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Methods of Molecular

Quantum Mechanics

An Introduction to ElectronicMolecular Structure

Valerio MagnascoUniversity of Genoa, Genoa, Italy


Quantum Mechanics


Quantum Mechanics

An Introduction to ElectronicMolecular Structure

Valerio MagnascoUniversity of Genoa, Genoa, Italy

This edition first published 2009

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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.

www.wiley.com

To my Quantum Chemistry students

Contents

Preface xiii

1 Principles 1

1.1 The Orbital Model 11.2 Mathematical Methods 2

1.2.1 Dirac Notation 21.2.2 Normalization 21.2.3 Orthogonality 31.2.4 Set of Orthonormal Functions 31.2.5 Linear Independence 31.2.6 Basis Set 41.2.7 Linear Operators 41.2.8 Sum and Product of Operators 41.2.9 Eigenvalue Equation 51.2.10 Hermitian Operators 51.2.11 Anti-Hermitian Operators 61.2.12 Expansion Theorem 61.2.13 From Operators to Matrices 61.2.14 Properties of the Operator r 71.2.15 Transformations in Coordinate Space 9

1.3 Basic Postulates 121.3.1 Correspondence between Physical Observables

and Hermitian Operators 121.3.2 State Function and Average Value

of Observables 151.3.3 Time Evolution of the State Function 16

1.4 Physical Interpretation of the Basic Principles 17

2 Matrices 21

2.1 Definitions and Elementary Properties 212.2 Properties of Determinants 232.3 Special Matrices 242.4 The Matrix Eigenvalue Problem 25

3 Atomic Orbitals 31

3.1 Atomic Orbitals as a Basis for Molecular Calculations 313.2 Hydrogen-like Atomic Orbitals 32

3.2.1 Choice of an Appropriate Coordinate System 323.2.2 Solution of the Radial Equation 333.2.3 Solution of the Angular Equation 373.2.4 Some Properties of the Hydrogen-like Atomic

Orbitals 413.2.5 Real Form of the Atomic Orbitals 43

3.3 Slater-type Orbitals 463.4 Gaussian-type Orbitals 49

3.4.1 Spherical Gaussians 493.4.2 Cartesian Gaussians 50

4 The Variation Method 53

4.1 Variational Principles 534.2 Nonlinear Parameters 57

4.2.1 Ground State of the Hydrogenic System 574.2.2 The First Excited State of Spherical Symmetry

of the Hydrogenic System 594.2.3 The First Excited 2p State of the Hydrogenic

System 614.2.4 The Ground State of the He-like System 61

4.3 Linear Parameters and the Ritz Method 644.4 Applications of the Ritz Method 67

4.4.1 The First 1s2s Excited State of the He-like Atom 674.4.2 The First 1s2p State of the He-like Atom 69

Appendix: The Integrals J, K, J0 and K0 71

5 Spin 75

5.1 The Zeeman Effect 755.2 The Pauli Equations for One-electron Spin 785.3 The Dirac Formula for N-electron Spin 79

6 Antisymmetry of Many-electron Wavefunctions 85

6.1 Antisymmetry Requirement and the Pauli Principle 856.2 Slater Determinants 87

viii CONTENTS

6.3 Distribution Functions 896.3.1 One- and Two-electron Distribution

Functions 896.3.2 Electron and Spin Densities 91

6.4 Average Values of Operators 95

7 Self-consistent-field Calculations and Model Hamiltonians 99

7.1 Elements of Hartree–Fock Theory for Closed Shells 1007.1.1 The Fock–Dirac Density Matrix 1007.1.2 Electronic Energy Expression 102

7.2 Roothaan Formulation of the LCAO–MO–SCFEquations 104

7.3 Molecular Self-consistent-field Calculations 1087.4 H€uckel Theory 112

7.4.1 Ethylene (N¼ 2) 1147.4.2 The Allyl Radical (N ¼ 3) 1157.4.3 Butadiene (N ¼ 4) 1197.4.4 Cyclobutadiene (N ¼ 4) 1207.4.5 Hexatriene (N ¼ 6) 1247.4.6 Benzene (N ¼ 6) 126

7.5 A Model for the One-dimensional Crystal 129

8 Post-Hartree–Fock Methods 133

8.1 Configuration Interaction 1338.2 Multiconfiguration Self-consistent-field 1358.3 Møller–Plesset Theory 1358.4 The MP2-R12 Method 1368.5 The CC-R12 Method 1378.6 Density Functional Theory 138

9 Valence Bond Theory and the Chemical Bond 141

9.1 The Born–Oppenheimer Approximation 1429.2 The Hydrogen Molecule H2 144

9.2.1 Molecular Orbital Theory 1459.2.2 Heitler–London Theory 148

9.3 The Origin of the Chemical Bond 1509.4 Valence Bond Theory and the Chemical Bond 153

9.4.1 Schematization of Valence Bond Theory 1539.4.2 Schematization of Molecular Orbital Theory 1549.4.3 Advantages of the Valence Bond Method 1549.4.4 Disadvantages of the Valence Bond Method 1549.4.5 Construction of Valence Bond Structures 156

CONTENTS ix

9.5 Hybridization and Molecular Structure 1629.5.1 The H2O Molecule 1629.5.2 Properties of Hybridization 164

9.6 Pauling’s Formula for Conjugated and AromaticHydrocarbons 1669.6.1 Ethylene (One p-Bond, n ¼ 1) 1699.6.2 Cyclobutadiene (n ¼ 2) 1699.6.3 Butadiene (Open Chain, n ¼ 2) 1719.6.4 The Allyl Radical (N ¼ 3) 1739.6.5 Benzene (n ¼ 3) 176

10 Elements of Rayleigh–Schroedinger Perturbation Theory 183

10.1 Rayleigh–Schroedinger Perturbation Equationsup to Third Order 183

10.2 First-order Theory 18610.3 Second-order Theory 18710.4 Approximate E2 Calculations: The Hylleraas

Functional 19010.5 Linear Pseudostates and Molecular Properties 191

10.5.1 Single Pseudostate 19310.5.2 N-term Approximation 195

10.6 Quantum Theory of Magnetic Susceptibilities 19610.6.1 Diamagnetic Susceptibilities 19910.6.2 Paramagnetic Susceptibilities 203

Appendix: Evaluation of m and « 212

11 Atomic and Molecular Interactions 215

11.1 The H–H Nonexpanded Interactionsup to Second Order 216

11.2 The H–H Expanded Interactions up to Second Order 22011.3 Molecular Interactions 225

11.3.1 Nonexpanded Energy Corrections up toSecond Order 226

11.3.2 Expanded Energy Corrections up toSecond Order 227

11.3.3 Other Expanded Interactions 23511.4 Van der Waals and Hydrogen Bonds 23711.5 The Keesom Interaction 239

12 Symmetry 247

12.1 Molecular Symmetry 247

x CONTENTS

12.2 Group Theoretical Methods 25212.2.1 Isomorphism 25412.2.2 Conjugation and Classes 25412.2.3 Representations and Characters 25512.2.4 Three Theorems on Irreducible

Representations 25512.2.5 Number of Irreps in a Reducible

Representation 25612.2.6 Construction of Symmetry-adapted

Functions 25612.3 Illustrative Examples 257

12.3.1 Use of Symmetry in Ground-stateH2O (1A1) 257

12.3.2 Use of Symmetry in Ground-stateNH3 (

1A1) 260

References 267

Author Index 275

Subject Index 279

CONTENTS xi

Preface

The structure of this little textbook is essentially methodological andintroduces in a concise way the student to a working practice in theab initio calculations of electronic molecular structure, giving a soundbasis for a critical analysis of the current calculation programmes. Itoriginates from the need toprovidequantumchemistry studentswith theirown personal instant book, giving at low cost a readable introduction tothe methods of molecular quantum mechanics, a prerequisite for anyunderstanding of quantum chemical calculations. This book is a recom-mended companion of the previous book by the author, ElementaryMethods of Molecular Quantum Mechanics, published in 2007 byElsevier,which containsmanyworked examples, and designed as a bridgebetween Coulson’s Valence and McWeeny’s Methods of MolecularQuantum Mechanics. The present book is suitable for a first-yearpostgraduate 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 applica-tion both in molecular orbital and valence bond theory, and to thestationary Rayleigh–Schroedinger perturbation methods, much attentionbeing given to the Hylleraas variational approximations, which areessential for studying second-order electric properties of molecules andmolecular interactions, as well as magnetic properties. In the last chapter,elements on molecular symmetry and group theoretical techniques arebriefly presented. Major features of the book are: (i) the consistent usefrom the very beginning of the system of atomic units (au), essential forsimplifying all mathematical formulae; (ii) the introductory use of densitymatrix techniques for interpreting the properties ofmany-body systems soas to simplify calculations involvingmany-electronwavefunctions; (iii) anintroduction to valence bond methods, with an explanation of the origin

of the chemical bond; and (iv) a unified presentation of basic elements ofatomic and molecular interactions, with particular emphasis on thepractical use of second-order calculation techniques. Though many ex-amples are treated in depth in this book, for other problems and theirdetailed solutions the readermay refer to the previous book by the author.The book is completed by alphabetically ordered bibliographical refer-ences, and by author and subject indices.Finally, I wish to thankmy sonMario for preparing the drawings at the

computer, and my friends and colleagues Deryk W. Davies and MicheleBattezzati for their careful reading of the manuscript and useful discus-sions. 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

andResearch (MIUR), under grant number 2006030944003, andAracneEditrice (Rome) for the 2008 publishing of what is essentially the Italianversion entitled Elementi di Meccanica Quantistica Molecolare.

Valerio MagnascoGenoa, 15 May 2009

xiv PREFACE

1Principles

1.1 THE ORBITAL MODEL

Thegreatmajority of the applications ofmolecular quantummechanics tochemistry are based on what is called the orbital model. The planetarymodel of the atom can be traced back to Rutherford (Born, 1962). Itconsists of a point-like nucleus carrying the whole mass and the wholepositive charge þZe surrounded byN electrons each having the elemen-tary negative charge �e and amass about 2000 times smaller than that ofthe proton and moving in a space which is essentially that of the atom.1

Electrons are point-like elementary particles whose negative charge isdistributed in space in the form of a charge cloud, with the probability offinding the electron at point r in space being given by

jcðrÞj2 dr ¼ probability of finding in dr the electron in state cðrÞð1:1Þ

The functions c(r) are called atomic orbitals (AOs, one centre) ormolecular orbitals (MOs, many centres) and describe the quantum statesof the electron. For (1.1) to be true, c(r) must be a regular (or Q-class)mathematical function (single valued, continuouswith its first derivatives,

1 The atomic volumehas 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


quadratically integrable) satisfying the normalization conditionðdr jcðrÞj2 ¼

ðdr c�ðrÞcðrÞ ¼ 1 ð1:2Þ

where integration is extended over the whole space of definition of thevariable r andwhere c�ðrÞ is the complex conjugate to c(r). The last of theabove physical constraints implies that c must vanish at infinity.2

It seemsappropriateat thispointfirst to introduce inanelementarywaytheessential mathematical methods which are needed in the applications, fol-lowed by a simple axiomatic formulation of the basic postulates of quantummechanics and, finally, by their physical interpretation (Margenau, 1961).

1.2 MATHEMATICAL METHODS

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

1.2.1 Dirac Notation

Function cðxÞ ¼ jci Y ketComplex conjugate c�ðxÞ ¼ hcj Y bra

�ð1:3Þ

The scalar product (see the analogy between regular functions andcomplex vectors of infinite dimensions) of c� by c can then be written inthe bra-ket (‘bracket’) form:ð

dx c�ðxÞcðxÞ ¼ hcjci ¼ finite number > 0 ð1:4Þ

1.2.2 Normalization

Ifhcjci ¼ A ð1:5Þ

thenwe say that the functionc(x) (the ket jci) is normalized toA (thenormof c). The function c can then be normalized to 1 by multiplying it by thenormalization factor N ¼ A�1=2.

2 In an atom or molecule, there must be zero probability of finding an electron infinitely far from

its nucleus.

2 PRINCIPLES

1.2.3 Orthogonality

If

hcjwi ¼ðdx c�ðxÞwðxÞ ¼ 0 ð1:6Þ

then we say that w is orthogonal (?) to c. If

hc0jw0i ¼ Sð6¼ 0Þ ð1:7Þ

then w0 and c0 are not orthogonal, but can be orthogonalized by choosingthe linear combination (Schmidt orthogonalization):

c ¼ c0; w ¼ Nðw0 � Sc0Þ; hcjwi ¼ 0 ð1:8Þ

whereN ¼ ð1� S2Þ�1=2 is the normalization factor. In fact, it is easily seenthat, if c0 and w0 are normalized to 1:

hcjwi ¼ Nhc0jw0 � Sc0i ¼ NðS� SÞ ¼ 0 ð1:9Þ

1.2.4 Set of Orthonormal Functions

Let

fwkðxÞg ¼ ðw1w2 . . .wk . . .wi . . .Þ ð1:10Þ

be a set of functions. If

hwkjwii 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 tobe orthonormal.

1.2.5 Linear Independence

A set of functions is said to be linearly independent ifXk

wkðxÞCk ¼ 0 with; necessarily; Ck ¼ 0 for any k ð1:12Þ

For a set to be linearly independent, it will be sufficient that thedeterminant of themetricmatrixM (seeChapter 2) be different fromzero:

MATHEMATICAL METHODS 3

detMki 6¼ 0 Mki ¼ hwkjwii ð1:13Þ

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,andwe can expand any function of that space into a linear combination ofthe basis functions. The expansion is unique.

1.2.7 Linear Operators

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

A½c1ðxÞþc2ðxÞ� ¼ Ac1ðxÞþ Ac2ðxÞA½ccðxÞ� ¼ cA½cðxÞ�

�ð1:14Þ

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

1.2.8 Sum and Product of Operators

ðAþ BÞcðxÞ ¼ AcðxÞþ BcðxÞ ¼ ðBþ AÞcðxÞ ð1:15Þ

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

ABcðxÞ 6¼ BAcðxÞ ð1:16Þ

where the inner operator acts first. If

AB ¼ BA ð1:17Þ

then the two operators commute. The quantity

½A; B� ¼ AB� BA ð1:18Þ

is called the commutator of the operators A; B.

4 PRINCIPLES

1.2.9 Eigenvalue Equation

The equation

AcðxÞ ¼ AcðxÞ ð1:19Þis called the eigenvalue equation for the linear operator A. When (1.19) issatisfied, the constant A is called the eigenvalue, the function c theeigenfunction of the operator A. Often, A is a differential operator, andthere may be a whole spectrum of eigenvalues, each one with its corre-sponding eigenfunction. The spectrum of the eigenvalues can be eitherdiscrete or continuous. An eigenvalue is said to be n-fold degenerate whenn different independent eigenfunctions belong to it.We shall see later thatthe Schroedinger equation for the amplitude c(x) is a typical eigenvalueequation, where A ¼ H ¼ TþV is the total energy operator (theHamiltonian), T being the kinetic energy operator and V the potentialenergy characterizing the system (a scalar quantity).

1.2.10 Hermitian Operators

A Hermitian operator is a linear operator satisfying the so-called ‘turn-over rule’:

hcjAwi ¼ hAcjwiðdx c�ðxÞðAwðxÞÞ ¼

ðdx ðAcðxÞÞ�wðxÞ

8<: ð1:20Þ

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 contin-uous part of the eigenvalue spectrum.Hermitian operators are �i@=@x, �ir, @2=@x2,r2, T ¼ �ð�h2=2mÞr2

and H ¼ TþV, where i is the imaginary unit (i2 ¼ �1),r ¼ ið@=@xÞþ jð@=@yÞþ kð@=@zÞ is the gradient vector operator,r2 ¼ r �r ¼ @2=@x2 þ @2=@y2 þ @2=@z2 is the Laplacian operator(in Cartesian coordinates), T is the kinetic energy operator for a particleof mass m with �h ¼ h=2p the reduced Planck constant and H is theHamiltonian operator.


1.2.11 Anti-Hermitian Operators

@=@x and r are instead anti-Hermitian operators, for which�cj @w

@x

�¼ �

�@c

@xjw�

hcjrwi ¼ �hrcjwi

8><>: ð1:21Þ

1.2.12 Expansion Theorem

Any regular (Q-class) function F(x) can be expressed exactly in thecomplete set of the eigenfunctions of any Hermitian operator3A. If

AwkðxÞ ¼ AkwkðxÞ; A� ¼ A ð1:22Þ

then

FðxÞ ¼Xk

wkðxÞCk ð1:23Þ

where the expansion coefficients are given by

Ck ¼ðdx0w�

kðx0ÞFðx0Þ ¼ hwkjFi ð1:24Þ

as can be easily shown by multiplying both sides of Equation (1.23) byw�kðxÞ and integrating.Some authors insert an integral sign into (1.23) to emphasize that

integration over the continuous part of the eigenvalue spectrum must beincluded in the expansion. When the set of functions fwkðxÞg is notcomplete, truncation errors occur, and a lot of the literature data fromthe 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 onfunctions) to matrices (acting on vectors; Chapter 2). Consider a finite

3 A less stringent stipulationof completeness involves the approximation in themean (Margenau,

1961).

6 PRINCIPLES

n-dimensional set of basis functions fwkðxÞgk ¼ 1; . . . ; n. Then, if A is aHermitian operator:

AwiðxÞ ¼Xk

wkðxÞAki ¼Xk

jwkihwkjAwii ð1:25Þ

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

Aki ¼ hwkjAwii ¼ðdx0 w�

kðx0ÞðAwiðx0ÞÞ ð1:26Þ

fAkig Y A ¼A11 A12 � � � A1n

A21 A22 � � � A2n

� � � � � � � � � � � �An1 An2 � � � Ann

0BB@

1CCA ¼ w�Aw ð1:27Þ

which is called the matrix representative of the operator A in the basisfwkg, and we use matrix multiplication rules (Chapter 2). In this way,the eigenvalue equations of quantum mechanics transform into eigen-value equations for the corresponding representative matrices. Wemust recall, however, that a complete set implies matrices of infiniteorder.Under a unitary transformation U of the basis functions w ¼

ðw1w2 . . .wnÞ:w0 ¼ wU ð1:28Þ

the representative A of the operator A is changed into

A0 ¼ w0�Aw0 ¼ U�AU ð1:29Þ

1.2.14 Properties of the Operator r

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

r ¼ i@

@xþ j

@

@yþ k

@

@zð1:30Þ


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

rF ¼ i@F

@xþ j

@F

@yþ k

@F

@zð1:31Þ

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

product

r � F ¼ @Fx@x

þ @Fy@y

þ @Fz@z

¼ div F ð1:32Þ

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

r �r ¼ r2 ¼ @2

@x2þ @2

@y2þ @2

@z2ð1:33Þ

is the Laplacian operator.From the vector product ofrby the vectorFweobtain a newvector, the

curl or rotation of F (written curl F or rot F):

r� F ¼

i j k

@

@x

@

@y

@

@z

Fx Fy Fz

��

��¼ curl F ¼ i curlx Fþ j curly Fþ k curlz F ð1:34Þ

a vector operator with components:

curlx F ¼ @Fz@y

� @Fy@z

curly F ¼ @Fx@z

� @Fz@x

curlz F ¼ @Fy@x

� @Fx@y

8>>>>>>>><>>>>>>>>:

ð1:35Þ

In quantummechanics, the vector product of the position vector rby thelinearmomentumvector operator�i�hr (see Section1.3) gives the angularmomentum vector operator L:

L ¼ �i�hr�r ¼ �i�h

i j k

x y z

@

@x

@

@y

@

@z

��

��¼ iLx þ jLy þ kLz ð1:36Þ

8 PRINCIPLES

with components

Lx ¼ �i�h y@

@z� z

@

@y

!; Ly ¼ �i�h z

@

@x� x

@

@z

!;

Lz ¼ �i�h x@

@y� y

@

@x

! ð1:37Þ

In the theory of angular momentum, frequent use is made of the ladder(or shift) operators:

Lþ ¼ Lx þ iLy ðstep-upÞ; L� ¼ Lx � iLy ðstep-downÞ ð1:38Þ

These are also called raising and lowering operators4 respectively.Angular momentum operators have the following commutation rela-

tions:

½Lx; Ly� ¼ iLz; ½Ly; Lz� ¼ iLx; ½Lz; Lx� ¼ iLy

½Lz; Lþ � ¼ Lþ ; ½Lz; L� � ¼ �L�½L2

; Lk� ¼ ½L2; L�� ¼ 0 k ¼ x; y; z

8><>: ð1:39Þ

The same commutation relations hold for the spin vector operator S(Chapter 5).

1.2.15 Transformations in Coordinate Space

We now give the definitions of the main coordinate systems useful inquantum chemistry calculations (Cartesian, spherical, spheroidal), therelations between Cartesian and spherical or spheroidal coordinates, andthe expressions of the volume element dr and of the operatorsr andr2 inthe new coordinate systems. We make reference to Figures 1.1 and 1.2.

(i) Cartesian coordinates (x,y,z):

x; y; z 2 ð�¥;¥Þ ð1:40Þdr ¼ dx dy dz ð1:41Þ

4 Note that the ladder operators are non-Hermitian.


r ¼ i@

@xþ j

@

@yþ k

@

@zð1:42Þ

r2 ¼ @2

@x2þ @2

@y2þ @2

@z2ð1:43Þ

(ii) Spherical coordinates (r,u,w):

rð0;¥Þ; uð0;pÞ; wð0;2pÞ ð1:44Þ

x ¼ r sin u cos w; y ¼ r sin u sin w; z ¼ r cos u ð1:45Þ

Figure 1.2 Cartesian and spheroidal coordinate systems

Figure 1.1 Cartesian and spherical coordinate systems

10 PRINCIPLES

dr ¼ r2dr sin u du dw ð1:46Þ

r ¼ er@

@rþ eu

1

r

@

@uþ ew

1

r sin u

@

@wð1:47Þ

r2 ¼ 1

r2@

@rr2

@

@r

� �þ 1

r21

sin u

@

@usin u

@

@u

� �þ 1

sin2u

@2

@w2

�

¼ r2r �

L2=�h2

r2ð1:48Þ

where er, eu, and ew are unit vectors along r, u, and w. InEquation (1.48):

r2r ¼

1

r2@

@rr2

@

@r

� �¼ @2

@r2þ 2

r

@

@rð1:49Þ

is the radial Laplacian and

L2 ¼ L � L ¼ ��h2

1

sin u

@

@usinu

@

@u

!þ 1

sin2u

@2

@w2

24

35

¼ ��h2@2

@u2þ cot u

@

@uþ 1

sin2u

@2

@w2

0@

1A

8>>>>>><>>>>>>:

ð1:50Þ

is the square of the angular momentum operator (1.36). For thecomponents of the angular momentum vector operator L wehave

Lx ¼ �i�h �sin w@

@u� cot u cos w

@

@w

� �ð1:51Þ

Ly ¼ �i�h cos w@

@u� cot u sin w

@

@w

� �ð1:52Þ

Lz ¼ �i�h@

@wð1:53Þ

Lþ ¼ �h expðiwÞ @

@uþ i cot u

@

@w

� �ð1:54Þ

L� ¼ �h expð�iwÞ � @

@uþ i cot u

@

@w

� �ð1:55Þ


(iii) Spheroidal coordinates ðm; n;wÞ:m ¼ rA þ rB

Rð1 � m � ¥Þ; n ¼ rA � rB

Rð�1 � n � 1Þ; wð0;2pÞ

ð1:56Þ

x ¼ R

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm2� 1Þð1� n2Þ

qcosw; y ¼ R

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm2� 1Þð1� n2Þ

qsin w;

z ¼ R

2ðmnþ 1Þ ð1:57Þ

dr ¼ R

2

� �3

ðm2 � n2Þ dm dn dw ð1:58Þ

r¼ 2

Rem

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2�1

m2�n2

s@

@mþen

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�n2

m2�n2

s@

@nþew

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm2�1Þð1�n2Þ

p @

@w

" #

ð1:59Þ

r2¼ 4

R2ðm2�n2Þ

� @

@mðm2�1Þ @

@m

� þ @

@nð1�n2Þ @

@n

� þ m2�n2

ðm2�1Þð1�n2Þ@2

@w2

� �ð1:60Þ

Equations (1.44)–(1.55) are used in atomic (one-centre) calculations,whereas Equations (1.56)–(1.60) are used in molecular (at least two-centre) calculations.

1.3 BASIC POSTULATES

Wenow formulate in an axiomaticway the basis of quantummechanics inthe form of three postulates.

1.3.1 Correspondence between Physical Obervablesand Hermitian Operators

In coordinate space, we have the basic correspondences

r ¼ ixþ jyþ kz ) r ¼ rp ¼ ipx þ jpy þ kpz ) p ¼ �i�hr

�ð1:61Þ

where i is the imaginary unit (i2 ¼ � 1) and �h ¼ h=2p is the reducedPlanck constant. More complex observables can be treated by repeated

12 PRINCIPLES

applications of the correspondences (1.61) under the constraint that theresulting quantum mechanical operators must be Hermitian.5 Kineticenergy andHamiltonian (total energy operator) for a particle ofmassm inthe potential V are examples already seen. We now give a few furtherexamples by specifying the nature of the potential energy V.

