave
Convergence of partial-wave expansions
Here we examine the convergence of the partial-wave
expansion (PVVE)
bound-state solution for
in
obtaining few-body systems. We consider examples of three-body systems consisting of p, n, and n (triton) or of p, n, and a hyperon A. The A is a neutral baryon with mass mAO 1116 MeV. It has spin 1/2, isospin 0 and strangeness -1. The NA interaction is not strong enough to form a bound state between a nucleon and a A. When one or few A's are embedded in a nucleus, they form a bound system called a A hypernucleus. The simplest hywhich is called a hypertriton pernucleus is the (p, n, A) system 3H, A and its ground state has P 1/2+. The binding energy of 3A H is 2.4 MeV from the three-particle threshold. As neither subsystem of pA nor nA forms a bound state, A couples weakly with p and n. In fact the A separation energy of 3A 1-1 is only about 0.2 MeV. Most of the binding energy of 3A 11 thus comes from that of the pn subsystem, namely the ground-state energy is just =
=
0.2 MeV below the threshold of deuteron to
attempt
to describe the
the coordinates
xI
=
rp
-
ground-state r, and X2
(d) +A. wave
(I'P
=
+
It would be natural
function Tf in terms of
Tn)/2
-
rA.
simplify the analysis we assume that the pn subsystem has the same spin and isospin state as d, that is spin 1 and isospin 0. This assumption, known to be a rather good approximation, leads us to To
the condition that Tf must be we assume
an even
that the total orbital
angular
function of xI. In addition momentum of the system is
zero, and that the three 1/2 spins are coupled to J 1/2. Under these in Tf be follows: can general expressed as assumptions =
Tf
=
f (XI
I
X2, XI
*
X2) [xi (pn)
x
X.L
(A)].LM,
2
2
where the space part f is a function Of X1, X2 and X1'X2. For Tf to be an even function of xI, f has to be an even function Of Xl'X2The Hamiltonian of the system is
given by
H=T+V
h2
a2
h2
192
---
21L, ax,
2
21L2 C9X2 2
+
Vpn
+
VpA + VnAI
Complements
234
where IL1
and A2 2MNMj1(2MN + IVIA). Only the central is included in the Hamiltonian. The pn interaction VP" in
=
potential the triplet u
1
=
MN12,
even
state is taken from the Minnesota
Table
(see
=
11.1).
The NA central
potential
potential [147]
with
is assumed to have
the form of Minnesota type
VNA=
x
I VR+I(I+Po')Vt+-(I-P -)V 2 2
(lu+ 1(2-u)P')
(11.20)
2
2
with
VR The
u
=
2
VORe-ISRT2,
value is set
Vt
equal
they
are
-Vfote-""
to 1.5. We
-Vfoe-S' 2. (11.21)
V,
1
assume
that the
pA
interaction
the same, though there is some evidence hi fact slightly different. The potential parameters are
and the nA interaction
that
=
are
3, 4 binding-energy data: The strength 109.8, and Vo, 200.0, Vot parameters in units of MeV axe VOR h3j-2 of in units the and are 0.7864, and 1.638, nt r,-,R 121.3, ranges 0.7513, respectively. This NA potential will be used in the next ms determined
so as
to fit A
=
=
=
=
=
=
=
Complement to calculate the binding energies of s-shell hypernuclei. The equation of motion for f (XIi X27 XI *X2) is obtained by substi0 and tuting Eq. (11.18) into the Schr6dinger equation (H E)Tf the to coordinates. the Owing symmetry propspin integrating over of and the function of the wave assumption VpA VnA, we only erty element matrix the evaluate need to of, say VnA. The matrix elspin ement of PI exchanging the spin coordinates of the nA pair can be calculated by rewriting the spin function of Eq. (11.18), expressed in (pn)A coupling order, to that of the p(nA) order by using the Racah coefficient in unitary form (see Complement 6.3): =
-
=
[Xj(pn)xX.j(A)j.iM= 2
2
1: U(-!-!.!1;IS)[X.L(p)xXs(nA)].jM 2222
2
2
S=0'1
Vf3_ 2
1
[X.L (p) x Xo (nA) I.LM + 2
2
2
[Xi(p)xXj(nA)jjM. 2
2
Then the desired matrix element is
&1(pn) xx.L(A)ji].LMjP'j[xj(pn) 2
2
2
x
X.L(A)jfljm) 2
2
2
(11.22)
C11.2
Convergence
of
235
partial-wave expansions
1 .1 .1 .1; is) 1.1.1; IS) U( 2222 1) S+I- U(I:2222
.1.
(11.23)
2
S
The
equation of motion for f reads h2
a2
h2
a2
-
2A2 19X2 2
21L1- axi 2
I
+V,A(IX2
x,
-
2
+
1, P)
Vp. (XI)
-
E
boundary condition that f VpA(r, P') is given by
with the Here
VpA (r, P')
VR +
as
f (XI
One has to note that P'
i
X2 i
(I
1 X2 +
2
XI'X2)
has to vanish for
1Vt + 4
4
VpA
+
3V,) (lu
=
0,
large
(11.24)
xl_ and X2-
1(2 u)P')
+
-
2
2
exchanges rp
XII,P
(11.25)
and rA, which induces the
transformation of the coordinates: x, --+ x1/2 X21 X2 --+ -3x,/4 X2/2. The potential V,,A(r, P') is defined in exactly the same manner,
-
-
where pr
now
and
3xj/4
ground-state energy and the solution
of Eq.
transforms x, and X2 to
xl_/2 + X2
-
X2/2,
respectively. We obtain the
The variational trial function may be chosen correlated Gaussians (CG) of Eq. (7.3):
as a
(11.24).
combination of the
K
f(XliX2iXI*X2)
=
ECkfFAk(X) +FPAkP(X)II
(11.26)
k=1
1
where the second term with P
function is
an even
(0
0
-1
)
function of xj, Le.., for A
assures
All A12
that the trial
A12 A22
PAP
-A12 All A22 -A12 It is interesting to analyse the paxtial-wave contents of the solution. For this purpose we calculate the quantity becomes PAP
2
C1
=(f(XliX27Xl*X2)lpllf(Xl7X27Xl*X2))
(11.27)
with
PI
=
I[yl(XI)
X
YI(X2)j00)([YI(Xl)
X
Yl(X2)1001-
(11.28)
236
Complements
The result is
given
in Table 11.8. The admixtures of
D, G andhigher
waves axe small and the PWE is very effective in the present case. The root-mean-square (rms) radius of the hypertriton is calculated to be
4.87 fm. The
fm,
distance between the proton and the neutron is 3.61 which is slightly smaller th an the corresponding distance (3-90 fin) rms
obtained for the free
deuteron,
and the
distance between the A
rms
and the center-of-mass of the proton and the neutron is as large as 9.98 fin. Therefore the hypertriton is barely bound by polarizing the deuteron slightly in order to increase the binding between the deuteron
and the A.
Table 11.8. Partial-wave
decomposition
1
0
2
4
C2
0.99913
0.79 X 10-3
0.6x 10-4
The trial function from the
of the
hypertriton
(11.26) gets higher paxtial-wave
wave
function
contributions
term X1'X2. The
parameters A12 determine the appropriate weights of various partial waves. It is of course possible to solve Eq. (11.24) by expanding f in partial waves: cross
f(X1iX2iXI'X2)
E
"'
A (XI X2) [YI (ill) i
X
YI (F2)] 00
.29)
I=even
Eq. (11.24) reduces to a coupled equation for fI(X1, X2)'s. Refer to Appendix A.5 for a method of evaluating the matrix element with this type of basis functions in the case that A (XI X2) is expanded as Then
7
a
(XlX2)le
combination of Gaussians
"
2 1
-bX2
2.Table 11.9 shows the
energy convergence in the partial waves included in the calculation. fin conformity with Table 11.8 the inclusion of S wave alone is a good
approximation to the hypertriton and bigher paxtial waves give modest contributions to the binding energy. Noncentral forces necessitate the inclusion of higher paxtial waves and the PWE will be
slowly convergent
in
general.
Even in the
case
of
potentials the energy convergence in the PWE in one particula,r Jacobi coordinate set depends on a system and an interaction. To central
examine the contribution of forces
we
tightly
bound than the
higher partial waves
in the
case
of central
example of the triton (nnp), which is more hypertriton. The energy has been calculated
consider another
C11.2 Table IL9.
Convergence
of
partial-wave expansions
Convergence of the hypertriton
energy in the
237
partial-wave
expansion (PWE). The CG indicates the result obtained with the correlated Gaussian of Eq. (11.26). The energy is from the three-particle threshold. The deuteron energy is -2.202 MeV.
CG
PWE
1.. E
(MeV)
by using the
0
2
-2.305
-2.358
wave
-2-378
function of type
(11.29),
where x, is the relative
distance vector between the two neutrons and X2 is the relative coordinate between the proton and the center-of-mass of the neutrons. The
spin-singlet state. Two potentials are used: One is the soft-core Volkov (VI) potential [1451 and the other is the repulsive-core ATS3 potential [1461. Table 11.10 compares the PWE results with other methods, the hyperspherical harmonics (HH) expansion method (see Sect. 5-2) and the CG calculation. The PWE gives fairly fast convergence for the V, potential and the inclusion of S and D waves already produces the energy that is close to the accurate value by the HH method. However, for the ATS3 potential the 6 are not convergence is rather slow and the partial waves of up to I
two neutrons
are
assumed to be in
a
=
sufficient to get the energy which is close to the value calculated with the CG basis (11.26). (The two neutrons are assumed to be in a spin-
singlet state, but, as the ATS3 potential is spin dependent, to allow them to be in a spin-triplet state as well leads to lower energy than the result of Table 11.10. The CG calculation including this possibility 100 dimension, gives -8.765 MeV (the rms radius is 1.67 fin) in K which is in perfect agreement with the energy [170] obtained by the =
Faddeev method
(see
Sect.
5.3).
vergence
explain why the ATS3 potential leads to the than the V, potential. The matrix element for
potential
between the two neutrons vanishes for the functions with
Here
we
slower
con-
the central
values, so the potential is not responsible for the mixingin of different partial waves. The potential between the neutron and the proton, however, induces the I-mixing. To see this, we note that the latter potential has the form V(Iax, + bX21) (for the triton case a 1, but it is extended to a general case.), which is a :E(1/2), b function Of X17 X2 and the angle,& between x, and X2. The potential can be expanded in terms of the multipole operators as follows different I
=
=
Complements
238
Convergence of the triton
Table 11.10. sion
(PWE).
The two neutrons
are
See Table 11.1 for the nucleon-nucleon
V,
energy in the
assumed to be in
partial-wave expanspin-singlet state.
a
potential.
[1451 HH[1691
PVVE
1niax E
(MeV)
ATS3
0
2
4
6
-8.005
-8.390
-8.447
-8.460
-8.4647
[146] CG
PVVE
Imax
E(MeV)
0
2
4
6
-2.948
-6.215
-7.210
-7.484
-7-616
00
V(Iax, +bX21)
=
EV1(XI,X2)P1(COS'6)
(11.30)
1=0
with VI (X1 i
X2)
=21+1 2
7r
fo
V(Iax,
+
bX2 I)PI (COS 6)Sin'0d#-
example, for the Gaussian potential of V(r) (see Eq. (6.53) or Eq. (A.81))
For
VI
Voe-t2T2
we
have
(X1 X2) i
=
where
=
(11-31)
Vo (21 + 1) c(l, ab) il (2 M2 jabjX1X2)
E(I, ab)
=
I for ab >
0, E(I, ab)
=
2 2 2) -g (a 2 xj+b X2,
(- 1)
1
(11.32)
for ab < 0. Since the
Legendre polynomial PI(cosO) is proportional to the tensor product [YI(i-1) x Yj(i2-)joo (see Eq. (6.54)) and has non-vanishing matrix elements between the functions of different partial waves, we find that the various multipole components contained in the potential bring about the mixing of the higher partial waves. The relative strength of the multipoles thus determines the extent to which each partial wave is contained in the solution. For a long-ranged potential (small p) the magnitude Of V1 (X1 X2) is determined by the factor IL21/ (21 1) 11, the soluto contributions main low that so partial waves give only tion. However, this is not the case for a short-ranged potential. The -
7
is, the larger the contribution of thehigher in the extreme case of V(I ax, + bX2 1) E.g., inultipole components. shorter the
potential
range
-
Quark
C11.3
Pauli effect in s-shell A
hypernuclei
239
(fk2/7)3/2 and letting 1L oc in Eq. J(ax, + bX2), by putting Vo (11.32) We get VI(X1, X2) ((21+ 1)/(47))J(Ialxl jbjX2)1(jabjXIX2), ,
=
-
-
which indicates that the
multipole strength is equally strong for arbi-
trary 1. The result of Table 11.10 the fact that the ATS3
repulsive-core part. Clearly the matrix
be understood in this way from contains the strong short-ranged
can
potential
element for the central
potential
between the
neutron and the proton vanishes for the functions of diffrent I values if they are expressed in terms of the different Jacobi coordinate set where
the first coordinate is chosen to the relative coordinate between the neutron and the proton. This consideration leads us to the following ansatz for the solution
(X(I), X(1)) [Y,(X(,)) YJ(X(1)) 100
f
1
1
2
X
2
1
+
X
fl(3) (X (3), (3)) [yJ(X(3))
X
1
+
where
fl(2) (X (2), (2)) [yJ(X(2)) 1
x('), x(') (i 1
=
2
X
X
1
2
1
2
1
1
y1I
(X(2))j 2
Yj(X 2(3))
00
100
(11-33)
1, 2, 3) stands for the ith Jacobi set (see Sect. 2.4).
This type of functions is used in the Faddeev method and the CRCG [26] method. It is known that this expansion converges much faster than the one using one particular Jacobi set, and the inclusion of low
partial waves
is
be sufficient to get accurate solutions (see In contrast to this approach, the CG basis
expected to
also Sect. 4.2.1 and
[30]).
particular Jacobi coordinate but takes account of the contribution of the high partial waves by the cross term 1412XVX2 in the exponent of the trial function. This ensures accurate solutions in the correlated Gaussian approach. It is noted that the calculation of matrix elements with high I values is computer time consuming (see Appendix A-5), whereas in the CG basis no such PWE is employed, which makes it possible to calculate matrix elements very quickly. of
Eq. (11.26)
uses
only
one
Quark Pauli effect in s-shell A hypernuclei 4 3 and 5A Ije are Called Among a few known A hypernuclei AH, 4jA 1, AIIe, s-shell hypernuclei because in the simplest version of the shell model 0 all the constituents can reside in an s orbit. By applying the L 11-3
-
=
240
Complements
correlated Gaussians of
Eq. (7.3),
of s-shell A
in order to reveal the
hypernuclei quark substructure of baryons of
in
we
calculate the
resolving
the
binding energies significant role of the long-standing problem
'He. Since the
pioneering work of [1711, the S-shell A hypernuclei have intriguing problems, among others, the anomalously small binding of 5AHe. According to a recent survey of hypernu5 clear physics [1721, "The anomalously small binding of AHe remains model calculations an enigma. Simple based upon AN potentials, parametrized to account for the low-energy AN scattering data and the binding energy of the A 3, 4 A-hypernuclei, overbindS A5He by offered several
=
2-3 MeV."
