#introduction To Tight Binding

Introduction to the tight binding approximation—implementation by diagonalisation

Anthony T Paxton Atomistic Simulation Centre School of Mathematics and Physics Queen’s University Belfast

http://titus.phy.qub.ac.uk/group/Tony/WSMS2009

WSMS–J¨ ulich, March 4, 2009

Outline 1. 2. 3. 4. 5.

Tight binding approximations Bond integrals and Slater Koster table Energy bands and DOS in the simple cubic structure Self consistent tight binding Tight binding in small molecules—some new results

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Energy bands in Ge

WSMS–J¨ ulich, March 4, 2009

pseudopotential

tight binding

free electron

One electron dispersion relations in germanium (after Harrison)

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Tight binding approximation—I

WSMS–J¨ ulich, March 4, 2009 The Bloch sum of s–orbitals at N sites {R}

1 X ik·R ψk (r) = √ e ϕs (r − R) N R obeys Bloch’s theorem because if R0 is another atomic site X ` 1 0 ik·R0 ik·R00 00 ´ ik·R0 ψk (r + R ) = √ e e ϕs r − R =e ψk (r) N 00 0 R =R−R If ψk is an eigenstate of the Schr¨ odinger equation H 0 ψk = εk ψk then its eigenvalue will be R ¯k H 0 ψk dr ψ εk = R ¯k ψk dr ψ

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WSMS–J¨ ulich, March 4, 2009

0

H =−

Tight binding approximation—II

~2 2 ∇ + Veff (r) 2m

Veff (r) =

X

00

v(r − R )

R00

In which case we have Z

2 3 Z ` 1 X ik·(R−R0 ) ~2 2 X 0 0´ 4 00 5 ¯ ψk H ψk = e ϕ ¯s r − R − ∇ + v(r − R ) ϕs (r − R) N 2m R,R0 R00 Z Z X ik·(R−R0 ) ` 0´ ¯k ψk = 1 e ϕ ¯s r − R ϕs (r − R) ψ N 0 R,R

The tight binding approximations 1. Neglect three centre integrals—i.e., those for which R 6= R0 6= R00 2. Neglect overlap integrals (except those for which R = R0 ) 3. Don’t even attempt to calculate the remaining integrals—treat them as disposable parameters of the TB model, retaining only nearest neighbours.

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Tight binding approximation—III

WSMS–J¨ ulich, March 4, 2009 Then we find Z εk = εs +

2 ϕ ¯s (r) 4

3 X

v (r − R)5 ϕs (r) +

R6=0

Z

e

ik·R

ϕ ¯s (r)v(r)ϕs (r − R)

R6=0

"

Z εs =

X

ϕ ¯s (r) −

# ~2 2 ∇ + v(r) ϕs (r) 2m

If we neglect the crystal field terms, X

εk = εs +

e

ik·R

h(R)

R6=0

+

+ R

h(R) = h = ssσ

So, given some hopping (or transfer) integrals and a crystal structure we can construct the energy bands.

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Bond integrals

WSMS–J¨ ulich, March 4, 2009

+

−

+

+

ssσ

−

−

+

ppπ − −

− +

+

+ −

+

−

+

+

pdσ

+

−

− + + − −

ddσ +

+ +

+

ppσ sdσ

+

spσ

−

+

+ −

− +

pdπ

−

+ − + +

+

+ − −

ddπ

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−

− −

− +

ddδ

The simple cubic s–band

WSMS–J¨ ulich, March 4, 2009

For example in the simple cubic s–band with lattice constant a {R} = { [±a, 0, 0] [0, ±a, 0] [0, 0, ±a] } εk = εs +

X

e

ik·R

h(R)

R6=0

= εs + 2ssσ (cos kx a + cos ky a + cos kz a) Plot this along high symmetry lines in the Brillouin zone, Γ: (000), R:

−(ε−εs) / ssσ

6

0

−6 R

Γ

X n(ε)

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π a (111),

X:

