Methods of molecular quantum mechanics; an introduction to electronic molecular structure

Methods of Molecular
Quantum Mechanics
An Introduction to Electronic
Molecular Structure

Valerio Magnasco
University of Genoa, Genoa, Italy

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Methods of Molecular
Quantum Mechanics

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Methods of Molecular
Quantum Mechanics
An Introduction to Electronic
Molecular Structure

Valerio Magnasco
University of Genoa, Genoa, Italy

www.pdfgrip.com


This edition first published 2009
Ó 2009 John Wiley & Sons, Ltd
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Library of Congress Cataloging-in-Publication Data
Magnasco, Valerio.
Methods of molecular quantum mechanics : an introduction to electronic molecular structure /
Valerio Magnasco.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-68442-9 (cloth) – ISBN 978-0-470-68441-2 (pbk. : alk. paper) 1. Quantum chemistry.
2. Molecular structure. 3. Electrons. I. Title.
QD462.M335 2009
541’.28–dc22
2009031405
A catalogue record for this book is available from the British Library.
ISBN H/bk 978-0470-684429 P/bk 978-0470-684412
Set in 10.5/13pt, Sabon by Thomson Digital, Noida, India.
Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall.

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To my Quantum Chemistry students

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Contents
Preface

xiii

1 Principles
1.1 The Orbital Model
1.2 Mathematical Methods
1.2.1 Dirac Notation
1.2.2 Normalization
1.2.3 Orthogonality
1.2.4 Set of Orthonormal Functions
1.2.5 Linear Independence
1.2.6 Basis Set
1.2.7 Linear Operators
1.2.8 Sum and Product of Operators
1.2.9 Eigenvalue Equation
1.2.10 Hermitian Operators
1.2.11 Anti-Hermitian Operators
1.2.12 Expansion Theorem
1.2.13 From Operators to Matrices

1.2.14 Properties of the Operator r
1.2.15 Transformations in Coordinate Space
1.3 Basic Postulates
1.3.1 Correspondence between Physical Observables
and Hermitian Operators
1.3.2 State Function and Average Value
of Observables
1.3.3 Time Evolution of the State Function
1.4 Physical Interpretation of the Basic Principles

1
1
2
2
2
3
3
3
4
4
4
5
5
6
6
6
7
9
12

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15
16
17


viii

CONTENTS

2 Matrices
2.1 Definitions and Elementary Properties
2.2 Properties of Determinants
2.3 Special Matrices
2.4 The Matrix Eigenvalue Problem

21
21
23
24
25

3 Atomic Orbitals
3.1 Atomic Orbitals as a Basis for Molecular Calculations
3.2 Hydrogen-like Atomic Orbitals
3.2.1 Choice of an Appropriate Coordinate System
3.2.2 Solution of the Radial Equation
3.2.3 Solution of the Angular Equation

3.2.4 Some Properties of the Hydrogen-like Atomic
Orbitals
3.2.5 Real Form of the Atomic Orbitals
3.3 Slater-type Orbitals
3.4 Gaussian-type Orbitals
3.4.1 Spherical Gaussians
3.4.2 Cartesian Gaussians

31
31
32
32
33
37

4 The Variation Method
4.1 Variational Principles
4.2 Nonlinear Parameters
4.2.1 Ground State of the Hydrogenic System
4.2.2 The First Excited State of Spherical Symmetry
of the Hydrogenic System
4.2.3 The First Excited 2p State of the Hydrogenic
System
4.2.4 The Ground State of the He-like System
4.3 Linear Parameters and the Ritz Method
4.4 Applications of the Ritz Method
4.4.1 The First 1s2s Excited State of the He-like Atom
4.4.2 The First 1s2p State of the He-like Atom
Appendix: The Integrals J, K, J0 and K0

53
53
57
57

61
61
64
67
67
69
71

5 Spin
5.1 The Zeeman Effect
5.2 The Pauli Equations for One-electron Spin
5.3 The Dirac Formula for N-electron Spin

75
75
78
79

6 Antisymmetry of Many-electron Wavefunctions
6.1 Antisymmetry Requirement and the Pauli Principle
6.2 Slater Determinants

85
85
87

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43
46
49
49
50

59


CONTENTS

ix

6.3 Distribution Functions
6.3.1 One- and Two-electron Distribution
Functions
6.3.2 Electron and Spin Densities
6.4 Average Values of Operators