The trial function is
given as
a
combination of the correlated Gaus-
sians:
CkAjFAk (X)XkJM?7kTMTli
TfJMTMT
(11.34)
k
where the operator A
antisymmetrizes the nucleon coordinates, XkjjU (Since L 0, the spin S is equal to the total angular momentum J), and 77kTMT is the isospin function. The spin and isospin functions are obtained by successive couplings, for examPle, XkJM= X(S12SI23 )JM [[[Xl/2 X XI/21SI2 X X1/2JSI23-IJTM with k representing a set of the intermediate spins S12, S1237 The optimal set was chosen for each matrix Ak. The matrix Ak containing is the
spin function
=
:::--:
...
....
nonlinear parameters
was
selected
by
the SVM.
The NEnnesota potential [1471 with u I was used for the NN interaction. The MA potential is the same as was used in the previous Complement. Table 11.11 lists the results of the calculation. The calculation reproduces the data well except for the case Of AHe, for which the theory overestimates the binding energy by about 1.9 MeV, =
consistent with what is mentioned above. Another is that
4 He A
is
bound than
4 H. A
thing to be noticed
strongly introducing a charge-dependent component as the IVA interaction, because then 4AHe would be less tightly bound because of the Coulomb repulsion. We have so far treated a baryon as a structureless particle and distinguished A from N. What happens if we take their quaxk substructure into consideration? Recently it has been shown in [173] that the anomalous binding problem of 5A He can be, at least partly, resolved more
It would be difficult to
understand this without
by considering
the
quark substructure
of the
baryons.
C11.3
Quark Pauli effect
in s-shell A
241
hypernuclei
Binding energies (in units of MeV) of s-shell hypernuclei. The separation energy BA of the hypernucleus AAX is defined by the difference of the binding energies, B(-A'1X)-B(-'1-1X). Table 11.1-1.
A
3
AH
A411
4AHe
5 He A
B(AX) A B(`1-1X)
2.38
10.62
9.91
34.93
2.20
8.38
7.71
29.95
BA(Cal.) BA(Exp.)
0.18
2.23
2.20
4.98
0.130.05
2.040.04
2.390.03
3.120.02
Let
us assume
for the sake of simplicity that
baryon is (u, d, d), A a
a
coinpos-
(u, d, s), (u, u, d), n particle of three quarks, p well. harmonic-oscUlator the in orbit in the Os and each quark moves with size parameter b. The spatial part of the three-quark baxyon is described by the function ite
=
=
OB
-`
(7rb2)
1
-1 4
exp
Z3
b2
( '3
(_ (P2 P2 P2)) (_ _b 2) O(int) 2b2
1
+
2
+
3
3
exp
-
where pi is the
=
2rI
(11.35)
B
quark's position vector,
rl
=
(P1
+ P2 +
P30
is the
is the spatial part of the intrinsic baryon's position vector, and 0(int) B of the function wave baryon depending on p, P27 (PI + P2)/2 P3* The value of b can be estimated to be about 0.86 fin by requiring that the function (11.35) reproduce the charge radius of the proton [1741. When the spatial part of the two-baryon wave function is described we interpret it as with the configuration of expf -3/(2b2)(r21 + r2)1, 2 the Os in all six orbit, exp(-p 2/(2b 2)), quarks move indicating that of the common harmonic-oscillator well. with size parameter b. By increasing the number of baryons, we thus expect that the manybaxyon wave function may receive a special constraint arising from the quark Pauli principle that any single-particle orbit can accommodate at most six quarks (three colors and up-down spins) for each flavor. It is easy to see that we have no apparent quark Pauli-forbidden 4 hypernuclei. However, this is not the case for states up to A 5 5He: Four nucleons of A He, when they are on top of one another, A have already six u-quarks and six d-quarks in the Os orbit, so neither u nor d quark of A can take the same Os orbit. This leads us to the conclusion that the five baryons cannot take a configuration of -
=
-
242
Complements
exp(-3/(2b 2) 1:5i=1_ r i2) forbidden state for
rated,
is
.
To be
5AHe,
more
specific, the (normalized) Paulibeing sepa-
with its center-of-mass motion
given by 5
TfPF (X)
=
JV
'(7b2)-3eXP(_ 4
-
3
3
_b2 Dr'
X5)2
(11.36)
i=1
where X5 is the center-of-mass coordinate of the five baryons. The normalization constant is given by JV (4M2 + M2)/(4MN + MA)2' =
N
A
where MN is the mass of N. Other Pauli-forbidden states sibly exist, but this is the simplest and most evident one. The solution TI obtained above for 5He A
Pauli-forbidden component of Eq. quence is very
was
might
pos-
found to contain the
(11.36) by only 0.44:'YD, but its conse-
If this Pauli-forbidden component is simply subtracted from the wave function Tf, the BA value would have been
from 4.98 MeV to 2.74
changed of
A5He!
The reduction is
pectation may be
significant:
a
principle
MeV,
anomalously
small
binding
because the reduction in the
calculation of the so-called variation
6.1),
no
due to the fact that the energy exvalue in TfF(x) is very large, i.e., 513.6 MeV. However, this premature conclusion from the viewpoint of the variational
mainly
binding energy is based on a before projection type (see Sect.
that
is, the variation has been done before the elimination of the Pauli-forbidden component is made from the trial function. To calculate the
binding energy more precisely by taldng into acquark effect, we have repeated the calculation by replacing the correlated Gaussian in Eq. (11.34) with the one that 'has count the
no
Pauli
Pauli-forbidden component:
FAk (X)
11.4 The
FAA: (X)
C nucleus
-
TfPF (X) (TfPF (X) IFAI, (X))
-
(11.37)
system of three alpha-particles example application of nonlocal potentials, we take up simple model for the 12 C nucleus, the 3a model. In this model 12C
As a
12
=
an
of the
as a
C11.4 The
12C
nucleus
as a
system of three alpha-particles
243
5
>4
3
Exp.
Pq
2
0.6
0.4
0.2
0.0
b
Fig.
[fml
separation energy BA of 5H A three-quark baryons
11.4. The A
parameter b of
containing
1.2
1.0
0.8
six protons and six neutrons is
as
a
function of the size
approximatedas
bosonic
a
system of three alpha-particles. The physical reason behind this picture is in the fact that the alpha-particle is a strongly bound, very stable system compared to light neighboring nuclei. One needs an energy of about 20 MeV to excite the of the
alpha-particle.
Because of this
nuclei tend to form
unique
consist-
feature, light subsystem ing of two protons and two neutrons, which is called an alpha-cluster. The alpha-cluster would never be identical to the alpha-particle but it may happen that a model descriptioii assun-Ang such subsystems can explain many properties of the light nuclei [60, 175, 59]. To calculate the binding energy of 12C in the 3a model, one needs to know the potential between two alpha-paxticles. It is hard to derive a potential which acts between composite particles through the underlying interactions of the particles composing the composite particles. Thus we use a phenomenological potential which successfully reproduces the alpha-alpha scattering phase shifts. The one employed here is an I-independent local potential consisting of both the nuclear some
paxt of Gaussian form and the Coulomb part
V(r)
-Vo
e
_P,
[1761:
4e2
2 =
a
erf (,3r),
+ r
(11-38)
244
Complements
where VO
122.6225
=
MeV, p 0.22:ftn-2, and scattering data very well
0.75 fin-'.
=
fits the
this
Though
to about Em
30 potential MeV and reproduces the 0+ resonance at 92 keV quite well, we have to note that it predicts "redundant bound states", two (-72.625 and
-25.617
I
=
2
in the I
MeV)
state, these bound states ftom the
0
=
Since the two
wave.
wave
and
one
(-21.999 MeV)
known to form
in the
bound
alpha-paxticles considered spurious and must be excluded are
no
axe
space. The existence of
configuration
=
system of two nuclei is due understood in the
to the Pauli
spurious
principle
and
states for
can
be
a
easily
version of the nuclear shell model. This is
simplest orthogonality condition model [1771 which succeeded to give a foundation to the deep local potential of type (11.38) from the microscopic theory of scattering between composite particles. Let us denote the spurious states as 0, 1 for I (r), (n 0 for I 0, and n 2). We require that the wave function TI, for the 3a system be free from the spurious components in the alpha-alpha pairs, that is, the basis of the
=
=
=
( Pnim(ri
-
rAff)
=
(11.39)
0-
Here ri is the position vector of ith alpha-particle. Unless this condition is imposed, the ground-state energy with the potential (11.38)
strongly overbound because
would be
the
alpha-clusters,
as
bosons,
would occupy the lowest possible states. An alternative, convenient approach to eliminate reasonably well the spurious components is to use
the nonlocal
Fini
1M ==
potential of projection operator type
I
(ri
rj)) (W,,,M (ri
-
and include it in the Hamilto-nian
-
as a
kind of
N
I
=
T
-
Vij(lri-rjl)+A
Here A is
a
wave
positive
E EI'i'!
(11.41)
j>i=l n1m
j>i=l
that if the
"pseudo potential" [178] N
E
Tn, +
(11.40)
rj) 1,
constant chosen to be very
large. The idea
is
function contains the spurious components, then the energy would be comparatively high. Therefore, the variational lowest solution would approach a state that has a negligible overlap with the
spurious components. The spurious bound
state
lar
O,am (r)
Cae- 2 a
can
be well
approximated
in the form
2
Y1. (r)
(11.42)
CIIA The
12
Then the nonlocal
C nucleus
potential
as a
245
system of three alpha-particles
summed
over m
takes the form
_Vnlra
f
ij
M
21+1
(rr') -47r
R
(cosv) E ca*ca, e-la 2
r2_ 1 af rf2
(11.43)
aa,
where V is the
between
angle
r
and
r.
The trial function for the
system was chosen as a linear combination of the correlated Gaussians, FA(x), of Sect. 7.2. The matrix element of the nonlocal potential (11.43) in this basis can be calculated with the use of Eq. (7.46). 3a
high sensitivity of the energy on A calls for numerical calculations with a high accuracy in order for the pseudo potential to play the role of the Pauli projector. It was found [1791 that the pro0 and 2 do not jection effect is so strong that the partial waves 1 The
=
contain bound states, and a contribution of more than 95 % to the ground-state energy is due to the partial wave I 4. Table 11.12 lists =
the
energies
of the
ground
state and the excited states obtained in
a
[1801 using the basis function of Eq. (11.33). h (X1 X2) is expanded as combinations of Gaussians 2 The calculation does not reproduce the experix11 x12 exp(-ax 1 _OX2) 2 mental energies very well. In [1801 the energies of the 2+1 and 41+ states
variational calculation The radial part
are
also
given. They
are
-3.77 and -2.25
MeV, respectively, which
compared to the experimental energies of -2.84 and 6.80 MeV. Here the discrepancy between theory and experiment is more serious: state is lower than that of the 0+1 state and the The energy of the 2+ , energy of the 4+1 state is much lower than the experimental energy. are
Note that there is
no
Pauli-forbidden state in the 1
makes the contribution of the
high partial
> 4 waves,
waves too much
which
important.
This calculation suggests that the 3a model with the local aa potential has only limited success and is not very realistic to reproduce the experimental energy spectra. This doe not, however, exclude the
Table 11.12. Energies of the 0+ states of 12C calculated in the 3a model [180]. The energy is measured from the threshold of the 3a breakup.
0+ 2
0+ 3
-3.38
-1.43
3.70
-7.27
0.38
3.03
0+ 1
(MeV) Exp. (MeV) Cal.
246
Complements
a
+
8Be,
a (
and
'Be(a,-y)12c.
Appendix
Matrix elements for
Gaussians
general
The matrix element for the correlated Gaussian with
arbitrary physically important potentials including central, tensor and spin-orbit components. A simple and straightforward method is presented to calculate the matrix element for the general correlated Gaussian-type geminals.
angular
momentum is obtained in
a
closed form for most
A.1 Correlated Gaussians We start with
Eq. (7.2)
to evaluate the matrix element for
a general generating functions g is given perform the operations prescribed in Eq.
The matrix element between the
case.
in Table 7.1 and
one
has to
(7.2). A.1.1
The
Overlap
overlap
of the basis functions
matrix element
can
be obtained
through
(fK" LM (u, A", X) 1 fKLM (u, A, x)) I
BKLBK-'L
ff
_ d9de'YLm(i;_)YLm(e')
d2K+L+2K+L
dA2K+LdAI2K'+L
3
X
detB
exp
[q A2 + q"A/2 + PAA/e. er]
where
Y. Suzuki and K. Varga: LNPm 54, pp. 247 - 298, 1998 © Springer-Verlag Berlin Heidelberg 1998
\,=0
,
(A. 1)
Appendix
248
B=A+A',
q=
2
fiB-lu,
B-lu ,
q
2
p
=; B-lu.
(A.2) To
perform
the
operation prescribed
Eq. (A.1),
in
we use
the
ex-
pansion
[qA2 + q, A/2 + pAA/ e. el]
exp
CO
00
CO
E 1: E H(n, q).ff(i , q) H(m, P) Vn+m y2n+m
fn
n=O n'=O m=O
(A-3) where
H(n, x)
is introduced to Xn
10
H(n, x)
for
n!
simplify the
notation and defined
non-negative integer
n
by
(A.4)
otherwise.
Differentiating with respect to A and Y, followed by A 0, 2K + L and gives non-vanishing contribution only when 2n + m 2K' + L, while the integration over the angles of e and e' 2n + m becomes nonzero if m is equal to L + 2k with non-negative integer k (see Eq. (6.18)): =
=
if
d iYLM(e-)yLM(,
di
1
j)*(e.eI)2k+L
K k and n Rewriting n overlap matrix element =
-
=
K'
-
k,
=
we
BI.-L
(A.5)
obtain the result for the
(fKIILM(UI, A', X) IfKLM(u, A, x)) 3
(2K+L)!(2K'+L)i
(27)N-1-
BKLBKIL
detB
2
min(K,K') X
E
H(K-k,q)H(K-k,q)H(L+2k,p)BkL- (A.6)
k=O
The B,,I value is given in Eq. (6.9). Note that the values of K and K' usually be chosen to be small. in practical cases, and then the sum
can
over
of K
k is limited to =
K'
=
just
a
few terms. In
0 the above result
particular, simplifies to
for the
special
case
A. 1 Correlated Gaussians
249
(ALM(Uli A i X) IfOLM(Ui A X)) 3
(2L + 1)!!