(a) The one-dimensional harmonic oscillator

If m is the mass of the oscillator of force constant k, then theHamiltonian is

H ¼ � �h2

2mr2 þ kx2

2ð1:62Þ

(b) The atomic one-electron problem (the hydrogen-like system)

If r is the distance of the electron of mass m and charge �e from anucleus of charge þZe (Z ¼ 1 will give the hydrogen atom), then theHamiltonian in SI units6 is

H ¼ � �h2

2mr2� 1

4p«0

Ze2

rð1:63Þ

To get rid of all fundamental physical constants in our formulae weshall introduce consistently at this point a system of atomic units7 (au)by posing

e ¼ �h ¼ m ¼ 4p«0 ¼ 1 ð1:64ÞThebasic atomic units of charge, length, energy, and time are expressedin SI units as follows:

charge; e e ¼ 1:602 176 462� 10� 19 C

length; Bohr a0 ¼ 4p«0�h2

me2¼ 5:291 772 087� 10�11 m

energy; Hartree Eh ¼1

4p«0

e2

a0¼ 4:359 743 802� 10�18 J

time t ¼ �h

Eh¼ 2:418 884 331� 10�17 s

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð1:65Þ

5 The quantities observable in physical experiments must be real.6 An SI dimensional analysis of the two terms of Equation (1.63) shows that they have the

dimension of energy (Mohr and Taylor, 2003): j�h2r2=2mj ¼ ðkgm2 s�1Þ2 m�2 kg�1 ¼kgm2 s�2 ¼ J; jZe2=4p«0rj ¼ C2 ðJ C� 2 mÞm�1 ¼ J.7 Atomic units were first introduced by Hartree (1928a).

BASIC POSTULATES 13

At the end of a calculation in atomic units, as we always shall do, theactual SI values can be obtained by taking into account the SI equiva-lents (1.65).

The Hamiltonian of the hydrogenic system in atomic units will thentake the following simplified form:

H ¼ � 1

2r2� Z

rð1:66Þ

Fromnowon,we shall consistently use atomic units everywhere, unlessexplicitly stated.(c) The atomic two-electron system

Two electrons are attracted by a nucleus of charge þZ. TheHamiltonian will be

H ¼ � 1

2r2

1 �1

2r2

2�Z

r1� Z

r2þ 1

r12¼ h1þ h2 þ 1

r12ð1:67Þ

where

h ¼ � 1

2r2 � Z

rð1:68Þ

is the one-electron Hamiltonian (which has the same functionalform for both electrons) and the last term is the Coulomb repulsionbetween the electrons (a two-electron operator). Z ¼ 2 gives the Heatom.(d) The hydrogen molecule-ion Hþ

2

This is a diatomic one-electron molecular system, where the electron issimultaneously attracted by the two protons at A and B. The Born–Oppenheimer Hamiltonian (see Chapter 9) will be

H ¼ hþ 1

R¼ � 1

2r2 � 1

rA� 1

rBþ 1

R¼ hA þV ð1:69Þ

where hA is the one-electron Hamiltonian (1.68) for atom A (withZ ¼ 1) and

V ¼ � 1

rBþ 1

Rð1:70Þ

is the interatomic potential between the hydrogen atom A and theproton B.

14 PRINCIPLES

(e) The hydrogen molecule H2

This is a diatomic two-electron molecular system. The Born–Oppenheimer Hamiltonian will be

H¼ h1þ h2þ 1

r12þ 1

R

¼ � 1

2r2

1�1

rA1� 1

rB1

!þ � 1

2r2

2�1

rA2� 1

rB2

!þ 1

r12þ 1

R

¼ hA1þ hB2þV ð1:71Þ

8>>>>>>><>>>>>>>:where hA1 and hB2 are the one-electron Hamiltonians (1.68) foratoms A and B (with Z¼ 1) and

V ¼� 1

rB1� 1

rA2þ 1

r12þ 1

Rð1:72Þ

is the interatomic potential between A and B.

1.3.2 State Function and Average Value of Observables

We assume there is a state function (or wavefunction, in general complex)Y(x,t) that describes in a probabilistic way the dynamical state of amicroscopic system. In coordinate space, Y is a regular function ofcoordinate x and time t such that

Yðx; tÞY�ðx; tÞ dx¼ probability at time t of finding at dx the system in stateY

ð1:73Þ

provided Y is normalized to 1:ðdxY�ðx; tÞYðx; tÞ ¼ 1 ð1:74Þ

where integration covers the whole space.The average value of any physical observable8A described by the

Hermitian operator A is obtained from

hAi ¼ÐdxY�ðx; tÞAYðx; tÞÐdxY�ðx; tÞYðx; tÞ ¼

ðdx A

Yðx; tÞY�ðx; tÞÐdxY�ðx; tÞYðx; tÞ ð1:75Þ

8 Its expectation value, that can be observed by experiment.

BASIC POSTULATES 15

where integration is extended over all space and A acts always on Y andnot on Y�. The last expression above shows that A is weighted with the(normalized) probability density YY�.

1.3.3 Time Evolution of the State Function

The state function Y is obtained by solving the time-dependentSchroedinger equation:

HYðx; tÞ ¼ i�h@Yðx; tÞ

@tð1:76Þ

a partial differential equation which is second order in the spacecoordinate x and first order in the time t. This equation involves theHamiltonian of the system H, so that the total energy E is seen to playa fundamental role among all physical observables.If the Hamiltonian H does not depend explicitly on t (the case of

stationary states), then, following the usual mathematical techniques, thevariables in Equation (1.76) can be separated by writingY as the productof a space function cðxÞ and a time function g(t):

Yðx; tÞ ¼ cðxÞgðtÞ ð1:77Þ

giving upon substitution

HcðxÞ ¼ EcðxÞgðtÞ ¼ g0 expð�ivtÞ

�ð1:78Þ

where E is the separation constant, g0 is an integration constant, andv ¼ E=�h. The first part of Equation (1.78) is the eigenvalue equation forthe total energy operator (the Hamiltonian) of the system, and cðxÞ iscalled the amplitude function. This is the Schroedinger equation that wemust solve or approximate for the physical descriptionof our systems.Thesecond equation gives the time dependence of the stationary state, whilegeneral time dependence is fundamental in spectroscopy. It is immediatelyevident that, for the stationary state, the probability YY� dx is indepen-dent of time:

Yðx; tÞY�ðx; tÞ dx ¼ jYðx; tÞj2 dx ¼ jcðxÞj2jg0j2 dx ð1:79Þ

16 PRINCIPLES

1.4 PHYSICAL INTERPRETATION OF THE BASICPRINCIPLES

The explanation of our, so far, unusual postulates is hidden in the natureof the experimental measurements in atomic physics. The experimentallyobserved atomicity of matter (electrons and protons, carrying the ele-mentary negative and positive charge respectively), energy (hn, Planck),linear momentum (h/l, De Broglie), and angular momentum (�h, Bohr)implies some limits in the measurements done at the microscopic level.The direct consequence of the ineliminable interaction between theexperimental apparatus and the object of measurement at the subatomicscale was shown byHeisenberg (1930) in his gedanken Experimente, andis embodied in his famous uncertainty principle, which for canonicallyconjugate quantities9 can be stated in the form

DxDpx h DEDt h ð1:80Þwhereh is the Planck constant,Dx is the uncertainty in themeasurement ofthe x-coordinate, and Dpx is the uncertainty resulting in the simultaneousmeasurement of the conjugate linear momentum. It can be seen that thecorresponding quantum mechanical operators do not commute:

½x; px� ¼ xpx � pxx ¼ i�h ð1:81Þwhereas operators that are not conjugate commute; say:

½x; py� ¼ xpy � pyx ¼ 0 ð1:82ÞThis means that we cannot measure with the same arbitrary accuracy

two conjugated dynamical variables whose quantum mechanicaloperators do not commute: the exact determination of the positioncoordinate of the electron would imply the simultaneous infinite in-accuracy in the determination of the corresponding component of thelinear momentum!As a consequence of Heisenberg’s principle, the only possible descrip-

tion of the dynamical state of a microscopic body is a probabilistic one,and the problem is now to find the function that describes such aprobability. This was achieved by Schroedinger (1926a, 1926b, 1926c,1926d), who assumed that particle matter (the electron) could be

9 In the sense of analytical mechanics.

PHYSICAL INTERPRETATION OF THE BASIC PRINCIPLES 17

described by the progressive wave in complex form:

Y ¼ A expðiaÞ ¼ A exp½2piðkx� ntÞ� ð1:83ÞwhereA is the amplitude and a the phase of amonochromatic plane waveofwavenumber k and frequency n, which propagates along x. Taking intoaccount the relations of De Broglie and Planck connecting wave-like andparticle-like behaviour:

k ¼ p

h; n ¼ E

hð1:84Þ

the phase of a matter wave could be written as

a ¼ 1

�hðpx�EtÞ ð1:85Þ

giving the wave equation for the matter particle in the form

Y ¼ A expi

�hðpx�EtÞ

� ¼ Yðx; tÞ ð1:86Þ

whichdefinesY as a function ofx and t at constant values ofp andE. Then,taking the derivatives ofY with respect to x and t, we obtain respectively

@Y@x

¼ i

�hpY whence ) p ¼ �i�h

@

@xð1:87Þ

which is our first postulate (1.61) for the x-component of the linearmomentum, and

@Y@t

¼ � i

�hEY whence ) E ¼ H ¼ i�h

@

@tð1:88Þ

the time-dependentSchroedingerEquation (1.76) giving the timeevolutionof the state functionY. Hence, the two correspondences (1.87) and (1.88),connecting linear momentum and total energy to the first derivatives ofthefunctionY,necessarily implythefundamentalrelationsoccurringinouraxiomatic proposition of quantum mechanics.As a consequence of our probabilistic description, in doing experiments

in atomic physics we usually obtain a distribution of the observableeigenvalues, unless the state function Y coincides at time t with theeigenfunction wk of the corresponding quantum mechanical operator, inwhich case we have a 100% probability of observing for A the value Ak.Such probability distributions fluctuate in time for all observables but theenergy,wherewe have a distribution of eigenvalues (the possible values of

18 PRINCIPLES

the energy levels) constant in time. These different cases are qualitativelydepicted in the plots of Figure 1.3.The formulation of quantummechanics of Section 1.3 is themost useful

for us, but not the only one possible. For instance, usingDirac’s approach,Troup (1968) showed the equivalence between our Schroedinger10 ‘wavemechanics’ (the continuous approach) and Heisenberg ‘matrix mechan-ics’ (the discrete approach), giving working examples in both cases (e.g.the harmonic oscillator and the hydrogen atom). However, this matter isoutside the scope of the present book.

Figure 1.3 Probability distribution of the physical observable A. Top row: definitevalue for the kth eigenvalue. Bottom row: fluctuation in time of the probabilitydistribution (left) and distribution of energy eigenvalues constant in time (right)

10 Based on the Schroedinger (1926a, 1926b, 1926c, 1926d) series of four papers entitled‘Quantisierung als Eigenwertproblem’. The equivalence between matrix and wave mechanics

was examined by Schroedinger (1926e) in an intermediate paper.

PHYSICAL INTERPRETATION OF THE BASIC PRINCIPLES 19

2Matrices

Matrices are the powerful algorithm connecting the differential equationsof quantum mechanics to equations governed by the linear algebraof matrices and their transformations. After a short introduction onelementary properties of matrices and determinants (Margenau andMurphy, 1956;Aitken, 1958;Hohn, 1964),we introduce specialmatricesand the matrix eigenvalue problem.

2.1 DEFINITIONS AND ELEMENTARY PROPERTIES

A matrix A of order m�n is an array of numbers or functions orderedaccording to m rows and n columns:

A ¼A11 A12 � � � A1n

A21 A22 � � � A2n

� � � � � � � � � � � �Am1 Am2 � � � Amn

0BB@

1CCA ð2:1Þ

and can be denoted by its ij element (i ¼ 1;2; . . . ;m; j ¼ 1; 2; . . . ;n) as

A ¼ fAijg ð2:2Þ

Matrix A is rectangular if n=m and square if n ¼ m. In squarematrices, elements with j ¼ i are called diagonal. To any square matrixA we can associate two scalar quantities: its determinant jAj ¼ detA(a number) and its trace trA ¼ P

iAii, the sum of all diagonal elements.


Valerio Magnasco


Two matrices A and B of the same order are equal if

B ¼ A Bij ¼ Aij for all i; j ð2:3Þ

Matrices can be added (or subtracted) if they have the same order:

A� B ¼ C Cij ¼ Aij � Bij ð2:4Þ

Addition and subtraction enjoy commutative and associativeproperties.Multiplying amatrixAbya complexnumber c impliesmultiplicationof

all elements of A by that number:

cA ¼ B Bij ¼ cAij ð2:5Þ

The product, rows by columns, of two (or more) matrices A by B ispossible if the matrices are conformable (the number of columns of Aequals the number of rows of B):

AB ¼ Cm�n n�p m�p

Cij ¼Xna¼1

AiaBaj; ABC ¼ Dm�n n�p p�q m�q

;

Dij ¼Xna¼1

Xpb¼1

AiaBabCbj

ð2:6Þ

Matrix multiplication is usually not commutative, the quantity

½A;B� ¼ AB�BA ð2:7Þ

being the commutator of A and B. If

½A;B� ¼ 0 ð2:8Þ

then matrices A and B commute.The product of more than twomatrices enjoys the associative property:

ABC ¼ ðABÞC ¼ AðBCÞ ð2:9Þ

The trace of a product of matrices is invariant under the cyclicpermutation of its factors.

22 MATRICES

2.2 PROPERTIES OF DETERMINANTS

Given a determinant |A| of order n (n rows and n columns), we call |Aij| theminor of |A|, a determinant of order ðn� 1Þ obtained from A by deletingrow i and column j, and by jaijj ¼ ð� 1Þiþ jjAijj the cofactor (signedminor) of |A|.The main properties of determinants will be briefly recalled here

(Aitken, 1958).

1. Adeterminant canbe expanded in an elementaryway in termsof anyof its rows or columns:

jAj ¼

A11 A12 � � � A1n

A21 A22 � � � A2n

� � � � � � � � � � � �An1 An2 � � � Ann

��

��¼

Xnj¼1

Aijjaijj ¼Xni¼1

Aijjaijj ð2:10Þ

the first being the expansion according to row i and the secondaccording to column j. The expansion of a determinant of order ngives n! terms.

2. A determinant changes sign if we interchange any two rows orcolumns.

3. A determinant is unchanged upon interchanging all rows andcolumns.

4. The product of |A| by a complex number c implies that only one row(or column) is multiplied by that number.

5. A determinant vanishes if two rows or two columns are identical.6. A determinant vanishes if all elements of a row or column are zero.7. If each element in any one row (or column) is the sumof two ormore

quantities, then the determinant can be written as the sum of two ormore determinants of the same order (distribution property). Forinstance: jaþ bcj ¼ jacj þ jbcj, where for short we write only thediagonal elements.

8. The determinant of the product of two or more square matrices ofthe same order is the product of the determinants of the individualmatrices:

jABCj ¼ jAj � jBj � jCj ð2:11Þ

PROPERTIES OF DETERMINANTS 23

2.3 SPECIAL MATRICES

1. The null matrix (even rectangular, all elements are zero):

0 ¼

0 0 � � � 0

0 0 � � � 0

� � � � � � � � � � � �0 0 � � � 0

0BBBB@

1CCCCA ð2:12Þ

0þA ¼ Aþ0 ¼ A A0 ¼ 0A ¼ 0 ðsame orderÞ ð2:13Þ

2. The diagonal matrix (order n):

L ¼

l1 0 � � � 0

0 l2 � � � 0

� � � � � � � � � � � �0 0 � � � ln

0BBBB@

1CCCCA lij ¼ lidij ð2:14Þ

where dij is the Kronecker delta.3. The identity matrix (order n):

1 ¼

1 0 � � � 0

0 1 � � � 0

� � � � � � � � � � � �0 0 � � � 1

0BBBB@

1CCCCA 1ij ¼ dij ð2:15Þ

1A ¼ A1 ¼ A ðsame orderÞ ð2:16Þ

4. The scalar matrix (order n):

L ¼

l 0 � � � 0

0 l � � � 0

� � � � � � � � � � � �0 0 � � � l

0BBBB@

1CCCCA ¼ l1 ð2:17Þ

24 MATRICES

5. Given a square matrix A of order n, we can construct the followingmatrices:

B ¼ A� Bij ¼ A�ij complex conjugate

B ¼ ~A Bij ¼ Aji transpose

B ¼ A� Bij ¼ A�ji adjoint

B ¼ A� 1 Bij ¼ ðdetAÞ� 1jajij inverse

8>>><>>>:

ð2:18Þ

When performed twice, the operations �, �, �, and � 1 restore theoriginal matrix. Products have the properties

ðABÞ� ¼ ~B~A ðABÞ� ¼ B�A� ðABÞ� 1 ¼ B� 1A� 1 ð2:19Þ

6. Given a square matrix A of order n, if

A ¼ A�; A ¼ ~A; or A ¼ A� ð2:20Þ

then we say that A is real, symmetric, or Hermitian (or self-adjoint)respectively.

7. If

A� 1 ¼ ~A A� 1 ¼ A� ð2:21Þ

then we say that A is orthogonal or unitary respectively.

2.4 THE MATRIX EIGENVALUE PROBLEM

A system of linear inhomogeneous algebraic equations in the n unknownsci (i ¼ 1; 2; . . . ;n) can be written in matrix form as

Ac ¼ b ð2:22Þ

if we introduce the matrices

A ¼A11 A12 � � � A1n

A21 A22 � � � A2n

� � � � � � � � � � � �An1 An2 � � � Ann

0BB@

1CCA ð2:23Þ

THE MATRIX EIGENVALUE PROBLEM 25

(i.e. the square matrix of coefficients)

c ¼c1c2� � �cn

0BB@

1CCA ð2:24Þ

(i.e. the column vector of the unknowns)

b ¼b1b2� � �bn

0BB@

1CCA ð2:25Þ

(i.e. the column vector of the inhomogeneous terms), and adopt matrixmultiplication rules.Matrix equation (2.22) can be interpreted as a linear transformation on

vector c, which is transformed into vector b under the action of matrixA.If A� 1 exists (detA=0), then the solution of system (2.22) is given by

c ¼ A� 1b ð2:26Þ

which is nothing but the well-known Cramer’s rule.When b is proportional to c through a number l, then

Ac ¼ lc ð2:27Þ

and we obtain what is known as the eigenvalue equation for the squarematrix A. By writing

ðA� l1Þc ¼ 0 ð2:28Þ

we obtain a system of linear homogeneous algebraic equations in theunknowns c, which has nontrivial solutions if and only if the determinantof the coefficients vanishes:

detðA� l1Þ ¼A11 � l A12 � � � A1n

A21 A22 � l � � � A2n

� � � � � � � � � � � �An1 An2 � � � Ann � l

��

��¼ 0 ð2:29Þ

26 MATRICES

Equation (2.29) is known as characteristic (or secular) equation of thesquare matrix A. It is an algebraic equation of degree n in l, with

l1; l2; . . . ; ln n roots ðthe eigenvalues of AÞc1; c2; . . . ; cn n column coefficients ðthe eigenvectors of AÞ

�ð2:30Þ

The whole set of the n eigenvalue equations for A

Ac1 ¼ l1c1; Ac2 ¼ l2c2; . . . ; Acn ¼ lncn ð2:31Þ

can be replaced by the full eigenvalue equation

AC ¼ CL ð2:32Þ

if we introduce the following square matrices of order n:

L ¼

l1 0 � � � 0

0 l2 � � � 0

� � � � � � � � � � � �0 0 � � � ln

0BBBB@

1CCCCA;

C ¼ ðc1c2 � � � cnÞ ¼c11 c12 � � � c1nc21 c22 � � � c2n� � � � � � � � � � � �cn1 cn2 � � � cnn

0BB@

1CCA ð2:33Þ

whereL is the diagonalmatrix of then eigenvalues andC is the rowmatrixof the n eigenvectors, a square matrix on the whole.If detC=0, thenC�1 exists and the squarematrixA can be brought to

diagonal form through the transformation

C� 1AC ¼ L ð2:34Þ

a process which is called the diagonalization of matrix A.If A is Hermitian

A ¼ A� ð2:35Þ

then eigenvalues are real and eigenvectors orthogonal, so that the com-pletematrix of the eigenvectors is a unitarymatrix (C� 1 ¼ C�), and (2.34)


therefore becomes:

C�AC ¼ L ð2:36Þ

Namely, a Hermitian matrix A can be brought to diagonal form by aunitary transformation with the complete matrix of its eigenvectors.We examine below the simple case of the 2� 2 Hermitian matrix A:

A ¼ 1 SS 1

� �ð2:37Þ

The secular equation is

1� l SS 1� l

�� ¼ 0 ð2:38Þ

giving upon expansion the quadratic equation in l:

l2 � 2lþ 1� S2 ¼ 0 ð2:39Þ

with the roots (the eigenvalues):

l1 ¼ 1� S; l2 ¼ 1þ S ð2:40Þ

We now turn to the evaluation of the eigenvectors.

(i) l1 ¼ 1� S

ð1� l1Þc1 þ Sc2 ¼ 0

c21þ c22 ¼ 1

(ð2:41Þ

We solve the homogeneous linear system (2.28) for the firsteigenvalue with the additional constraint of coefficients normal-ization:1

c2c1

� �1

¼ l1 � 1

S¼ � S

S¼ � 1 ) c2 ¼ � c1 ð2:42Þ

1 Solution of the homogeneous system is seen to give only the ratio c2/c1.

28 MATRICES

c21 þ c22 ¼ 2c21 ¼ 1 ) c1 ¼ 1ffiffiffi2

p ; c2 ¼ � 1ffiffiffi2

p ð2:43Þ

c1 ¼

1ffiffiffi2

p

� 1ffiffiffi2

p

0BBBB@

1CCCCA ð2:44Þ

(ii) l2 ¼ 1þ S

ð1� l2Þc1þ Sc2 ¼ 0

c21 þ c22 ¼ 1

(ð2:45Þ

We now solve system (2.28) for the second eigenvalue with theadditional constraint of coefficients normalization:

c2c1

� �2

¼ l2 �1

S¼ S

S¼ 1 ) c2 ¼ c1 ð2:46Þ

c21þ c22 ¼ 2c21 ¼ 1 ) c1 ¼ c2 ¼ 1ffiffiffi2

p ð2:47Þ

c2 ¼

1ffiffiffi2

p

1ffiffiffi2

p

0BBBB@

1CCCCA ð2:48Þ

It is left as an easy exercise for the reader to verify by direct matrixmultiplication that the complete matrix C of the eigenvectors

C ¼ ðc1 c2Þ ¼

1ffiffiffi2

p 1ffiffiffi2

p

� 1ffiffiffi2

p 1ffiffiffi2

p

0BBBB@

1CCCCA ð2:49Þ

is a unitary matrix

C�C ¼ CC� ¼ 1 ð2:50Þ


and that

C�AC ¼ L ð2:51Þ

More advanced techniques for the calculation of functions ofHermitian matrices and matrix projectors can be found elsewhere(Magnasco, 2007).

30 MATRICES

3Atomic Orbitals

3.1 ATOMICORBITALSASABASIS FORMOLECULARCALCULATIONS

We saw in Chapter 1 that AOs are one-electron one-centre functionsneeded for describing the probability of finding the electron at any givenpoint in space. They are, therefore, the building blocks of any theorythat can be devised inside the orbital model. In practical applications,we shall see in Chapter 4 that appropriate orbitals will be the basis of allapproximation methods resting on the variation theorem. A particulartype of AO is obtained from the solution of the atomic one-electronproblem, the so-called hydrogen-like atomic orbitals (HAOs). Even ifthe HAOs are of no interest in practice, they are important in that theyare exact solutions of the corresponding Schroedinger equation and,therefore, are useful for testing the accuracy of approximate calcula-tions. The great majority of quantum chemical calculations on atomsand molecules are based on the use of basis AOs that have a radialdependence different from that of the HAOs. They can be separated intotwo classes according to whether their decay with the radial variable r isexponential (Slater-type orbitals or STOs, by far the best) or Gaussian(Gaussian-type orbitals or GTOs).In the following, we shall first introduce the HAOsmostly with the aim

of (i) illustrating the general techniques of solution of one of the exactlysolvable Schroedinger eigenvalue equations and (ii) explaining from first


Valerio Magnasco


principles the origin of the quantum numbers (n, ‘, and m) that char-acterize these orbitals, and which arise from the regularity conditionsimposed upon the mathematical solutions. We shall move next toconsideration of STOs and GTOs, giving some general definitions andsome simple one-centre one-electron integrals which will be needed inChapter 4.

3.2 HYDROGEN-LIKE ATOMIC ORBITALS

HAOs are obtained as exact solutions of the Schroedinger eigenvalueequation for the atomic one-electron system:

Hc ¼ Ec H ¼ � 1

2r2 � Z

rð3:1Þ

whereZ is the nuclear charge, giving forZ ¼ 1; 2; 3;4; . . . the isoelectronicseriesH,Heþ , Li2þ , Be3þ , . . .. In what followswe shall go briefly throughthe solution of the eigenvalue equation, Equation 3.1. Solution of thesecond-order partial differential equation embodied in (3.1) necessarilyinvolves the steps outlined in Sections 3.2.1–3.2.5.