89
89
91
95

7 Self-consistent-field Calculations and Model Hamiltonians
7.1 Elements of Hartree–Fock Theory for Closed Shells

7.1.1 The Fock–Dirac Density Matrix
7.1.2 Electronic Energy Expression
7.2 Roothaan Formulation of the LCAO–MO–SCF
Equations
7.3 Molecular Self-consistent-field Calculations
7.4 H€
uckel Theory
7.4.1 Ethylene (N ¼ 2)
7.4.2 The Allyl Radical (N ¼ 3)
7.4.3 Butadiene (N ¼ 4)
7.4.4 Cyclobutadiene (N ¼ 4)
7.4.5 Hexatriene (N ¼ 6)
7.4.6 Benzene (N ¼ 6)
7.5 A Model for the One-dimensional Crystal

104
108
112
114
115
119
120
124
126
129

8 Post-Hartree–Fock Methods
8.1 Configuration Interaction
8.2 Multiconfiguration Self-consistent-field
8.3 Møller–Plesset Theory

8.4 The MP2-R12 Method
8.5 The CC-R12 Method
8.6 Density Functional Theory

133
133
135
135
136
137
138

9 Valence Bond Theory and the Chemical Bond
9.1 The Born–Oppenheimer Approximation
9.2 The Hydrogen Molecule H2
9.2.1 Molecular Orbital Theory
9.2.2 Heitler–London Theory
9.3 The Origin of the Chemical Bond
9.4 Valence Bond Theory and the Chemical Bond
9.4.1 Schematization of Valence Bond Theory
9.4.2 Schematization of Molecular Orbital Theory
9.4.3 Advantages of the Valence Bond Method
9.4.4 Disadvantages of the Valence Bond Method
9.4.5 Construction of Valence Bond Structures

141
142
144
145
148

150
153
153
154
154
154
156

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100
100
102


x

CONTENTS

9.5

9.6

Hybridization and Molecular Structure
9.5.1 The H2O Molecule
9.5.2
Properties of Hybridization
Pauling’s Formula for Conjugated and Aromatic
Hydrocarbons

9.6.1 Ethylene (One p-Bond, n ¼ 1)
9.6.2 Cyclobutadiene (n ¼ 2)
9.6.3 Butadiene (Open Chain, n ¼ 2)
9.6.4 The Allyl Radical (N ¼ 3)
9.6.5 Benzene (n ¼ 3)

162
162
164
166
169
169
171
173
176

10 Elements of Rayleigh–Schroedinger Perturbation Theory
10.1 Rayleigh–Schroedinger Perturbation Equations
up to Third Order
10.2 First-order Theory
10.3 Second-order Theory
10.4 Approximate E2 Calculations: The Hylleraas
Functional
10.5 Linear Pseudostates and Molecular Properties
10.5.1 Single Pseudostate
10.5.2 N-term Approximation
10.6 Quantum Theory of Magnetic Susceptibilities
10.6.1 Diamagnetic Susceptibilities
10.6.2 Paramagnetic Susceptibilities
Appendix: Evaluation of m and «

183

11 Atomic and Molecular Interactions
11.1 The H–H Nonexpanded Interactions
up to Second Order
11.2 The H–H Expanded Interactions up to Second Order
11.3 Molecular Interactions
11.3.1 Nonexpanded Energy Corrections up to
Second Order
11.3.2 Expanded Energy Corrections up to
Second Order
11.3.3 Other Expanded Interactions
11.4 Van der Waals and Hydrogen Bonds
11.5 The Keesom Interaction

215

12 Symmetry
12.1 Molecular Symmetry

247
247

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183
186
187
190

191
193
195
196
199
203
212

216
220
225
226
227
235
237
239


CONTENTS

xi

12.2 Group Theoretical Methods
12.2.1 Isomorphism
12.2.2 Conjugation and Classes
12.2.3 Representations and Characters
12.2.4 Three Theorems on Irreducible
Representations
12.2.5 Number of Irreps in a Reducible
Representation

12.2.6 Construction of Symmetry-adapted
Functions
12.3 Illustrative Examples
12.3.1 Use of Symmetry in Ground-state
H2O (1A1)
12.3.2 Use of Symmetry in Ground-state
NH3 (1A1)