(2v) N_I
(
4v
detB
2
)
L
(A.7)
P
A.1.2 Kinetic energy
The kinetic energy with the center-of-mass kinetic energy subtracted is given in Eqs. (2.10) and (2.11). It is simply written as FrAn-/2. The matrix element of the kinetic energy is then
Table 7.1
easily obtained by using
as
I
(fK"LM(UIjXjX)j ?rA7rjfKLM(UjAjX)) h2
(2r) N-I
2BKLBKIL
detB
(
d2K+L+2K+L
[R
X
dA2K+LdA/2K'+L
X
exp
3 2
+
if
de-- do
pA2
+
YLm (&) YLM (4 )
p/A/2
+
QAA'e-.e-f
[qA2 + q' A/2 + PAAte.e/
(A.8)
where R
M(AB-'A!A),
=
P'=
P
=
-i B-'AAAB-lu ,
-i!B-1A!AA!B-'u, Q
=
2
The matrix element of the kinetic energy manipulation similar to the overlap case:
B-'AAXB-lu. can
be obtained
I
rA7rjfKLM(u, A, x))
(fKfLM(Uf7A!jX)j 2
3
(2r)N-1
h2(2K+L)!(2K+L)!
x
detB
BKLBK"L
2
E f RH(K
k, q)H(K'
-
2
-
k, q)H(L + 2k, p)
k
+PH(K
-
k
-
1, q)H(K'
-
k, q')H(L + 2k, p)
(A.9) by using
a
250
Appendix
+P'H(K
k, q)H(K'
-
-
k
-
1, q)H(L -IF 2k, p)
+QH(K-k,q)H(K'-k,q')H(L+2k-l,p)lBkLThe
case
K'
with K
=
0 reduces to
a
(A.10)
simple result
very
(ALM(U17A x) I --iAw I fOLM (u, A, x)) 2 ,
h? 2
A.1.3 Next
(2,x)'V-1
(2L + 1)!! (R+LQp- )-
detB
4-x
3 2
)
L
(A.11)
P
Two-body interactions
we
derive the matrix element for the interaction of Eqs.
(2.3)
and
(11.1). To evaluate the matrix element of the operator expressed as a tensor product, it is convenient to make use of the famous WignerEckart theorem
spherical
[70, 71, 721, which states that a matrix element of a operator 0.-, between states with angular momenta
tensor
JM and XIW
coefficient and
be
can a
expressed
as
product of
a
reduced matrix element which is
z-components of the
(J'M'JO-AIJM)
angular
a
Clebsch-Gordan
independent of the
momenta:
(JMrILJYM-) =
,V2J'
V 110. 11 J).
(A.12)
+ I
Here the reduced matrix element is barred matrix element. The factor
expressed by the so-called double-
I/V -2-Y+I is factored out because
then the reduced matrix element is
symmetric
to the bra and ket
interchange
IEV 110. 11 X) provided of
OKIL
is
=
(-I)J+r.-J,(y 110. 11 J),
(A-13)
that the matrix element is real and the Hermitian
(0'1')
t
=
E(- 1)
"L
Or,, -i-L
with
a
phase
factor
E(E2
conjugate =
1).
By applying the Wigner-Eckart theorem we can express the matrix element for the orbital and spin angular momentum coupled wave function
as
follows:
(Tf(LI SI) JM IV(I'ri
-rj 1)
(Or ij (space) 0,,ij (spin)) ITf(LS) JM)
U(LnJS; L'S) V,f(2L' + 1) (2S + 1)
-
A.1 Correlated Gaussians
(L' 11 V(Iri
x
rj J)0,,jj (space)
-
251
11 L) (S' 11 0,,jj (spin) 11 S). (A.14)
potential V(r) is assumed to be a function of r only. Eq. (A.14) and its extension to a general coupled tensor operator is given in Exercise A.I. The reduced matrix element of the spatial part is obtained through Here the form factor of the
The derivation of
(LMn1ijL'M') (L' II V(I ri v/2--L'+I (L'IWIV(Iri
=
jV(r)(LM'jJ(rj case
uncoupled by
where
wave
-rj
(_1)r. 2S'
one uses
-
I I L)
-
r)O,,,,,j (space) ILM)dr.
the orbital and spin matrix element
(A.15)
angular momentum
(A.14)
can
be obtained
(0,,,j (space) 0,j (spin)) lTf(LS)JM)
ri 1)
-
U(LKJS'; LIS)
+ I
2S+1
x
-rj
function, the
R(L"S")JM IV(Iri
rj 1) 0,,i, (space)
(space) ILM)
=
In the
-
(LMLts1-tjLIML')(SMSrvjS,MS)
(TfL,MLS,ms, IV(Iri -rj J)0,st,,j (space)Oc,jj (spin jTfLmLsms) (A.16)
Since the spin matrix element is easily calculated as will be shown in Appendix AA, we will focus on the spatial matrix element and evaluate it for the most important components of the nuclear potential, that is, central, tensor and spin-orbit components.
(i) central
and tensor interactions
The operator
Or-A,j (space)
can
be I for the central
potential
and
Y2,,(r-i---r-j)
for the tensor operator. See Eq. (11.7). Thus both the central and tensor components can be treated by assuming the form
---rj) for 0,,,,,3. (space). Expressing ri
of Y,,,,, (,r-i
(N 7.1,
-
1)
we
x
1 column matrix
-
rj
as w (ij) x
Oj) defined iii Eq. (2.13)
and
with the
using Table
obtain
(fK'L'M'(UI74 x)IS(ri
-
rj
-
r)Y,,j,(rj
-
rj) IfKLM(U7 -47 X))
Appendix
252
'::::-
6)(fKLM(UIi W 7x)IJ(W(ij)X-r)jfKLM(u,A,x))
Yrg(
(27r)N-2C
1
2
1
e-fc7'Y-"(i )BKLBKL' X
if
de^- d,
detB
3 2
)
d 2K+L+2K'+L'
iYjLM(e,-) YJL, M,
dA2K+LdAI2K'+L'
XeXp[q,X2+ A/2+ Aye.e/+7Ae.,r+,Y/,XIe/.,r] (A-17) where C
w(ij)B-lw (ij)
c
I
^/2
q=q-
712
q'-
2c
w(ij) B-1u, =P-
2c
cw(ij)B-1U',
7
1'7YI.
(A.18)
C
All of these quantities
depend on i and j but we omit the labels i and j to simplify notations. The integration over the angles of e and e' can be performed by expanding exp (pAVe- e' + -/Ae- r + -l'.Ve,- r) in -
power series and
-
using the relation
(e.e )nj(e.T)n2(eI.,r)n3 n2 +n3
Lel=l
njn2n3
RL Lfn
[[YjL( )
X
Y
X
(A.19)
Y. 00
LL'r.
with Rnjn2n3 2 LIts
=
(_I)nl+n2+n3
Bnl-LI7 Bn2-12 12 Bn,3 -13 2
'1
2
2
3
111213
E(2L+:1:1 ) (2
V
r.
+
212+1
1)
C(1112; L)C(1113; L)C(1213; K)
U(IIIILK; L'12).
(A.20)
See Exercise A.2 for the derivation of this relation. In the
defining
I or 0, and 12 equation for R the sum over 11 is limited to ni, ni 2, and 13 have similar ranges, respectively. Possible values of L, L", and r, -
...'
A.1 Correlated Gaussians
for
a
given set of values of ni, n2, and
n3
limited
axe
253
by the conditions
that L takes the values nj +n2, nj +n2 2, ..., I or 0, and L' and r, take similar values given by ni, n3 and n2, n3, respectively. In addition, the -
is restricted to
sum over
L, L,
for
values of
given
and n2 + n3
-
would vanish.
K
r.
even
values of L + L+ K.
Conversely,
L, L', and K, all of the nj + n2 L, nj + n3 L, have to be non-negative and even, otherwise Rnin2n3 -
-
LL'r.
R has the symmetry: The reduced matrix element becomes
Clearly
(fK'L'(UIi -4 i X) 11 V(17'i
-
ri Dyt#
njn2n3
RLLII,,
pnjn3n2 "L'Ln
i -rj) 11 fKL (u, A, x)) 'R
Lf
(2K + L)! (2K'
+
L') 1
BKLBKILI
E
x
(
(27) N-2 detB
C.)2
H(ni, #)H(n2, -y)H(n3, 71) I(n2+n3+2,c)RnL,-n2n3 LIn
n,n2n3
2K + L
-
XH
nj
-
2K+L'-ni-n3
n2
2
2
(A.21) where the
I is defined in
Eq. (7. 10) potential for different pairs of particles be calculated with the above formula by changing only 01) in
integral
The matrix element of the can
Eq. (A. 18). We note
some
useful
applications
of
Eq. (A.21). For example,
calculate the matrix element of I ri-rj I' simply by putting r' and K 0. The two-paxticle correlation function can
one
V(r)
=
(fK-'Ll (u, A', x)
J (I ri
-
rj
a) Y,, (ri
-
rj)
fKL (u, A, x))
(A.22) easily calculated by taking J(r a) for V(r). K' We note again that the formula (A.21) simplifies for K 0, that is, the triple sum reduces to a single sum over, say ni (nl_ 0, min( L, L', (L + L' r,)/2)) and n2 and n3 have to satisfy L and nj + n3 L, respectively. In addition, in this nj + n2 njn2n3 calculated is R case simply by a term with 11 special nj, 12 n3 alone in Eq. (A.20). This simplicity will be used to obtain n2, 13 the matrix element of a density multipole operator. See Complement can
also be
-
=
=
=
-
=
=
=
=
A.2 for the details.
254
Appendix
(ii)spin-orbit The operator
angular
interaction
0,,,,,, (space)
momentum
((ri
rj)
-
1-
x
2h
and
and
(2.14)
trix element in Table 7.1 into
(fK"L"M"(Ufi Al7X)jV(jT'i
substituting
pj))
See
,.
x
and
Eq. (11. 9). -r,
with the
we
the
corresponding
ma-
i
obt
I'jj)1jzjjjfKLM(u, A, x)) (27)N-2
.1 Cr2
drV(r)e-2
-
is the orbital
1,
Eq. (7.2),
-
(pi
in terms of
x
Eqs. (2.13)
of
=
spin-orbit potential
Fij)x W)7r)
Expressing 1,_,,j use
1,1i,
for the
3 2
C) I
detB
BKLBK'Ll
d2K+L+2K'+L'
dA2K+LdA/2K'+LI q
x
exp
X
ifn*
+
X
qf A/2
r)t,
-
+
PAy e. ef + yXe.,r + -Y/Ale/. -
q'Af (e,
X
(A.23)
r),,
with 77
=
(MXB-lu,
Note that to derive
an
arbitrary
=
Eq. (A.23)
V(r) e-c('-a)2 (r for
q'
vector
x
a
a)dr
(MAB-luf. use
=
(A.24)
is made of the relation
(A.25)
0
provided V(r)
is
a
function of
r.
See
Eq.
(A-163). Before
performing the operation prescribed
that the matrix element vanishes in the has
parity
(_I)L
and because the
case
Eq. (A.23), we note 0 L. Because fKLM
in
of L
spin-orbit potential
does not
change
A. 1 Correlated Gaussians
parity, IL
-
L'I
has to be
even.
On the other
hand,
255
the tensorial char-
spin-orbit potential imposes the condition IL LI :5 1. Both the conditions are met only when L is equal to L'. This special result is entirely due to the unique feature Of fKLM and does not always hold for general wave functions. Combining Eq. (A.19) and the relation acter of the
i
fqX(e
-
r) 0
X
4vf2--x 3
-
77W (e'
rjqA[Yj(ii)
X
x
r)
Yj(i )Jj, -,qA[Yj(( )
x
YI(,P)JI, (A.26)
yields
the reduced matrix element
(fK,ILI (Ul A! X) I I V(I'ri I
I
as
-rj 1) Iij
I I fKL (u, A, x)) '3
4 V2--x
JLLI (-1)
3
x
E
q
L
(2K + L)!(2K' + L)!.
(27r)
( N-2C.)2 detB
BKLBKIL
l(n2+n3+3,c)H(nl,p)H(n2,,y)H(n3,-y')
n,n2n3
2K+L-nl-n2-1 H
xH 2
(2K'+
L
-
nj
-
n3,,,)
2
n., n2 ns
C(AI; L)U(LAII; 1L)RAL1
x
A
I(n2+n3+3,c)H(nj,#)H(n2, -y) H(n3 7f)
+77'
,
njn2n3
H
x
(
2K+L-nl 2
-n2
q)H(
2K+L-nj
njn2n3
C(AI; L)U(LA11; 1L)RLAI
-n3
-
1
2
(A.27)
straightforward and easy to follow. As a simple check of the above formula, the matrix element of the orbital angular momentum is calculated in Exercise A.3. It is again possible, however, to get a simpler formula by performing the'P integration first, as was done in Complement A. 1. This task is reserved for Exercise A.4. The above derivation is
256
Appendix
A.1.4
Density multipole operators
The matrix element of
density multipole operator plays a substantial role in investigating the properties of a system, e.g. the density, the deformation, the electromagnetic transition rates and the electron scattering form factors. The basic element of the Multipole operator a
takes the form
0,,,L, (space)
:---
f (Iri
xm
-
I)YI,
(A.28)
Note that the argument of the
density multipole operator is not ri but is correctly taken as ri xN, which is the single-particle coordinate measured from the center-of-mass coordinate. Comparison of Eqs. (2.12) and (2.13) immediately suggests that the matrix element of the density multipole operator ought to be calculated in exactly the same manner as that of the two-body potential. In fact the reduced -
matrix element
(fK"L-'(U/7 X7 X) 11 f (ITi
-
XNI)yn(Tii
-
N)
fKL (u, A, x))
XN
(A.29) be calculated
can
replacements
of
by Oj)
the --+
same
formula
as
0) inEq. (A.18)
Eq. (A.21) and of
with the trivial
V(r)
--+
f (r)
in
Eq.
(7.10). Just
as
the matrix element for the
lated from that of relation function
As
we
J(ri
can
xN
-
-
r),
and ri
be calcu-
the matrix element for the
also be derived ftom. that of
J(ri
-
rj
-
-
-
reduced to the
are
r)jfKLM(u,A,x)).
in fact obtained in the derivation
one
(fKfL'M" (U , X7 X) IJ(17VX (2K + L)! (2K'
+
BKLBK'L'
-
L)!
following Eq. (A.17),
r) IfKLM(u, A, x)) (2T,
)N-2C)
32
detB
E(LMr.M -MjLM)Y,,m, -m(i )* M
of type
This matrix element
the fi-nal result:
x
r).
