An-introduction-to-the-quantum-theory-for-chemists.pdf



PETER G. NELSON

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS AN EXPERIMENTAL APPROACH

2

An Introduction to the Quantum Theory for Chemists: An Experimental Approach 1st edition © 2019 Peter G. Nelson & bookboon.com ISBN 978-87-403-2732-8

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Contents

CONTENTS Acknowledgements

6

1 Introduction

7

2 Background

8

3

Emission spectra

9

4

Energies of atoms

11

5

Wave-like behaviour

16

6

Wave mechanics

19

7 Schrödinger’s theory of hydrogen atom

26

8 Dirac’s theory of hydrogen atom

30

9 Modified Schrödinger theory of hydrogen atom

34

10 Modified Schrödinger theory of other atoms

36

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Contents

11 Modified Schrödinger theory of molecules

46

12

Valence-bond (VB) theory

56

13

Ligand-field theory

60

14 Density functional theory (DFT)

62

15 Perspective on the quantum theory

64

16

65

Further reading

Questions

66

Answers

68

Appendix

69

5

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Acknowledgements

ACKNOWLEDGEMENTS I am very grateful to Dr. David Johnson for reading through the text and commenting extensively on it. His suggestions have led me to make considerable improvements. I have taken illustrations from the internet with my thanks. As far as I know, they are all free to copy, but if any are not, I apologize.

6

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Introduction

1 INTRODUCTION This little book is a brief introduction to the quantum theory for chemists. In it I keep as close to experiment as possible, and to ideas that are familiar to chemists. This approach takes me away from one based on mathematics. There is some mathematics, but I have kept this as simple as possible. The quantum theory is used in chemistry in two main ways. One is to make accurate calculations of the energies of chemical processes. These calculations are very complicated but are made possible by modern computers. Algorithms are now available for chemists to use. An example of such a calculation is that carried out on the sodium‒helium system at various pressures leading to the prediction that Na2He would be stable above 1.6 × 106 bar, a prediction verified by experiment. Chemists also use the results of approximate calculations to understand chemical phenomena and make tentative predictions. It is possible, for example, to explain the Periodic Table on the quantum theory, as we shall see in a later section. In this book, we shall keep the two uses of the quantum theory in chemistry in mind. We shall gradually build up the principles behind accurate calculations while also discussing some of the results from more approximate treatments.

7

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Background

2 BACKGROUND Physicists developed the quantum theory in the early decades of the 20th century to explain phenomena that could not be explained by classical physics. These included the discovery that atoms comprise a positively charged nucleus surrounded by electrons (Fig. 2.1). According to classical physics, an electron is attracted to a nucleus by a force that is inversely proportional to the square of the distance (Coulomb’s law) and should progressively lose energy and collapse on to the nucleus. That this does not happen means that, on an atomic scale, either Coulomb’s law breaks down (favoured by G.N. Lewis in his famous paper on atoms and molecules), or classical laws of motion break down, or both.

Figure 2.1 Simple model of an atom [David Johnson]

In Figure 2.1, the number of electrons (charge െ݁) is equal to the atomic number (ܼ), and the nuclear charge to ൅ܼ݁ . The simplest atom is that of hydrogen (ܼ ൌ ͳ)

8

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Emission spectra

3 EMISSION SPECTRA A phenomenon that helps to answer the question of what breaks down is the emission of light from a gas or vapour through which an electric discharge is passing (Figs. 3.1 and 3.2). This light can be analysed by passing it through a prism or grating in a spectrometer. A grating consists of a transparent material into which a large number of uniformly spaced wires have been embedded (about 300 per millimetre). A prism or grating bends light to different degrees according to its wavelength and splits it into its component colours. What is found is that the light is made up of narrow ranges of colour (“lines”) with dark spaces between them. This is shown in Figure 3.3 for a selection of elements. The spectra extend to the left into the ultraviolet and to the right into the infrared.

Figure 3.1 Apparatus for studying the passage of electricity through a gas [Shiksha Services]

Figure 3.2 Discharge of electricity through hydrogen [Wikimedia Commons]

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Emission spectra

Figure 3.3 Emission spectra of some elements compared with the spectrum of white light (1) [cnx.org] �

The scale in Figure 3.3 is of wavelength ( � )=in ångström units (Å = 10‒10 m). The � corresponding frequency is given by � � = � (3.1)

where � is the speed of light (2.998 × 108 m s‒1). Wavelength is determined by passing �= � light through a grating of known spacing (݀) and measuring the angle (ߠ) at which it is diffracted. Diffracted rays from adjacent slits reinforce each other when the extra distance � one has to travel (݀•‹ߠ) is equal to � =(Fig. 3.4). The emission spectra of metals are � obtained at elevated temperatures.

d

θ

� � sin �

Figure 3.4 Diffraction through slits

Now what in the gases or vapours of these elements are giving rise to the lines in their spectra? In the case of calcium and mercury, the answer must be their atoms because these elements are monatomic in the vapour. Sodium and hydrogen, on the other hand, are diatomic in the gas or vapour. Under the conditions of a discharge, however, the diatomic molecules will be to some extent dissociated, and from the general similarity in the spectra of these elements to those of calcium and mercury, we may conclude that these too are produced by atoms. 10

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Energies of atoms

4 ENERGIES OF ATOMS If atoms are responsible for the light that is emitted in a discharge tube, they presumably take electrical energy from the discharge and convert this into light. Since the light is restricted to certain frequencies, this means that constitutional energies of atoms (i.e. energies arising from within an atom as opposed to thermal motion of an atom as a whole) are restricted to certain values, light being emitted when the constitutional energy of an atom drops from one level to another (Fig. 4.1). The steps in energy are called “quanta” (plural of “quantum”, from Latin quantus, “how much”). This phenomenon brings us up against the problem we are addressing: according to classical physics, an atom can have any energy, and will progressively lose energy until its electrons collapse on to the nucleus.

Energy

light

Figure 4.1 Energy levels of an atom showing an energy drop and light produced

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Energies of atoms

To establish the energies an atom can have, we need to know the relationship between the drop in energy (‫ )’‘”†ܧ‬and the frequency of the light. This can be determined by means of the apparatus shown in Figure 4.2 (Franck and Hertz 1914).

Figure 4.2 Franck-Hertz apparatus [ND-RC]

In this, electrons are produced by passing a current through a filament (left). These are accelerated through the gas or vapour by a positively charged grid, the potential of which can be varied. Electrons passing through the grid are collected, and the current they produce measured at G. As the potential is increased, the current from the collector increases until, at a certain point, it drops. At this point the gas starts to emit one of the lines in its emission spectrum. Some examples are shown in Table 4.1, where ‫ ‡ܧ‬is the energy required to produce emission and ߥ is the frequency of the light emitted. The energy is given by ‫ Ž‡ܸݍ‬where ‫ ݍ‬is the charge ( ݁)) and ܸ‡Ž the electric potential.

What is happening in this experiment is that electrons collide with atoms in the tube, and impart energy to them. When the potential of the grid is low, the energy imparted by the electrons is low, and simply raises the thermal energy of the atoms. As the potential is increased, however, the energy imparted by the electrons goes up. When this energy is high enough to lift the atoms to a higher constitutional level, this excitation takes place. The atoms then emit the energy they have absorbed in this way as light and return to the lower level.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Energies of atoms

�em

Gas



�em �

He

31.7 × 10–19 J

4.79 × 1015 s–1

6.6 × 10–34 J s

Hg

7.8 × 10

J

1.18 × 1015 s–1

6.6 × 10–34 J s

Mg

4.3 × 10–19 J

6.56× 1014 s–1

6.6 × 10–34 J s

–19

Table 4.1 Energy required to produce emission and the frequency of the light emitted* *After Bolton, Patterns in Physics (1974)

The ratio in the last column is constant, and allows us to write the relation between energy drop and frequency of light emitted as �drop = ℎ� (4.1)

where �drop = ℎ�is a constant (6.626 × 10‒34 J s). This constant was introduced by Planck in another context (1900) and is called “Planck’s constant”.

Energy levels of hydrogen atom The simplest emission spectrum is that of hydrogen (Fig. 3.3). The wavelengths of the lines are assembled in Table 4.2, along with the corresponding frequencies. As the table shows, the frequencies are given by the simple equation �

� = � ��� −

�

��

� (� = 3, 4, 5 … )(4.2)

where ‫ ܤ‬ൌ = ܴܿ 3.2879 × 1015 s‒1. This relation was discovered by Balmer (1885). It is usually � � = expressed in terms of wavenumber (1/ �) in which case the constant is called the “Rydberg constant” (ܴ). The two constants are related by ‫ ܤ‬ൌ ܴܿ.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Energies of atoms

Wavelength/Å

Frequency/1014 s–1

Eq. (4.2)/1014 s–1

6564.7

4.5665

4.5665 (n = 3)

4862.6

6.1649

6.1648 (n = 4)

4341.6

6.9044

6.9046 (n = 5)

4102.9

7.3064

7.3064 (n = 6)

3971.2

7.5487

7.5488 (n = 7)

3890.1

7.7060

7.7060 (n = 8)

Table 4.2 Lines in the emission spectrum of hydrogen

Similar relations were discovered for light emitted in the ultraviolet (Lyman) and infrared (Paschen). These are respectively �

� = � ��� − �

� = � ��� −

�

�� �

��

� (� = 2, 3, 4 … )(4.3)

� (� = 4, 5, 6 … )(4.4)

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Energies of atoms

These formulae can be combined as follows �

� = � ���� −

�

��

� (� > �� )(4.5)

Thus, from equation (4.1), the changes in energy giving rise to the lines in the spectrum are �

�drop = ℎ� ���� −

�

��

�(4.6)

The energy levels of hydrogen atoms are therefore given by ��

� = − �� (� = 1, 2, 3 … ) (4.7) �

��

�

� � =drop � ���= − level � (� > ) > �� ): light being emitted when atoms from � �level (� �� � ��� ��� − ��to ��

�

�

��

�

�drop = � − �′ = − �� − �− ��� � = ℎ� ���� − ��

��

�

��

�(4.8) �

�

� = � �The − � (� is > called �� )�drop number a “quantum of is 2.1786� �× 10‒18 J (13.598 eV). = � − �′ number”. = − � −The �−value �� � = ℎ� � �� − ��� ��

��

� = − ��

�

�

�

�

On the model of Figure 2.1, the constitutional energy of an atom resides in the motion of the negatively charged electrons round the positively charged nucleus. On this model, therefore, equation (4.7) gives the different energies the one electron in a hydrogen atom can have. From this, the energy required to remove the electron completely from the atom (the “ionization potential”, I) is given by � = −� = ��� . For the atom in its ground state � ( = 1,),2,this (� 3 … is) 13.598 eV.

15

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave-like behaviour

5 WAVE-LIKE BEHAVIOUR If atoms are restricted in the energy they can have, they differ from, for example, a pendulum, which can have any amplitude. They are, however, similar to a stretched string, the vibrations of which are restricted to certain modes (Fig. 5.1). This raises the possibility that the motions of electrons are wave-like in character.