252
254
254
255
255
256
256
257
257
260

References

267

Author Index

275

Subject Index

279

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Preface

The structure of this little textbook is essentially methodological and
introduces in a concise way the student to a working practice in the
ab initio calculations of electronic molecular structure, giving a sound
basis for a critical analysis of the current calculation programmes. It
originates from the need to provide quantum chemistry students with their
own personal instant book, giving at low cost a readable introduction to
the methods of molecular quantum mechanics, a prerequisite for any
understanding of quantum chemical calculations. This book is a recommended companion of the previous book by the author, Elementary
Methods of Molecular Quantum Mechanics, published in 2007 by
Elsevier, which contains many worked examples, and designed as a bridge
between Coulson’s Valence and McWeeny’s Methods of Molecular
Quantum Mechanics. The present book is suitable for a first-year
postgraduate university course of about 40 hours.
The book consists of 12 chapters. Particular emphasis is devoted to the
Rayleigh variational method, the essential tool for any practical application both in molecular orbital and valence bond theory, and to the
stationary Rayleigh–Schroedinger perturbation methods, much attention
being given to the Hylleraas variational approximations, which are
essential for studying second-order electric properties of molecules and
molecular interactions, as well as magnetic properties. In the last chapter,
elements on molecular symmetry and group theoretical techniques are
briefly presented. Major features of the book are: (i) the consistent use

from the very beginning of the system of atomic units (au), essential for
simplifying all mathematical formulae; (ii) the introductory use of density
matrix techniques for interpreting the properties of many-body systems so
as to simplify calculations involving many-electron wavefunctions; (iii) an
introduction to valence bond methods, with an explanation of the origin

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xiv

PREFACE

of the chemical bond; and (iv) a unified presentation of basic elements of
atomic and molecular interactions, with particular emphasis on the
practical use of second-order calculation techniques. Though many examples are treated in depth in this book, for other problems and their
detailed solutions the reader may refer to the previous book by the author.
The book is completed by alphabetically ordered bibliographical references, and by author and subject indices.
Finally, I wish to thank my son Mario for preparing the drawings at the
computer, and my friends and colleagues Deryk W. Davies and Michele
Battezzati for their careful reading of the manuscript and useful discussions. In saying that, I regret that, during the preparation of this book,
DWD died on 27 February 2008.
I acknowledge support by the Italian Ministry for Education University
and Research (MIUR), under grant number 2006 03 0944 003, and Aracne
Editrice (Rome) for the 2008 publishing of what is essentially the Italian
version entitled Elementi di Meccanica Quantistica Molecolare.

Valerio Magnasco
Genoa, 15 May 2009

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1
Principles

1.1

THE ORBITAL MODEL

The great majority of the applications of molecular quantum mechanics to
chemistry are based on what is called the orbital model. The planetary
model of the atom can be traced back to Rutherford (Born, 1962). It
consists of a point-like nucleus carrying the whole mass and the whole
positive charge ỵZe surrounded by N electrons each having the elementary negative charge Àe and a mass about 2000 times smaller than that of
the proton and moving in a space which is essentially that of the atom.1
Electrons are point-like elementary particles whose negative charge is
distributed in space in the form of a charge cloud, with the probability of
finding the electron at point r in space being given by
jcrịj2 dr ẳ probability of finding in dr the electron in state cðrÞ
ð1:1Þ
The functions c(r) are called atomic orbitals (AOs, one centre) or
molecular orbitals (MOs, many centres) and describe the quantum states
of the electron. For (1.1) to be true, c(r) must be a regular (or Q-class)
mathematical function (single valued, continuous with its first derivatives,

1

The atomic volume has a diameter of the order of 102 pm, about 105 times larger than that of the
nucleus.

Methods of Molecular Quantum Mechanics: An Introduction to Electronic Molecular Structure
Valerio Magnasco
Ó 2009 John Wiley & Sons, Ltd

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2

PRINCIPLES

quadratically integrable) satisfying the normalization condition


2
dr jcrịj ẳ dr c rịcrị ẳ 1

1:2ị

where integration is extended over the whole space of definition of the
variable r and where cà ðrÞ is the complex conjugate to c(r). The last of the
above physical constraints implies that c must vanish at infinity.2
It seems appropriate at this point first to introduce in an elementary way the
essential mathematical methods which are needed in the applications, followed by a simple axiomatic formulation of the basic postulates of quantum
mechanics and, finally, by their physical interpretation (Margenau, 1961).

1.2

MATHEMATICAL METHODS

In what follows we shall be concerned only with regular functions of the
general variable x.

1.2.1

Dirac Notation


Function
cxị ẳ jci Y ket
Complex conjugate c xị ẳ hcj Y bra

1:3ị

The scalar product (see the analogy between regular functions and
complex vectors of infinite dimensions) of cà by c can then be written in
the bra-ket (‘bracket’) form:

1:4ị
dx c xịcxị ẳ hcjci ẳ finite number > 0

1.2.2
If

Normalization
hcjci ¼ A

ð1:5Þ

then we say that the function c(x) (the ket jci) is normalized to A (the norm
of c). The function c can then be normalized to 1 by multiplying it by the
normalization factor N ¼ Ầ1=2 .
2
In an atom or molecule, there must be zero probability of finding an electron infinitely far from
its nucleus.