-
(fK'L'Mf(U'jA!jx)jJ(iv-x give
cor-
-
already noted in the above derivations, both of ri xN rj are expressible in terms of the relative coordinates x as iv- being an appropriate 1 x (N 1) row matrix. There-
fore all of these matrix elements
we
can
have
Cvx with
was
density operator
L'
-1) -e -v/-2Ll + I
2
so
that
A.2 Correlated Gaussians with different coordinate sets
E
X
257
rn2+n3 Rnin2n3 H(nl, P) H(n2, -y) H(n8, -y') LLIn
nin2n3
2K+L X
H
-nI -n2
2K' + L'
-
H
2
n,
-
n3
2
(A.30) 7/,
given in exactly the same manner as in Eq. (A.18) with w(W being replaced by w. Here, r, takes the values
where c, 7,
p
are
L+L', L+L'-2,..., IL-L'I,
and it is of course restricted
by IAF-MI :!
The above matrix element becomes very simple for the special case K' of K 0 as was noticed in the previous subsection. Since the is.
=
=
applications, we show in Complement A.2 simplified to just a double sum.
matrix element has useful
that
Eq. (A.30)
can
be
A.2 Correlated Gaussians with different coordinate sets As the correlated Gaussians treated in the previous section have a very simple transformation property implied by Eq. (6.29), we as-
they are all expressed by a particular set of the relative coordinates x. However, the correlated Gaussians with the angular function OLM(X) given by Eq. (6.3) do not have such a nice property. Moreover, the use of different sets of the relative coordinates for these correlated Gaussians leads in general to a faster convergence because they allow us the possibility of describing naturally different types of correlations and asymptotics. The lineax transformation of the coordinate sets, however, leads to a formidable task even for a system of only four particles, because the function OLM(X) shows no simple transformation rule. See Appendix A.5 where the matrix elements for a three-body system are explicitly evaluated by transforming the corsumed that
related Gaussians from
one
Jacobi coordinate set to another. The aim
of this section is to outline ments for are
we
a
method of
calculating
the matrix ele-
the correlated Gaussians which
N-body system using expressed by different sets of the coordinates. By making use of Eq. (6.22) to generate the correlated Gaussian, have to evaluate the following matrix element for the operator 0: an
N-1
exp,
2
x-fXx)
11 Yij, j, (x ) j=1
Appendix
258
N-1
-1:Z.Ax)
1
0 exp
x
N-1
Yli.i (xi)
2
Climi,9ai
N
ali-Mi
01i
-
1
01j,i +Mi'
(-')Mj 01j,
3
C11 -mlj aal - ar'1j'+mi' j j j
97i j=1
(A.31)
(g(a'Jt';A!,x')J0Jg(aJt;A,x))
x
ri=O,-r =O 3
(-1)'YI-,,,(xi) to identity (Yl,,,(xi))* take care of the complex conjugation in the bra side. Note that the vector ti is defined by 2, i(l + i 2), -2-Fi) and likewise t!j by Here
(1
is made of the
use
T,
/2' i(I + T/2), -2,Tj j
complex conjugated.
transformation T trix element
x'
as
(A.31)
of g in the bra side should Assume that x' is related to x by a linear
The vector
j
not be
=
=
tj'
Tx. With the
between two
g's which
different sets of relative coordinates
Eq. (6.28) expressed in
of
use are
the
ma-
terms of
be reduced to
can
(g(a'lt'; X, x')J0Jg(aJt;A, x))
(g(Ta'lt';i XTx)101g(alt;A,x)), and
can
(A-32)
be obtained from Table 7.1 for most operators. 1, the matrix element
(A.32) takes, where v alt+ except for a trivial constant factor, the form e2 B' is matrix the and x given by (A+ TXT) (N 1) (N 1) Ta'It' For the unit operator 0
=
-IbB'V
=
,
-
-
To
simplify the
SN-1,
a.el
=
aNtIV
=
we
SN,
---,
rename
alt,
afN-,eN-l
=
expressed by a (2N 2) by 9Bs, where B is expressed in
6B'v
Then
notation
matrix B
can
be
-
aN-1tN-1 a2N-2t2N-2 82N-2-
-=
si,
x
(2N
-.-,
-
2) symmetric
terms of B' and T
as
follows
B
=
Thus the
(
B'
TB'
Bi TB'i
(A.33)
operation prescribed
in
Eq. (A.31)
can
be reduced to the
form N-
Oi",`
exp,
(2IBs
+ ibs
))
(A.34) .1=0'---'12M-2=0, 1=0,---,'2N-2=0
A.2 Correlated Gaussians with different coordinate sets
259
with
(A.35)
i9ail 0-Til-m Here the factor iv-s is included because it is needed in the
case
of the
potential energy matrix elements. See Table 7.1. It is worthwhile noting that the present formulation leads to a unified prescription of Eq. (A.34), which is independent of the choice of the relative coordinate sets but requires only a very simple calculation of the matrix B. It would be extremely tedious if one were to try to rewrite the anguN 1 lar function of fL'=I Yj . (x'.) in terms of those angular functions 3 conforming to the coordinates x. Using Leibniz's formula we can express Eq. (A.34) as 3
2
2
T
3
=1
Aj! (Ij
'XiAi
-
Ai)! (Ai
(1i Mj)! (1i
-
-
-
Mi)! mi
-
Ai
+
Ai)!
2N-2
Oj"
X
exp
(2 Os) -ri=O
2N-2
ai
X
ie,
(A.36) aj=O -ri=O
The last factor is
easily evaluated by using Eq. (6.58):
2N-2
11
ali-'Xi'mi-Ai e-vs i Ui=O -ri=O 2N-2
(A-37)
CIi-'XiMi-/-tiY1i-)LiMi-Ai(Wi)-
Possible values of Ai and [ii are determined by the conditions that O
vanish. Because of condition that Ai
-
assures
do not
that the coefficients
a special form of si-sj mentioned pi < 2Aj has also to be met.
below,
another
remaining factor, we note that ABs is quadratic both in the ai's and -ri's because of si sj -2aiaj (ri -r.j) 2. (S ince, for each aj, -ri appears at most in quadratic power, the number of To evaluate the
-
=
-
260
Appendix
differentiation with respect to -ri cannot exceed two times the number of differentiation with respect to aj, that is, /Xi pi :! 2Aj.) Therefore, -
.!Bs 2
when
e
these
are
contributes Thus
we
expanded in power series in both ai's and -Fi's, and when zero after the differentiation, only the term (.IgBs)Q 2
is
set to
2N-2
provided that Ej=j
2Q and
Ai
2M-2
2Q.
i=1
(-2)Q
19Bs a,\"/L'exp i
(2
Q!
Cti=O -ri=O
Q
2N-2
EBjjajaj(Tj-T.j)2
X
i=1
To have
1:2N-2 (Ai ILi)
have
(A-38)
i<j
aj=O -ri=o
non-vanishing contributions,
each of
Q
terms must be dif-
ferentiated with respect to aj, aj, and either twice with respect to 7i or -Fj, or once each with respect to both 7-i and -rj. This operation
yields 2Bij,
where the miniis
to -ri and -Fj. This
respect
sip
sign comes from the differentiation with is
denotedEij.
Therefore
we
obtain
2N-2
a,"' '14 exp
(2sBs)
ai=o -ri=O
(-4)Q Q! Q
(A-39)
Ejj, Bi,j,
x k=1
'k<jk
where the
extends
all the
possible combinations distributing the given multiple differentiation into pairs of differentiations. In fact the operation prescribed in Eq. (A.38) can be done easily by using the software Mathematica [1841. The case of a three-particle system is given in Co mplement A .3. sum
over
The matrix element for the kinetic energy can be obtained similarly. In fact the matrix element (A.32) for the kinetic energy takes the form relation
(9Cs)e12'BS,
but this
can
be dealt with
easily. Using
the
A.2 Correlated Gaussians with different coordinate sets
(Ws) exp
d
1Ms
(2 )
2
dp
( 19 (B +pC) s)
exp
261
(A.40)
2
P=O IgBs
(112N-2 aj " "Ws ing
one
of the
)
e2
j=1
Bi,,j,,'s
Eq. (A.39)
in
be obtained
can
by simply replac-
ai=O,ti=O
with
Ci,j,.
The
potential energy matrix element for the spin-orbit interaction requires a slight modification. Noting that the general form of the matrix element (A.32) for this case can be expressed as + ivis) (see (Sk)yeXP (19B.9 2
(2IMs
(Sk)jzeXP
7.1),
Table
+ iv-s
( -1)"
we
make
use
exp
ly(Wk)-M
(OWk)jj (21ABs ) exp
where the a,
spherical components a,,
- 2=(a.,
=
+
iay),
ical components a
( aa) .1
V-2
1 =
-72=
1
(
(
ao
of
aa
a
+i
-9a.
=
Da,
-67a--,
'9
aal
-!9Bs+fvs
e2
)ai=O,-ri=07
Eq. (A.37) with
lated ,r
by -
the
can
.
be obtained
a
-
(A.41) are
iay),
(a
defined
and the
-5a I o
=
Da-,
by spher-
given by
are
la
a
7Eo
I
"&a )-1
=
2N-2
( fjj=j 19i1Xi'lLi (Sk)g
by replacing YIk-Ak -k-Mk (Wk)
YIk-AkMk-Ak(Wk),
formula
which is
easily
calcu-
[70, 71]
Vaf (a) Yjm(et)
-Ti -1 (f(a)
I+ I
particular
we
f(a)
f (a)
+
[T
x
Yj-j-(et)]j,,.,
a
+ I
:21EI++11: In
a
spin-orbit interaction,
( aTwul,
gradient
vector
+ fvs
By using Eqs. (A.41) and (A.34)-(A.37),
-
-
aa,
I
differential operator V,, a
the matrix element for the
in
a
a
(219Bs
+ iv-s
!,, (a-v 2-
=
a
'9
aax
a,,, a-,
of
of the relation
-
1f (a)
[r
X
Y
(A.42)
a
obtain
Yjm(a)
aa A
-(21 + 1)
V I--11 (1/-t1mjI-Im+tt)Yj-jm+, ,(a).
(A.43)
Appendix
262
A.3 Correlated
Gaussian-type geminals general correlated Gaussian-type geminals calculated from the equation
The matrix element for the of
Eq. (6.26)
be
can
(W, BI k r) 10 1 f,,, (A, B, R, I
7
N
N
3
3
i=1
exp(
iP
IiZBR
t,. n ,,
p
p=1 j=1 q=1
-
qn,',
anip
ff H H H yt-i
3q
-
iR
-
2
liiBk 2
-
Pk)
(g(t+B'k;X+B',r)101g(t+BR;A+B,r)) tj=O' -=O
(A.44) It is crucial to know the
generating functions on i multiple differentiations. As
(tj,...,tN)
=
polynomial
as a
most
of the matrix element in the
dependence
of at most
can
be
degree
important operators 0
=
seen
2 in
=
from Table
exponential
J(fvr
I and
P
and
-
S).
%,...'t') 7.1, they
to do
appear
functions for the
The matrix element
for the Idnetic: energy has an additional factor which is again quadratic. The basic elements of the t and t' dependence include ti-tj, -ej--ej, ti.-ej,
and
S-ti, S-tj. As
the differentiation
the Cartesian coordinates no cross as
tip
and
can
%
,
be made
and
as
separately in each of
the basic elements have
terms in different directions of the Cartesian coordinates such
tiltj2,
x, y, and
possible
it is z
to do the needed differentiation in each of the
components.
The above considerations reduce the
operation prescribed in Eq. problem given in Exercise A.5. The for(A.44) mula developed in Exercise A.5 can be applied in each of the x, y and z components to complete the operations needed in Eq. (A.44). It is not difficult to extend the formula given in Exercise A.5 to get the matrix element for the Idnetic energy which has an additional polynomial of degree 2. In fact, Eq. (A.169) can be generalized to a relation which is valid for a general function: to the mathematical
( cNf (X1
,X2,...,
XK))
xl=o,---,XK=O
AA
Spin
matrix elements
263
N
(W#-N f (AA AA N!
f (AlZi A2Z,
27ri
A.4
Spin
AK
...
'O))'&=O i
AKZ)
(A.45)
dz.
ZN+I
IZ1=1
matrix elements
The interaction may depend not only on position coordinates but also on intrinsic degrees of freedom such as spin and flavor.. By represent-
ing the single-particle spin function by Im element is given by
(MI I o't, I m)
=
.1
V_3 (.12 m I IL 1
The matrix elements for
2
m!)
=
1/2),
a
basic matrix
(A.46)
-
spin operators, (S'AF 10,,,,, (spin) I SM),
axe
by using the Wigner-Eckaxt theorem (A-12). Here a usually reduced matrix element which is needed to evaluate the matrix element evaluated
is the
one
for Pauli matrix and it is
o-
2
given by
(A.47)
2
spin function discussed in Sect. 6.4 no technique involving angular momentum algebra is needed to obtain the spin matrix element. An elementary, direct method enables one to calculate the matrix element. We list the spin matrix element that appears in evaluating the nucleon-nucleon potential energy. For the
(Tr l M12 1 0"1 X
*
a2
I M1 M2 )
=
3(-1)M'1"_M1jMI+M27M/j+Mf2
("2 MI I MI1-?nl I "2 Tdi) ("2 M2 1 MI2 --M2
X a212m I MI M2 (MIMf 2 1 [01 1
("2 MI I Tnf1-MI
X
(1 R I_Ml "rnf2--M2 12 m).
X (Ul (M /M-I 2 I [r 1
+
2
a2)] 1M IM1 M2
Mf2
(A.48)
JM,MII+M2I-MI-M2
Tn 1) (-1 M2 1 Tnf2-M2
X
2
3
I 2
2
MI2
(A.49)
V3
E E qj=-I q2=-l
r
qj.
Appendix
264
IJ,,,,,n, (' 2
(I q,
I q2
2
ml_ 1
q2I
" 2
M) 1
+
Jmjm,('I2 M2 1 q2I .12 m) 2 1
I I m).