π a (100)

p–orbitals—I

WSMS–J¨ ulich, March 4, 2009

The three p-orbitals are ϕx (r − R), ϕy (r − R) and ϕz (r − R). The expansion of a trial wavefunction or Bloch sum is ψk (r) =

1 N

X

cα

α=x,y,z

X

e

ik·R

ϕα (r − R)

R

The Schr¨ odinger equation becomes a secular problem ˛ ˛ ˛ ˛ ˛ (εp − εk ) δ 0 + T 0 ˛ = 0 αα ˛ αα ˛ Tαα0 =

X

e

ik·R

Z ϕ ¯α (r)v(r)ϕα0 (r − R)

R6=0

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p–orbitals—II

WSMS–J¨ ulich, March 4, 2009

Z ϕ ¯α (r)v(r)ϕα0 (r − R) = h(R)

−

+

R h(R) = ppσ cos2 θ + ppπ sin2 θ

−

θ +

generally, if l, m and n are the direction cosines of R Z

“ ” 2 2 ϕx vϕx = l ppσ + 1 − l ppπ

Z ϕx vϕy = lm ppσ − lm ppπ Z ϕx vϕz = ln ppσ − ln ppπ

Bond Integrals

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WSMS–J¨ ulich, March 4, 2009

The Slater–Koster table, s–p block

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WSMS–J¨ ulich, March 4, 2009

The Slater–Koster table, sp–d block

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WSMS–J¨ ulich, March 4, 2009

The Slater–Koster table, d–d block

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The simple cubic p–band—I

WSMS–J¨ ulich, March 4, 2009

You don’t even need to diagonalise the secular matrix in the simple cubic crystal structure since it’s already diagonal!! Tαα0 = Tαα δαα0 and Txx = 2 ppσ cos kx a + 2 ppπ (cos ky a + cos kz a) Tyy = 2 ppσ cos ky a + 2 ppπ (cos kz a + cos kx a) Txx = 2 ppσ cos kz a + 2 ppπ (cos kx a + cos ky a) In the (100) direction, Γ → X, k = (kx 00) there are three bands, 8 > <2 ppσ cos kx a + 4 ppπ εk = εp + 2 ppπ + 2 (ppσ + ppπ) > :2 ppπ + 2 (ppσ + ppπ) 8 > <2 ppσ cos kx a ≈ εp + 2 ppσ > :2 ppσ

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The simple cubic p–band—II

WSMS–J¨ ulich, March 4, 2009

(ε−εp) / ppσ

2 0 −2 R

Γ

X n(ε)

8 > <2 ppσ cos kx a Γ → X : εk ≈ εp + 2 ppσ > :2 ppσ

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Bands of Strontium Oxide

WSMS–J¨ ulich, March 4, 2009

SrO lmf

SrO TB

0.8

0.4 0.6 Energy (Ry)

Energy (Ry)

0.2 0

0.4 0.2

−0.2 0 −0.4 Γ

M X

R

Γ XM R

−0.2 Γ

M X

8 > <2 ppσ cos kx a Γ → X : εk ≈ εp + 2 ppσ > :2 ppσ

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R

Γ XM R

Further. . .

WSMS–J¨ ulich, March 4, 2009 • d–bands • hybridisation • non orthogonal tight binding • total energy and force • self consistent tight binding • magnetic tight binding • small molecules • time dependent tight binding

”Fools rush in where angels fear to tread,” Alexander Pope 1711

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self consistent tight binding

WSMS–J¨ ulich, March 4, 2009

There are components of the electrostatic potential at site R due to all multipoles at sites R0 : VRL = e

2

X

˜ B RL R0 L0 QR0 L0

R0 L0

˛ ˛ ˜ is a sort of generalised Madelung matrix, proportional to ˛R − R0 ˛−1 for point charges. B They induce electrostatic shifts in the hamiltonian matrix elements which are on site and off diagonal: 0