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MATHEMATICAL METHODS

1.2.3

3

Orthogonality

If


hcjwi ẳ dx c xịwxị ẳ 0

1:6ị

then we say that w is orthogonal (?) to c. If
hc0 jw0 i ¼ Sð6¼ 0Þ

ð1:7Þ

then w0 and c0 are not orthogonal, but can be orthogonalized by choosing
the linear combination (Schmidt orthogonalization):
c ¼ c0 ;

w ẳ Nw0 Sc0 ị;

hcjwi ẳ 0

1:8ị

where N ¼ ð1 À S2 ÞÀ1=2 is the normalization factor. In fact, it is easily seen
that, if c0 and w0 are normalized to 1:
hcjwi ¼ Nhc0 jw0 À Sc0 i ¼ NS Sị ẳ 0

1.2.4

1:9ị

Set of Orthonormal Functions

Let
fwk xịg ẳ ðw1 w2 . . . wk . . . wi . . .Þ

ð1:10Þ

be a set of functions. If
hwk jwi i ¼ dki

k; i ¼ 1; 2; . . .

ð1:11Þ

where dki is the Kronecker delta (1 if i ¼ k, 0 if i 6¼ k), then the set is said to
be orthonormal.

1.2.5

Linear Independence

A set of functions is said to be linearly independent if
X
wk xịCk ẳ 0 with; necessarily; Ck ẳ 0 for any k

ð1:12Þ

k

For a set to be linearly independent, it will be sufficient that the
determinant of the metric matrix M (see Chapter 2) be different from zero:

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4

PRINCIPLES

detMki 6ẳ 0

Mki ẳ hwk jwi i

1:13ị

A set of orthonormal functions, therefore, is a linearly independent set.

1.2.6

Basis Set

A set of linearly independent functions forms a basis in the function space,
and we can expand any function of that space into a linear combination of
the basis functions. The expansion is unique.

1.2.7

Linear Operators

An operator is a rule transforming a given function into another function
^ satisfies
(e.g. its derivative). A linear operator A


^ xị ỵ c xị ẳ Ac
^ xị ỵ Ac
^ xị
Aẵc
1
2
1
2

^
^
Aẵccxị ẳ cAẵcxị

1:14ị

where c is a complex constant. The first and second derivatives are simple
examples of linear operators.

1.2.8

Sum and Product of Operators
^ ỵ Bịcxị
^
^
^
^
^ ỵ Aịcxị
A
ẳ Acxị
ỵ Bcxị
ẳ B

1:15ị

so that the algebraic sum of two operators is commutative.
In general, the product of two operators is not commutative:
^ Bcxị
^
^

^ Acxị
A
6ẳ B

1:16ị

where the inner operator acts first. If
^B
^
^ẳB
^A
A

1:17ị

then the two operators commute. The quantity
^ B
^B
^
^ ẳA
^ B
^A
ẵA;
^ B.
^
is called the commutator of the operators A;

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1:18ị

MATHEMATICAL METHODS

1.2.9

5

Eigenvalue Equation

The equation
^
Acxị
ẳ Acxị

1:19ị

^ When (1.19) is
is called the eigenvalue equation for the linear operator A.
satisfied, the constant A is called the eigenvalue, the function c the
^ Often, A
^ is a differential operator, and
eigenfunction of the operator A.
there may be a whole spectrum of eigenvalues, each one with its corresponding eigenfunction. The spectrum of the eigenvalues can be either
discrete or continuous. An eigenvalue is said to be n-fold degenerate when
n different independent eigenfunctions belong to it. We shall see later that
the Schroedinger equation for the amplitude c(x) is a typical eigenvalue
^ ẳH
^ ỵ V is the total energy operator (the
^ ¼T

equation, where A
^
Hamiltonian), T being the kinetic energy operator and V the potential
energy characterizing the system (a scalar quantity).