Figure 5.1 Modes of vibration of a stretched string �

= wavelength, (� = �

‫ = ܮ‬length of string)

[Chad Orzel, Forbes]

Now this possibility can be tested for a beam of electrons by passing it through a suitable grating. The grating has to be fine enough for diffraction to be possible. A suitable grating is provided by the atoms in a metal foil or crystal. Such a grating gives a diffraction pattern for X-rays (light of very low wavelength). When this experiment is carried out, it is found to give a diffraction pattern of the kind expected for waves (Fig. 5.2). This confirms our speculation that the motion of electrons is somehow wave-like.

Figure 5.2 Electron diffraction pattern [Free Patterns]

16

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave-like behaviour

Measurement of the pattern allows the wavelength of the beam to be determined and compared with the momentum (‫ ݌‬ൌ ݉‫ )ݒ‬of the electrons (݉ = mass, ‫ = ݒ‬speed). The latter � can be determined from the kinetic energy of the electrons ( � = �� � ) as given by the � and accelerating potential (� = ��el ). The relation between � = ��el‫ ݌‬is ��

� = �� (5.1)

Typical results are set out in Table 5.1. These show that the wavelength of the electron beam decreases as the momentum of the electrons is raised, and that ߣ‫ ݌‬is approximately constant, the same constant as in Table 4.1. We can therefore write �=

�

�

(5.2)

This is called the “de Broglie relation” after the physicist who first suggested it (1924). From equation (5.1), equation (5.2) can also be written �=

�

√���

=

�

���(���)

(5.3)

where ‫ ܧ‬is the total energy and ܸ the potential energy (e.g. of an electron in the field of other electrical charges).

17

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave-like behaviour

�



3.97 × 10–24 kg m s–1

1.65 × 10–10 m

6.6 × 10–34 J s

4.36 × 10–24 kg m s–1

1.49 × 10–10 m

6.5 × 10–34 J s

6.36 × 10–24 kg m s–1

1.06 × 10–10 m

6.7 × 10–34 J s

�=

�



Table 5.1 Wavelengths of electron beams diffracted by a crystal of nickel* *Davisson and Germer (1927)

Other small particles can be diffracted by gratings in a similar way, including atoms. What are the waves associated with particles? Physicists and philosophers still discuss this problem. One possibility is that they are in a sub-electronic medium that buffets small particles like waves on the sea buffeting a buoy, or water molecules buffeting a pollen grain in Brownian motion.

18

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

6 WAVE MECHANICS If small particles behave like waves, can we adapt the theory of waves to describe their motion? Consider the simple case of a particle confined to a one-dimensional box of length L, with ܸ = 0 and ‫= � = ܧ‬ inside ��el the box and ܸ= λ outside it. Let us�adapt the theory of waves in a �= stretched string to this problem, and restrict the values of � as in Figure 5.1: �=

��

(� = 1, 2, 3 …)(6.1)

�

From equation (5.3), this gives �=

�� �� (6.2)

����

The energy of the particle is therefore restricted to certain values (Fig. 6.1). This successfully reproduces what we have found for atoms (Sect. 4). The inverse dependence of ‫ ܧ‬on ݉ and ‫ ܮ‬means that the permitted energies are very much closer together for a macroscopic system than for one on an atomic scale.

Figure 6.1 Permitted energies of a particle in a box up to n = 4 and the waves associated with them (compare these with the waves in a stretched string in Figure 5.1) [Plus Magazine]

The corresponding equation for a particle in a three-dimensional box is ��

� = � ��� + �

� ��

���

+

��� ���

��

� �� (6.3)

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

This equation correctly predicts the properties of a monatomic gas under ordinary conditions when substituted into Boltzmann’s distribution law: �� �

=

e��� /��

∑� e��� /��

(6.4)

Here i numbers the different motions a particle can have, ܰ௜ is the number executing the��absolute temperature, motion i, ‫ܧ‬௜ is their energy, ܰ the total number of particles, � = el and ݇ Boltzmann’s constant. General equation We can derive a general equation for the motion of a particle in one dimension as follows. Consider again a wave in a stretched string. The amplitude of this is given by �(�, �) = �(�) sin 2π�� (6.5)

20

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

where �(�) = � sin

�π� �

(6.6)

In equation (6.5), the sine function determines the oscillation of the wave up and down, frequency = �(�) sin 2π�� being �(�,its �) = �(�) sinand 2π�� time (sine functions oscillate in value between +1 and ‒1). its The sine function in equation (6.6) determines the shape of the wave, and the�(�) factor = �,sin �π� � sin of the wave. It is the amplitude when (sin 2π��) = −1 the=“profile” magnitude. I shall call �(�) � and is the solid line in Figure 5.1; the broken line is when (sin 2π��) = −1. Continuing the analogy with waves in a stretched string, we can write the following equation for the profile of the wave for a particle in a box �(�) = � sin

�π�

�(�) = � sin

��π� �(���)

�

(6.7) �π�

where  corresponds to �(�) to .�. Introducing equation (5.3) into this gives  and= � sin �

� (6.8)

Double differentiation of this gives d� � d� �

+

�π� � ��

(� − �)� = 0 (6.9)

The corresponding equation for a particle moving in three dimensions is ∇� � +

�π� � ��

(� − �)� = 0 (6.10)

where ∇� (“del squared”) is the operator ∂�

∂�

∂�

∇� = ∂� � + ∂� � + ∂� � (6.11)

This equation can also be written ��

�π� �

∇� � + (� − �)� = 0(6.12)

The corresponding equation for two or more particles is ∑�

��

�π� ��

∇�� � + (� − �)� = 0 (6.13)

where i numbers the particles (1, 2, 3 etc.) and  is a function of all their positions (x1, y1, z1; x2, y2, z2; etc.). This is a general equation for systems whose energy does not change

21

�π� �

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

with time, and is called the “time-independent Schrödinger equation” after the physicist who devised it (Schrödinger 1926). It underlies much of quantum mechanics as applied to chemistry. The equation successfully reproduces the energy levels of the hydrogen atom [equation (4.7)] as we shall see in the next section. The function  is called the “wave function”. To describe a wave, it has to be everywhere finite, single-valued, and continuous. These conditions limit the solutions to the Schrödinger equation, and bring about quantization.

Figure 6.2 Erwin Schrödinger

Physical significance off 

From equation (6.5), the intensity of a wave in a stretched string is given by �(�, �) = �� (�, �) = �� (�) sin� 2π��(6.14)

�( �( � ) �, ��)�= � 2π�� sin� 2π�� (�,��) (�)�sin �)(�, =��). �= = This is positive for all�(�, �and

Now in wave mechanics, time variation is introduced into the Schrödinger equation in such � (�) �� (�, �) = sin �),= � �� (�) a way that, for systems with constant energy, the quantity corresponding to �(�, �),=�(�, does not vary with time (Box A): �(�, �) = � � (�)(6.15) �

From this, Born (1926) suggested that represents the probability of finding a particle =1 ∫� �(�)� d� � corresponds between and  andd � + d�.This means that, for a particle in a box, =1 ∫� �(�)� d� to the density of the image on a long-exposure photograph of the particle moving in the � for =n1= 1, 2 and 3 are displayed in Figure 6.3. The condition that the box. Plots of ∫� �(�)� d� particle should be found somewhere in the box

22

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

� ∫� �(�)� d� = 1(6.16)

fixes the value of � = in equation �2/�. (6.7). This process is called “normalization”. For a particle in a box, � = �2/�.

Figure 6.3 Wave functions and probability distributions for a particle in a box [Brane Space]

23

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

The probabilistic description of the motion of the particle means that there is uncertainty about the precise motion of the particle, i.e. where it is at a given time and how fast it is travelling. In other words, “it is impossible to specify simultaneously both the momentum and position of a particle”. This is the celebrated “Heisenberg uncertainty principle” (1927). The interpretation of  I have given here is the one usually adopted by chemists. Other interpretations have been proposed, which I will briefly discuss at the end (Sect. 15). Wave functions can be “complex”, i.e. contain the square root of minus one (i). In these cases,  is given by �� ∗ , where � ∗ is the “complex conjugate” of , i.e. , with -i in place of i;  is then real.

A useful property of wave functions is that they are “orthogonal” to each other, i.e. they ∗ satisfy ∫ �� ��� d� = 0 or ∫ �� ��� d� = 0 (� ≠ �� ). Box A Time dependence For a particle moving in one dimension, the time-independent Schrödinger equation [equation (6.9)] may be written: ��

d� �

− ��� � d� � + �� = �� (A1)

Now let the corresponding time-dependent equation be: ��

d� �

� ��

− ��� � d� � + �Ψ = ±i �π

where Ψ is a function of is satisfied by

or

��

(A2)

 and .. For a system whose energy does not change with time, this

�(�, �) = �(�)e‒i

�π�� �

� ∗ (�, �) = �(�)e+i



�π�� �

(A3)



(A4)

These equations can be derived from equations (A1) and (A2) by substituting

�(�)�(�). They give

�(�, �) =

�(�, �) = �(�, �)� ∗ (�, �) = �2 (�) (A5)

independent of t as required.

24

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Wave mechanics

Solving the Schrödinger equation In a small number of cases, the Schrödinger equation can be solved exactly. We have effectively seen this for a particle in a box. In most cases, however, the equation cannot be solved exactly, and approximate methods have to be used. One method is to omit the smallest term or terms in the potential energy and to solve the equation without them. The solutions can then be used to calculate the energy from the full equation. This is called “perturbation theory”, the small terms being treated as a perturbation. Another method is to choose a possible solution to the equation with variable parameters in it and to vary these parameters until they give the lowest energy (e.g. one might try the function � = �e��� and vary the parameter a). The lowest energy will be closest to the true energy. This is called the “variation principle”, and the method, the “variation method”. With skilful choice of functions and the use of modern computers, solutions very close to the true solution can be obtained in this way. We will give examples of the use of these methods later.

Simplified form of Schrödinger equation Equation (6.13) can be rewritten as follows: ��

− ∑� �π� � ∇�� � + �� = �� (6.17) �

This can be simplified to

where

Ĥ� = ��(6.18) ��

Ĥ = − ∑� �π� � ∇�� + � (6.19) ��

�

� Here Ĥ =is −a ∑mathematical � �π� � ∇� + � “operator” (it includes differentials) called the “Hamiltonian � operator”. This equation only holds for the correct wave function, but energies can be obtained from it with approximate functions by multiplying both sides by  (or � ∗ if is  complex) and integrating over all the coordinates to give what is effectively a mean value:

� = ∫ �Ĥ� dτ ( normalized)

(6.20)

� = ∫Here �Ĥ� dτ stands for dx1dy1dz1dx2 … We shall use this equation later. 25

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Schrödinger’s theory of hydrogen atom

7 SCHRÖDINGER’S THEORY OF HYDROGEN ATOM As noted in Section 2, the hydrogen atom is the simplest atom, with one electron (e) moving round a singly-charged nucleus (n). The potential energy in this case is the Coulomb energy of attraction between the electron and the nucleus: �=

(�)(��) �π�� �

(7.1)

Here  is the distance between the electron and the nucleus, and  is the permittivity of free space. The appropriate Schrödinger equation is equation (6.13) with � = 1 for the electron and � = 2 for the nucleus. This equation describes essentially two motions, the motion of the electron round the nucleus and the motion of the atom as a whole through space. These motions can be separated by introducing the reduced mass, ,, defined by �

�

=

�

�e

+

�

�n

(7.2)

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26

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Schrödinger’s theory of hydrogen atom

The motion of the electron round the nucleus is then given by equation (6.12) with � = �: ��

�π� �

��

∇� � + �� + �π� �� � = 0(7.3) �

Here  is a function of the position of the electron relative to the nucleus. Because �n ≫ �e , � ≈ �e .