HRL0 RL00 = UR ∆qR δL0 L00 +

X L

The Hubbard–U acts to resist charge accumulation. 0

H =H +H

Crystal field

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0

VRL ∆`0 `00 ` CL0 L00 L

Zirconia bands

WSMS–J¨ ulich, March 4, 2009 Fluorite (LDA)

Fluorite (TB) 0.2

1.4

t2

0

Energy (Ry)

Energy (Ry)

1.2 1 0.8

−0.2

e

−0.4 −0.6

0.6 −0.8 0.4 −1 0.2 XUK

Γ

X W

L

Γ

XUK

Rutile (LDA)

Γ

X W

L

Γ

Rutile (TB) 0.2

1.4

e

0

Energy (Ry)

Energy (Ry)

1.2 1 0.8

−0.2

t2

−0.4 −0.6

0.6 −0.8 0.4 −1 0.2 Γ

Z

R

A M

Γ

X

Γ

Z

R

A M

Γ

Z

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0 u.)

0.2

-0.2 -0.4

0.4

-0.005

T=50 K

Zirconia phase transition

WSMS–J¨ ulich, March 4, 2009

-0.01

0

0 5

10

15

ν (THz)

20

0.1

25

0.2

0.3

0.4

0.5

0.6

urfaces for a tetragonal cell with the a) section in the y ; x plane, (b) secS. Fabris, A. T. Paxton and M. W. Finnis, The isoenergetic contours are plotted FIG.6310. Temperature evolution of the free energy p FIG. 4.2 .Temperature dependence of the macroscopic order PRB, 094101 (2001) Ry/ZrO  projected arameter z . The symbols () are the results of thealong calcu-the order parameter direction h00i. (b) solid eye-line is extrapolated symbols () in arethe the 0 K calculations, the thick solid lin tions. The continuous theinresult of thethethermodynamic integration and corre gion near Tc where the large uctuations z make to the temperatures 50, 500, 1000, 1500, and 2000 K. eraging procedure inaccurate. ∆U (mRy) 1 0 -1 -2

2

0.6

0.4

0.5

δx

T

1

0

0.4-0.2 -0.4

-0.4

0

-0.2

0.2

0.4

δz

0.3 0.2

y of the 0 0.1K energy-expansion cont tetragonal0 cells with the tetrag800 1000 1200 1400 1600 1800 2000 2200 ections of the600corresponding energy T (K) ymmetry order-parameter directions (b)

∆F (mRy/ZrO2)

0.4

T=2000 K

1.5

0.2

<δ> (a.u.)

2

<0,0,δ> (a.u.)

0.5 0 -0.5 -1 -1.5 -2 -2.5 -0.6

T=0 K -0.4

-0.2 0 0.2 <0,0,δ> (a.u.)

0.4

0.6

FIG. 5. Calculation results of the frequencyFIG. squared  z2 vs. well in the internal energy along the =1.00 19 / 11. 27 Double

=1.01

1

TB model for water

WSMS–J¨ ulich, March 4, 2009

Finding TB parameters for water

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Water monomer

WSMS–J¨ ulich, March 4, 2009

q

q = δe p(δ)

d

θ

pind

−2q point charge model p(δ) = 2dδe cos pind

1θ 2

dipole polarisable model

= 5.65δ

2eδ = −αO (δ) 2 cos d

1θ 2

pexp = p(δ) + pind = 1.86

(Debye) ;

; (D)

αO (δ) ≈ 0.8 − 0.23δ → δ ≈ 0.9 ; δ

point dipole exp.