(A-50)
(IMIMI 1 2 i(T*0'2)(al)m+(r*0"1)(0'2)-raiMlM2)
f (-l)M2-M2 JM?M'j-MIrM2-M2
3
+
(-I)M,
1
-MI
JM7M2-M27MI-M'I'
(I2 mi I m---7n, 1
x
(al M12 I (r'0'1) (r 0*2) *
X
*1 2
Tn/1) (I2 M2 I m'--Tn2 2
MI
Tn2)
=
3
1 7n/1 -'rnl rMl-M"1 rM2-M'2 (I 2 TnI
2
7nf2)
(A-51)
(-J)MI-MI+M2-M2 2
Mf1 )(1
2
Tn2 1
TnI--Tn2j 2
I 2
MI)2
(A.52)
(al M2 3
[T X O'll I
[r X O'21112m IMI M2
X
(.12 mi 1 m--7n,
2
7nfl) ("2 M2 I Tnf2--Tn2
2
7nf2
7'ql 7q2(l qj+m'j-mj 1 q2+rn"2 -M2
X
12 m)
ql.=-l q2=-l x
(I q, I m'j-m,
ql+TWI--rni) (I q2 I m'2--M2
q2+m/2-M2) (A.53)
Here
r
is
a
3-dimensional vector and rn stands for its
spherical
com-
ponent Matrix elements for the isospin part same manner.
can
be obtained in
exactly the
A.5
with
Three-body problem
A.5
Three-body problem spin-orbit forces The matrix element for
tensor and
central,
with
spin-orbit forces
central,
265
tensor and
N-body system can be derived from the previous sections. In order to illusthat there alternative trate exist possibilities, here we use a sligthly different way of calculating the matrix elements for three-body problems with central, tensor and spin-orbit forces. In this approach the radial and angulax integrations are done separately and this involves the extensive use of angular momentum algebra. Needless to say, this approach cannot be easily pushed far. One may try to attempt to derive similar formulas for the four-body case, but to go beyond that general
formulas
in the
to be too tedious.
seems
choice
(i) a
an
presented
of basis functions
The trial function is taken in the form
TfJM,TTMT(123)
C,,,,A p ac(1, 23)
(A.54)
ac
with
expansion coefficients Q, and functions A o,,(I, 23) that are fully antisymmetric under the exchange of particles:
constructed to be
Aw,,, (1, 23)
Wc (1,
23)
oac(l, 32)
(the
construction of
+ W,,, (2, 3 1) + W,,c (3, -
(pac(2, 13)
-
12)
(pac(3, 21),
(A-55)
function is the same, but then the last three terms should have positive sign in the above
equation). function in
totally symmetric
a
The basis function a
Jacobi channel k
stands for
wave
W,,(k,pq) represents a specific (k 1, 2,3) (see Fig. 2.2). The =
wave
nota-
system where the particles p and q axe first (k, pq) connected and then the center-of-mass of the (pq) pair is connected
tion
with the nel, X (k)
particle ==
a
k. Thus the coordinate used in the kth Jacobi chan-
JXk, Ykb
is
given by MpT'p + Tnqrq
Xk
=
Tp
-
rq,
Yk
_
(A-56)
rk.
Mp +rnq Yk I in this section, instead of denote the first and second Jacobi coordinates.)
(Note
that
we use
f Xk
The basis function and isospin parts:
,
o,,,,(k,pq)
is
given by
a
(k)
Ix 1
product
I
X(k)I, 2
to
of space, spin
Appendix
266
'
FL'(kpq)
W,,r (k, pq)
where the
x
X'(k,pq)]jMj?7TtM,(k,pq), S
correlated Gaussian of the
is chosen to be
a
(-2 ;(-k)
Xk
spatial part
(A-57)
form
.FL'ML (k, pq)
=
exp
Ax(k)
2v+,\ 2n+l
Yk
YL' ML ") (iik, Vk) (A.58)
witil
'3
(
A=
62
6
0)
(A.59)
YI (Vk)ILML
(A.60)
-Y
and
L('ML Oik
-
k)
Y
The index
a
=
[YX(ik)
stands for
a
X
set of discrete labels
a
=
(Y, n, A, 1)
and
for the matrix A which contains the nonlinear paxameters 0, -y7 J. Here Y, n 0, 1, 2,..., while A is the orbital angular momentum of =
and q, and I is the orbital momentum between the center-of-mass of the (pq) pair and
the relative motion between the
angulax particle
the
particles p
k.
The index
c
denotes the labels
(L, s, S, t),
where L is the total
angular momentum, S the total spin, and s and t are the spin and the isospin of the (pq) pair, respectively. (The total isospin T
orbital
quantity, so that no T-mbdng is the trial function.) The spin and isospin parts axe thus
is here assumed to be
considered in
a
conserved
given by S
X S M, (k, pq)
=
,qTtMT(k,pq)
=
(k)] sm,
(A.61)
[?7t(pq) xqi(k)1TMT,
(A-62)
[X, (pq)
x
Xi 2
2
where Xsm (pq)
,qtm (pq)
[Xi (p)
x
X.1 (q)
2
[77.1 (p) 2
2
x
(A-63)
q.1 (q)jtm. 2
The function o,,, (k, qp) is obtained from p,,, (k, pq) by changing the sign of xk and the order of couplings in the spin and isospin functions. The sign change of xk results multiplication by the phase factor functions change as follows:
in
a
change of J to -J and spin and isospin
and the
A.5
Three-body problem with central,
XsrrL(qp)
(-1)1-'X ,,,(pq),
iltm(qp)
(-1)1-lqtm(pq). p,,,(k, qp)
Hence the function the
sign
of 6 and
267
spin-orbit forces
tensor and
(A.64)
V,,,(k,pq) by changing
is obtained from
by multiplying by
the.
phase
factor
(-I),\+'+t.
The kinetic energy operator with the center-of-mass kinetic. energy subtracted can. be written in terms of the relative coordinates as 3
2
0
0
A
-Tein
Trel
'k
k)
2rni
(k)
Yk
(A.65)
=Tk,
21j,2
where
(k) III
This
Mprn,
(Mp + Mq)Mk
(k) 2
mp + ni., +
Mp +Mq
(A-66)
Mk
is valid in any of the kth Jacobi coordinate set.
expression
basis
(ii) transformation of the
function
We, have to calculate the matrix elements between the basis functions To expressed in terms of different coordinate sets (1,23), (2,31), -
achieve this
one
has to transform the basis function from
coordinate set to another. This
can
be
one
..
Jacobi
accomplished by using
the
transformation Xk
Yk
)
T (kq)
where the matrices
(
X
(A.67)
q
Yq
connecting
the different Jacobi sets
U12
Tn2+rn3
T (21)
=
MS(7nl+M2+M3) (M2+M3)(tnl+M3)
MI
Mj+M3
M3
7n2+Tn3
T (31)
M2(Ml+M2+M3)
M1
(1712+M3)(M1+M2)
Tn1+M2
M3 __
tn I
I
+rn:3
T (32)
mj(mj+Yn2+Tn.3) (tnj+mj)(Tnj+tn2)
M2 "Ll
+TrL2
are
given by
268
Appendix MI -
M3+Mj
T(12)
TrIS(Ml+M2+M3) (7n3+7nI)(M2+7n3)
7
M2
M2+M3
MI
MI+M2
T(13) 1112(IILI +IIL2'1-11L.1)
?IL'j
(7rII+M2)(?tL2+7J'L.3)
"12 +7,11.3
_M2
MI+M2
T (23)
(A.68) MI(MI+M2+M3) (Tnl+Tn2)(M3+Ml)
M.4
M3+?nI
The space function TLc'M,, (k, pq), expressed in the kth Jacobi coordinate set, can be transformed to the one expressed in terms of the
qth Jacobi
coordinate set
as
follows:
TL'M (k, pq)=1: L3(kq) (ad),TLagr (q, kp), L 11
(A.69)
,
ry
where
,
MI,
(q, kp)
=
exp
(_2
)
and where a stands for 2P +
+ 2h +
The matrix A is
(A.67)
F
=
a
set
X
2V+X q
Yq2fz+Ty(Al) LM ,
2v + A + 2n + 1.
uniquely
(A.70)
(T/, ii, , 1), which has to satisfy the relation
(A.71)
determined from A
by
the
transformation
as
j_(kq) AT(kq) The the
Q, (X q, Yq
6(kq) (ad) L
in
(A.72) Eq. (A.69)
is the transformation coefficient of
polynomial part: 2v+.k X
k
Yk2n+Iy(AI) LML, (Xki Yk
EB(kq)(ad)X2F/+XY2Ft+Ty(XO (XV Yq -
=
L
q
LML,
(A.73)
a
To get this expression one has to make. use of Eq. (6.7) and to recouple, the product of the two angular functions (see Complement 6.3) by
A.5
with
Three-body problem
[y(1112) (:j,
)
12
X
central,
tensor and
spin-orbit
forces
269
y(1314-) (:j, )JLM 134
Ef 1121121314-134L y(113124) (:j LM
(A.74)
113124
113124
where the coefficient E is ficient C of
Eq. (6.11)
given by
a
unitary 9j symbol and the coef-
as
11 13 113
El'- 121121314134L 113124
12 14 124
112 134
Q11 13; 113) C(12 14; 124) (A-75) -
L
Note that the coefficient E may be defined
by the reduced
matrix
element -
-
121121314134L N 2L T 1 Ell 113124
(k '/2L + I (y(113124) LM
y(1112) (60, ) I 712
C
X
y(131A) I
)JLM)
,34
=
C ' ) 11 y(1314)( (-j)112+134-L(y(113124)(XA' ) Y(1112)(:j 134 112
=
(-1)13+lA (y(1112) (:j, 1
L
y(1314) ( C'
y(113124) L
12
34-
(A.76) which is
(YI
easily verified by using Eq. (A.147)
" 1) v/2k + I QW; k)
Y1,
Yk
and the relation
V21 + I C(kl'; 1). The transformation coefficient
(kq) BL (aa)
B(kq) (Cla) L
(A-77)
is then
expressed
as
nI &I- X2,XI1121L D"X D V1XIV2-X2 nj1jn212
X[
vl"*-1"2-"2 njIjn2I2
X
X
T(kq))21/1-+Ai(,p(kq) -
22
where the coefficient D is is constrained
)
kq) 12
21/2+1\2
(T(kq))
2n1+11
21
2n2+12
(A.78)
I
given in Eq. (6.8) and where the summation
by the conditions
2vl_ + A, + 21/2 + A2
=
27/ +
A,
2n, + 11 + 2n2 +12
=
2n +
11
Appendix
270
2vi + A, + 2n, + 11
=
X,
2 i+
2v2 + A2 + 2n2 +12
=
2Tz +
17
(A.79) and
the
triangular inequality for the angular momentum among (/XI7'X2iA)7 (11712 07 (Al 117 X) and (A2 12 1), respectively. The transformation to the pth Jacobi coordinate set can be per-
by
7
formed
well in
as
a
similar
The spin part of the
1
.
7
manner.
wave
function transforms
as
(-l)'21+'-SU(.! ISI; gs)X SIVIS (q, k-P)
X8S Ms (k, pq)
22
E(-I)'!+-SU(.!
-IS-!; 2 sft S Ms (p, qk).
2
The transformation of the
2
22
isospin part
is
given
in
(A.80)
exactly the
same
way-
The radial and
However,
no
such
appears in the
integration
angular integrations are separated in this approach. separation is made hi the factor e-6xk-yk which
exponent of the has to
one
function.FLIM,: (k, pq).
expand this
term into
Therefore in the
partial
waves
(see Eq.
(6.53)): ,-wx-Y
where
ii(x)
(6.48)
and
radial paxt
=
v"2-1-+I e(l, w) il (I w I xy) 01) 100 (--
4r
(A.81)
7
is the modified
E(I, w) one
=
I for
encounters
spherical Bessel. function of the first kind w rel="nofollow"> 0, 6(1, w) (-I)' for w < 0. In the of the form an integral =
cc
fo
I(n, 1, v, I w
2n+1+2
Y
Ivy 2
e-2
r-(2n)!! w I' -2 vn+l+a2 where
n
guerre
polynomial (10.15).
is
a
eXp
ij(jwjy)dy
(W2 ) L('+'!) ( W2) 2
n
2v
non-negative integer and
2v
Ln(')(x)
We define the
(A.82)
is the associated La-
following integral
I(v, n, 1, u, v, IwI; V) 0
fo JO
CO
X
2v+1+2
2n+1+2
Y
1
e-
UX2
_
IvY2
V(x)il(lwlxy)dxdy
Three-body problein
A.5
with
central,
tensor and
spin-orbit
forces
271
00
1
x
2u+1+2
IUx2 F
--
e
(A.83)
V(x)l(n, 1, v, lwlx)dx.
0
The one-dimensional
integration
the functional form of
ated. In the
special
V(x),
case
can
when
be
V(x)
I(v, n, 1, u, v, IwI; V=1) n
X
equation, depending on analytically or numerically evalu-
in the above
=
1
we
obtain
(2n)!!F (n + I +
8
V
F(k + U + + 1) 2 kl.(n k)!F(k + I +
2v
w2)
2 2
where F is the Gamma function
given
in
we axe
(k I q)
,
(A.84)
Eq. (7.8).
ready
to calculate the matrix elements. The
tween the functions in different Jacobi sets k and q
separately
k+v+l+A2
matrix elements
(iii)overlap Now
n+l+ 2
(w2)k
-
k=O
3
2
in the space,
=
spin and
can
overlap be-
be calculated
isospin parts:
( pc(k, pq) I Wycl (q, kp)) "I" 0
=
X
JLLI JSSI (.FL'ML (k, pq) I )7LMI (q, kp))
(X'SMS (k,pq)IX" SMS (q,kp))(,qtTm,(k,pq)lqt
m,(q,kp))(A.85)
The
spin part
is
easily obtained by using Eq. (A.80)
(X'SMS (k, pq) I X" SMS (q, kp))
=
(- 1)
" +'-S 2
as
.1 S U (.1 .1; 2 22
s's) -(A.86)
Likewise, the isospin part becomes
(,qtTM, (k, pq) lqt'TMT (q, kp)) The
ing
integration
=
( _j)-!+t-TU(I22ITI; 2 tt). 2
of the space part
the function in the bra to the
performed by transformdepending on the qth Jacobi
can
one
be
By substituting Eqs. (A.69) and (A-70) we obtain (the label the coordinate set is suppressed in the integral)
coordinate.
suffix q to
(A.87)
(-'FL'ML (k, pq) I
"
(q, kp))
=
E L3(kq) (aa) L
272
Appendix
X
if
x
2P+X+2Y'+)L" 2fz+[+2n'+I'- 2IUX2_lVy2_WX.,y e
y
(Y'X) oi, m) *Y(""/
X
LML
dx dy
LML
(A.88)
with
U= +o
=!+-t
V
The
angular integration
and
(A.74)
(YL(
easily by
(A-89) the
use
of
Eqs. (A.81)
can
ML
( C' m
d
L ML
_X-+ +I1,E(r w) Ety.OX'1"LL i.(Iwlxy). V2_x
4w
One
be done
=S+j'.
as
e- wX-Y
and then
can
w
(A.90)
easily verify that 1+1'-r, has to be even and non-negative, the integral formula (A.84) to obtain
one can use
(.FL'ML (k, pq) I
"' -1
LML
(q, kp))
4y"13(kq) Ir
,
X
I
X + 16 (M' W) EX"jj'-0,X'1 K Vf2-1
L
(P+vl+
"
LL
A+AI-K 2
2
'K,U, V,
IWI;
V=I) (A.91)
(iv) matrix elements of kinetic energy operator The next step is the calculation of the matrix elements of the kinetic energy operator
(k ITrel I q)
=
(W,, , (k, pq) ITq I W& d (q, kp)).