3.2.1 Choice of an Appropriate Coordinate System

The spherical symmetry of the potential energy V(r) suggests use of thespherical coordinates ðr; u;wÞ (see Figure 1.1). We have seen that in thiscase the Laplacian operator r2 separates into a radial Laplacian r2

r

and into an angular part which depends on the square of the angularmomentum operator L

2. In this way, it is possible to separate radial from

angular equations if we put

cðr; u;wÞ ¼ RðrÞYðu;wÞ ð3:2Þ

Upon substitution in (3.1) we obtain the following two separatedifferential equations:

d2R

dr2þ 2

r

dR

drþ 2 Eþ Z

r

� �� l

r2

� �R ¼ 0 ð3:3Þ

32 ATOMIC ORBITALS

L2Y ¼ lY ð3:4Þ

where l � 0 is a first separation constant.1 Equation 3.3 is the differentialequation determining the radial part of the HAOs and Equation 3.4 is theeigenvalue equation for the square of the angular momentum operatorL2determining the angular part of the orbitals. The latter equation is

found in general in the study of potential theory, the eigenfunctionsYðu;wÞ in complex form being known in mathematics as sphericalharmonics. At variance with the radial eigenfunctions R(r), whichare peculiar to the hydrogen-like system, theYðu;wÞ are useful in generalfor atoms.

3.2.2 Solution of the Radial Equation

The radial equation (3.3) has different solutions according to the value ofthe parameter E, the eigenvalue of Equation 3.1.E > 0 corresponds to the continuous spectrum of the ionized atom, its

eigenfunctions being oscillatory solutions describing plane waves. It is ofno interest to us here, except for completing the spectrumof theHermitianoperator H.E < 0 corresponds to the electron bound to the nucleus, with a discrete

spectrum of eigenvalues, the energy levels of the hydrogen-like atom asobserved from atomic spectra.It is customary to pose RðrÞ ¼ PðrÞ=r and

E ¼ � Z2

2n2ð3:5Þ

where n is a real integer positive parameter to be determined,2 and tochange the variable to

x ¼ Z

nr ð3:6Þ

1 We shall see later in this section that l ¼ ‘ð‘þ1Þ, where ‘ � 0 is the orbital quantum number.2 It is seen that (3.5) is nothing but the result in atomic units of Bohr’s theory for the hydrogenic

system of nuclear charge þZ.

HYDROGEN-LIKE ATOMIC ORBITALS 33

Therefore, the second-order ordinary differential equation (3.3) be-comes

d2P

dx2þ � 1þ 2n

x� ‘ð‘þ 1Þ

x2

� �P ¼ 0 ð3:7Þ

which must be solved with its regularity conditions in the interval0 � x � ¥ including the extrema.The study of the asymptotic behaviour of the function P(x) for the

electron far from or near to the nucleus shows that P(x) must have theform

PðxÞ ¼ expð� xÞx‘þ 1FðxÞ ð3:8Þ

where F(x) is a function to be determined from the solution of thedifferential equation

xd2F

dx2þ ð2‘þ 2Þ� 2x½ � dF

dxþ2ðn� ‘� 1ÞF ¼ 0 ð3:9Þ

Equation 3.9 is solved by a power series expansion (Taylor) in the xvariable:

FðxÞ ¼Xk

akxk ð3:10Þ

Upon substitution in (3.9), a two-term recursion formula is obtained forthe coefficients ak:

akþ 1 ¼2ðk� nþ ‘þ 1Þ

ðkþ1Þðkþ 2‘þ 2Þ ak k ¼ 0; 1;2; . . . ð3:11Þ

The study of the convergence of the power series (Equation 3.10) showsthat it converges to the function exp(2x). This solution is not physicallyacceptable since, in this case, P(x) would diverge at x ¼ ¥. So, theregularity conditions on the radial function P(x) require that the expan-sion Equation 3.10 should be truncated to a polynomial. This can be

34 ATOMIC ORBITALS

achieved if

ak =0; akþ 1 ¼ akþ 2 ¼ � � � ¼ 0 ð3:12Þ

which implies from (3.11) the necessary condition

k� nþ ‘þ 1 ¼ 0 ) kmax ¼ n� ‘� 1 ð3:13Þ

The physically acceptable radial solutions must, hence, include apolynomial of degree ðn� ‘� 1Þ at most.3 Equation 3.13 determines our,so far, unknown parameter n:

n ¼ kþ ‘þ 1 ) n ¼ ‘þ 1; ‘þ2; . . . ) n � ‘þ 1 ð3:14Þ

and, in this way, we obtain the well-known relation between principalquantum number n and orbital quantum number ‘.Coming back to RðxÞ ¼ PðxÞ=x; we see that our radial functions R(x)

will depend on quantum numbers n and ‘, beingwritten in un-normalizedform as

Rn‘ðxÞ ¼ expð� xÞx‘Xn� ‘� 1

k¼0

akxk ð3:15Þ

with

n ¼ 1; 2;3; 4; . . . ‘ ¼ 0;1; 2;3; . . . ; ðn� 1Þ ð3:16Þ

The functions in Equation 3.15 have ðn� ‘� 1Þ nodes, namely thosevalues of x for which the function changes sign. The detailed form of thefirst few radial functions can be readily obtained from (3.15) and (3.16)using the recursion formula (3.11) for the coefficients. It is found that

3 Mathematically speaking, these are related to the associated Laguerre polynomials L2‘þ1nþ ‘ ðxÞ

(Eyring et al., 1944).


n¼ 1; ‘¼ 0 R10 ¼ expð�xÞa0n¼ 2; ‘¼ 0 R20 ¼ expð�xÞða0þa1xÞ¼ expð�xÞð1�xÞa0n¼ 3; ‘¼ 0 R30 ¼ expð�xÞða0þa1xþa2x

2Þ

¼ expð�xÞ 1�2xþ 2

3x2

!a0

ns functions

8>>>>>>>>>><>>>>>>>>>>:

ð3:17Þ

n¼ 2; ‘¼1 R21 ¼ expð�xÞxa0n¼ 3; ‘¼1 R31 ¼ expð�xÞxða0þa1xÞ

¼ expð�xÞx 1� 1

2x

!a0

np functions

8>>>>>><>>>>>>:

ð3:18Þ

n¼ 3; ‘¼ 2 R32 ¼ expð�xÞx2a0 3d function ð3:19Þ

where a0 is a normalization factor. The plots ofRn‘ðxÞ versus x for 1s, 2s,2p, 3dHAOs are given in Figure 3.1. It is seen that the functionswith ‘¼0

0 0

R1s

R2p R3d

R2s

0

x/a0 x/a0

x/a0x/a0

0 0

n

Figure 3.1 Rn‘ radial functions for 1s, 2s, 2p, and 3d HAOs

36 ATOMIC ORBITALS

(called s) all have a cusp at the origin, while the functions with ‘¼ 1ðpÞ,‘¼2ðdÞ, ‘¼ 3ðfÞ, . . . are all zero at the origin, but with a differentbehaviour of their first derivative there. It is recommended that the readerverify that such radial functions are the correct solutions of the differentialequation (3.7) with PðxÞ¼ xRðxÞ.

3.2.3 Solution of the Angular Equation

We shall now give a short glance to the solution of the angular equa-tion (3.4), which follows much the same lines of solution as we have justseen for the radial equation. The first step is the separation of the partialdifferential equation in the angular variables u and w into two differentialequations, one for each variable, by posing

Yðu;wÞ ¼ QðuÞFðwÞ ð3:20Þ

This implies a second separation constant, which will for conveniencebe called m2. Of the two resultant total differential equations

d2Fdw2

þm2F ¼ 0 ð3:21Þ

and

1

sin u

d

dusin u

d

du

� �þ l� m2

sin2u

� �QðuÞ ¼ 0 ð3:22Þ

the first is the well-known equation of harmonic motion (Atkin, 1959),whose regular solutions in complex form are4

FmðwÞ ¼ 1ffiffiffiffiffiffi2p

p expðimwÞ m ¼ 0;�1;�2; . . . ð3:23Þ

and the second can be written in the form

d2Qdu2

þ cot udQdu

þ l� m2

sin2u

� �Q ¼ 0 ð3:24Þ

4

4 m is found to be integer from the single valuedness requirement imposed upon the functionF.


It is immediately evident from Equation 3.24 that the solutions Q willdepend on, besides l, the absolute value of the quantumnumber |m|, whileu ¼ 0 is a disturbing point in the interval of definition of the variable u

(0 � u � p), since sin u ¼ 0 there and the coefficients ofQ andQ0 diverge.A point like this is known in mathematics as a regular singularity (Ince,1963). Changing to the variable cos u ¼ xð� 1 � x � 1Þ, the asymptoticstudy of the resulting differential equation in the vicinity of the singularpoints x ¼ �1

ð1� x2Þ d2Qdx2

� 2xdQdx

þ l� m2

1� x2

� �Q ¼ 0 ð3:25Þ

is seen to have regular solutions when

QðxÞ ¼ ð1� x2Þm=2GðxÞ ð3:26Þ

where G(x) is a new function to be determined and we have putjmj ¼ m � 0 for short. Upon substitution we obtain for G(x) the differ-ential equation5

ð1� x2Þ d2G

dx2� 2ðmþ 1ÞxdG

dxþ l�mðmþ1Þ½ �G ¼ 0 ð3:27Þ

which is again solved by the power series expansion in the variable x:

GðxÞ ¼Xk

akxk ð3:28Þ

Proceeding in much the same way as we did for the radial equation,we obtain for the expansion coefficients the two-term recursionformula

akþ 2 ¼ðkþmÞðkþmþ 1Þ� l

ðkþ 1Þðkþ 2Þ ak k ¼ 0; 1; 2; . . . ð3:29Þ

5 Which is now free from singularities at x ¼ �1.

38 ATOMIC ORBITALS

where we obtain this time an even and an odd series. Study of theconvergence of the series shows that (3.28) is divergent at jxj ¼ 1, sothat once again the series must be truncated to a polynomial. This ispossible if

ak =0; akþ 2 ¼ akþ 4 ¼ � � � ¼ 0 ð3:30Þ

Namely, if the numerator of (3.29) vanishes, then

ðkþmÞðkþmþ 1Þ� l ¼ 0 ð3:31Þ

giving6

l ¼ ðkþmÞðkþmþ 1Þ k;m ¼ 0; 1;2; . . . ð3:32Þ

Posing

kþm ¼ ‘ a non-negative integer ð‘ � 0Þ ð3:33Þ

we obtain

‘ ¼ m; mþ 1; mþ 2; . . . ‘ � jmj � ‘ � m � ‘ ð3:34Þ

and we recover the well-known relation between angular quantumnumbers ‘ and m. Hence, we obtain for the eigenvalue of L

2

l ¼ ‘ð‘þ 1Þ ð3:35Þ

‘ ¼ 0;1; 2; 3; . . . ; ðn� 1Þ ð3:36Þ

m ¼ 0; �1; �2; �3; . . . ; �‘ ð3:37Þ

From (3.28) we have two polynomials, giving for the complete solutionof the angular equation (3.25)

6 Remember that we are using m for |m| � 0.


Q‘mðxÞ ¼ ð1� x2Þm=2Xð‘�mÞ=2

k¼0

a2kx2k þ

Xð‘�m� 1Þ=2

k¼0

a2kþ 1x2kþ 1

" #ð3:38Þ

where the first term in brackets is the even polynomial and the secondterm is the odd polynomial, whose degree is at most kmax ¼ ‘�mð� 0Þfor both.Using (3.38) together with the recursion formula (3.29) with

l ¼ ‘ð‘þ 1Þ, we obtain the following for the first few angular solutionsðx ¼ cos uÞ:

‘ ¼ 0; m ¼ 0 ‘�m ¼ 0 Q00 ¼ a0

‘ ¼ 1; m ¼ 0 ‘�m ¼ 1 Q10 ¼ xa1 / cos u

‘ ¼ 2; m ¼ 0 ‘�m ¼ 2 Q20 ¼ a0 þ a2x2 ¼ ð1�3x2Þa0

‘ ¼ 1; m ¼ 1 ‘�m ¼ 0 Q11 ¼ ð1� x2Þ1=2a0 / sin u

‘ ¼ 2; m ¼ 1 ‘�m ¼ 1 Q21 ¼ ð1� x2Þ1=2xa1 / sin u cos u

‘ ¼ 2; m ¼ 2 ‘�m ¼ 0 Q22 ¼ ð1� x2Þa0 / sin2u

‘ ¼ 3; m ¼ 2 ‘�m ¼ 1 Q32 ¼ ð1� x2Þxa1 / sin2u cos u

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð3:39Þ

where a0 and a1 are normalization factors for the even polynomials andodd polynomials respectively. It is seen that the angular solutions (3.39)are combinations of simple trigonometric functions proportional to theassociated Legendre polynomials Pm

‘ ðxÞ, well known in mathematics inpotential theory (Abramowitz and Stegun, 1965; Hobson, 1965) with aproportionality factor

Pm‘ ðxÞ ¼ ð� 1Þmþ ½ð‘þmÞ=2�Q‘mðxÞ ð3:40Þ

where ½ � � � � stands for ‘integer part of’, a factor anyway irrelevant from thestandpoint of the differential equation. The skilled reader can verify thesolutions (3.39) by direct substitution in the differential equation (3.25)with l ¼ ‘ð‘þ 1Þ.

40 ATOMIC ORBITALS

The polar diagrams of the angular functionsQ‘0 with7 ‘ ¼ 0; 1;2; 3 aresketched in Figure 3.2.

3.2.4 Some Properties of the Hydrogen-like Atomic Orbitals

The HAOs in complex form

cn‘mðr; u;wÞ ¼ Rn‘ðrÞY‘mðu;wÞ ð3:41Þ

satisfy simultaneously the following three eigenvalue equations:

Hcn‘m ¼ Encn‘m n ¼ 1;2; 3; . . .

L2cn‘m¼ ‘ð‘þ 1Þcn‘m ‘ ¼ 0;1; 2; 3; . . . ; ðn� 1Þ

Lzcn‘m¼mcn‘m m¼ 0;�1;�2;�3; . . . ;�‘

8>><>>: ð3:42Þ

where H is given by (3.1) and En by (3.5). Each eigenvalue is character-ized by one quantum number. So, the principal quantum number n

_+ +

z

= 1 = 0

+ ++__

_

+ _.zz

+ _

= 2 = 3

Figure 3.2 Schematic polar diagrams of the angular part of s, p, d, and f AOs withm¼ 0

7 It is seen that ‘ equals the number of nodal planes of the functions.


characterizes the energy levels and the orbital quantum number ‘ char-acterizes the square of the angular momentum, the so-called magneticnumber m, the z-component of the angular momentum.Now, a fundamental theorem of quantum mechanics (Eyring et al.,

1944) states that, when the same function is simultaneously an eigenfunc-tion of different operators, the corresponding operators commute in pairs;namely:

½H; L2� ¼ ½H; Lz� ¼ ½L2

; Lz� ¼ 0 ð3:43Þ

Operators commutingwith theHamiltonian are called constants of themotion, which means that energy, the square of the angular momentumand the z-component of the angular momentum can all be measured witharbitrary accuracy at the same time.The discrete spectrum of the first few energy levels of the bound

hydrogen-like electron up to n¼ 3 is given in Figure 3.3. It is seen that,

E < 0

n = 3

3s, 3p, 3d

g = 9

n = 2

2s, 2p

g = 4

n = 11s

g = 1

Figure 3.3 Schematic diagram of the energy levels of the hydrogen-like atom up ton¼ 3

42 ATOMIC ORBITALS

apart from the 1s ground state level, which is not degenerate, all excitedlevels are n2-fold degenerate. The degeneracy g of the excited energy levelsis partly removed in the many-electron atom.Lastly, we recall from first principles that

jcn‘mðrÞj2 dr ¼ probability of finding in dr the electron in state cn‘m

ð3:44ÞIntegrating over angles,we obtain the probability of finding the electron

inside a spherical shell of radius comprised between r and rþ dr, the so-called radial probability:

Pn‘ðrÞ dr ¼ ½Rn‘ðrÞ�2r2 dr ð3:45Þwhere Pn‘ðrÞ is the radial probability density.Figure 3.4 gives the radial probability density for the 1s ground state of

the H atom and the average value of the distance of the electron from thenucleus hri1s. It is seen that, while the probability density P1sðrÞ has amaximumat r ¼ a0 (the Bohr radius), hri1s ¼ 1:5a0 (the vertical bar in thefigure).

3.2.5 Real Form of the Atomic Orbitals

In valence theory, it is customary to use the real form for the angular partof the AOs, which is the same for all orbitals, even those that are not

P1s(r)

0 1 2 r/a0

Figure 3.4 Radial probability density for the H atom ground state


hydrogen-like. It is important to note that real AOs are no longereigenfunctions of the operator Lz, so that the last of the equations inEquation 3.42 is not true for real AOs. It should also be noted that inquantum mechanics the F-functions (3.23) in complex form are usuallygiven with the so-called Condon–Shortley phase (Brink and Satchler,1993):

Fþm ¼ ð� 1Þm expðimfÞffiffiffiffiffiffi2p

p m > 0

F�m ¼ expð� imfÞffiffiffiffiffiffi2p

p ¼ ð� 1ÞmðFþmÞ�

8>>>><>>>>:

ð3:46Þ

It is then possible to convert complex to realAOs using Euler’s formulafor imaginary exponentials:

expð�imwÞ ¼ cosmw� i sinmw ð3:47Þ

We obtain

Fcm ¼ ð� 1ÞmFþm þF�mffiffiffi

2p ¼ cosmfffiffiffi

pp

Fsm ¼ � i

ð� 1ÞmFþm �F�mffiffiffi2

p ¼ sinmfffiffiffip

p

8>>>>><>>>>>:

ð3:48Þ

where normalization is preserved during the unitary transformationðU� 1 ¼ U�Þ. The inverse transformation (from real to complexfunctions) is

Fþm ¼ ð� 1Þm Fcm þ iFs

mffiffiffi2

p

F�m ¼ Fcm � iFs

mffiffiffi2

p

8>>>><>>>>:

ð3:49Þ

Spherical harmonics in real form are called tesseral harmonics inmathematics (MacRobert, 1947), and in un-normalized form are

44 ATOMIC ORBITALS

written as

Y‘0; Yc‘m / Q‘m cosmw; Ys

‘m / Q‘m sinmw ðm > 0Þ ð3:50Þ

The first few (un-normalized) HAOs in real form are hence simplygiven by

1s / expð� crÞ ðc ¼ ZÞ2s / expð� crÞð1� crÞ ð2c ¼ ZÞ2pz / expð� crÞz2px / expð� crÞx2py / expð� crÞy

8>>>>>>><>>>>>>>:

ð3:51Þ

3s / expð� crÞ 1� 2crþ 2

3c2r2

!ð3c ¼ ZÞ

3pz / expð� crÞ 1� 1

2cr

!z

3px / expð� crÞ 1� 1

2cr

!x

3py / expð� crÞ 1� 1

2cr

!y

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

ð3:52Þ

3dz2 / expð� crÞ 3z2 � r2

23dxy / expð� crÞxy3dzx / expð� crÞzx3dyz / expð� crÞyz3dx2 � y2 / expð� crÞðx2 � y2Þ

8>>>>>>>><>>>>>>>>:

ð3:53Þ

where x, y, and z are given by Equation 1.45. Hence, our real AOs havethe same transformation properties of (x,y,z)-coordinates or of their


combinations. Replacing the radial polynomial part of theHAOsby rn� 1,the functional dependence of these real HAOs on ðr; u;wÞ is the same asthat of STOs with the orbital exponent c considered as a variableparameter, as we shall see below.

3.3 SLATER-TYPE ORBITALS

STOs were introduced long ago for the many-electron atoms bySlater (1930) and Zener (1930). Slater showed that for many pur-poses only the term with the highest power of r in the hydrogen-likeRn‘ðrÞ is of importance in practical calculations. Zener suggested thatreplacing the hydrogenic orbital exponent c ¼ Z=n by an effectivenuclear charge (Zeff¼Z � s)8 could account in some way for thescreening of the nuclear charge Z by the remaining electrons, givingin this way orbitals that are more diffuse than the original HAOs.Slater also gave some rules for estimating the screening constant s,which are today replaced by the variational estimation of the orbitalexponents c.Retaining only the highest ðn� ‘� 1Þ power of r of the polynomial

in (3.15), the dependence on ‘ is lost and we can write the general STO inreal form as

xn‘mðr; u;wÞ ¼ jn‘mi ¼ N expð� crÞrn� 1Yc;s‘mðu;wÞ ¼ RnðrÞYc;s

‘mðu;wÞð3:54Þ

where N is a normalization factor, Rn and Yc;s‘m are each individually

normalized to 1, and c > 0 is the orbital exponent considered as avariable parameter. Yc;s

‘m are the tesseral harmonics (3.50), which arebest expressed in terms of the associated Legendre polynomials Pm

‘ ðxÞ(x ¼ cos u) as

Yc;s‘mðu;wÞ ¼ NWP

m‘ ðxÞ

cosmw

sinmwm � 0

(ð3:55Þ

8 s ð< ZÞ was called by Zener the screening constant.

46 ATOMIC ORBITALS

As we have already said, the only difference with STOs from thehydrogenic AOs (3.51)–(3.53) is in the radial part Rn(r), which is nowindependent of ‘. This implies a wrong behaviour of the ns functionswith n > 1, which are now zero at the origin, lacking the characteristiccusp present in HAOs and which must be eventually corrected bySchmidt orthogonalization of the ns STO against all its inner sorbitals.Bearing in mind the characteristic integrals (Abramowitz and Stegun,

1965)

ð¥0

dr rnexpð� crÞ ¼ n!

cnþ 1ð3:56Þ

ð2p0

dwcos2 mw

sin2 mw

¼ pð1þ d0mÞ

ð1� d0mÞ

8<:

8<: ð3:57Þ

ð1� 1

dx ½Pm‘ ðxÞ�2 ¼

2

2‘þ 1

ð‘þmÞ!ð‘�mÞ! m ¼ jmj � 0 ð3:58Þ

the overall normalization factor N of the general STO is obtained as

N ¼ NrNW ¼ ð2cÞ2nþ 1

ð2nÞ!

" #1=22‘þ 1

2pð1þ d0mÞð‘�mÞ!ð‘þmÞ!

� �1=2ð3:59Þ

whereNr andNW are separate radial and angular normalization factors. Itmust be further recalled that the real spherical harmonics (tesseralharmonics) are orthogonal and normalized to 1:

ðdW Yc;s

‘0m0 ðWÞYc;s‘mðWÞ ¼ d‘‘0dmm0 ð3:60Þ

where W stands for the angular variables ðu;wÞ and dW ¼ sinu du dw.

SLATER-TYPE ORBITALS 47

It is left as an exercise for the reader to verify that the first fewnormalized STOs are

1s ¼ c30p

� �1=2expð� c0rÞ ð3:61Þ

s ¼ c5s3p

� �1=2expð� csrÞr ð3:62Þ

2pz ¼c5pp

!1=2expð� cprÞr cos u ð3:63Þ

with the matrix elements of the hydrogenic Hamiltonian being

1s � 1

2r2 � Z

r

��1s

� ¼ c20

2�Zc0 ð3:64Þ

s � 1

2r2 � Z

r

��s

� ¼ c2s

6� 1

2Zcs ð3:65Þ

2pz � 1

2r2 � Z

r

��2pz

� ¼ 1

2ðc2p �ZcpÞ ð3:66Þ

s � 1

2r2� Z

r

��1s

� ¼ Ss1s � c20

2þðc0 �ZÞ c0 þ cs

3

� �ð3:67Þ

where Ss1s is the nonorthogonality integral between 1s and s STOs:

Ss1s ¼ hsj1si ¼ffiffiffi3

p csc0 þ cs

2ðc0csÞ1=2c0þ cs

" #3ð3:68Þ

These formulae will be needed in the Chapter 4.

48 ATOMIC ORBITALS

3.4 GAUSSIAN-TYPE ORBITALS

Gaussian orbitals are largely used today in atomic and molecular com-putations because of their greater simplicity in computing multicentremolecular integrals. GTOs were introduced by Boys (1950) andMcWeeny (1950) mostly for computational reasons, but they are defi-nitely inferior to STOs because of their incorrect radial behaviour, both atthe origin9 and in the tail of the wavefunction. In today’s molecularcalculations it is customary to fit STOs in terms of GTOs, which requiresrather lengthy expansions with a large number of terms. Some failures ofGTOs with respect to STOs are briefly discussed in the Chapter 4.Most common Gaussian orbitals are given in the form of spherical or

Cartesian functions. We shall give some definitions here and a fewintegrals which are needed later.

3.4.1 Spherical Gaussians

For spherical GTOs it will be sufficient to consider only the radial part ofthe orbital, since the angular part is the same as that for STOs:

RnðrÞ ¼ Nn expð� cr2Þrn�1Nn ¼ 2nþ 1ð2cÞnþð1=2Þ

ð2n� 1Þ!! ffiffiffipp" #1=2

ðc > 0Þ

ð3:69Þwhere

ð2n� 1Þ!! ¼ ð2n� 1Þð2n� 3Þ . . . 3 1 ¼ ð2nÞ!2nn!