αO 3

model 0.33 0.45 —

ν1

e>0

pind αO (δ) =− p(δ) d3

ν2

(N → O → F → Ne) ν3

(˚ A )

—force constants (au)—

— 0.31 0.7

1.029 1.029 1.029

0.099 0.104 0.100

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1.002 0.847 1.062

αH2 O

Ecoh

(˚ A )

3

(Ry)

1.3 1.2 1.4

0.79 0.84 0.75

Rapid progress has recently been made in the precise determigeometry optimizations u nation of the equilibrium geometry and binding energy of the reported by Hobza et al. water dimer by ab initio methods.1h33 Apart from advances in using Dimer conventional basis WSMS–J¨ ulich, March 4, 2009 Water hardware and software the progress has been stimulated by energy, *E(FC), approache the availability of basis sets which (in principle) allow a sysaround [20.5 kJ mol~1, tematic approach to the basis set limit at a given level of [20.28 kJ mol~1 (ref. 25) theory, and by a growing consensus that the basis set super32). Corresponding MP2position error (BSSE) can rigorously be avoided by applying taken to represent the basis the counterpoise method (CP), so that the user can focus on a Relaxing the monomer geo good description of the interaction without having to worry kJ mol~1 (refs. 21 and 2 about the size of the BSSE. A survey of recent studies, giving valence correlation e†ects details of the geometries (cf. Fig. 1) obtained or explored, is mol~1.24,31,32 Thus the Ðn given in Table 1. point charge model all these contributions into non self consistent model dipole polarisable model A fairly complete set of results converging to the basis set mol~1. limit is now R availablerfat the rd MP2/CPα level of βtheory. At higher levels of the Xantheas21 has291 converged É ÉO equilibrium distance TB 95.4the OÉ 97.3 8.8◦ 97.6◦ Re Halkier et al.25,33 have s ◦ ◦ geometry, the step from M CCSD 291 95.7 96.4 5.5 124.4 monomer *E(FC) by amou for the aug-cc-pVDZ basi pVQZ, and their extrapola an increase to [0.27 kJ m mated the increase to [ monomer relaxation e†e lengthening is exaggerated expects a smaller relaxatio the coreÈvalence correction been studied. The aim of the present p CCSD(T) level of theory in W. et al., PCCP, 2227 (2000) Fig. Klopper 1 The equilibrium structure2, of the water dimer. result as is technically poss currently feasible to gene accurate as those availabl out CCSD(T) geometry o ¤ Dedicated to Professor Reinhart Ahlrichs on the occasion of his 60th birthday. aug-cc-pV5Z, with 572 fun 22 / 27

Water Tetramer

WSMS–J¨ ulich, March 4, 2009

—dipole moment (D)— monomer 1.82 dimer 1.97 1.98 trimer 2.22 2.23 2.24 tetramer 2.25 2.25 2.27 2.27 . . . bulk water ≈ 2.66 (point charge model)

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Polarisability of small molecules

WSMS–J¨ ulich, March 4, 2009 azulene

3

10 dipole moment (Debye)

dipole moment (Debye)

2 1 0 −1 −2 −3

p-nitroaniline

12

8

H

− O − O

+ N

H

6 4

O

2 −2

−1 0 1 electric field (GV/m)

0

2

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N+

− + N

O

−4

−2 0 2 electric field (GV/m)

H N H

4

Time dependent TB

WSMS–J¨ ulich, March 4, 2009 0.02 ∆q (electrons)

0.01 0 −0.01

π−bond current (µA)

30 20 10 0 −10 −20 −30

0

500

1000 1500 2000 2500 time (fs)

dρˆ 1 = [H, ρ] ˆ − Γ (ρˆ − ρˆ0 ) dt i~

770

;

780 time (fs)

790

jRR0 =

2e X HR0 L0 RL Im ρRL R0 L0 ~ 0 LL

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Ehrenfest dynamics

WSMS–J¨ ulich, March 4, 2009 0.02 ∆q (electrons)

0.01 0 −0.01

π−bond current (µA)

30 20 10 0 −10 −20 −30

0

MR

500

1000 1500 2000 2500 time (fs)

d2 R =− dt2

X

560

570 time (fs)

0

580

2ρRL R0 L0 ∇R HR0 L0 RL −

RL R0 L0 R0 6=R

X L

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QRL ∇R VRL − ∇R Epair

Acknowledgements

WSMS–J¨ ulich, March 4, 2009

Thanks to Catherine Walsh Alin Elena Jorge Kohanoff Tchavdar Todorov Mike Finnis

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