Equation (7.3) can be solved exactly. The result is that the energy is restricted to the following values:

where

�=−

�

��

�� �

(� = 1, 2, 3 … ) (7.4)

� = ��� �� = 2.1787 × 10‒18 J (13.598 eV) (7.5) �

This almost exactly reproduces equation (4.7).

Orbitals The solutions  of equation (7.3) are called “orbitals”. They give the probability of finding the electron at different points around the nucleus, and broadly reflect their motion (Sect. 6). The quantum number n determines the size of an orbital: the bigger n, the bigger the orbital, and the lower the binding energy. For � > 1, there are several motions having the same energy. This is called “degeneracy”. These motions are characterized by two subsidiary quantum numbers,  and  . The first of these can take the values � = 0, 1, 2 … (� − 1) and determines the shape of an orbital (more precisely, the angular momentum of the electron). Orbitals having � = 0 are spherically symmetrical and are called s orbitals. Orbitals having � = 1 are called p, those having � = 2 are called d, and those having � = 3, f. The labels derive from the names for series in alkali metal spectra. The second subsidiary quantum number takes the values �� = 0, ±1, ±2 … ± � and determines the directionality of an orbital (more precisely, the component of the angular momentum in a given direction). The number of orbitals for each value of  is therefore (2l + 1) . The various orbitals for n = 1, 2, and 3 are set out in Table 7.1 below. A selection are shown in Figure 7.1.

27

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

� > 1, 1

Schrödinger’s theory of hydrogen atom



No. of orbitals

Designation

0

 0

1

1s

0

0

1

2s

1

0, ±11 0,

3

2p

0

1

3s

0, ±11 0,

3

3p

5

3d

2

0 3

1 2

0,0, ±11,0, ±12

Figure 7.1 Schrödinger orbitals for a hydrogen atom (sections in x–z plane, probability densities represented by densities of dots) [Forgotten Planet]

Orbitals having different  values are not uniquely determined by equation (7.3) and can be combined. For example, p orbitals can be combined to give three of the kind shown in Figure 7.1 pointing in the x, y, and z directions: p� =

p� =

�

√� �

√�

(−p�� + p�� )(7.6) (p�� + p�� )(7.7)

p� = p�(7.8)

The factor

�

√�

comes from normalization (Sect. 6).

28

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Schrödinger’s theory of hydrogen atom

Exercise Show that the orbitals p� and p� have the same energy (E) as p�� and p�� . Use the simplified equation Ĥ� = �� [equation (6.18)]. ��

Figure 7.2 shows the energies of orbitals 1, 2, and 3. � = − �for � (� = 1, 2, 3 … ) 3s (1) 2s (1)

3p (3) 2p (3)

3d (5)

 1s (1) Figure 7.2 Energies of orbitals for a hydrogen atom on Schrödinger’s theory with numbers of orbitals in brackets

Hydrogen-like ions The above treatment can readily be extended to the hydrogen-like ions He+, Li2+, etc. For these, equation (7.1) becomes ܸ ൌ

ሺ௓௘ሻሺି௘ሻ ସɎఌబ ௥

(7.9)

where ܼ is atomic number and ܼ݁ nuclear charge. Equation (7.4) is then ‫ ܧ‬ൌ െ

from which

௓ మௐ ௡మ

(7.10) ଵ

ଵ

‫ ’‘”†ܧ‬ൌ ݄ߥ ൌ ܼ ଶ ܹ ቀ௡ᇲమ െ  ௡మቁ(7.11)

Lines corresponding to this equation have been observed for He+ (ܼ ൌ ʹ) , Li2+ (ܼ ൌ ͵), and Be3+ (ܼ ൌ Ͷ).

29

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Dirac’s theory of hydrogen atom

8 DIRAC’S THEORY OF HYDROGEN ATOM There are several indications that Schrödinger’s theory is not completely accurate. One is that, under high resolution, lines in the emission spectrum of hydrogen exhibit fine structure. For example, the line at about 6564.7 Å (Table 4.2) is a doublet, with a splitting of 0.14 Å. A second indication is that, when a beam of silver atoms is passed through a non-uniform magnetic field, it splits into two (Stern and Gerlach 1922). The same happens with hydrogen atoms (Phipps and Taylor 1927). This is not expected on Schrödinger’s theory. A third indication is the occurrence of surfaces in probability distributions for which ߰ ଶ ൌ Ͳ These surfaces are called “nodes”. Nodes occur, for example, in the distribution for a particle in a box when ݊ ൐ ͳ (Fig. 6.3). They also occur in orbitals for a hydrogen atom, e.g. the x‒y plane of 2p0 (Fig. 7.1). Now the question is, if the probability of a particle being at a node is zero, how does the particle get from one side of the node to the other? The answer to this question has to be that the particle travels infinitely quickly. This goes against the theory of relativity, which limits the speed of a particle to the speed of light (Box B).

30

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Dirac’s theory of hydrogen atom

Box B Relativity Einstein (1905) arrived at his theory of relativity by working out what effect the constancy of the speed of light in a medium has on how different observers see a moving object. The same results can be derived experimentally by employing the apparatus used to determine the ratio of mass to charge of an electron (see my e-book Introduction to Chemistry). This enables the mass (݉) of an electron to be determined at different speeds (ߥ). The results are shown in the Figure. The mass increases with speed, and tends towards infinity as the speed approaches the speed of light (c = 2.998 × 108 m s−1). 3.5

3

m/10-27 g

2.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

3

v/108 m s-1

Figure B1 Plot of electron mass against speed

When these results are fitted to simple equations, the best fit is obtained with the following equation, the same as that obtained by Einstein: ଵ

‫ ݕ‬ൌ ξଵି௫ మ

(B1)

where ‫ ݔ‬ൌ ‫ݒ‬Ȁܿ , ‫ ݕ‬ൌ ݉Ȁ݉଴ , and ݉଴ ൌ ݉ when ‫ ݒ‬ൌ Ͳ (the “rest mass”, 0.9109 × 10–27 g). From this equation, other equations can be derived, including equation (8.1) in the main text (see Appendix).

Dirac (1928) adapted Schrödinger’s theory to make it consistent with the theory of relativity. Schrödinger’s theory starts from equation (5.1). Dirac starts from the corresponding relativistic equation

31

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Dirac’s theory of hydrogen atom

ܶ ൌ ඥ‫݌‬ଶ ܿ ଶ ൅ ݉଴ଶ ܿ ସ (8.1)

From this he derives four simultaneous equations, the solutions of which give four functions ( ߰ଵ , ߰ଶ , ߰ଷ , and ߰ସ ), from which the probability distribution is calculated as

ߩ ൌ ߰ଵ ߰ଵ‫ כ‬+ ߰ଶ ߰ଶ‫ כ‬+ ߰ଷ ߰ଷ‫ כ‬+߰ସ ߰ସ‫(כ‬8.2)

States are characterized by four quantum numbers: ݊ , ݈ , ݆ , and ݉௝ . The first two are the same as on Schrödinger’s theory, and take the values ݊ ൌ ͳǡ ʹǡ ͵ ǥ and ݈ ൌ Ͳǡ ͳǡ ʹ ǥ ሺ݊ െ ͳሻ. ଵ The new quantum number ݆ takes the values ݆ ൌ ݈ േ ଶ , and ݉௝ takes the values ݉௝ ൌ ଵ ଷ േ ǡ േ ǥേ ݆ , totalling ሺʹ݆ ൅ ͳሻ values. The states for ݊ ൌ 1 and 2 are set out in Table 8.1. ଶ

ଶ

݊

݆

݈

1

ͳ ʹ

0 0

2

1

Table 8.1 Dirac orbitals for

݊ ൌ 1 and 2

ͳ ʹ ͵ ʹ ͳ ʹ

݉௝

No. of orbitals

ଵ ଵ ǡെଶ ଶ

2

ଵ ଵ ǡെଶ ଶ

͵ ͳ ͳ ͵ ǡ ǡെ ǡെ ʹ ʹ ʹ ʹ ଵ ଵ ǡെଶ ଶ

2

6

Dirac’s theory resolves the problems with Schrödinger’s theory discussed earlier. None of the probability distributions has nodes. The nearest they get are surfaces of very low probability, which the particle can cross with the speed of light. Also, the energy is given by equation (7.4) with a small addition that gives rise to the fine structure: ௐ

‫ ܧ‬ൎ െ ௡మ െ

ௐఈ మ ௡య

ቆ

ଵ

భ ௝ା మ

ଷ

െ ସ௡ቇ(8.3)

Here ߙ is the “fine-structure constant”: ௘మ

ߙ ൌ  ଶఢ

బ ௛௖

(8.4)

Figure 8.1 shows the energies of orbitals on Dirac’s theory for ݊ ൌ 1 , 2, and 3 (compare Fig. 7.2).

32

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Dirac’s theory of hydrogen atom

3s (2) 2s (2)

3p (6) 2p (6)

3d (10)

‫ܧ‬ 1s (2) Figure 8.1 Energies of orbitals for a hydrogen atom on Dirac’s theory with numbers of orbitals in brackets. Levels are split, but by amounts too small to be shown.

Figure 8.2 Paul Dirac

33

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of hydrogen atom

9 MODIFIED SCHRÖDINGER THEORY OF HYDROGEN ATOM Dirac’s theory of the hydrogen atom is mathematically complicated, and is not easily extended to other atoms. It does, however, suggest a simple modification of Schrödinger’s theory. Comparison of Figures 7.2 and 8.1 shows that there are twice as many orbitals on Dirac’s theory as on Schrödinger’s. To bring the two sets of numbers into line, we can add to Schrödinger’s quantum numbers ݊ , ݈ , and ݉௟ two further quantum numbers, s and ݉௦ ଵ ଵ taking the values ‫ ݏ‬ൌ ଶ and ݉௦ ൌ േ‫ ݏ‬ൌ േ ଶ . This is shown in Table 9.1. The additional quantum numbers are called “spin” quantum numbers, the spin being determined by s and its orientation by ݉௦. Analysis of Dirac’s orbitals, however, suggests that the numbers are associated more with a corkscrew motion, right-handed or left-handed. Orbitals having different values of ݉௦ are called “spin-orbitals”.