In the above
erator
can
(A.65)),
expression be expressed
and thus
we
(A.92)
used the fact that the kinetic energy opin different relative coordinate sets (see Eq.
we
defined Tr,,l in
conformity
with the coordinate
set in the ket. To calculate the matrix elements of the kinetic energy
operator
one
(X
first notes that
2v+.X
2n+l
Y
e-.21 ,6.2_:I_YY2_,E._y YLM
A.5
with
Three-body problem
30X2 X X
-2(
x
central,
02X4 + j2X2Y2
+
21/+)L-2
2n+1
Y
e
x
20(2v + A)X2 LM
e
1))
4 M 7F
13X2)
I
LM
+
+I
(2v
-
Ox 2)
x
proof of this
2Y(2z/ + 2/X +
EU(A-I1Ll;AK)C(11;K)Y('X-I
-V2A A
+
273
F3
1, _.L,3X2_ YY2_,5x.y 2
(2v + 2A +
X
-
spin-orbit forces
2_1 --Ox ilyy 2_,SX.y Y. (,Xj) 2
2v+X-1 2n+l+l
Y
tensor and
(A.93)
LM
relation is
given
in Exercise A.6. A similar
expression
holds for the second relative coordinate y. By the help of the above equation the effect of the kinetic energy operator applied on the basis function is
given by
a
linear combination of terms akin to the basis
function. The matrix element of the kinetic energy can therefore be given by repeatedly using the expression obtained for the overlap of the basis function.
(v) matTix
elements
The central V
=
=
of central potentials two-body interaction can be
V(I'r2
V(Xl-)
that is, the
7'3
-
+
1)
V(Irl
+
V(X2)
potential
is
+
-
T3
1)
+
V(jrI
as
-r2
1)
V(X3)
a sum
of terms
coordinate in different Jacobi sets. In for the
written
(A.94) depending on the first relative calculating the matrix element
case
(k I V(xp) I q)
=
( O,, (k, pq) I V(xp) I Wa,,, (q, kp)) -
JLLIJSSI(-'r-7LaML (k, pq) I V(xp) I.F&L (q, kp)) X
(X'SMS (k,pq)IX" SMS (q,kp))(TItTm,(k,pq)lqt
m,(q,kp)),
Appendix
274
(A.95) the
spin and isospin paxts
can
be calculated
as
before but the
spatial pth
functions in the bra and the ket have to be transformed into the
Jacobi coordinate set in which the potential is defined. The space paxt of the matrix element is calculated transformations k -+ p and q --+ p:
by using
the
"'
(FL'M,,(k,pq)IV(xp)l J,w, (q,kp)) L3(qp) (ala/) L
B(kp) (aa) L
(-'FL'M.,,(p,qk)IV(xp)IFL'
x
Here the matrices
corresponding
(A.96)
(p, qk)).
to the transformations
axe
defined
by T(kp) AT(kp)
A'
T(qP) A!T(qP)
=
(A.97)
respectively. By introducing +
U
one
+7
V
j+
W
(A.98)
&I
obtains
(.FL5'ML (p, q k) I V (xp) 1.FL 4 jr
X
IL
(p, q k))
/-2-r.+1,E(n, w) Et nOXI'P'LL
I(P+Vf+
+
ii+?P+ 2
2
K) U, V,
IWI;
V)
-
(A.99) (vi) matrix
elements
of tensor potentials through
The tensor interaction is defined
SP
2 /-4r
V -5 (Y2 (ip ) [O'q '
X
Crk] 2)
-
the tensor operator
(11.7) (A.100)
A.5
Three-body problem
with
central,
tensor and
275
spin-orbit forces
To calculate the matrix element of the tensor interaction
one
has to
same steps as for the central interaction, except that now part of the wave function has to be taken into account, and the
follow the the
spin spin part should as
the
one
potential.
also be transformed into the
The
use
of the
coordinate system theorem (A. 14) enables
same
Wigner-Eckart
to obtain
(kjV(xp)Spjq)
-=
(W,,(k,pq) IV(xp)Spl W,,,e(q, kp))
U(L2JS; LS) ',E47 VF(-2L+ 1) (2S' 1) 247r 7r r 5
+
x
II (,)7L' (k, pq) II V(xp) Y2 (x-) P
x
(X'S (k, pq) II [Cq
x
(nTtMT(k, pq) Iq tm, (q, kp)).
X
FL',f (q, kp))
O'kj 2 11 X'S/ (q, kp))
(A.101)
spatial part of the matrix element is calculated as before by transforming the basis functions to the pth Jacobi coordinate set Here the
(Jc'L'(k,pq) 11 V(xp)Y2( p-) Ij.FL,(q,kp))
13() (a& ) L'
13(kp) (ad) L
x
II FL',f (p, qk)), (YL' (p, qk) I I V(xp) Y2 (x-) P
where the last term is obtained
(JC L"(p, qk)
V( Xp) Y2 (Xp) ip
4ir
X
by using Eqs. (A.81) i
(A.102) and
(A.82)
as
(p, qk))
V2- r, + 1 c (r., w)
(YL' 1)
YO
0
Y2 (-
O)
YL
ii+?P+
X
2
2
r" U, V,
IWI;
V) (A.103)
Here u, v, w are defined in even and non-negative.
Eq. (A.98).
Note that
f+
r,
has to be
276
Appendix The reduced matrix element of the
using
6.3)
angular paxt is calculated by recoupling technique (see Complement
the
and
angular momentum Eq. (A.76). One obtains
YO (I
( Y.,L
K
YL(XI F) ( c
Y2 ( 0)
I
U(2n2r,; WO) Q2K; W) Kf
X
(YL
Y2
01
Ll
/2r,'+ 1
EV
2K+I
C(2r.; K)EX LXII,-'L2.
(A.104)
The
spin matrix element is also easily calculated by transforming the spin functions to the pth Jacobi system and by using Eqs. (A.147)
(A. 149)
and
as
(X'S (k, pq) 11 [Crq =
X
Uk]2 11 X" S" (q, kp))
E(-l).2k+g-SU(-nS-1; S) E(-1)-!+""-S'U(.! 2
22
X
(X S (p, qk)
[G'q
X
2
Ok] 2 11
22
XS P, (p, qk)),
2
(A.105)
where
(X S (p, qk) 11 [Uq
X
Uk12 11
XS P, (p, qk))
(2S + 1
(_I)s,,-St+s-g V X
(("221) 9 11 [O'q
X
29+1
U(Ii S2; S'9) 2
UkI 2 11 ("22'0 i
(A.106)
with
(('1'1)9 11 [O'q 22
X
4+ (vii) matrix The
Uk12 11 1
elements
spin-orbit force
22
2
2
1
1
1
1
2
2
8
2
(,V6) 2 =2v/_5JgjJ pj.
9
of spin-orbit potentials given through the operator (11.9)
is
(A.107)
277
A.5 Three-body problem with central, tensor and spin-orbit forces I
with
LV Sp -
As in the
Lp
=
-ixp
Sp
Vp,
=
2
(0'q+ffk)- (A.108)
of the tensor interaction, with the
case
Eckart theorem
(A.14)
(kjV(xp)Lp-Spjq)
one
_=
use
of the
Wigner-
obtains
( o,,,(k,pq) IV(xp)Lp-Spl W,,,,(q, kp))
U(L'lJS; LS)
r
+ 1) + -1)(2S' V-(2L x
x
(FE'(k,pq) 11 V(xp)Lp 11 FL, (q, kp))
(X'S (k, pq) I I Sp I I X'S, (q, kp)) (,qtTMT (k, pq) lqt'TMT (q, kp)).
(A.109) spatial part of the matrix element is calculated as before by transforming the basis functions to the pth Jacobi coordinate set
The
V (xp) Lp
(.FL' (k, p q)
L3(qp) (af&i) LI
L3L(kp) (aa) x
YL, (q, kp))
(.FL5 (p, qk) 11 V(xp)Lp 11 Jc'L",(p, qk)).
When the orbital
angular
momentum
(A.110)
operator
acts
on
the basis
(provided both the basis function and the operator in the same coordinate system) one obtains
function
pressed L
(x
21/+X 2n+1
Y
ij(x
X
A0 e--21- '3X2 _!IY _IYI _,5X_y YL(M
Y)X" +A Y2n+l e_21,3x
2v+,\ 2n+1
+X
Y
Note that the outer
vf2-[x
x
yll_,,
=
axe ex-
1)3X2
1 _
e- 2
'f
YY2
product i(x
X
(4-\,F2-x13)xyYj(,',1)
2_1
'YY
2_dX.y
YLI'Xj) (ic M
A0 -6"YLYL(M
y),, ).
is
equal
(A.111) to the tensor
Therefore the
product
spatial part
of
tained in
element, (.FL6(p, qk) 11 V(xp)L, 11 J:L` ,(p, qk)), can be obexactly the same way as in the central potential. It is given
by
of two terms: One is
the matrix
a sum
4vf2-7 J
47r 3
X
vf2-_K+I c (K, w)
(YL` r) ( C' m I I YO(Kn) ( C' m YI(I-I-) ( C' m I I YL(" F") (:t' o
278
Appendix
(P+1/1+
X1
n+W+ 2
K) U, V,
2
IWI;
V)
(A.112) and the other is
v/2 r. + 1 E (r,, w) (YL(
4-,T
YO(MM) (i
,
Fj+P+
X
fi+??+ 2
L
KI U1 V,
2
II
YL(fk'[')
IWI; V (A.113)
Here u, v,
defined in
Eq. (A.98). Note that in the first term and non-negative, and likewise in the second term I + 11 r. has to be even and non-negative. The angulax part in the first term can be obtained by using Eqs.
T+ P
+ 1
w
-
axe
r,
has to be
even
-
(A.74)
(A.76)
and
as
YO( MIS)
(YL
YI(l 1)
YL
(A.114)
pq
pq
The
angular part
(YL(
( b,
in the second term becomes
YO(K n)
YL(',"
L
i, L
L
YL(, "")
(A.115)
where
(YL(X"")
L
(_,)L-L" V
YLI 2-L+ 1
U
2XI +I
L 1; L'XI) (I
Xi,
i
L
YX, ( b)
(A.116) with
(Yl(;k) 11
L
11 Yl(.-b))
=
0(1 + 1)(21 + 1).
The spin matrix element action. The result is
can
be calculated
(A.117) as
in the tensor inter-
A.5
Three-body problem with central,
(x'S (k, pq) I I Sp I I
tensor and
spin-orbit
X'S'/ (q, kp))
-21+-SU('.!S.1; Sg) E(-I) 12+S'-S'U(.1 22
1
22
2
x
279
forces
(X S (p, qk) 11 Sv 11 X S-', (p, qk)),
S' 1;
(A.118)
where
(X'S (p, qk) I I Sp II X'S'f (p, qk)) S+I
S+IU (Iksi; Sg) =29+
-S,+S-
X
2
1
2
(A.119)
(("I22'1) 9O'q + Uk I I (I22'I) k)
with + O'k ((*"1) 22 9O'q
22
R)
8+1 E: 122 U(-! -191; '9
2+
206-JgJi j.
06-1) 2
((- 1)
-j;
+
1) (A.120)
280
Complements
Complements A. I Mabix elements
central
o
potentials
The matrix element of the central and tensor
potentials
can
be ob-
tained
through a step which is slightly different from the procedure presented in Appendix A. 1.3. Then the formula becomes simpler than Eq. (A.21). Here we outline the step which leads to this simpler result. Instead of using Eq. (A.19), we first perform the integration over the angle i6 in Eq. (A. 17), which leads to Tj I r2
(fK'LIMI (7Z, X, X) [
r)
YxIL ( ri
-
rj ) I fKLM (u,
A, x))
3
47r
(2r)N-2
BKLBKILI
detB
_.LCT2 e
X
x
[
A2
+
q/A/2 +
c) if
d& d
'
d2K+L+2K+L"
YLMWYLIMfO )*
exp
2
dA2K+LdA/2K'+Ll
AA/e. et] in (rV)Y"'j#Y)
(A.121)
Wi0 V
=
-tAe + -Y'Ye'.
(A.122)
If either -y or -y' vanishes, v is independent of the scalar product e-e', and the integration and the differentiation prescribed in Eq. (A.121) becomes very easy. When neither -y nor 2 exponent can be expressed in terms of v
qA2 + qiA/2 + pAA'e P
2 V
+
2-y-y' One first has to
-
-y'
is zero, the term in the
as
e'
(q_ 2-y
expand
A2
+
(qf
A 2) i,,,(rv) exp(2-y,,1V
A/2. 2-/
in terms of powers of
(see Eq. (6.48) for the expansion). Secondly, multiplied by the spherical haxmonic the
coupled
(A.123)
its
general term v expanded
must be
v
2n+n
in
The second step can [YL ( -_) x YL, (,E )] In this derivation one can avoid Eq. (6.7).
tensor form of
be done with the
use
of
recoupling the angular momenta, and hence would get the simplest possible formula for the matrix element.
CA. 1 Matrix elements of central potentials In the
with
x
make
following we work out the details for the central potential However, we shall not follow the above route exactly but
0.
=
a
281
little detour in order to make transparent the relation to the To this end we rewrite the left-hand side of
overlap matrix element. Eq. (A.123) as follows:
A2 + I A/2
+
pAA/ e. e/
qX2 + q'A12 + PAA/ e. e
=
2c
V2. (A.124)
V2
The term Hermite
2c
e
jo (VV) is expanded in
powers of
v
with the
use
of the
polynomials: V
V
2c)
exp
-
i o (r v)
=_
2c)
exp
1
-
2rv
(e'r
V
-rv)
e
-
00
V22 r) (V2c2)n. C
,,F2c Next
we
expand
exp q A2 +
q' A/2
n=0
exp
+
(2n + 1)!
H2n+l
c,
(qA2 + q'A/2 + pAAI e. ef)
p,\A'e
H(p -ra, qA2)H(pI
e'
-
-
) (2c V2)n -
2c
as
follows:
ni _
(2
n , q'A/2 )H(I
H(M' 2A2 )H(a, 7/2,X/2 )H(n
(,,2,)n
(A.125)
m
PPIIMMI
-
-
n
+
+
m
m', pAA'e-e)
a, 27-yAYe-e)
1: FPnPj (c, q, q, P, t,,Yf),X2p+'A/2p'+l (e-e')'
(A.126)
PPfj
with n!
Fpnp,,(c, q, q, p, 7,7')
1: H(p
=
-
m,
q)H(p
I -
M
Iq
MW
7
x
Here p, p, I,m, and m' are all of their values are determined m <
is
p, ne <
now
n+m-nz!
H(I -n+m+m',p) 2m+m'm!
(fK' L' MI (Uf, X) X) I V(I Vi
(n
-
m
-
ne)!'
(A.127)
non-negative integers and the
ranges
the conditions p + p' + I > n and + m! < n. The operation in Eq. (A. 121)
p', n I < m easily done, leading to -
ml 1.
in-m+m'
7
by
the result
-
'rj 1) 1 fKLM (U, A, X))
282
Complements
(2K + L)! (2K'
JLLIJMM'
+
L)!