¼ ð2nÞ!ð2nÞ!! ð� 1Þ!! ¼ 0!! ¼ 1

ð3:70Þ

is the double factorial.This normalization factor in (3.69) can be derived from the general

Gaussian integral

ð¥0

dr expð� cr2Þrn ¼ ðn� 1Þ!!ð2cÞðnþ 1Þ=2 sðnÞ ð3:71Þ

9 Spherical GTOs lack the characteristic cusp at the origin.

GAUSSIAN-TYPE ORBITALS 49

where

sðnÞ ¼ffiffiffip2

rfor n ¼ even; sðnÞ ¼ 1 for n ¼ odd ð3:72Þ

The normalized spherical Gaussian orbital will then be written as

jn‘mi ¼ Gðn‘m; cÞ ¼ N expð� cr2Þrn� 1Yc;s‘mðu;wÞ ð3:73Þ

with N ¼ NnNW.A few results for 1s GTOs are as follows:

Gð100; c0Þ � 1

2r2 � Z

r

��Gð100; cÞ

� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi72ðcc0Þ7=2ðcþ c0Þ5

s�Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32ðcc0Þ3=2ðcþ c0Þ2p

s

ð3:74Þ

and, if c0 ¼ c:

Gð100; cÞ � 1

2r2 � Z

r

��Gð100; cÞ

� ¼ 3

2c�Z

8c

p

� �1=2

ð3:75Þ

It is apparent that these formulae are sensibly less appealing than thosefound for STOs.

3.4.2 Cartesian Gaussians

The normalized Cartesian Gaussians are best written as

GuvwðcÞ ¼ N expð� cr2Þxuyvzw ¼ N exp½ � cðx2þ y2 þ z2Þ�xuyvzwð3:76Þ

whereu, v, andw are non-negative integers and the normalization integralis given by

50 ATOMIC ORBITALS

N ¼ ð4cÞUþVþW

ð2U� 1Þ!!ð2V�1Þ!!ð2W� 1Þ!!2c

p

� �3=2" #1=2

ð3:77Þ

where

2U ¼ uþ u0; 2V ¼ vþ v0; 2W ¼ wþw0 ð3:78Þ

GAUSSIAN-TYPE ORBITALS 51

4The Variation Method

In this chapter we present elements of the variation method, the mostpowerful technique for doing working approximations when the Schroe-dinger eigenvalue equation cannot be solved exactly. Applications involveoptimization of variational parameters, either nonlinear (orbital expo-nents) or linear (the Ritz method). To illustrate the method, examples onground and excited states of theH-like and theHe-like atomic systems areworked out in some detail.

4.1 VARIATIONAL PRINCIPLES

Let w be a normalizable regular trial (or variational) function. We definethe Rayleigh ratio as the functional1

« w½ � ¼ hwjHjwihwjwi ¼

Ðdx w�ðxÞHwðxÞÐdx w�ðxÞwðxÞ ð4:1Þ

where x are the electronic coordinates and H the Hamiltonian of thesystem. Then

«½w� � E0 ð4:2Þ

1 A function of function w(x).


Valerio Magnasco


is the Rayleigh variational principle for the ground state,E0 being the trueground state energy;

«½w� � E1 provided hc0jwi ¼ 0 ð4:3Þ

is the Rayleigh variational principle for the first excited state, provided thetrial function w is taken orthogonal to the true ground state function c0.The proofs of these statements are given in Magnasco (2007).Therefore, evaluation of the integrals in (4.1) under the suitable con-

straints of normalization and orthogonality gives the upper bounds tothe energy of the ground and excited states depicted in Figure 4.1. This isof fundamental importance in applications, since the variational energymust always lie above the true energy.It is easily seen that:

. The equality sign holds for the exact functions.

. If the variational function is affected by a first-order error, then theerror in the variational energy is second order. Therefore, energy is

EEnergy

(< 0)

ε [ϕ]

ε [ϕ]

E1

E0 E0

Figure 4.1 Energy upper bounds to ground (left) and first excited state (right)

54 THE VARIATION METHOD

always determined better than thewavefunction. The same is true forthe second-order energies of Chapter 10.

. The variational method privileges the regions of space near thenucleus, so that variationally determined wavefunctions may be notappropriate for dealing with the expectation values of operators thattake large values far from thenucleus, like the electric dipolemoment.

Variational approximations to energy and the wavefunction can beworked out simply by introducing some variational parameters {c} in thetrial function and then evaluating the integrals in the functional (4.1),giving in this way an ordinary function of the variational parameters {c},whichhas tobeminimizedagainst theparameters. For a singleparameter c(Figure 4.2)

ε (c)

0

cmin

c

εmin

Figure 4.2 Plot of the variational energy near the minimum versus c

VARIATIONAL PRINCIPLES 55

« w½ � ¼Ðdx w�ðx; cÞHwðx; cÞÐdx w�ðx; cÞwðx; cÞ ¼ «ðcÞ ð4:4Þ

d«

dc¼ 0 ) cmin ð4:5Þ

provided

d2«

dc2

!cmin

> 0 ð4:6Þ

In thisway,weobtain thebest approximation compatiblewith the formassumed for the approximate trial function. Increasing the number offlexible parameters increases the accuracy of the variational result.2

For N variational parameters, fcg ¼ ðc1; c2; . . . ; cNÞ, Equations 4.4and 4.5 must be replaced by

«½w� ¼ «ðc1; c2; . . . ; cNÞ ð4:7Þ

q«qc1

¼ q«qc2

¼ � � � ¼ q«qcN

¼ 0 ) cðbestÞf g ð4:8Þ

«ðbestÞ ¼ «ðc01; c02; . . . ; c0NÞ ð4:9Þ

wðbestÞ ¼ wðx; c01; c02; . . . ; c0NÞ ð4:10Þ

where fcðbestÞg ¼ ðc01; c02; . . . ; c0NÞ is the set of N optimized parameters.For working approximations, we must resort to some basis set of

regular functions (such as the STOs or GTOs of Chapter 3), introducingeither (i) nonlinear (orbital exponents) or (ii) linear variationalparameters.

2 Using appropriate numericalmethods, it is feasible today tooptimize variationalwavefunctionscontaining millions of terms, such as those encountered in the configuration interaction tech-

niques of Chapter 8.


4.2 NONLINEAR PARAMETERS

In this case, we cannot obtain any standard equations for the optimiza-tion, which must usually be done by numerical methods (e.g. see thesimple Ransil method, useful for functions having a parabolic behaviournear the minimum). The method is rather powerful in itself, but, whenmany parameters are involved, there may be some troubles with thedifficulty of avoiding spurious secondary minima in the energy hypersur-face, which would spoil the numerical results.In the following, we shall concisely apply this method to simple

variational functions containing a single variational parameter c ð> 0Þfor the one-electron and the two-electron atomic cases. TheH-like systemis particularly important, mostly for didactical reasons, since wecan compare our approximate results with the exact ones derived inChapter 3.

4.2.1 Ground State of the Hydrogenic System

We compare the relative accuracy of three variational functions, two ofSTO type and one GTO, each containing a single adjustable orbitalexponent c for the ground state of the hydrogen-like system having theone-electron Hamiltonian (in atomic units):

h ¼ � 1

2r2 � Z

rð4:11Þ

where Z is the nuclear charge (Z ¼ 1 for hydrogen). The expressions ofthe energy integral and those of the analytical optimization3 are collectedin Table 4.1 and the numerical results are given in Table 4.2.All functions in the first column of both tables are un-normalized, and

must be normalized at the end of each calculation. w1 is the single-parameter trial function having the correct functional form as the exactwavefunction c0. Optimization of the nonlinear variational parameter cyields the correct wavefunction and energy, corresponding to the equal-ity sign in (4.2).w2 is a Slater 2s,which lacks the cusp characteristic of thetrue ns hydrogenic wavefunctions (it is zero at the origin) and decreasestoo slowly in its tail because of r in its functional form. It gives the worst

3 The students are strongly recommended to do the calculations by themselves.

NONLINEAR PARAMETERS 57

energy result, with nomore than 75%of the truth. The third function w3

is the prototype of the 1s Gaussian function. Even if it has the wrongbehaviour at the origin (it has a zero derivative here) and decreasestoo quickly far from the nucleus, it nevertheless gives a fair result for theenergy, namely about 85% of the true value, which is 10% betterthan w2. However, increasing the number of 1s GTOs improves theenergy, but does not improve the wavefunction sufficiently; even takingN¼10 optimized 1s GTOs, it is still very different from the correctone.4 It is interesting to note fromTable 4.2 that the variational theoremtries to do its best to correct the inappropriate form of functions w2

and w3 by strongly increasing (cmin ¼ 1:5, the function contracts) ordecreasing (cmin ¼ 0:2829, the function expands) respectively the bestvalue for their orbital exponents. Taking c ¼ 1 in w2 would give«w ¼ �0:333 333Eh, which is about 67% of the true value. Hence,optimization of c improves energy by 8%.

Table 4.2 Numerical results of different variationalapproximations for the ground-state H atom

w cmin «(cmin)/Eh

w1 ¼ expð� crÞ 1 � 0.5

w2 ¼ expð� crÞr 1.5 � 0.375

w3 ¼ expð� cr2Þ 0.2829 � 0.4244

Table 4.1 Variational approximations to the ground state of thehydrogenic system

w «(c) cmin «min

w1 ¼ expð� crÞ c2

2�Zc Z �Z2

2

w2 ¼ expð� crÞr c2

6� Zc

2

3

2Z � 3

8Z2

w3 ¼ expð� cr2Þ 3

2c�Z

8c

p

� �1=2 8

9pZ2 � 4

3pZ2

4 The incorrect behaviour at the origin of 1s GTOs and the way of correcting for it are fully

discussed by Magnasco (2007).


4.2.2 The First Excited State of Spherical Symmetryof the Hydrogenic System

The first spherical excited state of the H atom is the 2s state, whoseenergy belongs to the four degenerate levels with n ¼ 2. As with all nsorbitals, the correct function has a cusp at the origin. The sphericalnormalized STO

s ¼ c5s3p

� �1=2

expð�csrÞr ð4:12Þ

is not orthogonal to the ground-state 1s function (orbital exponentc0 ¼ 1), with the nonorthogonality integral

S ¼ hc0jsi ¼ffiffiffi3

p cscs þ 1

2ffiffiffiffics

pcs þ 1

� �3

ð4:13Þ

and cannot be used as such in the variational calculation of the energyof the excited state. In fact, without the constraint of orthogonalitydemanded by (4.3), function s would give at most, as we have just seen, apoor bound to the ground-state energy.A convenient variational function for the first excited state of spherical

symmetry is obtained by first orthogonalizing s against c0 by the Schmidtmethod, giving the normalized trial function

w ¼ s� Sc0ffiffiffiffiffiffiffiffiffiffiffiffiffi1� S2

p hc0jwi ¼ 0 ð4:14Þ

which is now orthogonal to c0, as it must be. It is then easily seen that thenew variational function (4.14) gives the upper bound

«w ¼ w �1

2r2 � Z

r

��w

� �¼ hss þEpn � E2s ð4:15Þ

where

hss ¼ c2s6

� Zcs2

ð4:16Þ


is the energy of the nonorthogonal function (4.12) and

Epn ¼ S2ðhc0c0þ hssÞ� 2S hsc0

1� S2¼ � S

ðc0s� Sc20jhÞþ ðsc0 � Ss2jhÞ1� S2

> 0

ð4:17Þ

the term correcting for nonorthogonality. This repulsive term, called thepenetration (or exchange-overlap) energy, avoids the excited 2s electronfrom collapsing onto the inner 1s electron. It is easily seen by directcalculation that the nondiagonal matrix element in (4.17) is given by(compare with Equation 3.67)

hsc0¼ s � 1

2r2 � Z

r

��c0

� �¼ S

3ðc0�ZÞðc0 þ csÞ� 3

2c20

� ð4:18Þ

which, for c0 ¼ Z ¼ 1, reduces to

hsc0¼ � S

2ð4:19Þ

The behaviour of the three different components of the variationalenergy calculation for Z ¼ 1 are plotted qualitatively in Figure 4.3 as a

ε /Eh

Epn

0 cs

1.50.42

εϕ

hss

Figure 4.3 Plots of the variational energy components for H(2s) versus the cs orbitalexponent


function of the orbital exponent cs. The too attractive hss has its badminimum at cs ¼ 1:5 (see Table 4.2) and is contrasted by the repulsiveEpn so as to give the best variational minimum of «w ¼ �0:1234Eh

occurring for cs ¼ 0:4222. This result is within 98.7% of the exact result,E2s ¼ �0:125Eh. Even without doing orbital exponent optimizationðcs ¼ 0:5Þ, the remarkable result of «w ¼ �0:1192Eh would be obtained(about 95% of the exact value), showing the crucial importance ofSchmidt’s orthogonalization in assessing a reasonable variational approx-imation to the 2s excited state.

4.2.3 The First Excited 2p State of the Hydrogenic System

We can take as an appropriate variational function for this case thenormalized 2pz STO:

w ¼ c5pp

!1=2

expð�cprÞr cos u ð4:20Þ

orthogonal to c0 by symmetry. The orthogonality constraint is nowsatisfied from the outset, and simple calculation gives the upper bound(Equation 3.66):

«w ¼ 2pz � 1

2r2 � Z

r

��2pz

� �¼ 1

2ðc2p �ZcpÞ � E2p ð4:21Þ

giving upon cp optimization the exact value for the excited 2p energy levelof the H-like system:

d«wdcp

¼ cp� Z

2¼ 0 ) cpðbestÞ ¼ Z

2) «wðbestÞ ¼ �Z2

8ð4:22Þ

4.2.4 The Ground State of the He-like System

With reference to Figure 4.4, the two-electron Hamiltonian (in atomicunits) for the He-like system is

H ¼ h1þ h2 þ 1

r12; h ¼ � 1

2r2 � Z

rð4:23Þ


the sumof the twoone-electronhydrogenicHamiltonians and the electronrepulsion term (a two-electron operator). This last term hinders separa-tion of the two-electron Schroedinger eigenvalue equation into twoseparate equations, one for each hydrogenic electron. However, approx-imations are possible through the variation method using the simpletwo-electron product variational function:

wð1; 2Þ ¼ w1ð1Þw2ð2Þ ð4:24Þ

where the normalized one-electron functions are given by

w1s ¼c30p

� �1=2

expð�c0rÞ ð4:25Þ

Calculation then gives the variational bound

«w ¼ w1w2 h1 þ h2 þ 1

r12

��w1w2

� �¼ 2h1s1s þð1s2j1s2Þ ð4:26Þ

1

r12

2

r1

r2

O

Figure 4.4 The atomic two-electron system


namely:

«w ¼ 2c202

�Zc0

� �þ 5

8c0 ¼ c20� 2 Z� 5

16

� �c0 ð4:27Þ

In (4.26), the first term on the right is the one-electron matrix elementalready found for H, while the second is the two-electron repulsionintegral between the two spherical 1s charge distributions written incharge density notation:

h1s1 1s2jr�112 j1s1 1s2i ¼

ððdr1 dr2

½1sðr2Þ1s�ðr2Þ�r12

1sðr1Þ1s�ðr1Þ½ � ¼ ð1s2j1s2Þð4:28Þ

This new two-electron integral is evaluated in terms of the purely radialelectrostatic potential J1s(r1):

J1sðr1Þ ¼ðdr2

½1sðr2Þ1s�ðr2Þ�r12

¼ 1

r1� expð�2c0r1Þ

r1ð1þ c0r1Þ ð4:29Þ

when use is made of the one-centre Neumann expansion for 1=r12(Magnasco, 2007). Then, integral (4.28) is readily obtained by integrationin spherical coordinates:

ð1s2j1s2Þ ¼ðdr1 J1sðr1Þ½1sðr1Þ1s�ðr1Þ� ¼ 5

8c0 ð4:30Þ

Optimization of (4.27) with respect to c0 gives

c0 ¼ Z� 5

16ð4:31Þ

as best value for the orbital exponent and

«wðbestÞ ¼ � Z� 5

16

� �2

: ð4:32Þ

as best energy for the S(1s2) configuration of the He-like atom.


ForZ ¼ 2 (Heatom),Zeff ¼ Z�ð5=16Þ ¼ 27=16 ¼ 1:6875,and (4.32)gives «w(best)¼�2.847656Eh, which is more than 98% of the accuratevalue of�2:903 724Eh given by Pekeris (1958). Eckart (1930) gave as twononlinear parameter approximations to the ground-state energy of theHe atom the following improved ‘split-shell’ result:

« ¼ �2:875 661Eh c1 ¼ 2:183 171 c2 ¼ 1:188 531 ð4:33Þ

which is within 99% of Pekeris’s result. We end this section by noting:(i) that Equation 4.32 describes in part the ‘screening effect’ of the secondelectron on the nuclear charge Z,5 showing that the variation theoremaccounts as far as possible for real physical effects; (ii) that the calculatedenergy is not an observable quantity, so that comparison with experi-mental results is possible only through the values of the ionizationpotential, defined as

I ¼ «ðHeþ Þ� «ðHeÞ ð4:34Þ

Since the ionization potential I is smaller than the absolute energies ofeither the atom or the ion, its approximate values are affected by largererrors.

4.3 LINEAR PARAMETERS AND THERITZ METHOD

This famous method of linear combinations is due to the young SwissmathematicianRitz (1909) and, therefore, is usually referred to as theRitzmethod. From the variational point of view, flexibility in the trial functionis now introduced through the coefficients of the linear combination ofa given set of regular functions. Usually, the basis functions are fixed,but they can be successively optimized even with respect to the nonlinearparameters present in their functional form. We shall see that the Ritz

5 In the He atom, an electron near to the nucleus sees the whole nuclear charge Z; far from it, the

nuclear charge is ðZ�1Þ, as if it were fully screened by the other electron. The variational resultaverages between these two extreme cases, privileging the regions near to the nucleus (hence,

Zeff � 1:7, closer to 2 rather than 1).


method is intimately connected with the problem of the matrix diago-nalization of Chapter 2.We shall limit ourselves to consideration of a finite basis set of N

orthonormal functions x, the problem being best treated in matrix form.The Rayleigh ratio (4.1) can be written as

« ¼ HM�1 ð4:35Þ

where

H ¼ hwjHjwi; M ¼ hwjwi ð4:36Þ

If we introduce the set of N orthonormal functions as the rowmatrix

x ¼ ðx1x2 . . .xNÞ ð4:37Þ

and the corresponding set of variational coefficients as the columnmatrix

c ¼c1c2� � �cN

0BB@

1CCA ð4:38Þ

thenH andM in (4.36) can be written in terms of the (N�N) Hermitianmatrices H and M:

H ¼ c�x�Hxc ¼ c�Hc; M ¼ c�x�xc ¼ c�Mc ¼ c�1c ð4:39Þ

where M ¼ 1 is the metric matrix of the basis functions x. The matrixelements of matrices H and M are

Hmn ¼ hxmjHjxni; Mmn ¼ hxmjxni ¼ 1mn ¼ dmn ð4:40Þ

An infinitesimal first variation in the linear coefficients will induce aninfinitesimal change in the energy functional (4.35):

d« ¼ dH �M�1 �H �M�2dM ¼ M�1ðdH� «dMÞ ð4:41Þ

LINEAR PARAMETERS AND THE RITZ METHOD 65

where to first order in dc (a column of infinitesimal variation of coeffi-cients)

dH ¼ dc�Hcþ c�Hdc; dM ¼ dc�1cþ c�1dc ð4:42Þ

The necessary condition for « being stationary against arbitrary varia-tions in the coefficients yields the equation

d« ¼ 0 ) dH� «dM ¼ 0 ð4:43Þ

and in matrix form

dc�ðH� «1Þcþ c�ðH� «1Þdc ¼ 0 ð4:44Þ

Because matrix H is Hermitian, the second term in (4.44) is simplythe complex conjugate of the first, so that, since dc� is arbitrary, thecondition (4.43) takes the matrix form

ðH� «1Þc ¼ 0 ) Hc ¼ «c ð4:45Þ

which is the eigenvalue equation for matrix H (Equation 2.27). Thevariational determination of the linear coefficients under the constraintof orthonormality of the basis functions in the Ritz method is, therefore,completely equivalent to the problem of diagonalizing matrixH. Follow-ing what was said there, the homogeneous system (4.45) has nontrivialsolutions if and only if

jH� «1j ¼ 0 ð4:46Þ

The solution of the secular Equation 4.46 yields as best values for thevariational energy (4.35) the N real roots, which are usually ordered inascending order:

«1 «2 � � � «N ð4:47Þ

c1; c2; . . . ; cN ð4:48Þ

w1;w2; . . . ;wN ð4:49Þ


which are respectively the eigenvalues and eigenvectors of matrix H

and the best variational approximation to the eigenfunctions. The Ritzmethod not only gives the best variational approximation to the ground-state energy (thefirst eigenvalue«1), but also approximations to the energyof the excited states. A theorem due to MacDonald (1933) states furtherthat each of the ordered roots (4.47) gives an upper bound to the energy ofthe respective excited state.

4.4 APPLICATIONS OF THE RITZ METHOD

Anapplication to the first two excited states of theHe-like atomconcludesthis chapter.

4.4.1 The First 1s2s Excited State of the He-like Atom

We take as the orthonormal two-electron basis set the simple products ofone-electron functions

x1 ¼ 1s1 2s2; x2 ¼ 2s1 1s2 ð4:50Þ

which are assumed individually normalized and orthogonal.6 The (2�2)secular equation is

H11 � « H12

H12 H22 � «

�� ¼ 0 ð4:51Þ

with the matrix elements

H11 ¼ 1s1 2s2 h1 þ h2 þ 1

r12

��1s1 2s2

� �¼ h1s1s þ h2s2s þð1s2j2s2Þ ¼ E0þ J ¼ H22

ð4:52Þ

H12 ¼ 1s1 2s2 h1 þ h2 þ 1

r12

��2s1 1s2

� �¼ ð1s 2sj1s 2sÞ ¼ K ð4:53Þ

6 The appropriate 2s AO has the same form as w of Equation 4.14.

APPLICATIONS OF THE RITZ METHOD 67

Both two-electron integrals are written here in charge density nota-tion, J being called the Coulomb integral and K ðJÞ the exchangeintegral.The roots are

«þ ¼ H11 þH12 ¼ E0 þ JþK ð4:54Þ

«� ¼ H11 �H12 ¼ E0 þ J�K ð4:55Þ

whereas the correspondingwavefunctions have definite symmetry proper-ties with respect to electron interchange:

wþ ¼ 1ffiffiffi2

p ð1s 2sþ 2s 1sÞ

w� ¼ 1ffiffiffi2

p ð1s 2s� 2s 1sÞ ð4:57Þ

the first being symmetric, the second antisymmetric in the electroninterchange. The schematic diagram of the energy levels for the S(1s2s)excited state of the He-like atom is qualitatively depicted in Figure 4.5.The splitting between the two levels (degenerate in the absence of electronrepulsion) is just 2K; that is, twice the value of the exchange integral K.Variational optimization of the orbital exponent of the 2s function(Magnasco, 2007) for He (Z ¼ 2) gives cs ¼ 0:4822, which, used in

J + K

J - K

2 K

J

E0

(twofold degenerate)

Figure 4.5 Schematic diagram of the energy levels for the excited S(1s2s) state of theHe-like atom


conjunction with c0 ¼ 1:6875 for the 1s AO, yields the variational energybounds (E0 ¼ �2:348 01Eh, J ¼ 0:240 29Eh):

«þ ¼ �2:093 74Eh «� ¼ �2:121 68Eh ð4:58Þ

Either the excitation energies from the He ground state

D«þ ¼ 0:7539Eh D«� ¼ 0:7260Eh ð4:59Þ

or the splitting

2K ¼ 0:0279Eh ð4:60Þ

are seen to be in sufficiently good agreement with the experimental values(Moore, 1971) of 0.7560, 0.7282, and 0.0292 respectively. It is worthnoting that hydrogenic AOs (with cs ¼ 1) would give a wrong order forthe energy levels (Magnasco, 2007).

4.4.2 The First 1s2p State of the He-like Atom

The same considerations can be made for the first 1s2p excited state.Because of space degeneracy, in the absence of the electron interaction,there are now six states having the same energy, and precisely

x1 ¼ 1s2pz; x2 ¼ 2pz1s; x3 ¼ 1s2px; x4 ¼ 2px1s; x5 ¼ 1s2py; x6 ¼ 2py1s

ð4:61Þ

where we have omitted for brevity the electron labels (electrons alwaysarranged in dictionary order). We omit the details of the calculation,which follows strictly those already seen for the 1s2s state. We only saythat functions belonging to different (x,y,z) symmetries are orthogonaland not interacting with each other and with respect to all S states. The(6� 6) secular equation factorizes into three equivalent (2�2) blocks,whose matrix elements are obtained from the previous ones simply byreplacing 2s by 2p, E0 by E0

0, and J, K by J0; K0. The two roots are stilltriply degenerate, so that the sixfold degeneracy occurring in the one-electron approximation E0

0 is not completely removed by considerationof the electron repulsion. Complete removal of the residual degeneracy

APPLICATIONS OF THE RITZ METHOD 69

of the energy levels is accomplished only by taking into account spin andthe Zeeman effect in the presence of a magnetic field (Chapter 5). Theschematic diagram of the splitting of the energy levels for the P(1s2p)state of the He-like atom is given in Figure 4.6. The evaluation inspherical coordinates of the integrals J, K, J0, and K0 is sketched in theAppendix.Variational optimization of the orbital exponent of the 2p functions

(Magnasco, 2007) for He (Z ¼ 2) gives cp ¼ 0:4761, which, used inconjunction with c0 ¼ 1:6875 for the 1s AO, yields the following varia-tional energy bounds (E0

0 ¼ �2:313 99Eh, J0 ¼ 0:236 67Eh):

«þ ¼ �2:072 37Eh «� ¼ �2:082 26Eh ð4:62Þ

In this case, too, either the excitation energies from the He groundstate

D«þ ¼ 0:7753 Eh D«� ¼ 0:7654Eh ð4:63Þ

or the splitting

2K ¼ 0:0099Eh ð4:64Þ

are in reasonably good agreement with the experimental values (Moore,1971) of 0.7796, 0.7703, and 0.0093 respectively. However, from aquantitative point of view, it is seen that the error in the variational

J ′ + K ′

′

J ′ - K′

2 K

J ′

E0′

(sixfold degenerate)

Figure 4.6 Schematic diagram of the energy levels for the excited P(1s2p) state ofthe He-like atom


calculation of the excited P state is larger than that resulting for theexcited S state. The splitting between the P levels is about one order ofmagnitude smaller than that observed for the S levels.