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34

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

݊ 1

2

Modified Schrödinger theory of hydrogen atom

݉௦

݈

݉௟ 0

±

0

0

±

1

0

±

0

Table 9.1 Spin-orbitals for ݊ ൌ 1 and 2

±

0, ±11

݉௟ ൅ ݉௦ ଵ ଵ ǡെଶ ଶ ଵ ଵ ǡെଶ ଶ ଵ ଵ ǡെଶ ଶ

͵ ͳ ͳ ͵ ǡ ǡെ ǡെ ʹ ʹ ʹ ʹ

No. of orbitals 2 2

6

Spin-orbitals (χ) are derived from Schrödinger orbitals by multiplying ߰ by ߙ for ݉௦ ൌ ଵ ± or ߚ for ݉௦ ൌ െ ଶ , where ߙ and ߚ are functions of a spin coordinate (ߦ) satisfying the normalization equations ‫ ߙ ׬‬ଶ †ߦ ൌ ‫ߚ ׬‬ଶ †ߦ ൌ ͳ and the orthogonality equation ‫ ߦ†ߚߙ ׬‬ൌ Ͳ.

35

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

10 MODIFIED SCHRÖDINGER THEORY OF OTHER ATOMS For atoms and ions containing two or more electrons, the appropriate Schrödinger equation for the motion of the electrons round the nucleus is equation (6.13) with ݉௜ ൌ ߤ : σ௜

௛మ

଼Ɏమ ఓ

‫׏‬ଶ௜ ߰ + ሺ‫ ܧ‬െ ܸ ሻ߰ = 0(10.1)

The potential energy is given by ௓௘ మ

ܸ ൌ  െ σ௜ ସɎఌ

బ ௥೔

௘మ

൅ σ௜ σ௝வ௜ ସɎఌ

బ ௥೔ೕ

(10.2)

where ‫ݎ‬௜௝ is the distance between electron i and electron j. Equation (10.1) can be solved by various approximate methods. The simplest is to neglect the interelectronic repulsion terms in the equation. These are smaller than the nuclear attraction terms. The solutions then have the form ߰ ൎ ߮ƒ ሺͳሻ߮„ሺʹሻ ǥ(10.3)

where the electrons (numbered 1, 2 …) are in orbitals (lettered a, b …) satisfying the Schrödinger equation for ௛మ

଼Ɏమ ఓ

௓௘ మ

‫׏‬ଶ ߮ + ቀ‫ ܧ‬൅ ସɎఌ ௥ቁ ߮ = 0(10.4) బ

Each electron is then in its own orbital and its energy is given by equation (7.7): ‫ ܧ‬ൌ െ

௓ మௐ ௡మ

(10.5)

This is called “the orbital approximation”.

This version of it is very crude. Electrons do repel each other. A refinement is to treat the repulsion between each electron as having the effect of reducing the positive charge on the nucleus: ௛మ

଼Ɏమ ఓ

‫׏‬ଶ ߮ + ሾ‫ ܧ‬൅

ሺ௓ିௌ ሻ௘ మ ସɎఌబ ௥

ሿ߮ = 0(10.6)

36

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

S is called the “screening constant” and ሺܼ െ ܵሻ the “effective nuclear charge” (ܼ‡ˆˆ ). The energy is now ‫ ܧ‬ൌ െ

మ ௓‡ˆˆ ௐ

௡మ

(10.7)

Values of ܼ‡ˆˆ can be inferred from ionization energies derived from emission spectra (Sect. 4).

Thus, for example, on this approximation, a helium atom (ܼ ൌ ʹ) has in its ground state two electrons in a 1s orbital. This is called its “electronic configuration” and written 1s2. The energy required to remove an electron (the first ionization energy) is ‫ܫ‬ଵ ൌ ͳǤͺͳܹǤ This is less than the energy required to remove the second electron (‫ܫ‬ଶ ൌ ܼ ଶ ܹ ൌ Ͷܹ) reflecting the repulsion between them. The lower value corresponds to ܼ‡ˆˆ ൌ ͳǤ͵ͷǤ

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When we come to the next atom (lithium, ܼ ൌ ͵), we have to introduce a new principle. This is because, in its ground state, this does not have three electrons in a 1s orbital. The first ionization energy is ‫ܫ‬ଵ ൌ ͲǤ͵ͻܹǤ If ݊ ൌ 1 , this corresponds to ܵ ൌ ʹǤͶ, which exceeds the number of other electrons to do the screening (two). The third electron thus has to go into an orbital with ݊ ൌ ʹ, in which case ܵ ൌ ͳǤͺ.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

The new principle is that electrons cannot have the same motion. In other words, two or more electrons cannot occupy the same spin-orbital. This is called “the Pauli exclusion principle”. Thus, in a helium atom, if one electron occupies the 1s+ spin-orbital (the 1s ଵ orbital with ݉௦ ൌ ൅ ଶ), the other has to occupy the 1s–. Likewise, in lithium, the third electron has to go into a higher spin-orbital. For a given value of n, screening constants increase in the order s > p > d > f. Thus for lithium, the 2s spin-orbitals have a lower energy than the 2p. This is shown in Figure 10.1. The ground-state configuration of a lithium atom is thus 1s22s1. 3s (2)

3p

3d

(6)

(6)

2p

2s

(6)

(2) ‫ܧ‬ 1s (2)

Figure 10.1 Energies of spin-orbitals for a lithium atom with numbers of spin-orbitals in brackets

We are now in a position to work out the electronic configurations of other atoms. This can be done by using the “building-up principle”. This gives the order of filling of spin-orbitals in going from one atom to another. This is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s … Note that the 3d spin-orbitals are out of line. Some configurations are irregular because the balance between nuclear attraction and interelectronic repulsion is very close for some atoms. Configurations for atoms up to ܼ ൌ ͵͸ in their ground states are given in Table 10.1, with irregular ones in red.

38

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

Atom

Z

Configuration

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1s1 1s2 1s22s1 1s22s2 1s22s22p1 1s22s22p2 1s22s22p3 1s22s22p4 1s22s22p5 1s22s22p6 1s22s22p63s1 1s22s22p63s2 1s22s22p63s23p1 1s22s22p63s23p2 1s22s22p63s23p3 1s22s22p63s23p4 1s22s22p63s23p5 1s22s22p63s23p6 1s22s22p63s23p64s1 1s22s22p63s23p64s2 1s22s22p63s23p64s23d1 1s22s22p63s23p64s23d2 1s22s22p63s23p64s23d3 1s22s22p63s23p64s13d5 1s22s22p63s23p64s23d5 1s22s22p63s23p64s23d6 1s22s22p63s23p64s23d7 1s22s22p63s23p64s23d8 1s22s22p63s23p64s13d10 1s22s22p63s23p64s23d10 1s22s22p63s23p64s23d104p1 1s22s22p63s23p64s23d104p2 1s22s22p63s23p64s23d104p3 1s22s22p63s23p64s23d104p4 1s22s22p63s23p64s23d104p5 1s22s22p63s23p64s23d104p6

Table 10.1 Electronic configurations of atoms up to

ܰ‘—–‡” 1 2/0* 1 2 3 4 5 6 7 8/0* 1 2 3 4 5 6 7 8/0* 1 2 3 4 5 6 7 8 9 10 11 12/2* 3 4 5 6 7 8/0*

ܼ ൌ ͵͸ and number of outer electrons

*Number if outer shell is taken to be that of the preceding atom/next atom.

Configurations can be simplified by representing the configuration of the preceding inactive gas by its symbol in bold, and by enclosing the inner electrons in square brackets. This gives for bromine, for example, [Ar3d10]4s24p5. Note how the 3d electrons have become part of the core.

39

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

Table 10.1 can be extended to heavier atoms. Silver atoms used in the Stern-Gerlach experiment have a similar configuration to copper, viz. [Kr4d10]5s1. With their one outer electron, they behave in this experiment like hydrogen atoms. The configurations in Table 10.1 correlate with the Periodic Table and Lewis’s model of the atom (Lewis 1916). Thus, for example, the inactive gases (high-lighted) all have complete shells. The alkali metals have one more electron than this, which they can lose to form M+ ions. The halogens have one electron less, and can gain an electron to form X‒ ions. For further details, see my e-book Introduction to Chemistry. Configurations of ions do not follow the same order of filling as atoms. For cations the order becomes 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 5s … with the 3d spin-orbitals in line. Thus, for example, while the configuration of an iron atom is [Ar]4s23d6, the configuration of a Fe2+ ion is [Ar]3d6 and of a Fe3+ ion is [Ar]3d5.

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40

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

Indistinguishability of electrons For many atoms and ions, equation (10.3) needs to be modified to take account of the indistinguishability of electrons. Consider a helium atom with the configuration 1s12s1. For this, equation (10.3) gives ߰ሺͳǡʹሻ ൎ ߮ͳ• ሺͳሻ߮ʹ• ሺʹሻ or ߮ʹ• ሺͳሻ߮ͳ•ሺʹሻ(10.8)

from which

ଶ ሺ ሻ ଶ ሺ ሻ ଶ ଶ ߩሺͳǡʹሻ ൎ ߮ͳ• ሺͳሻ߮ͳ• ሺʹሻ(10.9) ͳ ߮ʹ• ʹ RU߮ʹ•

These functions distinguish the electrons: the first places electron 1 in the 1s orbital and electron 2 in the 2s; the second does the reverse. This problem can be overcome by taking linear combinations

or

ଵ

ሾ߮ͳ• ሺͳሻ߮ʹ•ሺʹሻ + ߮ʹ• ሺͳሻ߮ͳ• ሺʹሻሿ(10.10)

ଵ

ሾ߮ͳ• ሺͳሻ߮ʹ•ሺʹሻ ‒ ߮ʹ• ሺͳሻ߮ͳ• ሺʹሻሿ(10.11)

߰ଵ ሺͳǡʹሻ ൎ

ξଶ

߰ଶ ሺͳǡʹሻ ൎ

ξଶ

from which

ଵ

or

ߩଵ ሺͳǡʹሻ ൎ ሾ߮ͳ• ሺͳሻ߮ʹ• ሺʹሻ + ߮ʹ• ሺͳሻ߮ͳ• ሺʹሻሿ2(10.12) ଶ

ଵ

ߩଶ ሺͳǡʹሻ ൎ ଶ ሾ߮ͳ• ሺͳሻ߮ʹ• ሺʹሻ ‒ ߮ʹ• ሺͳሻ߮ͳ• ሺʹሻሿ2(10.13)

In equations (10.12) and (10.13), electrons 1 and 2 can now be interchanged without ଵ and ͳ come from the normalization (Sect. 6). changing the function. The factors ξଶ

ʹ

The two states described by equations (10.10)‒(10.13) I consider further below.