BKLBK-'L
K+K+L
2
(2
E
detB
J(n, c)
n=O
min(K,K")
E
X
Fk7-k KI-k L+2k (c, q, q
A'Yi
^//) BkL
(A.128)
i
k=O
where
J(n, c)
is the
integral defined by I
J(n, c)
= -
-V/7-r-(2n + 1)! "0
f V(jx)e-X2HI (X)H2n+l(X)dX-
x
In the
overlap
case
of
V(r)
=
matrix element
comes nonzero
(A.129)
C
0
if and
I the formula
(A.6),
only if n
(A. 128)
reduces to that of the
because then the
=
0 and
integral J(n, c)
be-
FF2k Kf -k L+2k (c, q, q, p, -Y, -y')
k, q)H(K' k, q')H(L + 2k, p). Though this relationship is not apparent in Eq. (A.21), both equations should give the same result for the central potential. For the special case with K K' 0, Eq. (A.128) reduces to a much simpler form which essentially includes a double sum: reduces to
H(K
-
-
=
(fOLM (U =
I
=
A! 7 X) I V(ITi -rj 1) 1 fOLM (u, A, x)
(fOLM(W,X,X)jfOLM(u,A,x)) L
X
1
D
77
n
E (L-n)! ( cp )
J(n, c),
(A.130)
n=O
where the
overlap (fOLM(UIiWiX)jfOLM(u,Ax)) is given by Eq. a particularly nice feature for power (A.7). k law potentials V(r) because then the c-dependence of J(n, c), r The above formula has =
,
now
denoted Jk (n,
c),
is factored out
as
follows:
k
Jk (n C)
=
(2)
2
Vir(2n + 1)!
C
(2) C
00
-2
k 2
n
1
7,=, E
M=0
fo
X
k+I
ai2n+l
( dX2n+1
2% -X
-e
(-l)m22n-2m+l mI(2n-2m+I)!
-V
n-m
+
dx
k+3). 2
(A.131)
CA.2 Matrix elements of
283
density multipoles
j are contained only in c and -y-y'lcp, this factorization makes the summing up of Eq. (A.130) over i and j much faster. Especially for the Coulomb force (k -1), we get Since the
indices i and
particle
=
2c
J-1- (n, c)
(-l)n (2n + 1) n!
(A-132)
-
-x
potentials the route given below Eq. (A.123) leads following simpler result including only a single sum:
For other types of us
to the
(f0LM(UIiWiX)jV(jri
-
rjj)jf0LM(UiA7X))
(2L + 1)!!
(27r)N-1
4v
detB
L
)3 1: 2
n=O
I
L!
n!(L
77
-
n)!
n L-n
c
00
2n+2 X
V'r(2n + 1)!!
10 V(eX) X2n+2
e-'2dx.
(A.133)
C
particular simplicity occurs for a Gaussian potential as e-'2, we obtain perform the n sum easily. For V(r) A
one can
then
=
(ALM (U X X) I eXP I_ X(r, i
i
_,rj)21 IfOLM(U7 Ai X)) 3
(2L + 1)!!
(21r)N-1,
4v
detB
3
f
-I
)2
+
2r,) (P C
L
7,y + C
i+
(A.134) of density multipoles Here we Eq. (A.30) takes a simple form for the special case K' 0. As we have seen in the chapters of applications, the of K 0 is, with the use of its fall general A correlated Gaussian with K matrix and u vector, a very useful basis function. The simplified form A.2 MatTix elements show that
=
=
=
of the matrix elements
was
in fact used in
our
calculations. TrI the
and
case
in order to have
even 0, L nj n2 has to be non-negative the term from contribution -n2) /2, q). H((2K+L-ni non-vanishing the other On and even. Likewise Lf n3 has to be non-negative nj of Eq. (A.20) becomes nonhand, the recoupling coefficient Rnin2n3 LLIr.
of K
=
-
-
a
-
-
L' and n2 + n3 K vanishing provided that ni + n2 L, n, + n3 are all non-negative and even. Therefore we are led to. the result that L the summation in Eq. (A.30) is restricted to the case of nj + n2 L n and L. By renaming nj as n, we obtain n2 and nj + n3 -
-
-
=
=
=
-
284
n3
Complements
=
L'
n, where
-
from 0 to
n runs
min(L, L').
limited
by the condition that n2 + n3 non-negative and even and also K > max(IL are
the
-
triangular relation
n
=
-
Possible values of L + L'
LJ, IM
2n
-
-
of the Clebsch-Gordan coefficient
-
is
is
be
M'j) (L M K AV-
due to
MIL'IW). Furthermore, L-n L-n RLL'r. n
n2
=
L
11 :!
-
12
note that under these conditions
we
is contributed
13
n,
n3
=
by
=
single
a
L'
-
n
in
term with
11
=
RnLjn2n3 L-lis
ni
Eq. (A.20). This
=
n,
12
=
=
is because
13 :5 n3 and 11 + 12 > L, 11 + 13 ! L. Using the ni, well-known formulas for the Clebsch-Gordan coefficients (10 P 0 1 L 0) < n2,
and the Racah coefficient L-n L r,; L'
U(n
[751
L-n)
(2L 2n+ 1)! (2L' 2n)! (L +L' is) 1 (L +L' + n + 1) 1 (2L + 1)! (2L')! (L + L' 2n is) 1 (L +L' 2n + r, + 1)! -
-
-
-
-
-
(A-135) we
obtain the
following
result:
(fOLIMI'(Ul X, X)IJ(17VX -r)jfOLM(U, A, X) (2T) M-2
(
=
min(L,Ll)
2
3
2
1
E pnL-nI/ IL-nrL+L'-2n
cl e-'Ycr
detB
n=O
L+L'-2n
1:
x
ZLL'nn (L M r, M
-MJL'M)
r.=max(jL-L'J, IM-M,'I) x
where
r.
Y. M'
--M
('0 *'
(A-136)
+ L + L' has to be
ZLLfnr.
-
(-1)
2
even
(L-L'+x)
and the coefficient
F 2L
ZLL'nr.
is
given by
(2 +1) 47r
T L, + L' + -+I)! JL-K)!(L K
-
x
(2n)!! (L + V
F L-(LL --+L"5; L
x
-
2n
-
n)!! (L + L'
-1i ((L ---LL++-'KS
-
2n +
x
+
1)!! (A-137)
CA.3 Overlap matrix elements for
With one can
an
three-particle system
a
appropriate choice of the (N
-
I)-dimensional
285
vector
w
calculate various matrix elements. To calculate the matrix
J(ri
element of
-
xN
-
r),
one
only needs
to set
w
equal
to
w(') which
Eq. (2.12). The values of c, p, -y, -/' are obtained from Eq. w(ij) with being replaced by 0). In exactly the same manner,
is defined in
(A.18)
one can
also calculate the matrix element of J(ri -rj -r)
the vector
w('j),
defined in
Eq. (2.13),
for
w.
By using
by employing
the well-known
formula
J(iv-x -,r)
J(Ifvxl
-
r)
Y"11
r2 r.
WX
(A.138)
Y"tt
it
Eq. (A.136) immediately enables one to obtain the matrix element of the density multipole operator J(ji7vxj r)Y,,,,(w` _x). The matrix element for the central potential, V(j ri -rj 1), is obtainable from Eq. (A. 136) as well. By integrating the equation multiplied 0 contributes to by V(r) over r, we see that only the term with n the matrix element of the central potential. It is easy to see that the integration leads us to the same result as Eq. (A.133). -
=
Overlap matrix elements of the correlated Gaussians for a threeparticle system We tabulate in Table A.1 results of the calculation for the quantity A.3
/2N-2
61, i9aili (9-rili-mi
exp
( -2IS-Bs)
(A.139)
three-particle system (N 3) assun-Ang li < 2. The vector si is defined by aj(1 2), -2-ri). The calculation was done with i 2, i(l + Mathematica [1841. The quantity apparently has a symmetry property with respect to the interchange of li, mi +-+ 1j, mj. E.g., if one calculates this quantity for a given set of angular momenta, then the quantity corresponding to a set (111 MI 13 M3 12 M2 14 M4) can easily be obtained from it by simply interchanging the suffices 2 and 3 of the B matrix. Note that the table can be used for the four-paxticle system as well if one of the angular momenta is restricted to zero. for the
=
1
,
,
1
1
1
286
Complements
Table A.I. Tabulation of vector si is defined
Eq. (A.139) for
the
three-particle system. The
*Tj 2, i(l +7,i 2), -2-ri).
The matrix B is a 4x4 by ai (1 symmetric matrix and its elements are denoted a B12, b B137 C B34. The tenth to twelfth colunu3.s give the B23, eB241 f B14, d types and the coefficients of terms needed to construct the solution, where each coefficient must be multiplied by a factor given in the ninth column of the corresponding entry. For example, in the case of 11 12 MI M2 -2, 14 2, 13 2, M3 2, M4 -2, the quantity of Eq. (A. 139) is given by 12288 x (3b 2e2 + 3c2d2 + 12bcde). -
=
=
=
=
=
=
=
11
MI
12
M2
=
=
=
13
M3
=
=
=
=
14
M4
-4
f
--
0
0
0
0
1
1
1
-1
0
0
0
0
1
0
1
0 192
f2
0
0
0
0
2
2
2
-2
1
0
0
0
0
2
1
2
-1
-1
0
0
0
0
2
0
2
0
1
32
ef
0
0
1
1
1
1
2
-2
6
0
0
1
1
1
0
2
-1
-3
0
0
1
1
1
-1
2
0
1
0
0
1
0
1
0
2
0
2
0
0
1
0
1
-1
2
1
-1
0
0
1
-1
1
-1
2
2
1
16
I
1
1
0
1
0
1
-1
1
0
1
0
1
0
1
0
af
be
cd
1
1
1
(Continued
on
the next
page.)
CA.3
(Continued
Overlap
matrix elements for
a
three-particle system,
-
from Table A. I.)
1536
def
0
0
2
2
2
0
2
-2
-2
0
0
2
2
2
-1
2
-1
3
0
0
2
1
2
1
2
-2
2
0
0
2
1
2
0
2
-1
-1
0
0
2
0
2
0
2
0
2
256
af2
bef
cdf
I
1
1
2
0
2
-2
-6
-6
1
1
1
2
-1
2
-1
9
9
1
1
0
2
1
2
-2
1
1
1
0
2
0
2
-1
6
1
1
1
-1
2
2
2
-2
-3
1
1
1
-1
2
1
2
-1
3
1
1
1
-1
2
0
2
0
-3
2
2
-2
3
3
-6 -6 3
-1
-1
1
0
1
0
2
1
0
1
0
2
1
2
-1
-3
-3
-3
1
0
1
0
2
0
2
0
3
4
4
1
0
1
-1
2
2
2
-1
1
0
1
-1
2
1
2
0
1
-2
-1
1
-1
2
2
2
0
-1
-1
-1
1
-1
2
1
2
1
1
1
b2e2 acdf
C2d2
3
3
3
12288
a2f2 abef
bede
2
2
2
2
2
-2
2
-2
2
2
2
1
2
-1
2
-2
-3
2
2
2
0
2
0
2
-2
3
2
2
2
0
2
-1
2
-1
2
1
2
1
2
0
2
-2
2
1
2
1
2
-1
2
-1
12
-6 2
2
2
-3
-3
6
-2
4
3
3
3
3
6
4
-5
-5
3
3
3
6
6
6
-2
2
2
1
0
2
2
0
0
2
2
0
0
2
2
-3
-1
0
287
Exercises
288
Exercises
Eq. (A.14).
A.I. Derive
Noting the definition of the scalar product (11.2), expanding the angular momentum coupled ket functions, e.g., Solution.
(LS) JM)
E
=
in
Eqs. (11.1) and
states of bra and
(Lm, SM2 I JM) Lrni) I SM2)
(A.140)
TaIM2
and
using the Wigner-Eckart theorem (A. 12),
((L'S') JM 1
we
have
(0,, (space) 0,,, (spin)) (LS) JM) -
1) 1'(Lm, SM2 I JM) (LW1 SW2 I JM) AMIM2M'MI 1 2
X
-vF2-L'+1 X
The
use
(L' 11 0,,(space)
(SM2K --PISIMf2) (S' 11 0,,(spin) V12--S'+1
of
a
L)
(A.141)
S).
symmetry property of Clebsch-Gordan coefficients +
(SM21S --AISIMI) 2 enables
one
(-1)IS
V
2S+1
(KILSIMf2 I SM2)
to rewrite the above matrix element
((L'S') JIVI 1 (-W
X
=
(A.142)
as
(0,, (space) Q, (spin)) I (LS) JM) -
(V 11 0,,(space) 11 L)(S' 11 0,,(spin) 11 S) + + 1) -1)(2S V/(-2L'
E
(Lm1r,/LJL'm'I)(L'm'IS'm'2JJM)
JL'ra1M2?-afIM2r x
(Lm1SM2JJM)(K.ASITnI2JSTn2)-
(A.143)
sum over A, Tn1 , M2 , MfJ , TnI2 of the product of the four ClebschGordan coefficients is just the recoupling coefficient U(Lr.JS';L"S)
The
(see Eq. (6.67)).
This
complets
the derivation.
Exercises
289
A generalization to the matrix element of a coupled tensor operator be made similarly. Let us calculate the matrix element
can
((L',5 )XM'j [01(space)
x
Ox(spin)j,,, J(LS)JM
(LIMI1 SIM/2 I JM) (Lm, SM2 I JM) (1mAp I rw) MIMMIM2MA 2 1
(Lmj1mjL'm ,f2--L'+
(L'
01(space) 11 L)
(SMALI S/ M2I) (S' V -2-Sl+1
OX(spin) 11 S).
11
The
sum over
?n j, m'2
Tn1 , M2 i M7 A
,
of the
(A-144)
product of the five Clebsch-
Gordan coefficients
(LmISM2 I JM) (1mXMjnv) (LmIlmILal) MIM2MtlMflM'2 X
can
be taken
by
(STn2l\l-tlS'M2)(VM'IS'M'21XM')
means
(JMr,vIYM')
of the
unitary 9j symbol
L
S
J
I
A
K
L'
S'
X
(A.145)
as
(A-146) _
_
multiply Eq. (A.145) by (JMr'VjYAF) and sum over M and v with M' being fixed, then what one has is a sum of the product of six Clebsch-Gordan coefficients involving nine angular momenta, and it is nothing but the 9j symbol in unitary form given in Eq. (A.146) (see Eq. (6.78)). Then with the help of the orthogonality relation of the Clebsch-Gordan coefficients Eq. (A.145) itself has to be equal to (A.146). Substituting this result into Eq. (A.144) and using To derive this result
the definition of the reduced matrix element
((L'S')X 11 [01(space)
x
one
obtains
0,\(spin)l,, 11 (LS)J) L
S
J
2JI + 1
(2LI
+
1) (2SI + 1)
1
A
P
S,
-
x
(L'
01 (space)
L) (S' I I 0,\ (spin)
S).