APPENDIX: THE INTEGRALS J, K, J0 AND K0

The two-electron one-centre integrals J, K, J0, and K0 occurring in thecalculationof the first excited states ofHe canbe evaluatedas one-electronintegrals in spherical coordinates once the appropriate electrostaticpotentials are known. With reference to the 1s, s, and 2pz STOs definedby Equations 3.61–3.63, the electrostatic potentials are evaluated usingthe one-centre Neumann expansion for 1=r12, giving

J1s2ðr1Þ ¼ðdr2

½1sðr2Þ�2r12

¼ 1

r11� expð�2c0Þð1þ c0r1Þ½ � ð4:65Þ

Js1sðr1Þ ¼ðdr2

½sðr2Þ1sðr2Þ�r12

¼ 12

ðc0þ csÞ3c0

3cs5

3

!1=21

r1� expð�2rÞ 1þ 4

3rþ 2

3r2

!" #

ð4:66Þ

J1s2pzðr1; uÞ ¼ðdr2

½1sðr2Þ2pzðr2Þ�r12

¼ 8c3=20 c

5=2p

ðc0þ cpÞ3cos u

1

r21� expð�2rÞð1þ2rþ2r2þ r3Þ �ð4:67Þ

where 2r ¼ ðc0þ cs;pÞr1. It is important to note that the radial part ofthe potentials can be also evaluated in spheroidal coordinates bychoosing r1 ¼ fixed as Roothaan (1951a) did. Once the potentials areknown, the basic two-electron integrals needed are easily evaluated inspherical coordinates. The results are simple functions of the orbitalexponents:

ð1s2j1s2Þ ¼ 5

8c0 ð4:68Þ

APPENDIX: THE INTEGRALS J, K, J0 AND K0 71

ðs2j1s2Þ ¼ cs2� 3c0þ cs

2

csc0þ cs

!5

ðs 1sj1s2Þ ¼ c0þ cs3

Ss1s 1� 6c0þ cs3c0þ cs

c0þ cs3c0þ cs

!324

35

ðs 1sjs 1sÞ ¼ 44c30c

5s

ðc0þ csÞ7

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð4:69Þ

ð2p2z j1s2Þ ¼cp2

� 3c0þ cp2

cpc0þ cp

!5

ð1s 2pzj1s2pzÞ ¼28

3

c30c5p

ðc0þ cpÞ7

8>>>>>><>>>>>>:

ð4:70Þ

where the nonorthogonality integral Ss1s ¼ h1sjsi ¼ S is given byEquation 3.68.Then:

J ¼ ð2s2j1s2Þ ¼ ð1� S2Þ� 1½ðs2j1s2Þþ S2ð1s2j1s2Þ� 2Sðs 1sj1s2Þ�K ¼ ð1s 2sj1s 2sÞ ¼ ð1� S2Þ� 1½ðs 1sjs 1sÞþ S2ð1s2j1s2Þ� 2Sðs 1sj1s2Þ�

(

ð4:71Þ

J0 ¼ ð2p2z j1s2Þ; K0 ¼ ð1s 2pzj1s 2pzÞ ð4:72Þ

As a final example, we give below the calculation of J0 and K0.

(i) Evaluation of J0

J0 ¼ ð2p2z jJ1s2Þ ¼4

3c5p

ð¥0

dr expð�2cprÞ½r3� expð�2c0rÞðr3þ c0r4Þ�

¼ 4

3c5p

3� 2

ð2cpÞ4� 3� 2

ð2c0þ2cpÞ4þ c0

4� 3� 2

ð2c0þ2cpÞ5

24

35

8<:

9=;

¼ cp2

� 3c0þ cp2

cpc0þ cp

!5

ð4:73Þ


(ii) Evaluation of K0

K0 ¼ ð1s2pzjJ1s2pzÞ¼c3=20 c

5=2p

p

ðdrexp �ðc0þcpÞr

��rcosuJ1s2pzðr;cosuÞ

¼ 512

3

c30c5p

ðc0þcpÞ7ð¥0

dr expð�2rÞ½r�expð�2rÞðrþ2r2þ2r3þr4�

¼ 512

3

c30c5p

ðc0þcpÞ71

22� 1

42�2

2

43�2

3�2

44� 4�3�2

45

0@

1A

¼ 1

6

c30c5p

ðc0þcpÞ7ð256�64�64�48�24Þ¼28

3

c30c5p

ðc0þcpÞ7

ð4:74Þ

APPENDIX: THE INTEGRALS J, K, J0 AND K0 73

5Spin

In this chapter, starting from the Zeeman effect, we introduce elements ofPauli’s formal theory for one- and two-electron spin,with a generalizationto the Dirac formula for N-electron spin.

5.1 THE ZEEMAN EFFECT

The study of atomic spectra under high resolution reveals a new feature ofthe electron: its spin. Evenorbitally nondegenerate levels, like the1sgroundstate of the H atom, are split into a doublet under the action of a magneticfield B,1 the splitting being linear in the field strength (Zeeman effect).To explain this effect, Pauli (1926) postulated for the electron

the existence of two spin states, a and b, satisfying the eigenvalueequations

Sza ¼ 1

2a; Szb ¼ �1

2b ð5:1Þ

where

S ¼ iSxþ jSy þ kSz ð5:2Þ

1 For a magnetic field of 1T, the splitting for a ground-state H atom is about 105 times smaller

than the separation of the first orbital levels.


Valerio Magnasco


is the spin vector operator (analogous to the orbital angular momentumvector operator L, but whose observable values can be half-integeras well). It is assumed that S and its components satisfy the samecommutation rules as those of the angular momentum operators.2

The two spin states are assumed to be kets normalized to 1 andorthogonal to each other:

hajai ¼ hbjbi ¼ 1; hajbi ¼ hbjai ¼ 0 ð5:3Þ

It may sometimes be useful to associate with spin the formal variable s(as distinct from r, the triplet of coordinates specifying the position inspace of the electron) and write Equation 5.3 formally as

ðdsa�ðsÞaðsÞ ¼

ðdsb�ðsÞbðsÞ ¼ 1;

ðdsa�ðsÞbðsÞ ¼

ðdsb�ðsÞaðsÞ ¼ 0

ð5:4Þ

An electron with spin in a uniform magnetic field B (Bx ¼ By ¼ 0;Bz ¼ B) acquires a potential energy

VðsÞ ¼ �mS �B ¼ gebeBSz ð5:5Þ

where ge � 2 is the intrinsic electron g-factor (the so-called anomalyof spin), be is the Bohr magneton (the unit of magnetic moment), given by

be ¼e�h

2mc¼ 9:274 015� 10�24 J T�1 ð5:6Þ

and mS, the magnetic moment operator associated with the electronspin S, is given by

mS ¼ �gebeS ð5:7Þ

The total one-electron Hamiltonian including spin will be

hðr; sÞ ¼ h0ðrÞþ gebeBSz ð5:8Þ

2 ½Sx; Sy� ¼ iSz; ½Sy; Sz� ¼ iSx; ½Sz; Sx� ¼ iSy; ½S2; Sk� ¼ 0; k ¼ x; y; z.

76 SPIN

Let us now introduce a space orbital xl(r) and the two spin-orbitals

clðxÞ ¼ clðr; sÞ ¼ xlðrÞaðsÞ�clðxÞ ¼ �clðr; sÞ ¼ xlðrÞbðsÞ

(ð5:9Þ

satisfying the eigenvalue equations

h0cl ¼ «lcl; h0�cl ¼ «l�cl

Szcl ¼1

2c; Sz�cl ¼ � 1

2�cl

8>><>>: ð5:10Þ

The energy level «l hence has a twofold spin degeneracy, which isremoved in the presence of the field. Using these two spin-orbitals as anorthonormal basis, the Ritz method shows that the matrix representativeof the Hamiltonian h over the functions ðcl

�clÞ is already diagonal, the(2� 2) secular equation, therefore, having the roots

«1 ¼ «l þ 1

2gebeB

«2 ¼ «l � 1

2gebeB

8>>><>>>:

ð5:11Þ

which give a Zeeman splitting linear in the field B:

D« ¼ «1 � «2 ¼ gebeB ð5:12Þ

Transitions from the bottom level «2 to the upper one «1 are possible bymagnetic dipole radiation (Dixon, 1965), so that the electron spin can bereoriented by a photon of energy:

hn ¼ gebeB ð5:13Þ

a process originating what is known as electron spin resonance (ESR). Asimilar process can occur for nuclei of spin 1

2 (1H, 13C), with the order of

levels reversed. For 1H:

bN ¼ e�h

2mPc¼ 1

1836

e�h

2mc¼ 1

1836be ð5:14Þ

THE ZEEMAN EFFECT 77

is the nuclear magneton and gN ¼ 5:585. Connected to the nuclear spinequivalent of (5.13) is the nuclear magnetic resonance (NMR), an experi-mental technique of great importance nowadays in structural organicchemistry.

5.2 THE PAULI EQUATIONS FORONE-ELECTRON SPIN

Until now nothing has been said about the x and y components of thevector operator S. If we introduce the spin ladder operators

Sþ ¼ Sx þ iSy; S� ¼ Sx� iSy ð5:15Þ

(i being the imaginary unit, i2 ¼ �1),3 then the analogy with orbitalangular momentum suggests for spin the two-step ladder of Figure 5.1,where the operators Sþ and S� respectively step-up (raise) or step-down(lower) the spin functions upon which they act. Therefore, it is intuitivethat

Sþa ¼ 0 ðtop of the ladderÞ; S�a ¼ b

Sþb ¼ a; S�b ¼ 0 ðbottom of the ladderÞ

(ð5:16Þ

By adding and subtracting the corresponding equations, it is easily seenthat the two spin states a and b do satisfy the equations

Sxa ¼ 1

2b; Sya ¼ 1

2ib; Sza ¼ 1

2a

Sxb ¼ 1

2a; Syb ¼ �1

2ia; Szb ¼ � 1

2b

8>>><>>>:

ð5:17Þ

which are known as Pauli’s equations for the one-electron spin andwhichare of fundamental importance for thewhole theory of spin, even inmany-electron systems.Since

S2 ¼ S � S ¼ S

2

x þ S2

y þ S2

z ð5:18Þ

3 Sþ and S� have the commutation properties ½Sz; Sþ � ¼ Sþ ; ½Sz; S�� ¼ �S�; ½S2; S�� ¼ 0.

78 SPIN

using Equations 5.17 twice it is easily seen that

S2a ¼ 3

4a; S

2b ¼ 3

4b ð5:19Þ

so that the two spin states a and b have the same eigenvalue with respectto the operator S

2(S(S þ 1) with S ¼ 1

2, doublet) and opposite eigen-value (MS ¼ 1

2 ; MS ¼ � 12) with respect to Sz. The name doublet (two

independent spin functions) comes from the so-called spin multiplicity2Sþ1.

5.3 THE DIRAC FORMULA FOR N-ELECTRON SPIN

For N ¼ 2 electrons, we can write the 22 ¼ 4 products of two-electronspin functions (which are already eigenfunctions of the operator Sz):

aa; ab; ba; bbMS ¼ 1 0 0 �1

�ð5:20Þ

where electron labels are always assumed in dictionary order, namely:

a1a2; a1b2; b1a2; b1b2 ð5:21Þ

and, therefore, are omitted for brevity.

α, Ms = 1/2

S+ S–

β, Ms = –1/2S = 1/2

Figure 5.1 The two-step ladder for spin 12

THE DIRAC FORMULA FOR N-ELECTRON SPIN 79

The proper spin eigenstates for the N ¼ 2 electron spin problem willsatisfy the eigenvalue equation

S2h ¼ SðSþ 1Þh ð5:22Þ

with total spin S ¼ 0 (singlet) or S ¼ 1 (triplet). Tofind the spin eigenstatesforN ¼ 2 it will be sufficient to act upon the set (5.20) with the operatorS2, finding its matrix representative S2 over the orthonormal basis set of

the four product functions, and diagonalizing it to get its eigenvalues andeigenvectors.Noting that

S2 ¼ S � S ¼ ðS1 þ S2Þ � ðS1 þ S2Þ ¼ S

2

1 þ S2

2þ 2S1 � S2 ð5:23Þ

where S2

1 and S2

2 are one-electron spin operators and S1 � S2 a two-electronspin operator, we have for the scalar product

S1 � S2 ¼ ðiSx1 þ jSy1 þ kSz1Þ � ðiSx2 þ jSy2 þ kSz2Þ

¼ Sx1Sx2þ Sy1Sy2þ Sz1Sz2

ð5:24Þ

Repeated application of Pauli’s equations (5.17) shows, for instance,that

ðS1 � S2Þab ¼ 1

4þ 1

4

� �ba� 1

4ab ð5:25Þ

so that

S2ab ¼ 3

4þ 3

4� 2

4

� �abþ 2

1

4þ 1

4

� �ba ¼ abþba ð5:26Þ

Completing the process for all product functions (5.20), we obtain formatrix S2 the block-diagonal form:

S2 ¼2 0 0 00 1 1 00 1 1 00 0 0 2

0BB@

1CCA ð5:27Þ

80 SPIN

The (2� 2) inner block gives the secular equation:

1� l 11 1� l

�� ¼ 0 l ¼ SðSþ 1Þ ð5:28Þ

having the roots

l4 ¼ 0 ) S ¼ 0

l2 ¼ 2 ) S ¼ 1

(ð5:29Þ

Solving the associated linear homogeneous system gives

h4 ¼1ffiffiffi2

p ðab�baÞ ðS ¼ 0Þ

h2 ¼1ffiffiffi2

p ðabþbaÞ ðS ¼ 1Þ

8>>>><>>>>:

ð5:30Þ

for the two eigenvectors. Therefore, we obtain theN ¼ 2 spin eigenstates(Na is the number of electrons with spin a and Nb that of electrons withspin b, with Na þNb ¼ N) collected in Table 5.1.In conclusion, we see that the eigenstates of the two-electron spin

problem have definite symmetry properties under the interchange 1$2(or a$b), the triplet being symmetric and the singlet antisymmetricunder electron (spin) interchange.From Equation 5.26 we further note that we can write for S

2

S2 ¼ Iþ P12 ¼ Iþ Pab ð5:31Þ

Table 5.1 Eigenstates of the two-electron problem

Eigenstates g S MS ¼ ðNa �NbÞ=2h1 ¼ aa 1 1

h2 ¼ 1ffiffiffi2

p ðabþbaÞ 1 0

h3 ¼ bb 1 �1

h4 ¼ 1ffiffiffi2

p ðab�baÞ 0 0


a formula due to Dirac (1929), which can be easily generalized to theN-electron spin problem as (L€owdin, 1955)

S2 ¼ N

4ð4�NÞIþ

Xk < l

Pkl ð5:32Þ

where I is the identity operator and Pkl interchanges spin state kwith spinstate l. Dirac’s formula (5.32) allows one to avoid repeated use of Pauli’sequations (5.17) and to calculate in a very simpleway the effect of S

2upon

many-electron (spin) functions. In the case of Slater determinants, whichwe shall meet later on, the operator (5.32) will act only upon the spin partof the spin-orbital functions, leaving the space orbitals unchanged.The eigenstates of S

2for the N-electron spin problem are states of

definite total spin S ð 0Þ satisfying simultaneously the two eigenvalueequations

S2h ¼ SðSþ 1Þh; Szh ¼ MSh ð5:33Þ

where

S ¼ 0; 1; 2; . . . ;N

2forN ¼ even

S ¼ 1

2;3

2;5

2; . . . ;

N

2forN ¼ odd

8>>><>>>:

ð5:34Þ

MS ¼ Na �Nb

2ð5:35Þ

In terms of the spin multiplicity 2Sþ 1, we have

N ¼ even ) singlets; triplets; quintets; . . . ð5:36Þ

N ¼ odd ) doublets; quartets; sextets; . . . ð5:37Þ

The number fNS of linearly independent spin eigenstates of givenS for the N-electron spin system is given by a formula due to Wigner(1959):

f NS ¼ ð2Sþ 1ÞN!

N

2�S

� �!N

2þ Sþ1

� �!

ð5:38Þ

82 SPIN

As an example, for the three-electron system, Wigner’s formula gives

N ¼ 3; S ¼ 1

2) f 31=2 ¼ 2

N ¼ 3; S ¼ 3

2) f 33=2 ¼ 1

8>>><>>>:

ð5:39Þ

so that we have two distinct doublets (four functions) and one quartet(four functions as well).


6Antisymmetry ofMany-electron

Wavefunctions

The effects of symmetry on the many-electron wavefunctions arise fromthe physical identity and, therefore, from the indistinguishability of theelectrons, and are of fundamental importance in the treatment of themany-particle systems. Use of suitable density functions allows us to passfrom the abstract 4N-dimensional mathematical space of the N-electronwavefunctions to the three-dimensional þ spin space where physicalexperiments are done (McWeeny, 1960).

6.1 ANTISYMMETRY REQUIREMENTAND THE PAULI PRINCIPLE

Let x1 and x2 be two fixed points in the space–spin space and Y(x1,x2)a normalized two-electron wavefunction. Then, since electrons are iden-tical particles, we have from first principles that

jYðx1; x2Þj2 dx1 dx2 ¼ probability of finding electron 1at dx1 and electron 2 at dx2

¼ jYðx2; x1Þj2 dx1 dx2 ¼ probability of finding electron 2at dx1 and electron 1 at dx2

8>><>>: ð6:1Þ


Valerio Magnasco


Therefore, it follows that

jYðx2;x1Þj2 ¼ jYðx1; x2Þj2 ) Yðx2;x1Þ ¼ �Yðx1;x2Þ ð6:2Þ

and the wavefunction must be symmetric (þ sign) or antisymmetric(� sign) in the interchange of the space–spin coordinates of the twoelectrons.The Pauli principle states that ‘electrons must be described only by

antisymmetric wavefunctions’, namely:

Yðx2;x1Þ ¼ �Yðx1; x2Þ ð6:3Þ

which isPauli’s antisymmetry principle in the formgiven byDirac (1929).This formulation includes the well-known exclusion principle forelectrons in the same orbital with the same spin:

Yðx1; x2Þ ¼ clðx1Þclðx2Þ�clðx1Þclðx2Þ ¼ 0 ð6:4Þ

where, as usual, we take electrons in dictionary order and interchangespin-orbitals only. It is, instead, allowed to put two electrons in the sameorbital with different spin:

Yðx1; x2Þ ¼ clðx1Þ�clðx2Þ� �clðx1Þclðx2Þ

¼ clðx1Þ �clðx1Þclðx2Þ �clðx2Þ

�� ¼ jclðx1Þ�clðx2Þj

ð6:5Þ

where the Pauli-allowed wavefunction can be written in the form of adeterminant of order 2 with electrons as rows and spin-orbitals ascolumns.1 This determinant is known as Slater determinant, since Slater(1929) was the first to introduce this notation in his quantummechanicaltreatment of atomic multiplets without using group theory.If the spin-orbitals are assumed to be orthonormal, then we shall write

the normalized Slater determinant within a vertical double bar as

Yðx1;x2Þ ¼ 1ffiffiffi2

p jclðx1Þ�clðx2Þj ¼ jjclðx1Þ�clðx2Þjj ð6:6Þ

1 Recall that we use cl to denote a spin-orbital with spin a and �cl a spin-orbital with spin b.

86 ANTISYMMETRY OF MANY-ELECTRON WAVEFUNCTIONS

6.2 SLATER DETERMINANTS

Generalizing to the N-electron system, we shall write the N-electronwavefunction Y satisfying Pauli’s principle as

Yðx1;x2; . . . ; xNÞ ¼ 1ffiffiffiffiffiffiN!

pc1ðx1Þ c2ðx1Þ � � � cNðx1Þc1ðx2Þ c2ðx2Þ � � � cNðx2Þ� � � � � � � � � � � �

c1ðxNÞ c2ðxNÞ � � � cNðxNÞ

��

��¼ jjc1c2 � � � cNjj ð6:7Þ

a Slater determinant of order N, where rows denote space–spin coordi-nates of the electrons (x ¼ rs) and columns the spin-orbital functions.If the latter are orthonormal, then Y of Equation 6.7 is normalized to 1:

hYjYi ¼ðdx1 dx2 . . . dxNY�ðx1;x2; . . . ;xNÞYðx1;x2; . . . ;xNÞ ¼ 1 ð6:8Þ

The elementary properties of determinants introduced in Chapter 2show that (6.7) can be equally well written as

Yðx1; x2; . . . ; xNÞ ¼ 1ffiffiffiffiffiffiN!

pc1ðx1Þ c1ðx2Þ � � � c1ðxNÞc2ðx1Þ c2ðx2Þ � � � c2ðxNÞ� � � � � � � � � � � �

cNðx1Þ cNðx2Þ � � � cNðxNÞ

��

��ð6:9Þ

where we now choose spin-orbitals as rows and electrons as columns,having interchanged rows with columns in the original definition (6.7).Furthermore, it is easily seen that Y does satisfy Pauli’s antisymmetry

principle:

Yðx2;x1; . . . ; xNÞ ¼ 1ffiffiffiffiffiffiN!



��

��¼ �Yðx1;x2; . . . ; xNÞ ð6:10Þ

since this is equivalent to interchange of two rows in the determinant (6.7)and the determinant changes sign.

SLATER DETERMINANTS 87

Also, Pauli’s exclusion principle is satisfied by (6.7):

Yðx1;x2; . . . ;xNÞ ¼ 1ffiffiffiffiffiffiN!



��

��¼ 0 ð6:11Þ

since (6.11) now has two identical columns and the determinant vanishes.For theN-electron system, the wavefunction Y is antisymmetric if it is

left invariant by an even number of electron interchanges (permutations)and it changes sign with an odd number of interchanges. The probabilitydensity (as the Hamiltonian operator) is instead left unchanged after anynumber of interchanges among the electrons.2

A single Slater determinant is often sufficient as a first approximationfor closed-shell systems (S ¼ MS ¼ 0), but some components of the open-shell systems, like theMS ¼ 0 components of the triplet states 1s2s(3S) ofHe or 1sg1suð3Sþ

u Þ of H2, require a linear combination of Slater deter-minants to get a state of definite S.A few atomic and molecular examples are given below.

(i) Ground state of the He atom (N ¼ 2)

Yð1s2; 1SÞ ¼ jj1s 1�sjj S ¼ MS ¼ 0

(ii) Ground state of the Be atom (N¼ 4)

Yð1s22s2; 1SÞ ¼ jj1s 1�s 2s 2�sjj S ¼ MS ¼ 0

(iii) Ground state of the H2 molecule (N¼ 2)

Yð1s2g ;

1Sþg Þ ¼ k1sg1s�gk S ¼ MS ¼ 0

where

1sg ¼ aþ bffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2S

p a ¼ 1sA; b ¼ 1sB ð6:12Þ

is the bondingmolecular orbital doubly occupied by the electrons.

2 Recall that we simply interchange spin-orbitals between the electrons kept in dictionary order.


(iv) Ground state of the LiH molecule (N¼ 4)

Yð1s2 2s2; 1SþÞ ¼ jj1s 1�s 2s 2�sjj S ¼ MS ¼ 0

where 1s and 2s are the first two MOs obtained, in a firstapproximation, by the linear combination of the basic AOs (1s,2s) on Li and (1s) on H.

All these electronic states are described by a single Slater determinant. But:

(v) First excited 1s2s state of the He atom (N¼ 2)

Yð1s 2s; 3SÞ ¼

jj1s 2sjj S ¼ 1; MS ¼ 1

1ffiffiffi2

p jj1s2�s jjþ jj1�s 2sjj� �S ¼ 1; MS ¼ 0

jj1�s 2�s jj S ¼ 1; MS ¼ � 1

8>>>><>>>>:

Only the components of the triplet with jMSj ¼ S are described bya single Slater determinant, whereas the component withMS ¼ 0requires two Slater determinants.

6.3 DISTRIBUTION FUNCTIONS

The passage from the abstract mathematical space of the N-electronwavefunction Y to the physical four-dimensional space including spin isdone by the distribution functions. Since the Hamiltonian operator for amany-electron atom or molecule contains only symmetrical sums of one-electron and two-electron operators, the most important distributionfunctions for uswill be those involvingone and two electrons respectively.