41

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

Spin-orbitals To obtain spin-orbitals, equation (10.3) or its combinations have to be multiplied by spin functions. For two electrons, these are ߪଵ ሺͳǡʹሻ ൌ ߙሺͳሻߙሺʹሻ(10.14)

ߪଶ ሺͳǡʹሻ ൌ ߙሺͳሻߚሺʹሻ(10.15)

ߪଷ ሺͳǡʹሻ ൌ ߚሺͳሻߙሺʹሻ(10.16) ߪସ ሺͳǡʹሻ ൌ ߚሺͳሻߚሺʹሻ(10.17)

Of these, ߪଵ and ߪସ do not distinguish the electrons, but ߪଶ and ߪଷ do, and must be replaced by linear combinations: ߪା ሺͳǡʹሻ ൌ ߪି ሺͳǡʹሻ ൌ

ଵ

ξଶ ଵ

ξଶ

ሾߙሺͳሻߚሺʹሻ ൅ ߚሺͳሻߙሺʹሻሿ(10.18) ሾߙሺͳሻߚሺʹሻ െ ߚሺͳሻߙሺʹሻሿ(10.19)

As shown in Box C, functions ߪଵ, ߪା, and ߪସ correspond to a total spin of one, function ߪି to a total spin of zero. In the latter, the electrons are paired, in the former, unpaired. Box C Simplified treatment of spin To calculate the total spin for a system of two electrons, we define a mathematical operator å by the equation åఈ ఈ

where ‫ݏ‬

ൌ

åఉ ఉ

ଵ

ൌ ‫ݏ‬

(C1)

ൌ ଶ . Thus for ߪଵ ሺͳǡʹሻ, the total spin is given by

total spin ൌ

ሺåభ ାåమሻఙభ ሺଵǡଶሻ ఙభ ሺଵǡଶሻ

Similarly for ߪା, ߪି, and ߪସ .

ൌ

ሺåభ ାåమ ሻఈሺଵሻఈሺଶሻ ఈሺଵሻఈሺଶሻ

ൌ

ሾåభ ఈሺଵሻሿఈሺଶሻାఈሺଵሻሾåమ ఈሺଶሻሿ ఈሺଵሻఈሺଶሻ

ൌ ʹ‫ ݏ‬ൌ ͳ (C2)

General Pauli principle We have seen that the Pauli exclusion principle requires the ground state of the helium atom to be 1s2 with the two electrons paired. The total wave-function for this is obtained by combining equation (10.3)

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

߰ሺͳǡʹሻ ൎ ߮ͳ• ሺͳሻ߮ͳ• ሺʹሻ(10.20)

with ߪି [equation (10.19)] to give ߯ሺͳǡʹሻ ൎ

ଵ

ξଶ

߮ͳ• ሺͳሻ߮ͳ•ሺʹሻሾߙሺͳሻߚሺʹሻ െ ߚሺͳሻߙሺʹሻሿ(10.21)

This equation differs from the equations that would be obtained by combining equation (10.20) with the other spin functions (ߪଵ, ߪା, and ߪସ) in that it changes sign when electrons 1 and 2 are interchanged. It is said to be “asymmetric” with respect to this interchange. This leads to the general Pauli principle: wave functions of electron systems are asymmetric with respect to interchange of electrons. We can now return to the 1s12s1 configuration of the helium atom. To satisfy the general Pauli principle, equation (10.10) must be combined with ߪି to produce a single state (“singlet”), whereas equation (10.11) must be combined with ߪଵ, ߪା, and ߪସ to produce a “triplet”. Thus from equations (10.10) and (10.19), ଵ

߯•‹‰Ž‡– ሺͳǡʹሻ ൎ ଶ ሾ߮ͳ•ሺͳሻ߮ʹ• ሺʹሻ + ߮ʹ• ሺͳሻ߮ͳ• ሺʹሻሿሾߙሺͳሻߚሺʹሻ െ ߚሺͳሻߙሺʹሻሿ(10.22)

while from equations (10.11), (10.14), (10.17), and (10.18),

43

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

߯–”‹’Ž‡– ሺͳǡʹሻ ൎ

ଵ

ξଶ

Modified Schrödinger theory of other atoms

- ߙሺͳሻߙሺʹሻ

(10.23)

- ߚሺͳሻߚሺʹሻ

(10.25)

ሾ߮ͳ• ሺͳሻ߮ʹ•ሺʹሻ ‒ ߮ʹ•ሺͳሻ߮ͳ• ሺʹሻሿ-

ଵ

ξଶ

ሾߙሺͳሻߚሺʹሻ ൅ ߚ ሺͳሻߙሺʹሻሿ

(10.24)

Energies For the 1s12s1 configuration of the helium atom, which is lower in energy, the singlet or the triplet? Spectroscopy indicates the triplet, but this can be derived by substituting equations (10.22) and (10.23), (10.24), or (10.25) into the modified form of equation (6.20): ‫ ܧ‬ൌ ‫† ߯>߯ ׬‬ɒ (10.26)

Here > for a helium atom is, from equations (10.1) and (10.2), ௛మ

ʹ݁ʹ Ͳ ‫ͳݎ‬

௛మ

> ൌ െ ଼Ɏమఓ ‫׏‬ଵଶ െ ଼Ɏమ ఓ ‫׏‬ଶଶ െ ͶɎߝ

This can be simplified to

ʹ݁ʹ Ͳ ‫ʹݎ‬

െ ͶɎߝ

݁ʹ (10.27) Ͳ ‫ʹͳݎ‬

൅ ͶɎߝ

݁ʹ (10.28) Ͳ ‫ʹͳݎ‬

> ൌ >ଵ ൅ >ଶ ൅ ͶɎߝ



where > is the Hamiltonian operator for an electron moving in the field of the nucleus.

The results are:

  ‫ –‡Ž‰‹•ܧ‬ൎ ‫ܧ‬ଵ• ൅ ‫ܧ‬ଶ• ൅ ‫ ܬ‬൅ ‫(ܭ‬10.29)

where

  ‫ –‡Ž’‹”–ܧ‬ൎ ‫ܧ‬ଵ• ൅ ‫ܧ‬ଶ• ൅ ‫ ܬ‬െ ‫(ܭ‬10.30)



 ‫•ͳܧ‬ ൌ ‫(߬† •ͳ߮ > •ͳ߮ ׬‬10.31) 

 ‫•ʹܧ‬ ൌ ‫(߬† •ʹ߮ > •ʹ߮ ׬‬10.32)

‫ܬ‬ൌቀ

௘మ

ସɎఌబ ௘మ

ቁ‫׭‬

మ ሺଵሻఝమ ሺଶሻ ఝͳ• ʹ•

‫ ܭ‬ൌ ቀସɎఌ ቁ ‫׭‬ బ

௥భమ

†߬ଵ †߬ଶ(10.33)

ఝͳ•ሺଵሻఝʹ• ሺଵሻఝͳ•ሺଶሻఝʹ•ሺଶሻ ௥భమ

†߬ଵ †߬ଶ(10.34)

  and ‫•ʹܧ‬ are negative quantities, ‫ ܬ‬and ‫ ܭ‬are positive. Thus, if the orbitals are the same ‫•ͳܧ‬ in the two states, ‫ –‡Ž’‹”–ܧ‬is more negative than ‫–‡Ž‰‹•ܧ‬. This result remains true if the orbitals

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of other atoms

are optimized for each state. ‫ ܬ‬represents the classical energy of repulsion between the ଶ ଶ charge distributions ߮ͳ• ݁ and is called the “Coulomb energy”. ‫ ܭ‬is a non-classical ݁ and ߮ʹ• quantity, and is called the “exchange energy”. It is this that gives, when the orbitals are the same, the high-spin state a lower energy than the low-spin.

Self-consistent field (SCF) calculations In the previous section, I did not specify the orbitals. For simplicity, they could have been the orbitals for the He+ ion. To improve on these, we can use the variation method (Sect. 6). Suitable functions are chosen for ߮ͳ• and ߮ʹ• containing variable parameters. The parameters in ߮ʹ• are first varied to minimize the energy, then the parameters in ߮ͳ•, then again the parameters in ߮ʹ•, and so on, until the results do not change. At this point, the fields created by ߮ͳ• and ߮ʹ• are self-consistent. The best possible energy obtained in this way is less negative than the true value because of the limitations of the orbital approximation. Electrons do not move in orbitals as if the other electrons are not there. Rather, they tend to keep away from each other. Their motions are accordingly correlated. The energy arising from this is called the “correlation energy”. This can be estimated by comparison with similar systems, or computed by a calculation beyond the orbital approximation.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

11 MODIFIED SCHRÖDINGER THEORY OF MOLECULES Since, in general, nuclei move more slowly than electrons, we can treat the electrons in a molecule as moving round fixed nuclei. This is called the “Born-Oppenheimer approximation”. The appropriate Schrödinger equation in this case is again equation (10.1), i.e. σ௜

௛మ

଼Ɏమ ఓ

‫׏‬ଶ௜ ߰ + ሺ‫ ܧ‬െ ܸ ሻ߰ = 0(11.1)

Hydrogen molecule-cation The simplest molecule is the hydrogen molecule-cation, ଶା , with one electron moving round two nuclei. For this, equation (11.1) simplifies to ௛మ

଼Ɏమ ఓ

‫׏‬ଶ ߰ + ሺ‫ ܧ‬െ ܸ ሻ߰ = 0(11.2)

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

with ௘మ

ܸ ൌ  െ ସɎఌ

బ ௥

௘మ

െ ସɎఌ

బ ௥

௘మ

൅ ସɎఌ

బ ௥

(11.3)

where A and B label the nuclei, ‫ ݎ‬is the distance of the electron from nucleus A, ‫ݎ‬ from nucleus B, and ‫ ݎ‬the distance between the two nuclei. The last term represents the internuclear repulsion and is a constant in the calculation. Equation (11.2) can be solved exactly, and gives a series of energy levels and orbitals (“molecular orbitals”, MO) in the same way as the Schrödinger equation for the hydrogen atom. The equation can also be solved approximately by a method that can be used for other molecules where an exact solution is not possible. This is based on the idea that, when the electron is near one nucleus, its motion will be similar to that in an atom centred on that nucleus. This suggests the approximation ߰ ൎ ܿ ߮ ൅ ܿ ߮(11.4)

where ߮ is the wave-function for a hydrogen atom centred on nucleus A and ߮ on B (these may be 1s, 2s, 2p etc.); ܿ and ܿ are numerical coefficients whose values have to be determined. This approximation involves a linear combination (LC) of atomic orbitals (AO) and is called the “LCAO approximation”. Equation (11.4) gives the probability distribution as ߩ ൎ ܿଶ ߮ଶ ൅ ʹܿ ܿ ߮ ߮ ൅ ܿଶ ߮ଶ(11.5)

In the case of the ଶା ion, symmetry requires ܿଶ in this equation to be equal to ܿଶ or ܿ ൌ േܿ . The value of ܿ comes from the normalization condition ‫ ߬†ߩ ׬‬ൌ ͳ(11.6)

Substituting equation (11.5) into this gives ܿ ൌ

ଵ

ඥሺଶേଶௌሻ

(11.7)

where ܵ ൌ ‫ ߬† ߮ ߮ ׬‬. This is called the “overlap integral” as it measures the degree of overlap of ߮ and ߮ . It varies in value between zero (no overlap) and one (complete overlap). For most molecules at their equilibrium internuclear distance, S is small (0.2‒0.3), but for the ଶା ion and H2 molecule it is higher (0.59 and 0.73 respectively). The atomic orbitals ߮ and ߮ therefore give two molecular orbitals ߰ା ൎ

ఝ ାఝ

ඥሺଶାଶௌሻ

(11.8)

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

and ߰ି ൎ

ఝିఝ

ඥሺଶିଶௌሻ

(11.8)

From equation (6.20), these give

and

‫ܧ‬ା ൎ

ఈାఉ

(11.9)

‫ ିܧ‬ൎ

ఈିఉ

(11.10)

ଵାௌ

ଵିௌ

where ߙ ൌ ‫† ߮> ߮ ׬‬ɒ ൌ ‫† ߮> ߮ ׬‬ɒ and ߚ ൌ ‫† ߮> ߮ ׬‬ɒ ൌ ‫† ߮> ߮ ׬‬ɒ . . The integral ߙ is called a “Coulomb integral” and is negative; it is approximately equal to the energy of an electron in orbital ߮ of a hydrogen atom. The integral ߚ is called a “resonance integral” and is also generally negative. For the ଶା ion at its equilibrium internuclear distance, calculations give ߚ ൌ ͲǤͺͺߙ . With S = 0.59, the values of ‫ܧ‬ା and ‫ ିܧ‬are accordingly 1.18ߙ and 0.29ߙ respectively. ‫ܧ‬ା is therefore more negative than the energy of an electron in orbital ߮ of a hydrogen atom, while ‫ ିܧ‬is less negative. Consequently, ߰ାଶ is called a “bonding orbital” and ߰ିଶ an “antibonding orbital”.