1.2.10

Hermitian Operators

A Hermitian operator is a linear operator satisfying the so-called turnover rule:
8
^ ẳ hAcjwi
^
< hcjAwi


1:20ị

^
^
: dx c xịAwxịị
ẳ dx Acxịị
wxị
The Hermitian operators have the following properties:
(i) real eigenvalues;
(ii) orthogonal (or anyway orthogonalizable) eigenfunctions;
(iii) their eigenfunctions form a complete set.
Completeness also includes the eigenfunctions belonging to the continuous part of the eigenvalue spectrum.
^ ẳ h2 =2mịr2
Hermitian operators are i@=@x, ir, @ 2 =@x2 , r2 , T

^
^
and H ẳ T ỵ V, where i is the imaginary unit (i2 ¼ À1),
r ẳ i@=@xị ỵ j@=@yị ỵ k@=@zị is the gradient vector operator,
r2 ẳ r r ẳ @ 2 =@x2 ỵ @ 2 =@y2 ỵ @ 2 =@z2 is the Laplacian operator
^ is the kinetic energy operator for a particle
(in Cartesian coordinates), T
^ is the
of mass m with h ¼ h=2p the reduced Planck constant and H
Hamiltonian operator.

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6

1.2.11

PRINCIPLES

Anti-Hermitian Operators

@=@x and r are instead anti-Hermitian operators, for which
8



>
< cj @w ẳ @c jw
@x

@x
>
:
hcjrwi ẳ hrcjwi

1.2.12

1:21ị

Expansion Theorem

Any regular (Q-class) function F(x) can be expressed exactly in the
^ If
complete set of the eigenfunctions of any Hermitian operator3A.
^ xị ẳ Ak w xị;
Aw
k
k
then
Fxị ẳ

X

^ ẳ A
^
A

wk xịCk

1:22ị

1:23ị

k

where the expansion coefficients are given by

Ck ẳ dx0 wk x0 ịFx0 ị ẳ hwk jFi

ð1:24Þ

as can be easily shown by multiplying both sides of Equation (1.23) by
wÃk ðxÞ and integrating.
Some authors insert an integral sign into (1.23) to emphasize that
integration over the continuous part of the eigenvalue spectrum must be
included in the expansion. When the set of functions fwk ðxÞg is not
complete, truncation errors occur, and a lot of the literature data from
the quantum chemistry side is plagued by such errors.

1.2.13

From Operators to Matrices

Using the expansion theorem we can pass from operators (acting on
functions) to matrices (acting on vectors; Chapter 2). Consider a finite
3
A less stringent stipulation of completeness involves the approximation in the mean (Margenau,
1961).

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MATHEMATICAL METHODS

7

^ is a
n-dimensional set of basis functions fwk ðxÞgk ¼ 1; . . . ; n. Then, if A
Hermitian operator:
^ i xị ẳ
Aw

X

wk xịAki ẳ

k

X

^ ii
jwk ihwk jAw

1:25ị

k

where the expansion coefficients now have two indices and are the
elements of the square matrix A (order n):

^
^ i x0 ịị
1:26ị
Aki ẳ hwk jAwi i ẳ dx0 wk x0 ịAw
0

A11
B A21
fAki g Y A ẳ B
@
An1

A12
A22

An2






1
A1n
A2n C
^
C ẳ w Aw
A
Ann

1:27ị

^ in the basis
which is called the matrix representative of the operator A
fwk g, and we use matrix multiplication rules (Chapter 2). In this way,
the eigenvalue equations of quantum mechanics transform into eigenvalue equations for the corresponding representative matrices. We
must recall, however, that a complete set implies matrices of infinite
order.
Under a unitary transformation U of the basis functions w ¼
ðw1 w2 . . . wn ị:
w0 ẳ w U

1:28ị

^ is changed into
the representative A of the operator A
†^ 0
A0 ¼ w0 Aw
ẳ U AU

1.2.14

1:29ị

Properties of the Operator r

We have seen that in Cartesian coordinates the vector operator r (the
gradient, a vector whose components are operators) is defined as
(Rutherford, 1962)
rẳi

@
@
@
ỵj
ỵk
@x
@y
@z

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1:30ị


8

PRINCIPLES

Now, let F(x,y,z) be a scalar function of the space point P(r). Then:
rF ẳ i

@F
@F
@F
ỵj
ỵk
@x
@y
@z

1:31ị

is a vector, the gradient of F.
If F is a vector of components Fx, Fy, Fz, we then have for the scalar
product
rF ẳ

@Fx @Fy @Fz
ỵ
ỵ
ẳ div F
@x
@y
@z

ð1:32Þ

a scalar quantity, the divergence of F. As a particular case:
r r ẳ r2 ẳ

@2
@2
@2
ỵ
ỵ
@x2 @y2 @z2

1:33ị

is the Laplacian operator.
From the vector product of r by the vector F we obtain a new vector, the
curl or rotation of F (written curl F or rot F):