6.3.1 One- and Two-electron Distribution Functions

The distribution functions are the diagonal elements of the one- and two-electron density matrices (McWeeny, 1960) defined as

r1ðx1; x01Þ ¼ N

ðdx2 dx3 . . . dxN Yðx1;x2; . . . ; xNÞY�ðx01; x2; . . . ;xNÞ

ð6:13Þ

DISTRIBUTION FUNCTIONS 89

r2ðx1; x2;x01; x02Þ ¼ NðN� 1Þðdx3 . . . dxN Yðx1;x2; x3; . . . ; xNÞ

� Y�ðx01; x02; x3; . . . ; xNÞð6:14Þ

where the first set of variables in r1 and r2 comes from Y and the secondfromY�. Functions (6.13) and (6.14) have only a mathematical meaning,and are needed when a differential operator (like r2) acts on the rs.3 Allphysical meaning is carried instead by their diagonal elements ðx01 ¼ x1Þand ðx01 ¼ x1; x

02 ¼ x2Þ, which are the one- and two-electron distribution

functions

r1ðx1; x1Þ ¼ N

ðdx2 dx3 . . . dxN Yðx1; x2; . . . ; xNÞY�ðx1; x2; . . . ; xNÞ

ð6:15Þ

r2ðx1; x2; x1; x2Þ ¼ NðN� 1Þðdx3 . . . dxN Yðx1; x2;x3; . . . ;xNÞ

� Y�ðx1; x2; x3; . . . ; xNÞ ð6:16Þ

having the following conservation properties:

ðdx1 r1ðx1;x1Þ ¼ N ð6:17Þ

the total number of electrons and

ðdx2 r2ðx1;x2; x1;x2Þ ¼ ðN� 1Þr1ðx1; x1Þ;ðdx1 dx2 r2ðx1; x2; x1; x2Þ ¼ NðN�1Þ ð6:18Þ

the total number of indistinct pairs.The physical meaning of the distribution functions (6.15) and (6.16) is

as follows:

r1ðx1;x1Þ dx1 ¼ probability of finding an electron at dx1 ð6:19Þ

3 We recall that the operator r2 acts only on Y and not on Y�.


r2ðx1;x2;x1;x2Þdx1 dx2 ¼ probability of finding an electron at dx1 and;simultaneously; another electron at dx2

ð6:20Þ

In this way, the distribution functions take into account from thebeginning the fact that electrons are physically identical particles. Thetwo-electron (or pair) distribution function is very important in the studyof the correlation between electrons (McWeeny, 1960), but this point willnot be analysed further here. Instead, we shall further examine in moredetail the properties of the one-electron distribution function.

6.3.2 Electron and Spin Densities

In its most general form, the one-electron distribution (or density) func-tion (6.15) can be written by separating its space from spin parts asfollows:

r1ðr1s1; r1s1Þ ¼ raa1 ðr1; r1Þaðs1Þa�ðs1Þþ rbb1 ðr1; r1Þbðs1Þb�ðs1Þ

þ rab1 ðr1; r1Þaðs1Þb�ðs1Þþ rba1 ðr1; r1Þbðs1Þa�ðs1Þð6:21Þ

where the rs are now space functions only. By integrating over spin, thelast two terms in (6.21) vanish, because of the orthogonality of the spinfunctions, and we are left with the spinless quantities

raa1 ðr1; r1Þ ¼ ra1ðr1; r1Þ; rbb1 ðr1; r1Þ ¼ rb1ðr1; r1Þ ð6:22Þ

having the following evident physical meaning:

ra1ðr1; r1Þ dr1 ¼ probability of finding at dr1 an electron with spin a

ð6:23Þ

rb1ðr1; r1Þ dr1 ¼ probability of finding at dr1 an electron with spin b

ð6:24Þwith ð

dr1 ra1ðr1; r1Þ ¼ Na ð6:25Þ


the number of electrons with spin a and

ðdr1 r

b1ðr1; r1Þ ¼ Nb ð6:26Þ

the number of electrons with spin b. Therefore, the sum

Pðr1; r1Þ ¼ ra1ðr1; r1Þþ rb1ðr1; r1Þ ð6:27Þ

is the electron density, as observed, for example, from the X-ray diffrac-tion spectra of polycyclic hydrocarbons (Bacon, 1969), with

Pðr1; r1Þ dr1 ¼ probability of finding at dr1 an electron with either spin

ð6:28Þ

whereas the difference

Qðr1; r1Þ ¼ ra1ðr1; r1Þ� rb1ðr1; r1Þ ð6:29Þ

is the spin density, with

Qðr1;r1Þdr1 ¼ probability of finding at dr1 an excessof spinaover spinb

ð6:30Þ

P and Q satisfy the following conservation relations:

ðdr1 Pðr1; r1Þ ¼ Na þNb ¼ N ð6:31Þ

the total number of electrons and

ðdr1 Qðr1; r1Þ ¼ Na �Nb ¼ 2MS ð6:32Þ

where

MS ¼ Na �Nb

2ð6:33Þ


is the eigenvalue of the Sz operator

SzY ¼ MSY ð6:34Þ

Asa simple example, let us consider nowthe two-electronwavefunctionY describing the bond between atoms A and B arising from the doubleoccupancy of the (normalized) bonding MO fðrÞ:

Yðx1;x2Þ ¼ jjf�fjj ¼ fðr1Þfðr2Þ 1ffiffiffi2

p aðs1Þbðs2Þ�bðs1Þaðs2Þ½ � ð6:35Þ

with

fðrÞ ¼ xAðrÞcA þ xBðrÞcB ¼ xAðrÞþ lxBðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2 þ 2lS

p ð6:36Þ

where xA and xB are two basis AOs centred on A and B respectively,normalized but not orthogonal to each other, and l ¼ cB=cA is thepolarityparameter of the MO. The integral

S ¼ hxAjxBi ¼ðdr x�

AðrÞxBðrÞ ð6:37Þ

describes the overlap between the AOs xA and xB and, therefore, is calledthe overlap integral.4

Using definition (6.15) omitting for brevity the suffix 1, useless at themoment, the integration over dx2 gives the one-electron distributionfunction

r1ðx;xÞ ¼ fðrÞf�ðrÞ½aðsÞa�ðsÞþbðsÞb�ðsÞ� ð6:38Þ

so that

ra1ðr; rÞ ¼ rb1ðr; rÞ ¼ fðrÞf�ðrÞ ¼ Rðr; rÞ ð6:39Þ

4 S depends on the internuclear distance R, being one at R ¼ 0 and zero at R ¼ ¥.


The coefficients of aa� and bb� are equal for the doubly occupied MOand are usually denoted by R in MO theory. We then obtain for theelectron and spin densities of our bond orbital

Pðr; rÞ ¼ ra1ðr; rÞþ rb1ðr; rÞ ¼ 2Rðr; rÞ ¼ 2fðrÞf�ðrÞ ð6:40Þ

Qðr; rÞ ¼ ra1ðr; rÞ� rb1ðr; rÞ ¼ 0 ð6:41Þ

as expected for an MO doubly occupied by electrons with opposite spinand MS ¼ 0.The electron density (6.40) can be further analysed in terms of elemen-

tary contributions from the AOs, giving the so-called population analysis,which shows how the electrons are distributed between the differentatomic orbitals in the molecule. We obtain

Pðr; rÞ ¼ qAx2AðrÞþ qBx

2BðrÞþ qAB

xAðrÞxBðrÞS

þ qBAxBðrÞxAðrÞ

Sð6:42Þ

from (6.36), where x2AðrÞ and x2

BðrÞ are the atomic densities andðxAðrÞxBðrÞÞ=S and ðxBðrÞxAðrÞÞ=S are the overlap densities, all normal-ized to 1, while the coefficients

qA ¼ 2

1þ l2 þ 2lS; qB ¼ 2l2

1þ l2 þ 2lSð6:43Þ

are the atomic charges and

qAB ¼ qBA ¼ 2lS


are the overlap charges. The charges are normalized so that

qA þ qBþ qABþ qBA ¼ 2þ 2l2 þ4lS

1þ l2 þ2lS¼ 2 ð6:45Þ

the total number of electrons in the bond orbital fðrÞ.For a homopolar bond, l ¼ 1:

qA ¼ qB ¼ 1

1þ S; qAB ¼ qBA ¼ S

1þ S ð6:46Þ


so that for S > 0, in the molecule, the charge on atoms is decreased,electrons being transferred in the intermediate region between nuclei to anextent described by qAB and qBA. This reduces internuclear repulsion andmeans bonding.For a heteropolar bond, l=1, and we define gross charges on A and B

as

QA ¼ qA þ qAB ¼ 2þ 2lS


QB ¼ qBþ qBA ¼ 2l2 þ 2lS


and formal charges on A and B as

dA ¼ 1�QA ¼ l2 � 1


dB ¼ 1�QB ¼ � l2 � 1


If l > 1, then dA ¼ d > 0 and dB ¼ � dA ¼ � d < 0 and we have thedipole Aþ dB� d (e.g. the LiH molecule). Further examples for the 1Sþ

g

ground state and the 3Sþu excited triplet state of the H2 molecule can be

found elsewhere (Magnasco, 2007). It is seen there that, in a repulsivestate such as triplet H2 (or ground-state He2), electrons escape fromthe interatomic region determining repulsion (antibonding) between theinteracting atoms, the so-called Pauli repulsion.Generalization of this analysis to theN-electron determinant is known

as Mulliken population analysis (Mulliken, 1955).

6.4 AVERAGE VALUES OF OPERATORS

With the aid of the density functions we can easily evaluate the averagevalues of any symmetrical sum of one- and two-electron operators:

YjXNi¼1

OijY* +

¼ðdx1 O1r1ðx1; x01Þjx0

1¼x1

ð6:51Þ

AVERAGE VALUES OF OPERATORS 95

YjXN

i;j¼1ðj=iÞOijjY

* +¼

ðdx1 dx2 O12r2ðx1x2;x1x2Þ ð6:52Þ

where Y is the N-electron wavefunction normalized to 1 and Oi and Oij

are the one- and two-electron operators respectively. In (6.51) we haveused the one-electron density matrix (6.13) instead of the one-electrondistribution function, leaving the possibility that O1 couldbe adifferentialoperator (liker2), which acts on the first set of variables in r1 but not onthe second. The notation in (6.51) should by now be clear: first, let theoperator O1 act on r1, then put x01 ¼ x1 in the resulting integrand andintegrate over x1. In (6.52) we have assumed that the operator O12 is asimple multiplier (like the electron repulsion 1/r12).The electronic Hamiltonian He contains two such symmetrical sums,

and its average value over theN-electron wavefunctionY can be writtenas

hYjHejYi ¼ YXNi¼1

hi þ 1

2

XNi;jðj=iÞ

1

rij

��Y

* +

¼ðdx1 h1r1ðx1; x01Þjx0

1¼x1

þ 1

2

ðdx1 dx2

1

r12r2ðx1x2;x1x2Þ

ð6:53Þ

where

h1 ¼ � 1

2r2

1 þV1; V1 ¼ �Xa

Za

ra1ð6:54Þ

is the one-electron bare nuclei Hamiltonian, V1 is the attraction of theelectronbyall nuclei of charge þZa in themolecule, and1/r12 the electronrepulsion.Hence, the electronic energy Ee consists of the following three terms:

Ee ¼ðdx1 � 1

2r2

1r1ðx1; x01Þjx01¼x1

� �þ

ðdx1 V1 r1ðx1; x1Þ

þ 1

2

ðdx1 dx2

1

r12r2ðx1x2;x1x2Þ

ð6:55Þ

which have the following simple, physically transparent, interpretation.The first term in (6.55) is the average kinetic energy of the electrondistribution r1, the second is the average potential energy of the electron


distribution r1 in the field provided by all nuclei in the molecule, and thethird is the average electronic repulsion of an electron pair described bythe pair function r2.Lastly, it is important to stress that the integral in (6.51) would involve

N(N!)2 4N-dimensional integrations, which are reduced to a simple four-dimensional space–spin integration by use of the density matrix! Generalrules for the evaluation of the integrals in (6.55) in the general case of twodifferent Slater determinants Y and Y0 built from orthonormal spin-orbitals were first given by Slater (1931) and recast in density matrixform by McWeeny (1960).

AVERAGE VALUES OF OPERATORS 97

7Self-consistent-field

Calculations and Model

Hamiltonians

Hartree–Fock (HF) theory was developed in the early 1930s by thetheoretical physicists Hartree (1928a, 1928b) and Fock (1930) to dealwith the quantum mechanical problem of many-particle atomic andmolecular systems. Initially developed by Hartree for complex atoms ina numerical form using as Y the simple product of N spin-orbitals, thetheory was put on a more sound basis by Fock, who first solved thevariational problem of optimizing the orbitals in a determinantal Yobeying Pauli’s antisymmetry principle. HF theory amounts essentiallyto finding the best form for the orbitals inside the so-called independentparticle model (IPM), where an electron in the molecule is assumed tomove in the average field provided by all nuclei and the sea of all otherelectrons. Besides the Coulomb potential J due to all electrons, theantisymmetry requirement imposed by Fock upon the wavefunction Yoriginates a nonlocal exchange potential K.Since no correlationbetween the electrons is providedby thismodel, the

HF energy is used to define exactly the correlation energy of the system as

Correlation energy ¼ Exact energy of the nonrelativistic Hamiltonian

�HF energy

ð7:1Þ


Valerio Magnasco


The correlation energy amounts to about 1 eV per electron pair andis quite difficult to account for, as we shall briefly outline in Chapter 8.We shall introduce, first, the essential lines of the HF theory for closedshells, with its practical implementation by Hall (1951) and Roothaan(1951b) yielding the so-called self-consistent-field (SCF)method inside theRitz linear-combination-of-atomic-orbitals (LCAO) approach, and, next,the LCAOmethod devised by H€uckel (1931) to deal in a topological waywith the p electrons of conjugated and aromatic hydrocarbons. H€uckeltheory has recently been used by the author to introduce an elementarymodel of the chemical bond (Magnasco, 2002, 2003, 2004a, 2005).

7.1 ELEMENTS OF HARTREE–FOCK THEORYFOR CLOSED SHELLS

Let

Y ¼ jjc1c2 . . .cNjj hcijcji ¼ dij S ¼ MS ¼ 0 ð7:2Þ

be a normalized single determinant wavefunction of doubly occupiedorthonormal spin-orbitals {ci}. We shall introduce consistently the fol-lowing notation:

N¼ number of electrons ¼ number of spin-orbitals {ci(x)}, x ¼ rs;i ¼ 1;2; . . . ;N

n ¼ N=2 number ofdoubly occupied spatialMOs {fi(r)}, i ¼ 1;2; . . . ;nm ¼ number of basic spatial AOs {xm(r)}, m ¼ 1; 2; . . . ;mm � nm ¼ n means minimal basism > n usually includes polarization functions (e.g. 3d, 4f, 5g, . . . on C,

N, O, F atoms and 2p, 3d, 4f, . . . on H atoms, as explained inSection 7.3).

We then see that the many-electron single determinant wavefunctionY has the following properties.

7.1.1 The Fock–Dirac Density Matrix

All physical properties of the system are determined by the fundamentalphysical invariant r, a one-electron bilinear function in the coordinates x

100 SELF-CONSISTENT-FIELD CALCULATIONS

and x0, defined as

rðx; x0Þ ¼XNi¼1

ciðxÞc*i ðx0Þ ð7:3Þ

which is called the Fock–Dirac density matrix, and whose formalmathematical properties are

ðdx rðx;x0Þjx0¼x ¼

ðdx rðx; xÞ ¼ N conservation ð7:4Þ

ðdx00 rðx;x00Þrðx00; x0Þ ¼ rðx; x0Þ idempotency ð7:5Þ

r0ðx; x0Þ ¼ rðx;x0Þ invariance ð7:6Þ

where r0 is the result of any unitary transformation among its spin-orbitals.Mathematically speaking, rðx; x0Þ is the kernel of the integral operator

rðxÞ ¼ðdx0 rðx; x0ÞPxx0 ð7:7Þ

which transforms a function F(x) into

rðxÞFðxÞ ¼ðdx0 rðx;x0ÞPxx0FðxÞ ¼

ðdx0 rðx;x0ÞFðx0Þ ð7:8Þ

a new function of x, namely

rF ¼ðdx0 rðx;x0ÞFðx0Þ ¼

ðdx0

XNi¼1

ciðxÞc*i ðx0ÞFðx0Þ ¼

XNi¼1

jciihcijFi

ð7:9Þ

wherewe introduced theDirac notation for the last term. In otherwords, ris a projection operator in Fock space which projects out of the arbitraryregular function F(x) its N-components along the basic vectors jcii,i ¼ 1; 2; . . . ;N.

ELEMENTS OF HARTREE–FOCK THEORY FOR CLOSED SHELLS 101

In a short notation, properties (7.4)–(7.6) can be symbolically writtenas

trr ¼ N; r2 ¼ r; r0 ¼ r ð7:10Þ

It is often said that Equation 7.3 provides a spin-orbital representationof the projector r.

7.1.2 Electronic Energy Expression

The general expression for the electronic energy of the many-electronsystem was given in Chapter 6 as

Ee ¼ðdx1 h1r1ðx1; x01Þjx01¼x1

þ 1

2

ðdx1 dx2

1

r12r2ðx1x2;x1x2Þ ð7:11Þ

But, Lennard-Jones (1931) showed that in HF theory

r1ðx; x0Þ ¼ rðx; x0Þ ð7:12Þ

r2ðx1x2; x01x02Þ ¼rðx1; x01Þ rðx1;x02Þrðx2; x01Þ rðx2;x02Þ

��

¼ rðx1; x01Þrðx2; x02Þ� rðx1;x02Þrðx2;x01Þð7:13Þ

Hence, in HF theory, the electronic energy assumes the characteristicone-electron form (no electron correlation) IPM):

Ee ¼ðdx hrðx;x0Þjx0¼x þ

1

2

ðdx ½ JðxÞ� KðxÞ� rðx;x0Þjx0¼x ð7:14Þ

where both one-electron and two-electron components of Ee are ex-pressed in terms of the fundamental invariant r and of Coulomb andexchange one-electron potentials:

Jðx1Þ ¼ðdx2

rðx2; x2Þr12

ð7:15Þ


Kðx1Þ ¼ðdx2

rðx1; x2Þr12

Px1x2 ð7:16Þ

The exchange potential K is an integral operator with kernelrðx1;x2Þ=r12 and J is a multiplicative operator that involves the diagonalelement of the Fock–Dirac density matrix.In HF theory it is customary to introduce the one-electron Fock

operator

FðxÞ ¼ hþ JðxÞ� KðxÞ ð7:17Þ

where

h ¼ � 1

2r2 þV; V ¼ �

Xa

Za

rað7:18Þ

Therefore:

FðxÞ ¼ � 1

2r2þVeff; Veff ¼ Vþ J� K ð7:19Þ

The Fock operator is the sum of the kinetic energy operator and theeffective potential Veff felt by each electron, due to the nuclear attractionand the Coulomb-exchange field of all electrons. In terms of the Fockoperator (7.17), it is seen that

Ee r½ � ¼ðdx FðxÞrðx;x0Þjx0¼x �

1

2

ðdx ½JðxÞ� KðxÞ�rðx; x0Þjx0¼x ð7:20Þ

where the average electron repulsion must now be subtracted in order toavoid counting it twice.Hall (1951) and Roothaan (1951b) have shown independently that the

best variational energy1 for theHFdeterminant (7.2) is obtainedwhen thespin-orbitals satisfy the HF equations:

FðxÞciðxÞ ¼ «iciðxÞ i ¼ 1; 2; . . . ;N ð7:21Þ

1Subject to the constraint of keeping orthonormal the spin-orbitals)method of Lagrange

multipliers as applied to the energy functional (7.20).

ELEMENTS OF HARTREE–FOCK THEORY FOR CLOSED SHELLS 103

where F is the same for all electrons. Despite their simple aspect, (7.21)are complicated integro-differential2 equations in which the operatorF depends on the {ci} for which the equation should be solved, so thatthey must be solved by iteration)hence the SCF method startingfrom any convenient initial guess.3 The iteration must be stoppedwhen the spin-orbitals obtained as solutions of the HF equations donot differ appreciably from those used in the construction ofFðrÞ ) Ffcig. This is usually done by putting a convenient thresholdon the energies.

7.2 ROOTHAAN FORMULATION OF THE LCAO–MO–SCF EQUATIONS

Eliminating spin from FðxÞ ¼ FðrsÞ, we obtain the spinless Fockoperator:

FðrÞ ¼ hþ 2JðrÞ� KðrÞ ð7:22Þ

where

Jðr1Þ ¼ðdr2

Rðr2; r2Þr12

ð7:23Þ

Kðr1Þ ¼ðdr2

Rðr1; r2Þr12

Pr1r2 ð7:24Þ

are spinless Coulomb and exchange potentials (space only), and

Rðr1; r2Þ ¼ raðr1; r2Þ ¼ rbðr1; r2Þ ¼Xocci

fiðr1Þf*i ðr2Þ ð7:25Þ

is the Fock–Dirac density matrix for closed shells, the summation in thelast term of (7.25) being over all occupied orbitals (i ¼ 1;2; . . . ;n).

2K is the integral operator (7.16) while r2 is a differential operator.3Usually, H€uckel orbitals.


The HF equations for the occupied spatial MOs hence become

FðrÞfiðrÞ ¼ «ifiðrÞ i ¼ 1;2; . . . ; n ¼ N

2 ð7:26Þ

R has the projection properties of a density matrix in the space ofoccupied MOs:

trR ¼ n; R2 ¼ R; R0 ¼ R ð7:27Þ

Introducing a basis of m ð� nÞ AOs, if C is the rectangular (m�n)matrix of the LCAO coefficients

fiðrÞ ¼Xmm¼1

xmðrÞcmi; fðrÞ ¼ xðrÞC ð7:28Þ

then the Roothaan pseudoeigenvalue equations for the ith MO are bestformulated in matrix form as

Fci ¼ «iMci i ¼ 1; 2; . . . ; n ð7:29Þ

and for the whole set of the n occupied MOs

FC ¼ MCe ð7:30Þ

where

F ¼ x�Fx ð7:31Þ

is the (m�m) matrix representative of the spinless Fock operator over them basis functions x ¼ ðx1x2 . . . xmÞ

M ¼ x�x ð7:32Þ

is the (m�m) metric matrix of the atomic basis, and

e ¼«1 0 � � � 0

0 «2 � � � 0

� � � � � � � � � � � �0 0 � � � «n

0BB@

1CCA ð7:33Þ

ROOTHAAN FORMULATION OF THE LCAO–MO–SCF EQUATIONS 105

is the (n� n) diagonal matrix of the eigenvalues pertaining to occupiedMOs.The density matrix R in the atomic basis will be

Rðr; r0Þ ¼ fðrÞfðr0Þ� ¼ xðrÞCC�xðr0Þ� ¼ xðrÞRxðr0Þ�; R ¼ CC�

ð7:34Þ

where R is the (m�m) matrix representative of the Fock–Dirac densitymatrix over the AO basis x (hence follows the analysis of the electrondistribution in the molecule)Mulliken population analysis, a general-ization of the simple example given in Section 6.3). The projectionoperator properties of matrix R in the AO basis can be written as

trRM ¼ n; RMR ¼ R ð7:35Þ

Roothaan’s equations (7.29) are solved iteratively from the pseudose-cular equation

jF� «Mj ¼ 0 ð7:36Þ

which is an algebraic equation in « having as m roots the MO orbitalenergies

«1; «2; . . . ; «nocc

j «nþ 1; «nþ 2; . . . ; «munocc

ð7:37Þ

illustrated in Figure 7.1.According to Hund’s rule, for nondegenerate levels, in the molecular

ground state the first n levels are occupied by electrons with opposite spin(bonding levels, «i < 0) and the remaining ðm� nÞ levels are unoccupied(empty, antibonding levels, «i > 0). The highest occupied MO (HOMO)is fn (orbital energy «n) and the lowest unoccupiedMO (LUMO) is fnþ 1

(orbital energy «nþ 1).The Roothaan SCF electronic energy is given in matrix form as

Ee ¼ 2

ðdr FðrÞRðr; r0Þjr0¼r �

ðdr GðrÞRðr; r0Þjr0¼r

¼ 2trFR� trGR

ð7:38Þ


where

GðrÞ ¼ 2JðrÞ� KðrÞ ð7:39Þ

is called the Roothaan matrix of total electron interaction. The matrixelements of (7.38) over AOs are

Rmn ¼Xocci

cmic�ni ð7:40Þ

Fmn ¼ hmn þGmn; hmn ¼ hxmjhjxni ð7:41Þ

Gmn ¼Xl

Xs

2Rls ðxnxmjxlxsÞ�1

2ðxlxmjxnxsÞ

� �ð7:42Þ

. εm

..

.

εn+1,φn+1 LUMO

εn,φn HOMO

.εn-1

..

.

ε1

.

Figure 7.1 Diagram of the orbital energies in an SCF calculation

ROOTHAAN FORMULATION OF THE LCAO–MO–SCF EQUATIONS 107

where the two-electron integrals are written in the charge densitynotation:

ðxnxmjxlxsÞ ¼ðdr1 dr2

xlðr2Þx�sðr2Þ

r12xnðr1Þx�

mðr1Þ ð7:43Þ

Under appropriate simplifying assumptions in the matrix elementsabove, all LCAO approximations that can be derived from HF theory,such as H€uckel, extended H€uckel, Pople’s complete neglect of differentialoverlap (CNDO), intermediate neglect of differential overlap (INDO),and so on, are easily obtained.