(A.147)
Exercises
290
In and
special by using
cases one can
(S' 11111 S)
=
set A
=
0
or
1
=
0 in the above formula
JSS, A/2S +I
derive useful formulas
as
(A.148)
given below:
((L'S')J' 11 01(space) 11 (LS)J)
jssl(_I)L-J+J-L' V1-2LI-j,-+1U(SLXl;JL) +I
(L' 11 01(space) 11 L .
x
((L'S')X 11 OX(spin) JLLI
(A.149)
(LS)J)
V TS-,+-, U(LSJA; JS) (S'
Ox (spin)
S).
(A.150) A.2. Derive
Solution.
Eq. (A.19).
Using Eq. (6.18),
we
have
(e. e ) ni (e. r) n2 (ef.,r) n3 rn2 +n3
Y,
Y,
E
BkllBk2l2Bk3l3
2ki+1,,=nj. 2k2+12=n2 2k3+13=n3
(-1)11+12+13-v/'(211
I)T(2172 1) (2K3 +T) [Y11 ( -)
[y12 (' -) Xy12 ('0100 [Y13 0 ) As
was
done in Exercise 6.1
IY12 0
X
Y12 (f9]OO [y13 0 12 13
12 13
0
X
0
X
Y13 ('06'
(see Eq. (6-106)),
)
X
y13 0
0100
(A.151)
we
recouple the product
0
Q12 13; "0 11Y12 (0-)
2K+1 +
Y711 (
)100
-
212
X
1) (213
+
1)
Q12 13; K)
X
Y13 0
)Ir-
X
Yr-(f6)100
Exercises
1[Y12 (P-)
X
X
Y13
0 )I
r-
X
YK M 1 00'
(A.152)
The last step is to combine the two scalar
1YII (10
X
Y11 (0
IRY11 (P-)
X
Y13 (
)100 RY12 ( -)
X
0 )10
[Y12
Y11
X
EELI111101213r-r- [[YL(&)
X
I)JrX
291
products X
Y13
Lf( I&
Y
follows:
YK00100
0 )]KlrX
as
YIs (j
X
Yr-00100 (A.153)
)100,
LL'
given by Eq. (A.75). When one of the angular momenta is zero, the 9j symbol is reduced to the Racah coefficient (see Eq.(6.86)). Using this simplification in the coefficient E and substituting the above results completes the derivation.
where the coefficient E is
A.3. Calculate the matrix element of the orbital
IAjj
angular
momentum
for the correlated Gaussians.
Solution. The matrix element of tween the
gration
lAij
be obtained from that be-
can
generating functions of Table 7.1 or by performin the intein Eq. (A.23) with V(r) being set to unity. The result
over r
is
(fK'L'M'(Ul
i
A
10) 1 IjLjj I fKLM (u, A, x)) 3
BKLBKILI
ff
(
(2v) M-1 detB
2
)
1
*Y, +,q,-Y) c
d2K+L+2K+L'
dMi YLm ( _) YL, m, (o )
AAexp
dA2K+LdX12K-+L'
[q A2 + q1 A/2 + pAYe-e'] i(e x e),, (A.154)
The
integration
over
_ and
o
can
be done
by using the formula
ff de^_d4! YLm( -_)YL,m,(, )*(i(e e),,(e-e')') x
_2 4
1
e=I,eI=1
110111, d,T V '2__Ll+1 (LMIjLjL'M) E Bklv _21+IEW 2k+l=n
Exercises
292
(A.155) Clearly I has to be L I and also L' 1. In order that a possible value L' or IL L'I 2 has to be satisfied. The latter of 1 exists, either L case is obviously impossible because the condition IL L'I < I has to =
=
-
-
be met. Thus
we
equal, otherwise
reach the
the
conclusion that L and L' must be
same
would vanish.
integral
7r(2L
Using
3(L 1) 3(L 7(L2L 47(2L 7F(2L 1)'
3L
QL-I 1; L)
+
QL+1 1; L)
____
-
47r(2L + 1)'
+
-
2+L
U(L-llLl;Ll)=
V:2LL+
U(L+l IM; L1)
2
(A.156) together
with the relation 2k + 2L + 1
BkL-1
=
2k+L we can r
2k
BkLi
Bk-I L+I
Bk-Li (A.157)
show that 11
JJ dM4 YLm( -)YL,Iv,(i )*(i(eJLLI VFL (L In order for the
and
2k+L
+
x
1) (LMly I LM)
integral not
to
vanish,
e'),(e-e)' (A.158)
Bn+I-L
n+I n
+1
2
-
L must be
non-negative
even.
Employing this leads to
Eq. (A.6)
result in a
Eq. (A. 154)
and
compaxing its result with
solution
(fKILIMI (u!, X, X) 11pi, IfKLM(u, A, x))
05LL'1\1L(L + 1) (LMI[tILM) 1 X
-
('Y?71 + 7177) (fKI LM (W, X, X) I fKLM (u, A, x)).
(A.159)
PC
As
a
check of the above
formula,
one
the matrix element of the total orbital 2
7
I:i<j IlLij (see
^11,q)
can
be
Therefore,
Eq. (2.22)).
easily
we
The
done with the
sum-ma
use
of
may
use
angular i n,
this to calculate
momentum
over
i <
LIL
=
j, of .1C (-y?7' +
Eq. (2.15), yielding just
expect that the desired matrix element becomes
z2 Lp.
Exercises
(fK"LfM' (ul A!, x) ILIIfKLM(UA, X)
--`
,
x
JLLI -
fL-(L + 1)
(LM11-ilLM)(fK'LM(WiA iX)IfKLM(UIAIX))-
On the other
hand,
reach the
we can
same
result
293
(A-160)
through the Wigner-
Eckart theorem:
(a'L'M'IL,IaLM) JLLf
:::
JLLI'
(LM IAILM) (aL 11 ,F2L + I
L
11 aL)
(LM 1/ilLM) VIL(L + 1) (aL 11111 aL) ,V -2-L+1
+ 1)(aLMIaLM), JLLI (LM lpILM) VFL-(L
where the labels tions. Here
the
and a' stand for the other labels of the
used the fact that
we
angular
a
(A.161)
momentum of the
L,,
wave
does not
wave
func-
change the magnitude of
function because it is
a
generator
of the rotation group. A.4. Calculate the matrix element of the
the method of Solution. The
spin-orbit potential following
Complement A.I.
integration
over
(fK'LIM" (U A X) I V(I'ri i
7
P is first done in
Eq. (A.23), leading to
-ri 1) IlLij I fKLM (Ui
A X)) 3
00
(,yq +'y
ff
de^-
AA/exp
n)Idr
r2 V(r) e-
47r
IBKLBK"L' ( Cr2
di YLm( -)Ypm, (i )-
(27r)N-2C2 detB
d2K+L+2K'+L' d/\2K+Ld/X/2K+L'
[qA2 + q/A/2 + #AA'e. e] i(e x e%,rii(rv) V
X=0'XI=0 e=I,ef=I
(A.162) Here
v
is the
length of the
vector
v
defined
by Eq. (A.122) and
use
is
made of the formula
f 0"(b
x
r),,&
=
47rrii(ar)(b a
x
a),,,
(A.163)
Exercises
294
can
r), ,
-Nf2i [b
=
V.L3'ir [b
rl
x
v2 term e-2c
The
and
proved by using Eqs. (6.53)
be
which
Yj- (i6)]
x
'i, (rv) is expanded
(6.54)
noting (b
x
I,
done in
as was
-
and
Complement
V
A.I: CO
2 V
exp( 2c) -
1
r
-ii(rv)
2
2n+2
(6cr) C
H2n+3
X
the
Eq. (A.158),
of
use
( 6r) C c
+
2(2n + 3)H2n+l
expansion (A.126) and integrating
the
By using
(V2c)
3
(2c) - ' r n=0 (2n + 3)1
v
n
-rj 1) lij
(
V
K+K+L-1
I:
(7771 + 71,q) C
I
H, (X) H2n+3 (X)
F-x(2n + 3)!
V
+ 2 (2n +
E
32
(2 detB
10 V(jX) e-X2
3) H2n+l (x)
C
I dx
min(K,K) X
with
00
n+I -
n=O
X
4
II fKL(u, A, x))
JLLI 'FL(L+,)(2L+,)(2K+L)!(2K'+L)! BKLBK-L 1
- and
over
obtain
we
(fKf L' (u', A, x) I I V(I'ri
X
(A.164)
-
1
Fk-kKI-kL+2k-I (c, q, qr
BkL. L+2k
k=O
(A.165) In the
when
case
V(r)
=
I
to the matrix element. The function
then becomes
L+2k P
we can
H(K
confirm that
A.5. For
a
=
...,
(X1-i
XK) ---7
matrix, calculate
n
=
0 contributes
Fr2k KI-k L+2k-1 (CI q, q, p, -y, -y)
k, q) H(W
-
k, q) H(L
Eq. (A.165) reduces
given function f (X1, X2
f (X1, X2, where:i
-
term with
only the
to
XK)
+
2k, p) Therefore .
Eq. (A-159).
of the form
=1 TcAx + 6,
XK)7
2
(bl,..., bK),
and A is
a
K
x
K
symmetric
Exercises
,)ni
11==02
ef(Xl,X27---,XK
OX,ni Solution. A solution
(A-166)
7XK::--O*
be obtained with the
can
use
of Leibniz's formula
we
show another way to solve this
+
-+AK
as was
done in A.2. Here
Let
operator L denote
an
...
295
problem.
a L
A,
A2
+
(All
Where
e' 'Cef (xI
7
...
7
7XK)
...
*
AK) =
is
an
auxiliary parameter.
ef (x'+6,
ef(XI
7
...
(A-167)
19XK
(9X2
XI
7XK)
'
7
...
Then
we
have
7XK+IMK)
exp1UXV2 (2 +
W + Ax
11)
-
(A-168)
the above equation in power equal to EK i==l ni. Expanding series of 0, collecting the terms of the Nth power in 0, and putting 0 (i =. 1, K) leads to Xi Let N be
=
...,
(f-Nef(xl
7
...
)XK)
[N 21 =
)
Xj=O .... 7XK=O
N!
E k! (N
(1U ) k( b) N-2k. -
-
k=O
2k)!
2
(A.169) problem of finding a solution to Eq. (A. 166) is thus reduced to a simpler problem of expanding the right-hand side of Eq. (A.169) in are expanded as follows: the Ai's. To this end ( b)N-2k and (WA)k 2 The
(N
( b) N-2k
11! 12!
k
2k)! IK.
(Albl)", (AA) 12 k!
(I)k
AA)
2
-
Mll! M12!
2
7nKK!
(XKbK) 1K,
(All A,2)mll
Mij
x
(2Al2AlA2 )M12
Collecting those obtain
a
solution
...
x
(AKKAK 2)MKK.
terms which have powers of
A,nj A2 n2
(A-170) AK nK
we
Exercises
296
K
ani -e
f(XI,X27
....
9X,ni i2L
XK))x
1=07
...
IXK=O
K
k
(')
E 11 ni!H(mij, Aii)H(li, bi)
2
k=O
i=1
Mij K
XH
H(mij, 2Aij),
(A.171)
j>i=l
where
and
mij's
li's
are
all
non-negative integer and
must
satisfy the
following relations K
K
mij
=
k,
mij +
j>i=l
=
(i
ni
K).
1,
(A. 172)
j=1
A.6. Derive
Eq. (A.93).
Solution. First x
E mji + li
(
2 x
we
note that
(Al) '+Ae-2IpX2_,5X.y YLM 41
M)
(,Axe- .j'8X2_jx.Y) X2v+Ay(,Xl) 2
LM
+
e-12 jax 2-,6X.y A.'X 21/+Ay(,\I) LM PC
+2
(Vxe-2.jpX2_jx`.Y) (VXX
M)
2v+,X
-
AI) YL(M P9M
The differentiation in the second term of the
above equation
(AxX
2v+,\
can
be
right-hand side
performed by using Eq. (6.59)
AO
I
The third term
can
be
reduced, by
means
to
(Vxe- .IpX2_jX.y 2
of the
as
AI) YL(M
2v(2z/ + 2A + I)x 2v+,X-2 YL(M PC M (A.42),
(A.173)
VXX
2v+A
AI) YL(M (ic'
of the
(A.174) gradient formula
Exercises
X2v+X-l e
F+' OX
1
oX2 -,6x.,y
(2v + 2A + 1)
1
-A-+I 2A+1
297
X
YX-1 ('' )4
X
Y' Ml LM
2v[[(- x-Jy)XYX+I( 3)11\Xyl( )ILM (A.175)
By using the angular momentum algebra the equation is reduced to
first term in the round
bracket of the above
[[(-OX
-
JY)
,rL3
X
Y)'-' POD'
X
Y' M] LM
OxC(1A-I;A)YL(A')(:t, ) M
+ Sy
A
U(A-1 I L 1; A r,) C(11; r)Y(X-l LM
2A+lt XYL
X1) M
Jy E U(A
r.)
( O,
-
11 L 1; A
K) C(11; r,) YLM
K
(A.176) where
use
Similarly
is made of
Eq. (A.156)
for
C(l A
-
1;
A)
C(A
-
11; A).
the second term becomes
11(- X-*X YA+'(" )IXX Y1MILM XY(,X1) L M PC M I
yE U(A+I ILI; Ais)C(11; r,)YL(A+1r')(1b, M
The contribution of the first term is
(A.177)
easily obtained by using
Exercises
298
2 e- .l#X2_,5X.y
1
+
02X2 + j2y2 + 2,3dx -,y) e-'2,8X
where the last term
involving
the scalar
2
product
-,5x.y
(A.178)
1
x
-
y
can
be reduced
to
X.
Y Y(Ai) LM
=
,
E
-v -3[x
x
y]oo0'1) LM
U(11 ; 0"0 Ry
X
[X
X
Ily
X
YA-1 4A A
X
Y' MI LM
_X
Ry
X
Y +'P
X
YI(MILM*
-V
3
Y
X
X
1WILM
Y
r.=Al
2,X + I
(A.179)
right-hand side of the above equation appear Eqs. (A.176) and (A.177), respectively. Combining these results we obtain the desired equality (A.93) The two terms of the
Ax
(X
2 Ll +,X
l,3X2_,6x.y
e-2
AI) YL(M
30X2 +,32X4 + j2X2Y2 + 2zj(2i/ + 2A +
2 -2Sx 2v+A-l y e-LPx
I
X
x
):+
TA
I
in
1)) 2
X
1 _
-
20(271 + A)X2
2v+,X-2
2
_,SX.y
LpX2_jX.y
e- 2
V/
YL(M
7F
3
(2y + 2A + 1_ OX2)
EU(A-IlLl;Aiz)C(11;K)YL(A-l)(. c, ) M
Is
'A+I -
2A +I
x
(2y
_
OX2)
I:U(A+IILI;Ar,)C(11;r,,)YL K
M
(A.180)
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