All the orbitals of a hydrogen atom give rise to molecular orbitals. The lowest energy is obtained with 1s orbitals. The molecular orbitals obtained from 1s orbitals are shown in Figure 11.1 and their energies in Figure 11.2. The bonding orbital is designated ɐଵ• and the ‫ כ‬, where σ signifies that the orbital is symmetrical around the internuclear antibonding ɐଵ• axis, as would be expected from combining s orbitals. The ground state of an ଶା ion is therefore ɐଵଵ• .

Figure 11.1 Molecular orbitals obtained by combining 1s orbitals [Interactive Student Tutorial]

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Energy

Modified Schrödinger theory of molecules

1sA

‫כ‬ ɐଵ•

1sB

ɐଵ•

Figure 11.2 Energies of molecular orbitals obtained by combining 1s orbitals

Hydrogen molecule and other Period 1 homonuclear diatomic molecules and molecular ions The hydrogen molecule, like the helium atom, can be treated in the orbital approximation ଶ . The molecule thus has two bonding [equation (10.3)]. The ground state is then ɐଵ• electrons, twice as many as ଶା . Their respective bond numbers are therefore 1 and ½. This is reflected in their experimental dissociation energies, 4.48 and 2.65 eV respectively. There is a similar correlation for other Period 1 homonuclear diatomic molecules and their ions (Table 11.1).

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

Molecule

Configuration

Number

D/eV

ଶା

ɐଵଵ•

½

2.65

1

4.48

ଶି

ଶ ‫כ‬ଵ ɐଵ• ɐଵ•

½ ½

2.47

‡ଶ

ଶ ‫כ‬ଶ ɐଵ• ɐଵ•

0

0.00

ଶ

‡ା ଶ

ଶ ɐଵ•

ଶ ‫כ‬ଵ ɐଵ• ɐଵ•

Table 11.1 Configurations and bond numbers of Period 1 homonuclear diatomic molecules and their ions compared with their dissociation energies (D)

Period 2 homonuclear diatomic molecules and molecular ions Other homonuclear diatomic molecules and molecular ions can be treated by combining atomic orbitals. For Period 2 molecules, the 1s orbitals are inner orbitals, so overlap between ‫ כ‬molecular orbitals, but them is small. As we have seen, they combine to give ɐଵ• and ɐଵ• these will differ little in energy. For Period 2 molecules, they will be full, and contribute no net bonding. The next atomic orbitals are 2s and 2p. To a first approximation, 2s orbitals combine to give ɐଶ• and ɐ‫כ‬ଶ• orbitals, 2pz orbitals (z being along the internuclear axis) combine to give ‫כ‬ ‫כ‬ ‫ כ‬orbitals, and 2p and 2p orbitals combine to give Ɏ ɐଶ’ and ɐଶ’ ଶ’ೣ, Ɏଶ’೤, Ɏଶ’ೣ, Ɏଶ’೤ and x y orbitals. The Ɏ orbitals are formed by p orbitals overlapping sideways and are symmetrical above and below a plane containing the nuclei. Because the p orbitals overlap sideways, Ɏ bonds are weaker than σ bonds. The resulting energy-level diagram is shown in Figure 11.3.

Figure 11.3 Molecular orbitals formed from 2s and 2p atomic orbitals (first approximation, holding for elements from O to Ne) [Bobcat Chemistry]

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

A complication is that both 2s and 2pz orbitals give σ orbitals, and, for the early elements in Period 2, their energies are close. Consequently, lower energies are obtained by taking linear combinations of these orbitals, and using the variation principle to determine the coefficients. The results are shown in Figure 11.4.

Figure 11.4 Molecular orbitals for Li2 to Ne2 [Open Text (modified)]

From Figure 11.4, we can assign ground-state configurations to the molecules from Li2 to Ne2 as shown in Table 11.2. Also in the table are their bond numbers and dissociation energies, showing a high degree of correlation between them. The last column gives the number of unpaired electrons. Species with unpaired electrons are paramagnetic. The theory correctly predicts that B2 and O2 have this property. The theory is therefore an improvement on that behind Figure 11.3, which wrongly predicts that B2 is diamagnetic and C2 paramagnetic.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

Molecule

Configuration

Number

D/eV

Unpaired

‹ଶ

ɐଶଶ•

1

1.06

0

0

0.61

0

1

~3.0

2

2

6.32

0

3

9.76

0

2

5.11

2

1

1.60

0

0

0.00

0

‡ଶ

ɐଶଶ• ɐ‫כ‬ଶ ଶ•

ଶ

ଵ ଵ ɐଶଶ• ɐ‫כ‬ଶ ଶ• Ɏʹ’ೣ Ɏʹ’೤

ଶ

ଶ ଶ ଶ ɐଶଶ• ɐ‫כ‬ଶ ଶ• Ɏʹ’ೣ Ɏʹ’೤ ɐଶ’

ଶ

ଶ ଶ ɐଶଶ• ɐ‫כ‬ଶ ଶ• Ɏʹ’ೣ Ɏʹ’೤

ଶ

ଶ ଶ ଶ ‫כ‬ଵ ‫כ‬ଵ ɐଶଶ• ɐ‫כ‬ଶ ଶ• ɐଶ’ Ɏʹ’ೣ Ɏʹ’೤ Ɏଶ’ೣ Ɏଶ’೤

‡ଶ

ଶ ‫כ‬ଶ ଶ ଶ ‫כ‬ଶ ‫כ‬ଶ ɐଶଶ• ɐ‫כ‬ଶ ଶ• ɐଶ’ Ɏʹ’ೣ Ɏʹ’೤ Ɏଶ’ೣ Ɏଶ’೤ ɐଶ’

ଶ

ଶ ଶ ଶ ‫כ‬ଶ ‫כ‬ଶ ɐଶଶ• ɐ‫כ‬ଶ ଶ• ɐଶ’ Ɏʹ’ೣ Ɏʹ’೤ Ɏଶ’ೣ Ɏଶ’೤

Table 11.2 Configurations and bond numbers of Period 2 homonuclear diatomic molecules compared with their dissociation energies (D)

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A similar table can be drawn up for the corresponding ions. Of interest are those occurring in compounds. These are listed in Table 11.3. Ion

Configuration

ଶଶି

ଶ ଶ ଶ ɐଶଶ• ɐ‫כ‬ଶ ଶ• Ɏʹ’ೣ Ɏʹ’೤ ɐଶ’

ଶ ଶ ଶ ‫כ‬ଵ ɐଶଶ• ɐ‫כ‬ଶ ଶ• ɐଶ’ Ɏʹ’ೣ Ɏʹ’೤ Ɏଶ’ೣ

ଶା

ଶ ଶ ଶ ‫כ‬ଶ ‫כ‬ଵ ɐଶଶ• ɐ‫כ‬ଶ ଶ• ɐଶ’ Ɏʹ’ೣ Ɏʹ’೤ Ɏଶ’ೣ Ɏଶ’೤

ଶି

ଶ ଶ ଶ ‫כ‬ଶ ‫כ‬ଶ ɐଶଶ• ɐ‫כ‬ଶ ଶ• ɐଶ’ Ɏʹ’ೣ Ɏʹ’೤ Ɏଶ’ೣ Ɏଶ’೤

ଶି ଶ

Number

D/eV

Unpaired

3

(8.69)a

0

2½

6.48

1

1½

4.09

1

1

(1.49)b

0

Table 11.3 Configurations and bond numbers of some Period 2 homonuclear diatomic molecular ions compared with their dissociation energies a

Bond energy in H2C2. b Bond energy in H2O2.

Hydrogen-helium cation The hydrogen-helium cation, HHe+, is isoelectronic with the hydrogen molecule. It is formed in discharge tubes containing hydrogen and helium. It differs from the hydrogen molecule in having nuclei with different charges. This means that we cannot put ܿଶ equal to ܿଶ in equation (11.5). Instead we have to express the energy in terms of ܿ and ܿ and use the variation method to get the best value. A simple alternative is to substitute equation (11.4) into equation (6.18), multiply the result by ߮ or ߮ , and integrate over all the coordinates. Multiplication by ߮ gives ܿ ߙ ൅ ܿ ߚ ൌ ‫ ܧ‬ሺܿ ൅ ܵܿ ሻ(11.11)

Multiplication by ߮ gives

ܿ ߚ ൅ ܿ ߙ ൌ ‫ܧ‬ሺܵܿ ൅ ܿ ሻ(11.12)

These give

௖

௖

ఈ ିா

ఉିாௌ

 ൌ െ ఉିாௌ ൌ െఈ

from which

 ିா

(11.13)

ሺߙ െ ‫ ܧ‬ሻሺߙ െ ‫ ܧ‬ሻ ൌ ሺߚ െ ‫ܵܧ‬ሻଶ(11.14) 53

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Modified Schrödinger theory of molecules

Equation (11.14) can be readily solved in the special case that ߙ ൌ ߙ. This then gives

ሺߙ െ ‫ܧ‬ሻଶ ൌ ሺߚ െ ‫ܵܧ‬ሻଶ(11.15)

from which equations (11.9) and (11.10) follow.

To see what happens when ߙ ് ߙ, we let B be more electronegative than A, and set (by way of illustration and to simplify the arithmetic) ߙ ൌ  െ͵ units, ߙ ൌ െ͸ units, ߚ ൌ െʹ units, and ܵ ൎ Ͳ. Equation (11.14) is then ሺെ͵ െ ‫ ܧ‬ሻሺെ͸ െ ‫ ܧ‬ሻ ൌ ሺʹሻଶ(11.16)

units. Substituting these values into equation the solutions of which are ‫ ܧ‬ൌ െ͹ units and െʹ—‹–• ଵ ௖ (11.13) gives ൌ ൅ʹ and െ ଶ respectively. The normalized wave-functions are therefore ௖

and

߮ା ൎ

ఝ ାଶఝ

߮ି ൎ

ଶఝ ିఝ

ξହ

ξହ

(11.17)

(11.18)

respectively. The first solution thus corresponds to the bonding orbital and is concentrated on B; the second solution corresponds to the antibonding orbital, and is concentrated on A. The bond is accordingly polar and the molecule has a dipole moment. The energy levels are as shown in Figure 11.5.