7.3 MOLECULAR SELF-CONSISTENT-FIELDCALCULATIONS

The AOs used as a basis in quantum chemical calculations are theSTOs and GTOs introduced in Chapter 3. Calculations are affectedby errors arising (i) from the very nature of the basis set (we sawthat STOs are definitely superior to GTOs) and (ii) from the insuffi-ciency of the basis set, the so-called truncation errors. We now brieflyreview the different types of bases most used in today molecularcalculations.A minimal basis set involves those AOs which are occupied in the

ground state of the constituent atoms. An extended set includespolarization functions, namely those AOs that are unoccupied in theground state of the atoms, say 3d and 4f AOs for C, N, O, F and 2pand 3d AOs for H.In general, we speak of single zeta (SZ) for a minimal set, double zeta

(DZ) when each AO is described by two functions, triple zeta (TZ) wheneach AO is described by three functions, and so on. When polarizationfunctions (P) are included, we have correspondingly DZP, TZP or DZPP,TZPP (Van Duijneveldt-Van de Rijdt and Van Duijneveldt, 1982), de-pending onwhether the polarization functions are only on the heavy atomor on the H atom as well.GTO bases are often of the redundant Cartesian type:

GuvwðrÞ ¼ Nxuyvzwexpð� cr2Þ ð7:44Þ


whose number is given by the binomial coefficient

Lþ 2L

� �¼ 1

2ðLþ 1ÞðLþ 2Þ

if L ¼ uþ vþw.To reduce the large number of primitives usually needed in GTO

calculations it is customary to resort to contracted GTOs, where eachfunction is the sum of a certain number of primitives, each contractionscheme being specified by fixed numerical coefficients. The best basis is, ofcourse, uncontracted.As an example, a contraction scheme used for LiHbyTunega andNoga

(1998) is based on the following spherical GTOs for Li and H:

ð14s 8p 6d 5fj12s 8p 6d 5fÞ ) ½11s 8p 6d 5fj9s 8p 6d 5f� ð7:45Þ

The 204 primitives (103 GTOs on Li, 101 on H) are contracted to 198functions (100 GTOs on Li, 98 GTOs on H), with the polarizationfunctions left uncontracted. This gives a rather moderate contraction.Another example of a more sensible contraction can be taken from

Lazzeretti and Zanasi (1981) in their roughlyHFCartesianGTO calcula-tion on H2O, which includes polarization functions on O and H:

ð14s 8p 3d 1fj10s 2p 1dÞ ) ½9s 6p 3d 1fj6s 2p 1d� ð7:46Þ

The 110 GTO primitives are here reduced to 91 contracted GTOs,giving an SCF molecular energy of �76:066 390Eh. A more extendedbasis forH2Owas the ½13s 10p 5d 2fj8s 4p 1d� contractedCartesianGTObasis set recently usedbyLazzeretti (personal communication, 2004) in anSCF calculation on H2O, giving a molecular energy of � 76:066 87Eh,only 0:63� 10� 3Eh above the HF limit of � 76:067 50Eh estimated byRosenberg and Shavitt (1975) for H2O.Use is sometimes made of even-tempered (or geometrical) sequences of

primitives, where the orbital exponents ci are restricted by

ci ¼ abi i ¼ 1; 2; . . . ;m ð7:47Þ

with a and b fixed and different for functions of s, p, d, f, . . . symmetry.Thus, the number of nonlinear parameters (orbital exponents) to beoptimized in a variational calculation is drastically reduced.

MOLECULAR SELF-CONSISTENT-FIELD CALCULATIONS 109

Also widely used today are the GTO bases introduced by Pople andcoworkers in the different versions of GAUSSIANprogrammes: STO-nG,6–31G, 6–31G

�, 6–31G

��, to denote, respectively, (i) n-GTOs to represent

a single STO, (ii) sixGTOs for the inner-shell þ split-valenceGTOs, threeinner þ one outer, (iii) the samewith additional polarization functions onthe heavy atom, and (iv) the same with additional polarization functionson the H atoms.For use in calculations going beyond HF, Dunning (1989) introduced

well-balanced correlation-consistent polarized valenceGTObasis sets forthe first-row atoms from B through to Ne (Table 7.1) and H (Table 7.2),denoted by the acronym cc-pVXZ, where X ¼ D;T;Q; . . . (double, triple,quadruple, . . .) indicates the number of functions in the original basis set,as shown below (Tables 7.1 and 7.2).These bases were further elaborated by Woon and Dunning (1995)

by extending the correlation-consistent polarized valence basis sets(cc-pVXZ) to include core-valence correlation effects giving (cc-pCVXZ)bases for the same atoms B through to Ne. For polarizabilities, additionaldiffuse functions were added, giving the so-called augmented basis setsaug-cc-pCVXZ. It must be admitted that such acronyms look ratheresoteric to the uninitiated reader!An atomic and a molecular example are given in Tables 7.3 and 7.4,

where a comparison is made between different STO andGTOSCF resultsfor ground-state Neð1SÞ and HFð1Sþ Þ at the experimentally observedinternuclear distance Re ¼ 1:7328a0. The GTO bases in Table 7.3 aregeometrical basis sets, whereas the GTO basis sets in Table 7.4 are ofCartesian type.

Table 7.1 Correlation-consistent polarized basis sets for first-row atoms B throughto Ne

Primitive (sp) set contracted Polarization set

cc-pVDZ (9s4p) [3s2p] (1d)cc-pVTZ (10s5p) [4s3p] (2d1f)cc-pVQZ (12s6p) [5s4p] (3d2f1g)

Table 7.2 Correlation-consistent polarized basis sets for H

Primitive (s) set contracted Polarization set

cc-pVDZ (5s) [2s] (1p)cc-pVTZ (6s) [3s] (2p1d)cc-pVQZ (7s) [4s] (3p2d1f)


4In the top row of Table 7.4 the STOs were numbered individually since not all d and f functions

on F or H were included in the calculation.

Table 7.3 Comparison between STO,aGTOb andHF/2Dc SCF results (atomic units)for the ground state of the Ne atom

STO basis/AO m SZ 4 DZ 8 HF 23 HF/2D

Ee � 127.812 2 � 128.53511 � 128.547 05 � 128.547 13«i/1s � 32.662 13 � 32.759 88 � 32.772 48 � 32.772 4542s � 1.732 50 � 1.921 87 � 1.930 43 � 1.930 3922p � 0.561 72 � 0.841 43 � 0.850 44 � 0.850 411

GTO basis/AO m Small 24 Intermediate 28 Large 37

Ee � 128.464 724 � 128.528 123 � 128.543969«i/1s � 32.761 82 � 32.757 62 � 32.765 522s � 1.926 82 � 1.922 26 � 1.925 872p � 0.841 96 � 0.842 87 � 0.846 30

aClementi and Roetti (1974): SZ (1s1p), DZ (2s2p), HF (8s5p).bClementi and Corongiu (1982): small (9s5p), intermediate (10s6p), large (13s8p).c Sundholm et al. (1985): (8s5p).

Table 7.4 Comparison between STO,a–c GTOd and HF/2De,f SCF results (atomicunits) for the ground state of the HF molecule at Re ¼ 1:7328a0

STO basis/AO m SZa 6 �DZPPb 16 HFc 244 HF/2De,f

m 0.584 0.852 95 0.764 0 0.756 076f

E � 99.479 � 100.057 54 � 100.070 30 � 100.07082f

«i/1s � 26.260 1 � 26.306 17 � 26.294 28 � 26.294 57e

2s � 1.484 9 � 1.610 68 � 1.600 74 � 1.600 99e

3s � 0.594 0 � 0.774 59 � 0.768 10 � 0.768 25e

1p � 0.469 7 � 0.657 86 � 0.650 08 � 0.650 39e

GTO basis/AO m STO-3Gd 6 aug-ccpVDZd 34 aug-ccpVTZd 80

m 0.506 95 0.759 83 0.756 19E � 98.570 775 � 100.034422 � 100.061868«i/1s � 25.900 028 � 26.308 360 � 26.296 3062s � 1.471 216 � 1.607 876 � 1.602 1263s � 0.585 187 � 0.770 810 � 0.768 2281p � 0.464 165 � 0.650 201 � 0.650 247

aBallinger (1959): SZ (2s3p|1s).bClementi (1962): �DZPP (5s5p2d|2s2p).cCade and Huo (1967): HF (5s8p3d2f|3s2p1d).dLazzeretti and Pelloni (personal communication, 2006): STO-3G [2s1p|1s], aug-ccpVDZ[4s3p2d|3s2p], aug-ccpVTZ [5s4p3d2f|4s3p2d].e Sundholm (1985).f Sundholm et al. (1985).

MOLECULAR SELF-CONSISTENT-FIELD CALCULATIONS 111

The tables show that the one-term approximation (SZ) is totallyinsufficient either for the atomic or the molecular case, giving largeerrors in total energy and orbital energies. The two-term (DZ) approx-imation improves both energies, but to reach the HF level a sensiblylarger number of functions is needed. The comparison between STO andGTOresults forNe shows howmuch larger the number ofGTOsmust beto get an accuracy comparable to that of the corresponding STOs.Nevertheless, the nearly HF GTO basis for Ne, containing about twiceas many functions as those of the corresponding STOs, is still in error byabout 3� 10� 3Eh for the electronic energy and evenmore for the orbitalenergies.In the molecular HF case (Table 7.4), it is seen that the SZ approxima-

tion is very poor, giving a molecular energy of about 0:592Eh (371 kcalmol�1!) above the HF limit of the last column and a charge distributionwhich severely underestimates the electric dipole moment m. The nearlyDZPP approximation of column 3 yields instead a sensible overestima-tion of the dipole moment. The nearly HF STO results of column 4 (Cadeand Huo, 1967; also see McLean and Yoshimine (1967b)) are in fairlygood agreement with the accurate HF/2D results (the ‘benchmark’) of thelast column, based on the two-dimensional numerical quadrature of theone-electron HF equation, and, therefore, free from any basis set andtruncation errors.The same considerations are almost true for the GTO calculations of

the bottom part of Table 7.4. STO-3G calculations are useless either forenergy or dipole moment. The aug-ccpVDZ basis45 overestimates dipolemoment less than the corresponding STO set, but gives a worse energy.The aug-ccpVTZ basis56 gives a good dipole moment, but still under-estimates the energy, which was not unexpected since these Dunningsets are devised particularly for calculation of electric molecularproperties.

7.4 H€UCKEL THEORY

H€uckel theory amounts to a simple LCAO–MO theory of carbon pelectrons in conjugated and aromatic hydrocarbons whose s skeleton

5aug¼ one s, p, d diffuse functions on F, one s, p on H.6aug¼ one s, p, d, f diffuse functions on F, one s, p, d on H.


is assumed planar.67 Each carbon atom contributes an electron in its2pp AO, AOs being assumed orthonormal and the coefficients deter-mined by the Ritz method. The elements of the H€uckel matrix H aregiven in terms of just two negative unspecified parameters, namely thediagonal a (the Coulomb integral) and the nearest-neighbour off-diagonal b (the resonance or bond integral), simply introduced in atopological way as

Hmm ¼ a ð7:48Þ

Hmn ¼ bdm;m1 ð7:49Þ

Smn ¼ dmn m; n ¼ 1; 2; . . . ;N ð7:50Þ

H€uckel theory, therefore, is a noniterative MO theory distinguishingonly between linear (open) or closed (ring) chains.It is convenient to introduce the notation

a� «

b¼ � x; « ¼ aþ xb ð7:51Þ

D« ¼ «�a ¼ xb ) D«b

¼ x ð7:52Þ

so that xmeasures the p bond energy in units of b (x > 0 means bonding,x < 0 antibonding). The total p bond energy in H€uckel theory78 is thensimply

DEp ¼Xocci

D«i ð7:53Þ

while the p charge density in terms of the AOs is given by twiceEquation 7.34 (Section 7.2).Henceforth, we shall denote by DN the H€uckel determinant of order

N, considering in detail the case of ethylene, the allyl radical, linear and

7s--p separation is possible in this case.8It is assumed that nuclear repulsion is essentially cancelled by the negative of the electron

repulsion (Equation 7.20).

HUCKEL THEORY 113

cyclic butadiene, hexatriene and benzene, taken as representatives oflinear chains withN ¼ 2; 3; 4;6 and rings (closed chains) withN ¼ 4; 6.For these systems, we shall solve in an elementary way for both eigen-values and eigenvectors of theH€uckelmatrixH, obtaining in thisway theorbital energy levels and the schematic shapes of the MOs shown inFigures 7.2–7.7. General, andmore powerful, techniques of solution aredue to Lennard–Jones (1937) and will be used later in Section 7.5.

7.4.1 Ethylene (N¼ 2)

The H€uckel secular equation for N ¼ 2 is

D2 ¼ �x 11 � x

�� ¼ x2� 1 ¼ 0 ð7:54Þ

LUMO-1-√2

x

LUMO

HOMO0

1 HOMO√2

N = 2 N = 3

LUMO-0.618

-2

x

LUMO-1.618

HOMO0

HOMO

21.618

0.618

N = 4(Chain)

N = 4(Ring)

Figure 7.2 H€uckel orbital energies for a linear chain withN ¼ 2; 3; 4 and a ring withN ¼ 4


whose roots (eigenvalues, left in the top row of Figure 7.2) are x ¼ 1,while the normalized eigenvectors (theMOs of the top row of Figure 7.3)are given by the (2 � 2) unitary matrix

C ¼ ðc1c2Þ ¼

1ffiffiffi2

p � 1ffiffiffi2

p

1ffiffiffi2

p 1ffiffiffi2

p

0BBBB@

1CCCCA ð7:55Þ

7.4.2 The Allyl Radical (N ¼ 3)

The H€uckel secular equation for N¼3 is

D3 ¼�x 1 01 � x 10 1 � x

�� ¼ � xðx2 �1Þþ x ¼ � xðx2 � 2Þ ¼ 0 ð7:56Þ

N = 2

φ1

-.

-.

-..

+

φ2

+..+ .+ ..+ .-

-..+ .+

N = 3

-- - - +- -+

φ1 φ3φ2

N = 4

-

+.

-.+

-.+

-

+.

-

+.

-.+

+.-

+

-.

φ1 φ2

-.

+.

+.

φ3

-.

+.

-.+

+.

-.

φ4

+ + - +

+ - - + - + -

Figure 7.3 H€uckel MOs for a linear chain with N ¼ 2; 3; 4

HUCKEL THEORY 115

whose roots (right in the top row of Figure 7.2) are x ¼ ffiffiffi2

p; 0, with the

normalized eigenvectors (second row of Figure 7.3) given by the (3� 3)unitary matrix

C ¼ ðc1c2c3Þ ¼

1

2

1ffiffiffi2

p 1

2ffiffiffi2

p

20 �

ffiffiffi2

p

2

1

2� 1ffiffiffi

2p 1

2

0BBBBBBBBB@

1CCCCCCCCCA

ð7:57Þ

We now give below the explicit calculation of the eigenvectors corre-sponding to each eigenvalue.H€uckelMOs (the eigenvectors) are obtainedby solving in turn for each eigenvalue the linear homogeneous systemassociated with the secular Equation 7.56 with the additional constraintof coefficient normalization.

(a) x1 ¼ ffiffiffi2

p

1: � ffiffiffi2

pc1 þ c2 ¼ 0

2: c1�ffiffiffi2

pc2 þ c3 ¼ 0

3: c2 �ffiffiffi2

pc3 ¼ 0

4: c21þ c22 þ c23 ¼ 1

8>>>><>>>>:

ð7:58Þ

1: gives: c2 ¼ ffiffiffi2

pc1

3: gives: c3 ¼ 1ffiffiffi2

p c2 ¼ c1

so that it is immediately obtained that

c21 þ 2c21 þ c21 ¼ 4c21 ¼ 1 ) c1 ¼ 1

2; c2 ¼

ffiffiffi2

p

2; c3 ¼ 1

2ð7:59Þ

which is the first column of matrix (7.57).(b) x2 ¼ 0

c2 ¼ 0

c1 þ c3 ¼ 0

c21 þ c22 þ c23 ¼ 1

8><>: ð7:60Þ


giving

c2 ¼ 0; c3 ¼ �c1; c21þ c21 ¼ 2c21 ) c1 ¼ 1ffiffiffi2

p ; c2 ¼ 0; c3 ¼ � 1ffiffiffi2

p

ð7:61Þwhich is the second column of matrix (7.57).

(c) x3 ¼ � ffiffiffi2

p

1:ffiffiffi2

pc1 þ c2 ¼ 0

2: c1 þffiffiffi2

pc2 þ c3 ¼ 0

3: c2þffiffiffi2

pc3 ¼ 0

c21 þ c22 þ c23 ¼ 1

8>>><>>>:

ð7:62Þ

1: gives now : c2 ¼ � ffiffiffi2

pc1

3: gives : c3 ¼ � 1ffiffiffi2

p c2 ¼ c1

so that it is immediately obtained that

c21þ 2c21 þ c21 ¼ 4c21 ¼ 1 ) c1 ¼ 1

2; c2 ¼ �

ffiffiffi2

p

2; c3 ¼ 1

2ð7:63Þ

which is the third column of matrix (7.57).

It is of interest to evaluate charge and spin density distributions inthe allyl radical, asssuming that the unpaired electron has a spinS ¼ MS ¼ 1

2

� . It is convenient to write explicitly the three MOs as

f1 ¼1

2ðx1þ

ffiffiffi2

px2þx3Þ; f2 ¼

1ffiffiffi2

p ðx1�x3Þ; f3 ¼1

2ðx1�

ffiffiffi2

px2þx3Þ

ð7:64Þ

We then have the following for the ra and rb components of thedistribution function:

ra ¼ f21þf2

2 ¼1

4x21þ

2

4x22þ

1

4x23þ

2

4x21þ

2

4x23 ¼

3

4x21þ

2

4x22þ

3

4x23

ð7:65Þ

HUCKEL THEORY 117

rb ¼ f21 ¼

1

4x21þ

2

4x22þ

1

4x23 ð7:66Þ

This gives for the electron density

PðrÞ ¼ ra þ rb ¼ x21þ x2

2þ x23 ð7:67Þ

so that the charge distribution of the p electrons is uniform (one electrononto each carbon atom), as expected for an alternant hydrocarbon,89 andfor the spin density we have

QðrÞ ¼ ra � rb ¼ 1

2x21 þ

1

2x23 ð7:68Þ

According to (7.68), in the H€uckel spin density the a unpaired electronis 1

2 on atom 1 and 12 on atom 3, being zero at the central atom. This is

contrary to the ESR experimental observation that some a spin at the endatoms induces some b spin at themiddle atom. This wrong result is due tothe lack of any electron correlation in the wavefunction, which belongs tothe class of IPM functions. Both (7.67) and (7.68) satisfy the appropriateconservation relations:910 trP ¼ 3 (the total number of p electrons) andtrQ ¼ 1 (the unpaired p electron of a spin).The p bond energy in allyl is 2

ffiffiffi2

p ¼ 2:828 (units ofb), comparedwith 2for the p bond energy of the ethylenic double bond (the prototype of thedouble bond). The difference

DEpðallylÞ�DEpðethyleneÞ ¼ 2:828� 2 ¼ 0:828 ð7:69Þ

is called the delocalization energy of the double bond in the allylradical. It corresponds to a stabilization of the conjugated p system inthe radical.

9Alternant hydrocarbons are conjugatedmolecules inwhich the carbonatoms canbedivided into

two sets, crossed and circled, such that no two members of the same set are bonded together. All

molecules considered here are alternant hydrocarbons, inwhich energy levels occur in pairs, with

a p bond energyx, and coefficients of the pairedMOs which are either the same or change sign(Murrell et al., 1985).10Equations 6.31 and 6.32.


7.4.3 Butadiene (N ¼ 4)

The H€uckel secular equation for the linear chain with N ¼ 4 is

D4 ¼

� x 1 0 0

1 � x 1 0

0 1 � x 1

0 0 1 �x

��

��¼ x4 �3x2 þ1 ¼ 0 ð7:70Þ

where we have marked in boldface the top right and the bottom leftelements that differ from those of the closed chainwhichwill be examinedin the next point. Equation 7.70 is a pseudoquartic equation that can beeasily reduced to a quadratic equation by the substitution

x2 ¼ y ) y2 � 3yþ 1 ¼ 0 ð7:71Þ

having the roots

y ¼ 3 ffiffiffi5

p

2¼

y1 ¼ 3þ ffiffiffi5

p

2¼ 2:618

y2 ¼ 3� ffiffiffi5

p

2¼ 0:382

8>>>>><>>>>>:

ð7:72Þ

So, we obtain the four roots (left in the bottom row of Figure 7.2)

x1 ¼ ffiffiffiffiffiy1

p ¼ 1:618; x2 ¼ ffiffiffiffiffiy2

p ¼ 0:618HOMO

;

x3 ¼ � ffiffiffiffiffiy2

p ¼ � 0:618LUMO

; x4 ¼ � ffiffiffiffiffiy1

p ¼ � 1:618

ð7:73Þ

with the first two being bonding levels and the last two antibondinglevels.The calculation of theMO coefficients (the eigenvectors corresponding

to the four roots above) proceeds as we did for allyl, and the MOs (third

HUCKEL THEORY 119

and fourth row of Figure 7.3) are

f1 ¼ 0:371x1 þ 0:601ðx2 þ x3Þþ 0:371x4

f2 ¼ 0:601x1 þ 0:371ðx2 � x3Þ� 0:601x4

f3 ¼ 0:601x1 � 0:371ðx2 þ x3Þþ 0:601x4

f4 ¼ 0:371x1 � 0:601x2 þ 0:601x3 � 0:371x4

8>>>>><>>>>>:

ð7:74Þ

with the first two being bonding MOs (f2 ¼ HOMO) and the last twoantibonding MOs (f3 ¼ LUMO).Proceeding as we did for allyl, it is easily seen that the electron charge

distribution is uniform (one p electron onto each carbon atom, alternanthydrocarbon) and the spin density is zero, as expected for a state withS ¼ MS ¼ 0, since the two bonding MOs are fully occupied by electronswith opposite spin. The delocalization energy for linear butadiene is

DEpðbutadieneÞ� 2DEpðethyleneÞ ¼ 4:472� 4 ¼ 0:472 ð7:75Þ

and, therefore, is sensibly less than the conjugation energy of the allylradical.

7.4.4 Cyclobutadiene (N ¼ 4)

The H€uckel secular equation for the square ring with N ¼ 4 is

D4 ¼

� x 1 0 1

1 � x 1 0

0 1 � x 1

1 0 1 � x

��

��¼ x4 � 4x2 ¼ x2ðx2 � 4Þ ¼ 0 ð7:76Þ

where the boldface elements are the only ones differing from those of thelinear chain (1 and 4 are now adjacent atoms). The roots of Equation 7.76are x1 ¼ 2, x2 ¼ x3 ¼ 0 (doubly degenerate), and x4 ¼ �2 (right in thebottom row of Figure 7.2).


It is of interest to solve in some detail the homogeneous systemoriginating (7.76) because of the presence of degenerate eigenvalues.

(a) x1 ¼ 2 The homogeneous system to be solved is

1: � 2c1þ c2þ c4 ¼ 0

2: c1� 2c2þ c3 ¼ 0

3: c2� 2c3þ c4 ¼ 0

4: c1þ c3 �2c4 ¼ 0

c21þ c22 þ c23 þ c24 ¼ 1

8>>>>>>><>>>>>>>:

ð7:77Þ

Subtracting 4. from 2. gives � 2c2 þ 2c4 ¼ 0 ) c4 ¼ c2.Then

1: gives: �2c1þ 2c2 ¼ 0 ) c2 ¼ c1

2: gives: � c1 þ c3 ¼ 0 ) c3 ¼ c1

4c21 ¼ 1 ) c1 ¼ c2 ¼ c3 ¼ c4 ¼ 1

2

so that we finally obtain

f1 ¼ 1

2ðx1 þ x2 þ x3 þ x4Þ ð7:78Þ

the deepest bonding MO, sketched in the diagram of the top row ofFigure 7.4. To simplify the drawings, the 2pp AOs in the MOs areviewed from above the zxmolecular plane, so that only the signs of theupper lobes are reported.(b) x2 ¼ x3 ¼ 0 (twofold degenerate eigenvalue)The system is

1: c2 þ c4 ¼ 0 ) c4 ¼ � c2

2: c1 þ c3 ¼ 0 ) c3 ¼ � c1

c21 þ c22 þ c23 þ c24 ¼ 1

8><>: ð7:79Þ

Equations 3 and4are useless, since they give the same result. Therefore,we have three equations and four unknowns, and there is one degree offreedom left, which means that one unknown can be chosen in anarbitrary way.