Energy

߮A

ɐ‫כ‬ ɐ

߮B

Figure 11.5 Energy levels arising by combining s orbitals having different energies

An approximate value for the dipole moment in our illustrative example can be obtained as follows. From equation (11.17) and ߮ ߮ ൎ Ͳ, ߩା ൎ

ఝమ ାସఝమ(11.19) ହ

ଵ

ସ

The fraction of time an electron spends on atom A is therefore and on B, . The difference ହ ହ for two electrons gives the dipole moment of the molecule: ଷ

ߤ ൎ ʹ ቀହቁ ݁‫(ݎ‬11.20) 54

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Modified Schrödinger theory of molecules

Other heteronuclear diatomic molecules The treatment of HHe+ can be extended to other heteronuclear diatomic molecules. Of particular interest to chemists are CN‒ and NO+ ions. These are isoelectronic with N2 and contain polar triple bonds.

Polyatomic molecules Polyatomic molecules can be treated in the same way as diatomic molecules, by fixing the positions of the nuclei, and combining atomic orbitals centred on these nuclei. The result is an energy-level diagram like that in Figure 11.6. ======

(antibonding)

‒‒‒‒‒‒‒

(antibonding)

Energy

====== 4H 1s

====== C 2p ‒‒‒‒‒‒‒ C 2s

====== (bonding)

‒‒‒‒‒‒‒ (bonding)

Figure 11.6 Molecular-orbital energy-level diagram for methane. The eight valence electrons occupy the bonding orbitals.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Valence-bond (VB) theory

12 VALENCE-BOND (VB) THEORY This is an earlier theory than molecular orbital theory described above, and is used particularly in describing the properties and reactions of organic compounds.

Hydrogen molecule VB theory supposes that the bond in the H2 molecule is formed by an electron in the 1s orbital of hydrogen atom A pairing up with an electron in the 1s orbital of B. Since the electrons are indistinguishable, this gives ߰ ൎ ܰሾ߮ ሺͳሻ߮ ሺʹሻ ൅ ߮ ሺʹሻ߮ ሺͳሻሿ(12.1)

where is ሺthe constant. To satisfy the general Pauli principle, this wave ߰ ൎ ܰሾ߮ ͳሻ߮normalization  ሺʹሻ ൅ ߮ ሺʹሻ߮  ሺͳሻሿ function is coupled with spin function ߪି [equation (10.19)].

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Valence-bond (VB) theory

Equation (12.1) may be compared with the corresponding MO expression [from equation (11.4)]: ߰ ൎ ܰሾ߮ ሺͳሻ ൅ ߮ ሺͳሻሿሾ߮ ሺʹሻ ൅ ߮ ሺʹሻሿ(12.2)

This differs from equation (12.1) in having terms in ߮ ሺͳሻ߮ ሺʹሻ and ߮ ሺͳሻ߮ ሺʹሻ , i.e. it gives more weight to both electrons being on the same atom.

Water molecule In terms of the orbitals ʹ’௫, ʹ’௬, and ʹ’௭ (Sect. 7), the outer configuration of an oxygen atom is ʹ• ଶʹ’ଵ௫ ʹ’ଵ௬ ʹ’ଶ௭. The atom can therefore form electron-pair bonds with two hydrogen atoms, ʹ’ଵ௫ coupling with the 1s electron of one and ʹ’ଵ௬ coupling with the 1s electron of the other. Since ʹ’௫ and ʹ’௬ are 90° apart, this predicts an H—O—H angle of 90°. The observed angle is 104.5°, showing that the treatment is only approximate.

Methane molecule A carbon atom has the outer configuration ʹ• ଶ ʹ’ଵ௫ ʹ’ଵ௬ . By the reasoning of the last section, we might expect it to form CH2. To describe CH4, it is necessary, first, to promote a 2s electron to the 2p shell to give the atom the configuration ʹ•ଵʹ’ଵ௫ ʹ’ଵ௬ ʹ’ଵ௭ . The orbitals have then to be combined in such a way as to give four identical orbitals (h) directed towards the corners of a tetrahedron. These combinations are: ଵ

Šଵ ൌ ሺ•൅’௫ ൅ ’௬ ൅ ’௭ ሻ(12.3) ଶ

ଵ Šଶ ൌ ሺ•ԟ’௫ ԟ ’௬ ൅ ’௭ ሻ(12.4) ଶ

ଵ

Šଷ ൌ ଶ ሺ•ԟ’௫ ൅ ’௬ ԟ’௭ ሻ(12.5) ଵ Šସ ൌ ଶ ሺ•൅’௫ ԟ’௬ ԟ’௭ ሻ(12.6)

These orbitals can couple with the 1s orbitals of four hydrogen atoms to form CH4. The first process is called “promotion”, the second, which is mathematical, is called “hybridization”. The corresponding energy level diagram is shown in Figure 12.1, and may be compared with Figure 11.6. Both approaches have their merits. Figure 12.1 brings out the equivalence of the bonds; Figure 11.6 ties in with the photoelectron spectrum which has two peaks in the appropriate energy range with intensities in the ratio 1:3.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Valence-bond (VB) theory

======

(antibonding)

====== C 2p Energy

‒‒‒‒‒‒‒ C 2s ====== 4H 1s

====== (bonding)

Figure 12.1 Valence-bond energy-level diagram for methane.

Other hybrid orbitals are set out in Table 12.1. Exercise Show that the hybrids in equations (12.3) – (12.6) are normalized and orthogonal. You can assume that the functions s, px, py, and pz have these properties. Use the conditions ‫ ߰ ׬‬ଶ †߬ ൌ ͳ and ‫߰߰ ׬‬Ԣ†߬ ൌ Ͳ (Sect. 6). Type

Components

Shape

Angle

sp

s, pz

linear

180°

sp2

s, px, py

trigonal planar

120°

sp3

s, px, py, pz

tetrahedral

109.5°

Table 12.1 Some hybrid orbitals

Ethylene molecule VB theory can readily be applied to other organic molecules. For example, ethylene can be formulated with ʹ•ʹ’௫ ʹ’௬ hybrids on the carbon atoms (Table 12.1). These are at 120° and form the ɐ bonds: H

H

C—C H

H 58

AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Valence-bond (VB) theory

The ʹ’௭ orbitals are at 90° to these, and combine to form a π bond. Benzene molecule The Kekulé structures for benzene (German benzol) can be formulated in a similar way to ethylene.

These structures can be mathematically combined to give a structure with a lower energy and with all the C—C bonds the same: ߰ ൌ ܰሺ߰ଵ ൅ ߰ଶ ሻ(12.7)

Following Linus Pauling, the effect of combining valence bond structures is called “resonance”, and the structures that are combined are called “resonance structures”.

Oxygen molecule This can be formulated in the same way as ethylene, only with lone pairs in place of the bonds to hydrogen. This gives a bond number of two, as does MO theory (Table 11.2). Unlike MO theory, however, it predicts that all the electrons will be paired, whereas MO theory predicts that two will be unpaired. The paramagnetism of oxygen confirms the latter.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Ligand-field theory

13 LIGAND-FIELD THEORY Ligand-field theory is a theory of transition-metal compounds. These have d electrons in their valence shells. There are five d orbitals, with ݉௟ ൌ Ͳǡ േͳǡ േʹ (Sect. 7). These can be combined to give the orbitals shown in Figure 13.1.

Figure 13.1 Diagrams showing the shapes of d orbitals [chemeddl]

The orbitals in Figure 13.1 combine with orbitals of ligands to give molecular orbitals. How they combine depends on the shape of the metal-ligand system. If it is octahedral, as

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Ligand-field theory

it usually is, the d orbitals split into two groups. The orbitals †௭ మ and †௫ మି௬మ (labelled eg) point directly towards the ligands and interact with them to give bonding and antibonding MOs. The bonding orbitals are occupied by ligand electrons, leaving the antibonding orbitals available for metal d electrons. The remaining d orbitals (– ଶ‰) point between the ligands, and either do not interact with them or interact relatively weakly. The orbitals available for metal d electrons are therefore as shown in Figure 13.2. —— —— (from ‡‰ ) —— —— —— (– ଶ‰ )

߂‘…–

Figure 13.2 Orbitals available to metal d electrons in an octahedral species

The splitting in Figure 13.2 is called the “ligand-field splitting”. This gives rise to a band or bands in the absorption spectrum, from which it can be calculated. For an octahedral species, the splitting is denoted ߂‘…– . If ߂‘…– is small, the metal d electrons distribute themselves among the orbitals available to them in the same way as in a free metal ion in the same oxidation state. If, however, ߂‘…– is large, the electrons fill the lower level preferentially. This leads to the formation of high-spin or low-spin species. Consider, for example, the ferric complexes [FeF6]3‒ and [Fe(CN)6]3‒. The corresponding ହ free ion (Fe3+) has five 3d electrons, which occupy the 3d orbitals singly (total spin ൌ ). ଶ The complex [FeF6]3‒ has the same arrangement as the free ion and is high-spin. On the other hand, the complex [Fe(CN)6]3‒ has its 3d electrons in the lower set of orbitals, leaving ଵ only one unpaired electron (total spin ൌ ). It is therefore low-spin. This difference shows ଶ itself in the magnetic properties of the two complexes. Ligand-field theory rationalizes a great deal of transition-metal chemistry.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Density functional theory (DFT)

14 DENSITY FUNCTIONAL THEORY (DFT) Calculations on many-electron systems can be simplified by making use of the theorem that the properties of such systems are determined by the overall electron density, ߩ (Hohenberg and Kohn 1964). This makes the energy a “functional” of ߩ, i.e. a function of a function ( ߩ being a function of position). This means that the energy can be written ‫ ܧ‬ሾߩሿ ൌ ܶሾߩሿ ൅  ܸ‡ ሾߩሿ ൅ ܸ‡‡ ሾߩሿ(14.1)

Here ܶ is the total kinetic energy of the electrons, ܸ‡ is the total energy of attraction between the electrons and the nuclei, and the ܸ‡‡ total energy of repulsion between the electrons. The last term can be further written ܸ‡‡ ሾߩሿ ൌ ‫ܬ‬ሾߩሿ ൅ ‫ ܧ‬ሾߩሿ(14.2)

where ‫ ܬ‬is the classical energy of repulsion between the electrons (the Coulomb energy) and ‫ ܧ‬covers exchange and correlation (Sect. 10).

Now according to a second Hohenberg-Kohn theorem, the lowest value of ‫ ܧ‬ሾߩሿ is the true energy. This provides a variation method for determining ‫ܧ‬. This is simpler than the variation method for solving the Schrödinger equation because, while ߰ depends on the coordinates of all the electrons, ߩ only depends on ‫ݔ‬, ‫ݕ‬, and ‫ݖ‬. In what follows, I will represent ‫ݔ‬, ‫ݕ‬, ‫ ݖ‬by ߬. To minimize equation (14.1), we need expressions for the terms in it. For ܸ‡ and ‫ܬ‬, we can use the classical equations ௘మ

and

ܸ‡ ሾߩሿ ൌ െ ቀସɎఌ ቁ σ ‫׬‬ ௘మ

ଵ

బ

‫ܬ‬ሾߩሿ ൌ ቀସɎఌ ቁ ଶ ‫׭‬ బ

௓ఘሺఛሻ ௥

ఘሺఛభ ሻఘሺఛమ ሻ ௥భమ

†߬ (14.3)

†߬ଵ †߬ଶ(14.4)

In equation (14.3),  labels the nuclei and ‫ ݎ‬is the distance of any given point in the electron density distribution from nucleus . In equation (14.4), the numbers 1 and 2 specify two different points in the electron density distribution, a distance ‫ݎ‬ଵଶ apart.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Density functional theory (DFT)

For ܶ it is usual to go back to orbitals (Kohn and Sham 1965). From equation (6.12), ௛మ

ܶ ൌ െ ଼Ɏమ௠

‡

‫׏‬మ ట ట

(14.5)

However, workers are trying to find simpler functionals for ܶ, inspired by that for a free electron gas: ଷ

మ

ఱ

ܶሾߩሿ ൌ ଵ଴ ሺ͵Ɏଶ ሻయ ‫ߩ ׬‬య †߬ (14.6)

Finally, a suitable functional has to be found for ‫ ܧ‬The simplest is to approximate it to that for a free electron gas:

‫ ܧ‬ሾߩሿ ൎ ‫ ߝ ׬‬ሺߩሻߩ†߬(14.7)

where ߝ is the exchange-correlation per electron in a homogeneous gas of constant density. This is called the “local-density approximation”. This reproduces energies with an accuracy of 10‒20%, and bond lengths with an accuracy of 2%. Other functionals have been proposed, which improve the accuracy.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Perspective on the quantum theory

15 PERSPECTIVE ON THE QUANTUM THEORY The quantum theory is remarkably successful. It has been tested in many ways and so far, has passed every test. Yet the physical basis of the theory remains obscure, even after many years of enquiry. Scientists differ widely over their understanding of it. In particular, some scientists worry that the theory describes physical systems like a single hydrogen atom probabilistically. They wonder whether there are “hidden variables” determining the outcome of quantum events. I hinted at this when I introduced the idea of a sub-electronic medium in Section 5. One sceptic was Einstein, who argued, “God does not play dice with the universe”. As I mentioned in Section 7, physicists have suggested other interpretations of ߰ from the one I have presented here. I survey these in my article: “How Do Electrons Get Across Nodes? A Problem in the Interpretation of the Quantum Theory”, – Journal of Chemical Education, 1990, Vol. 67, pp. 643–647.

Of particular interest is the proposal of Bohm (1952). He recast Schrödinger’s equation so that the kinetic energy is now classical and the potential energy non-classical. The latter has a term added to it called the “quantum potential”. The result for a hydrogen atom is that the electron in the ground state is stationary. This outcome is quite close to Lewis’ picture of electrons in atoms and molecules being in fixed positions and Coulomb’s law breaking down (Sect. 2). Bohm and Vigier (1954) associate the quantum potential with the motion of a sub-electronic fluid, but other physicists dismiss it as a construct of the mathematics. More work on the quantum theory remains to be done!

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Further reading

16 FURTHER READING There are several textbooks on the application of the quantum theory to chemistry. These include Peter W. Atkins and Ronald S. Friedman, Molecular Quantum Mechanics, 4th edn., Oxford University Press, 2004; Christopher J. Cramer, Essentials of Computational Chemistry, 2nd edn., Wiley, 2004. Applications of basic theories are also given in advanced textbooks of organic chemistry and inorganic chemistry.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Questions

QUESTIONS The questions are graded; Questions 7‒10 are more difficult. 1. The spectrum of sunlight consists of the colours of the rainbow with series of dark lines. Some of these lines (C, F, G) satisfy equation (4.2). What are they?

[Wikipedia] ଵ

2. The potential energy of a simple harmonic oscillator is given by ܸ ൌ ଶ ݇‫ ݔ‬ଶ, where ݇ is a constant (the “force constant”). Write down the Schrödinger equation for this system. 3. What is the electronic configuration of (a) a Co atom, (b) a Co2+ ion, (c) a Co3+ ion? 4. Describe the bonding in the acetylene molecule according to VB theory.

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Questions

5. Predict the number of unpaired electrons possessed by the cobaltic complexes [CoF6]3‒ and [Co(CN)6]3‒. You may assume that the splittings are similar to those for the corresponding ferric complexes. 6. Calculate the dissociation energy of the ଶା ion in the LCAO approximation using the values ߙ ൌ െͳ͵Ǥͳ eV, ߚ ൌ െͳͳǤͶ eV, and ܵ ൌ ͲǤͷͻ. మ 7. The function ߰ ൌ ܰ‡ି௔௫ is a solution to the Schrödinger equation in Question 2, Ɏ † †௙ where ܽ ൌ ௛ ξ݉݇. Determine ‫ ܧ‬for this solution. [Hint: ‡௙ ൌ ‡௙ ] †௫ †௫ 8. Derive a possible wave function for a particle on a ring of radius r from equation (6.7) for a particle in a box. Let x be distance round the circle, hence ߠ ൌ ‫ݔ‬Ȁ‫ݎ‬. [Hint: make sure ߰ is single valued.] ߠ r

‫ݔ‬

9. Calculate the energy of the wave-function in Question 8. 10. The ଷା ion occurs in hydrogen discharges and interstellar space. It is triangular in shape (angles 120°). Calculate the energy of the bonding molecular orbital formed by adding 1s orbitals centred on the three nuclei (A, B, and C).

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Answers

ANSWERS 1. The lines arise from hydrogen atoms in the sun absorbing energy from the sun’s light. Compare Table 4.2, where C, F, and G are the first three lines. †మ ట

2. †௫ మ +

଼Ɏమ ௠ ቀ‫ܧ‬ ௛మ

ͳ ʹ

െ ݇‫ ʹݔ‬ቁ ߰ = 0

3. (a) [Ar]4s23d7; (b) [Ar]3d7; (c) [Ar]3d6 4. The carbon atoms are •’௭ hybridized to form the ɐ bonds; the ’௫ and ’௬ orbitals combine to form Ɏ௫ and Ɏ௬ bonds at 90° to the ɐ bonds and to each other. 5. [CoF6]3‒, 4; [Co(CN)6]3‒, 0 6. Use equation (11.9), ‫ܧ‬ሺሻ ൌ  െͳ͵Ǥ͸ eV (Sect. 4), and ‫ܧ‬ሺ ା ሻ ൌ Ͳ. These give ‫ ܦ‬ൌ ‫ܧ‬ሺሻ ൅ ‫ܧ‬ሺ ା ሻ െ ‫ܧ‬ሺଶା ሻ ൎ െͳ͵Ǥ͸ െ ሺെͳͷǤͶሻ ൌ ͳǤͺ eV. The experimental value is 2.65 eV. ଵ ௞ ଵ ଵ 7. ൌ ଶ ݄ߥ, where ߥ ൌ ଶɎ ට௠ . The general solution is ൌሺ݊ ൅ ଶሻ݄ߥ (݊ ൌ Ͳǡ ͳǡ ʹ ǥ) . 8. In the case of a one-dimensional box, we had to fit a whole number of half wavelengths in the box (Fig. 6.1). For a particle confined to a circle, fitting a whole number of half wavelengths into the circumference makes the wave continue double valued. To avoid this, round the circle with the opposite sign, rendering ߰ ൌ …‘• ݊ߠ a whole number of full wavelengths must be fitted round the circumference. This ௫ gives ߰ ൌ •‹ ݊ ൌ •‹ ݊ߠ with ݊ ൌ ͳǡ ʹ ǥ Another possible solution is ߰ ൌ …‘• ݊ߠ ௥ ᇲ with ݊ ൌ Ͳǡ ͳǡ ʹ ǥ These can be combined using complex numbers to give ߰ ൌ ݁ ‹௡ ఏ with ݊ ᇱ ൌ Ͳǡ േͳǡ േʹǡ ǥ ௡మ ௛మ 9. Use equation (6.9) with V = 0. This gives ‫ ܧ‬ൌ ଼Ɏమ ூ , where I (the moment of inertia) = mr2. 10. The orbital is ߰ ൎ ܰሺ߮ ൅ ߮ ൅ ߮ ሻ. Substitute this into equation (6.18), multiply ఈାଶఉ the result by ߮, and integrate. Answer: ‫ ܧ‬ൎ ଵାଶௌ

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Appendix

APPENDIX Relativity Equations We start from equation (B1) in Box 1. This describes how the mass (m) of an object varies with its speed (‫)ݒ‬: ݉ൌ

௠బ

మ

ටଵିೡమ ೎

(1)

where ݉଴ is the rest mass and c the speed of light. Squaring this equation gives:

‫ ݒ‬ଶ ݉ଶ ൅ ܿ ଶ ݉଴ଶ ൌ ܿ ଶ ݉ଶ(2)

We shall require the differential of this:

݉‫ ݒ†ݒ‬൅ ‫ ݒ‬ଶ †݉ ൌ ܿ ଶ †݉ (3)

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AN INTRODUCTION TO THE QUANTUM THEORY FOR CHEMISTS: AN EXPERIMENTAL APPROACH

Appendix

Now according to classical mechanics, the force (F) required to increase the speed of an object is given by: F = rate of change of momentum =

†௣ †௧

(4)

where ‫ ݌‬ൌ ݉‫ݒ‬. The corresponding increase in energy (E) is given by:

dE = force × distance = ‫(ݔ†ܨ‬5)

Combining equations (4) and (5) gives: †௣

†௫

†‫ ܧ‬ൌ ቀ †௧ ቁ †‫ ݔ‬ൌ ቀ †௧ ቁ †‫ ݌‬ൌ ‫ ݌†ݒ‬ൌ ‫†ݒ‬ሺ݉‫ݒ‬ሻ ൌ ݉‫ ݒ†ݒ‬൅ ‫ ݒ‬ଶ †݉ (6)

Combining equations (3) and (6) thus gives:

†‫ ܧ‬ൌ ܿ ଶ †݉ (7)

Integrating this from E = 0 when m = 0 gives Einstein’s celebrated equation,

‫ ܧ‬ൌ ݉ܿ ଶ (8)

This equation can be recast by first squaring it:

‫ ܧ‬ଶ ൌ ݉ଶ ܿ ସ ൌ ݉଴ଶ ܿ ସ ൅  ሺ݉ଶ ԟ ݉଴ଶ ሻܿ ସ(9)

From equation (5), ݉ଶ ԟ ݉଴ଶ ൌ

௠మ௩ మ ௖మ

(10)

Substituting this into equation (9) gives ‫ ܧ‬ଶ ൌ ݉଴ଶ ܿ ସ ൅  ݉ଶ ‫ ݒ‬ଶ ܿ ଶ ൌ ݉଴ଶ ܿ ସ ൅  ‫݌‬ଶ ܿ ଶ(11)

Since the energy E is entirely kinetic, the square root of this gives equation (8.1) in the text: ܶ ൌ ඥ‫݌‬ଶ ܿ ଶ ൅ ݉଴ଶ ܿ ସ (12)

70