HUCKEL THEORY 121

With reference to the coordinate system of Figure 7.4, we firstchoose c4 ¼ c1 as additional arbitrary constraint. We then immedi-ately obtain

f2 ¼ f2x ¼1

2ðx1 � x2 � x3 þ x4Þ / x ð7:80Þ

a function having the same transformation properties as thex-coordinate (left diagram in the second row of Figure 7.4). yz is anodal plane for this MO. Next, we choose c4 ¼ � c1 as additionalarbitrary constraint, therefore obtaining

f3 ¼ f2y ¼1

2ðx1þ x2 �x3 �x4Þ / y ð7:81Þ

a function having the same transformation properties as the y-co-ordinate (right diagram in the second row of Figure 7.4). zx is now anodal plane for this MO. f2x and f2y are the pair of HOMOsbelonging to the doubly degenerate energy level «2 ¼ «3; which

++y

12. .x

+ +

+ +

3 4. .φ1

+_ +

+_ _ _

+_

φ2~ x φ3~ y

+_

+ _φ4

+

Figure 7.4 Coordinate system and H€uckel MOs for cyclobutadiene


transform as the pair of basic vectors ex and ey of the symmetry D2h

(see Chapter 12) to which the s skeleton of cyclobutadiene belongs.They are, therefore, orthogonal and not interacting, as can beimmediately seen:

hf2xjf2yi ¼1

4hx1� x2� x3 þx4jx1 þ x2 � x3� x4i

¼ 1

4ð1� 1þ 1� 1Þ ¼ 0

ð7:82Þ

hf2xjHjf2yi ¼1

4hx1 � x2 � x3 þ x4jHjx1 þ x2 � x3 � x4i

¼ 1

42ðaþb�bÞ� 2ðaþb�bÞ½ � ¼ 0

ð7:83Þ

It is also seen that they belong to the same eigenvalue « ¼ a:

«2x ¼ 1

4hx1 �x2 � x3 þ x4jHjx1 � x2 � x3 þ x4i

¼ 1

4ð4a�2bþ 2bÞ ¼ a

ð7:84Þ

«2y¼1

4hx1þx2�x3�x4jHjx1þx2�x3�x4i¼

1

4ð4aþ2b�2bÞ¼a

ð7:85Þ

(c) x4 ¼ � 2Now, the solution of the homogeneous system

1: 2c1þ c2 þ c4 ¼ 0

2: c1 þ2c2 þ c3 ¼ 0

3: c2 þ2c3 þ c4 ¼ 0

4: c1 þ c3 þ 2c4 ¼ 0

c21 þ c22 þ c23 þ c24 ¼ 1

8>>>>>><>>>>>>:

ð7:86Þ

does not present any problem. The calculation proceeds exactly in thesame way as we did for the first eigenvalue, and the last antibondingMO (LUMO) is found to be

f4 ¼1

2ðx1 �x2 þ x3 � x4Þ ð7:87Þ

HUCKEL THEORY 123

f4 is sketched in the diagramof the bottom row of Figure 7.4 and showsthe existence of two nodal planes orthogonal to each other.1011

The delocalization energy for cyclobutadiene is

DEpðcyclobutadieneÞ� 2DEpðethyleneÞ ¼ 4�4 ¼ 0 ð7:88Þ

so that, in H€uckel theory, the p system of cyclobutadiene has zerodelocalization energy, which is a rather unexpected result.

7.4.5 Hexatriene (N ¼ 6)

The H€uckel secular equation for the linear chain with N ¼ 6 is

D6 ¼

� x 1 0 0 0 0

1 � x 1 0 0 0

0 1 � x 1 0 0

0 0 1 �x 1 0

0 0 0 1 � x 1

0 0 0 0 1 � x

��

��¼ 0 ð7:89Þ

where we have marked in boldface the top right and the bottom leftelements that differ from those of the closed chain which corresponds tothe benzene molecule. We obtain the sixth-degree equation

D6 ¼ x6 � 5x4þ 6x2� 1 ¼ ðx3 þ x2� 2x�1Þðx3 � x2 � 2xþ 1Þ ¼ 0

ð7:90Þ

whose solutions are expected to occur in pairs of opposite sign, since onlyeven powers of x appear in it. It has the following roots,1112 written inascending order:

x1 ¼ 1:802; x2 ¼ 1:247; x3 ¼ 0:445;

x4 ¼ � 0:445; x5 ¼ � 1:247; x6 ¼ �1:802

(ð7:91Þ

11As a general rule, the number of nodal planes increases for the higher p orbitals, while the

deepest bondingMOhas no nodal planes (except for themolecular plane,which is common to all

molecules considered here).12These cubic equations are not easily solved. The roots were evaluated using the Lennard-Jones

formula (7.105).


These roots are symmetrical in pairs about x¼ 0, as it should be for analternant hydrocarbon. In the factorized form of Equation 7.90, x2, x4,and x6 are solutions of the first cubic equation and x1, x3, and x5 of thesecond cubic equation.The p energy levels of hexatriene are sketched in the left diagram of

Figure 7.5. The delocalization energy for linear hexatriene is

DEpðhexatrieneÞ� 3DEpðethyleneÞ ¼ 2ð1:802þ 1:247þ 0:445Þ� 6

¼ 6:988� 6 ¼ 0:988

ð7:92Þ

We now turn to benzene, the closed chain corresponding tohexatriene.

-2

x

1 802

LUMO-1.247

-1

-1.802

LUMO

0

-0.445

HOMO

HOMO0.445

11.247

21.802

N = 6(Chain)

N = 6(Ring)

Figure 7.5 H€uckel orbital energies for linear chain (hexatriene) and ring (benzene)for N ¼ 6

HUCKEL THEORY 125

7.4.6 Benzene (N ¼ 6)

The H€uckel secular equation for the closed chain (ring) with N ¼ 6 is

D6 ¼

� x 1 0 0 0 11 � x 1 0 0 00 1 � x 1 0 00 0 1 �x 1 00 0 0 1 � x 11 0 0 0 1 � x

��

��¼ 0 ð7:93Þ

where the boldface elements are the two differing from those of the linearchain (1 and 6 are nowadjacent atoms). By expanding the determinantweobtain a sixth-degree equation in x that can be easily factorized into thethree quadratic equations:1213

D6 ¼ x6 � 6x4 þ 9x2 � 4 ¼ ðx2 � 4Þðx2 � 1Þ2 ¼ 0 ð7:94Þwith the following roots, written in ascending order:

x ¼ 2; 1; 1; � 1; � 1; �2 ð7:95ÞBecause of the high symmetry of the molecule, two levels are now

doubly degenerate. The calculation of the MO coefficients can be doneusing the elementary algebraic methods used previously for the allylradical and cyclobutadiene.With reference to Figure 7.6, a rather lengthy

2

y

.13 ..

4 6

x

...

5

Figure 7.6 Numbering of carbon atoms in benzene and coordinate system

13The same factorization canbe obtainedusing hexagonal symmetry. In hexatriene the symmetryis lower and only two cubic equations are obtained, equivalent to using the symmetry plane

bisecting the molecule.


calculation shows that the real MOs are

f1 ¼1ffiffiffi6

p ðx1þ x2þ x3 þx4 þ x5 þ x6Þ

f2 ¼1

2ðx1 �x3 � x4 þ x6Þ / x

f3 ¼1ffiffiffiffiffiffi12

p ðx1 þ 2x2 þ x3 � x4 � 2x5 � x6Þ / y

f4 ¼1ffiffiffiffiffiffi12

p ðx1 � 2x2 þ x3 þ x4 � 2x5 þ x6Þ / x2 � y2

f5 ¼1

2ðx1 �x3 þ x4 � x6Þ / xy

f6 ¼1ffiffiffi6

p ðx1� x2þ x3 �x4 þ x5 � x6Þ:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð7:96Þ

The first degenerate MOs transform as (x,y) and are bonding(HOMOs); the second degenerate MOs transform as (x2 � y2, xy) andare antibonding (LUMOs). These degenerate MOs have one and twonodal planes respectively. The highest antibonding MO, f6, has threenodal planes. The six real MOs for the p electron system of benzene aresketched in the drawings of Figure 7.7, where the same assumptions weremade as for cyclobutadiene.The ra and rb components of the distribution function for benzene are

equal:

ra ¼ rb ¼ f21 þf2

2 þf23

¼ x21

1

6þ 1

4þ 1

12

!þ x2

2

1

6þ 4

12

!þ x2

3

1

6þ 1

4þ 1

12

!

þx24

1

6þ 1

4þ 1

12

!þ x2

5

1

6þ 4

12

!þ x2

6

1

6þ 1

4þ 1

12

! ð7:97Þ

so that

ra ¼ rb ¼ 1

2ðx2

1 þ x22 þ x2

3þ x24þ x2

5 þx26Þ ð7:98Þ

HUCKEL THEORY 127

We then have

PðrÞ ¼ ra þ rb ¼ x21 þ x2

2 þ x23 þ x2

4 þ x25 þ x2

6 ð7:99Þ

for the electron density and the charge distribution of the p electrons isuniform (one electron onto each carbon atom), as expected for analternant hydrocarbon; whereas the spin density is zero:

QðrÞ ¼ ra � rb ¼ 0 ð7:100Þ

as it must be for a singlet state.

Figure 7.7 H€uckel MOs for the p electrons in benzene (real form)


The p bond energy (units of b) for benzene is

DEp ¼ 2� 2þ 4� 1 ¼ 8 ð7:101Þ

When the p bond energy of three ethylenes

3DEpðethyleneÞ ¼ 3� 2 ¼ 6 ð7:102Þ

is subtracted from (7.101), we obtain the following for the delocalizationenergy of the three double bonds in benzene:

DEpðbenzeneÞ� 3DEpðethyleneÞ ¼ 8� 6 ¼ 2 ð7:103Þ

This is the highest value obtained so far among themolecules studied. Thisenergy lowering is responsible for the great stability of the benzenemolecule, where the three aromatic p bonds are fully delocalized andbear no resemblance at all to three ethylenic double bonds. Furtherstability in benzene arises from the fact that it has no strain in its s

skeleton.It is also of interest to compare thepbond energyof benzenewith that of

hexatriene. We have

DEpðbenzeneÞ�DEpðhexatrieneÞ ¼ 8� 6:988 ¼ 1:012 ð7:104Þ

so that ring closure in benzene is favoured, leading to further stabilizationwith respect to the open chain.

7.5 A MODEL FOR THE ONE-DIMENSIONALCRYSTAL

Increasing the number of interacting AOs increases the number of result-ing MOs. For the CNHNþ2 polyene chain the MO levels, which alwaysrange between aþ 2b and a� 2b, become closer and closer up totransforming inbands (a continuous successionofmolecular levels)whichare characteristic of solids.Using the general formula

«k ¼ aþ 2b cosp

Nþ 1k k ¼ 1; 2; . . . ;N ð7:105Þ

A MODEL FOR THE ONE-DIMENSIONAL CRYSTAL 129

derived byLennard-Jones (1937)1314 for the orbital energy of the kthMO inthe N-atom linear polyene chain, the following results are easilyestablished.

(i) First level (k ¼ 1):

«1 ¼ aþ 2b cosp

Nþ 1lim

N!1«1 ¼ aþ 2b ð7:106Þ

Last level (k¼N):

«N ¼aþ2bcospNNþ1

¼aþ2bcosp

1þ 1

N

limN!1

«N ¼a�2b ð7:107Þ

The difference between two successive levels is

D«¼ «kþ1�«k ¼ 2b cosp

Nþ1ðkþ1Þ�cos

pNþ1

k

" #

¼ �4b sinp2

2kþ1

Nþ1sin

p2

1

Nþ1

ð7:108Þ

where use was made of the trigonometric identity

cosa�cosb¼ �2sinaþb

2sin

a�b

2ð7:109Þ

Hence, for N!1, D«!0 and we have formation of a continuousband of molecular levels. The limiting values aþ2b and a�2b arereached asymptotically when N!1.(ii) ForN!1, therefore, the polyene chain becomes the model forthe one-dimensional crystal. We have a bonding band with energyranging from aþ 2b to a and an antibonding band with energyranging from a to a� 2b, which are separated by the so called Fermilevel, the top of the bonding band occupied by electrons. It isimportant to note that using just one b, equal for single and doublebonds, there is no band gap between bonding and antibonding levels.

14See also McWeeny (1979).


If we admit jbdj > jbsj, as reasonable and done by Lennard-Jones(1937) in his original study, then we have a band gap D ¼ 2ðbd �bsÞ,which is of great importance in the properties of solids. Metals andcovalent solids, conductors and insulators, and semiconductors canall be traced back to the model of the infinite polyene chain extendedto three dimensions (McWeeny, 1979).

A MODEL FOR THE ONE-DIMENSIONAL CRYSTAL 131

8Post-Hartree–Fock Methods

In this chapter we shall briefly introduce some methods which, mostlystarting from the uncorrelated HF approximation, attempt to reachchemical accuracy (1 kcalmol�1 or less) in the quantum chemical calcula-tion of the atomization energies. We shall outline first the basic principlesof configuration interaction (CI) and multiconfiguration SCF (MC-SCF)techniques, proceeding next to some applications of the so-called many-body perturbation methods, mostly the Møller–Plesset second-orderapproximation to the correlation energy (MP2), which is the startingpoint of the more efficient methods of accounting for correlation effectsdirectly including the interelectronic distance in thewavefunction, such astheMP2-R12 and CC-R12methods of the Kutzelnigg group. The chapterends with a short introduction to density functional theory (DFT).

8.1 CONFIGURATION INTERACTION

Given a basis of atomic or molecular spin-orbitals, we construct a linearcombination of electron configurations in the form of many-electronSlater determinants, with coefficients determined by the Ritz method,to give the CI wavefunction:

Yðx1; x2; . . . ;xNÞ ¼Xk

Ykðx1; x2; . . . ;xNÞCk ð8:1Þ

When all possible configurations arising from a given basis set areincluded, we speak of the full-CI wavefunction. It should be recalled that


Valerio Magnasco


only configurations of given S and MS belonging to a given molecularsymmetry, have nonzero matrix elements of the molecular Hamiltonian.Even if the method is, in principle, exact if we include all configurations,expansion (8.1) converges usually quite slowly and the number ofconfigurations becomes rapidly very large, involving up to millions ofdeterminants.1

This is due to the difficulty of the wavefunction (8.1) in accounting forthe cusp condition for each electron pair (Kutzelnigg, 1985):

limrij ! 0

1

Y0

@Y@rij

� �¼ 1

2ð8:2Þ

which is needed to keep the wavefunction finite when rij ¼ 0 in presenceof the singularities of the Coulomb terms in the Hamiltonian.Following Kato (1957), Kutzelnigg (1985) has shown that the Y

expanded in powers of the interelectronic distance rij

Y ¼ Y0ð1þ arij þbr2ij þ cr3ij þ � � � Þ ð8:3Þ

satisfies this condition near the singular points for the pair of electronsi and j when a ¼ 1

2. In fact:

@Y@rij

¼ Y0ðaþ2b rij þ � � � Þ ð8:4Þ

1

Y0

@Y@rij

¼ aþ 2b rij þ � � � ð8:5Þ

limrij !0

1

Y0

@Y@rij

� �¼ a ¼ 1

2ð8:6Þ

where a is a constant, with a ¼ 1

2if i and j are both electrons, a ¼ �ZB if i

is a nucleus of charge þZB and j an electron.Using He as a simple example, Kutzelnigg (1985) showed that use of

a starting wavefunction of the type

Y0 1þ 1

2r12

� �ð8:7Þ

1 Special techniques are required for solving the related secular equations of such huge dimen-

sions (Roos, 1972).

134 POST-HARTREE–FOCK METHODS

whereY0 is the simple product two-electron hydrogen-like wavefunctionfor ground-state He, gives a cusp-corrected CI expansion rapidly con-vergentwith the biexcitationswith ‘ ¼ 0;1; 2; 3; 4; . . . (s2, p2, d2, f2, g2, . . .type): just 156 interconfigurational functions up to ‘ ¼ 4 giveE ¼ �2:903 722Eh, roughly the same energy value obtained by includingabout 8000 interconfigurational functions with ‘ � 6 in the ordinaryCI expansion starting from Y0. The accurate comparison value, due toFrankowski and Pekeris (1966), is E ¼ �2:903 724 377 033Eh, a‘benchmark’ for the He atom correct to the last decimal figure (picohar-tree). Of course, use of the wavefunction (8.7) as a starting point in theCI expansion (8.1) involves the more difficult evaluation of unconven-tional one- and two-electron integrals.

8.2 MULTICONFIGURATION SELF-CONSISTENT-FIELD

In thismethod,mostly due toWahl and coworkers (Wahl andDas, 1977),both the form of the orbitals in each single determinantal function and thecoefficients of the linear combination of the configurations are optimizedin a wavefunction like (8.1). The orbitals of a few valence-selectedconfigurations are adjusted iteratively until self-consistency with thesimultaneous optimization of the linear coefficients is obtained. Themethod predicts a reasonable well depth in He2 and reasonable atomiza-tion energies (within 2 kcalmol�1) for a fewdiatomics, such asH2, Li2, F2,CH, NH, OH and FH.2

8.3 MØLLER–PLESSET THEORY

Since the Møller–Plesset approach is based on Rayleigh–Schroedinger(RS) perturbation theory, which will be introduced to some extent only inChapter 10, it seems appropriate to give a short r�esum�e of it here.Stationary RS perturbation theory is based on the partition of the

Hamiltonian H into an unperturbed Hamiltonian H0 and a smallperturbation V, and on the expansion of the actual eigenfunction c andeigenvalue E into powers of the perturbation, each correction beingspecified by a definite order given by the power of an expansion parameterl. For themethod to be applied safely, it is necessary (i) that the expansion

2 He2, F2 and NH are not bonded at the SCF level.

MØLLER–PLESSET THEORY 135

converges and (ii) that the unperturbed eigenfunction c0 satisfies exactlythe zeroth-order equation with eigenvalue E0. While E1, the first-ordercorrection to the energy, is the average value of the perturbation overthe unperturbed eigenfunction c0 (a diagonal term), the second-orderterm E2 is given as a transition (nondiagonal) integral in which state c0

is changed into state c1 under the action of the perturbation V. Furtherdetails are left to Chapter 10.Møller–Plesset theory (Møller and Plesset, 1934) starts from E(HF)

considered as the result in first order of perturbation theory, EðHFÞ ¼E0 þE1, assuming as unperturbed c0 the single determinant HF wave-function, and as first-order perturbation the difference between theinstantaneous electron repulsion and its average value calculated at theHF level. Therefore, it gives directly a second-order approximation tothe correlation energy, since by definition

EðcorrelationÞ ¼ EðtrueÞ�EðHFÞ ð8:8Þwhen E(true) is replaced by its second-order approximation

EðtrueÞ � E0þE1 þE2 ¼ EðHFÞþE2ðMPÞ ð8:9ÞHence:

EðtrueÞ�EðHFÞ � E2ðMPÞ ¼ EðMP2Þ¼ second-order approximation to the correlation energy

ð8:10Þ

It is seen that only biexcitations can contribute to E2, since mono-excitations give a zero contribution for HF Y0 (Brillouin’s theorem).Comparison of SCF and MP2 results for the 1A1 ground state of the

H2O molecule (Rosenberg et al., 1976; Bartlett et al., 1979) shows thatMP2 improves greatly the properties (molecular geometry, force con-stants, electric dipole moment) but gives no more than 76% of theestimated correlation energy.

8.4 THE MP2-R12 METHOD

This is a Møller–Plesset second-order theory, devised by Kutzelnigg andcoworkers (Klopper and Kutzelnigg, 1991),3 which incorporates the

3 Presented at the VIIth International Symposium on Quantum Chemistry, Menton, France,

2–5 July 1991.


linear r12-dependent term into MP2 and MP3. Difficulties with the newthree-electron integrals occurring in the theory are overcome in terms ofexpansions over ordinary two-electron integrals for nearly saturated basissets. Fairly good results, improving upon ordinaryMP2, are obtained forthe correlation energies in simple closed-shell atomic and molecularsystems (H2, CH4, NH3, H2O, HF, Ne) using extended sets of GTOs.

8.5 THE CC-R12 METHOD

The coupled-cluster (CC)method is a natural infinite-order generalizationofmany-bodyperturbation theory (MBPT), ofwhichMøller–PlessetMP2was the second-order approximation. InMBPT, starting from a referencewavefunction Y0, multiple excitations from unperturbed occupied (occ)orbitals to unoccupied (empty) ones are considered. The theory wasdeveloped for use in many-body physics mostly in terms of ratherawkward4 secondquantization anddiagrammatic techniques (McWeeny,1989).In the CC method, the exact wavefunction is expressed in terms of

an exponential form of the variational wavefunction,5 where a clusteroperator expðTÞ acts upon a single-determinant reference wavefunctionY0.

6 In the full CCSDT model, the cluster operator is usually truncatedafter T3 (triple excitations).The CC-R12 method incorporates explicitly the interelectronic dis-

tance r12 into the wavefunction by replacing T by S ¼ Tþ R, where Ris the r12-contribution to the double excitation cluster operator. Theoperator R

kl

ij creates unconventionally substituted determinants in whicha pair of occ orbitals i,j is replaced by another pair of occ orbitals l,kmultiplied by the interelectronic distance r12.TheCCSDT-R12method devised byNoga andKutzelnigg (1994) is the

best available today for the computation of the atomization energies ofsimplemolecules.7 CCSDT-R12 calculations on ground-stateNH3,H2O,FH, N2, CO, F2 at the experimentally observed geometries, using nearlysaturated, well-balanced (spdfgh|spdf) GTO basis sets, give atomizationenergies in perfect agreement with the experimental spectroscopic data

4 At least for ordinary quantum chemists.5 German, ansatz.6 Possibly HF, in which case the contribution of monoexcitations vanishes because of Brillouin’s

theorem.7 It is actually (2007) in progress the extension of the theory to the calculation of second-order

molecular properties, such as frequency-dependent polarizabilities.

THE CC-R12 METHOD 137

(Noga et al., 2001). It is hoped that in this way it will be possible to obtain‘benchmarks’ in the calculation of atomization energies, at least for thesmall molecules of the first row.

8.6 DENSITY FUNCTIONAL THEORY

DFTwas initially developed byHohenberg andKohn (1964) and byKohnand Sham (1965), and is largely used today by the quantum chemicalcommunity in calculations on complex molecular systems. It must bestressed that DFT is a semiempirical theory accounting in part for electroncorrelation.The electronic structure of the ground state of a system is assumed to be

uniquely determined by the ground state electronic density r0(r), anda variational criterion is given for the determination of r0 and E0 froman arbitrary regular function r(r). The variational optimization of theenergy functional E[r] constrained by the normalization condition:

E½r� � E½r0�;ðdr rðrÞ ¼ N ð8:11Þ

shows that the functional derivative8 is nothing but the effective one-electron Kohn–Sham Hamiltonian h

KS:

dE½r�drðrÞ ¼ � 1

2r2 þVeffðrÞ ¼ h

KSðrÞ ð8:12Þ

where

VeffðrÞ ¼ VðrÞþ JðrÞþVxcðrÞ ð8:13Þ

the effective potential at r is the sum of the electron–nuclear attractionpotential V, plus the Coulomb potential J of the electrons of density r,plus the exchange-correlation potential Vxc for all the electrons. It is seenthat the effective potential (8.13) differs from the usual HF potential bythe undetermined correlation potential inVxc. SinceVxc cannot be definedexactly, it can only be given semiempirical evaluations. Most used inapplications is the Becke–Lee–Yang–Parr (B-LYP) correlation potential.Kohn–Sham orbitals fKS

i ðrÞ, i ¼ 1; 2; . . . ;n, are then obtained from the

8 The Euler–Lagrange parameter l of the constrained minimization.


iterative SCF solution of the corresponding Kohn–Sham eigenvalueequations, much as we did for the HF equations of Chapter 7:

hKSðrÞfiðrÞ ¼ «ifiðrÞ i ¼ 1;2; . . . ; n ð8:14Þ

where hKS

is the one-electron Kohn–Sham Hamiltonian (8.12).With the best functionals available to date it is possible to obtain bond

lengths within 0.01A�for the diatomic molecules of the first-row atoms,

and atomization energies within about 3 kcalmol�1, at a cost which issensibly lower than MP4 or other equivalent calculations.

DENSITY FUNCTIONAL THEORY 139

9Valence Bond Theory

and the Chemical Bond

In this chapter we shall consider, first, elements of the Born–Oppenheimerapproximation, concerning the separation in molecules of the motion of theelectrons from that of the nuclei. It will be seen that, by neglecting smallvibronic terms, the nucleimove in the field provided by the nuclei themselvesand the molecular charge distribution of the electrons, determining what iscalled a potential energy surface, a function of the nuclear configuration.Next, we shall introduce the study of the chemical bond by considering

the simplest two-electron molecular example, the H2 molecule. It will beseen that the single configurationMOapproach fails to describe the correctdissociation of the molecule in ground-state H atoms because of thecorrelation error. A qualitatively correct description of the bond dissocia-tion in H2 is instead provided by the Heitler–London (HL) theory, wheredifferent electrons are allotted to different atomic orbitals, the resultingwavefunction for the ground state then being symmetrized with respect toelectron interchange in order to satisfy Pauli’s antisymmetry principle.HL theory may be considered as introductory to the so-called valence

bond (VB) theory of molecular electronic structure, where localizedchemical bonds in molecules are described in terms of covalent and ionicstructures. The theory is considered at an elementary level for givingqualitative help in studying the electronic structure of simple molecules,in a strict correspondence between quantum mechanical VB structuresand chemical formulae. The importance of hybridization is stressed in


Valerio Magnasco


describing bond stereochemistry in polyatomicmolecules, with particularemphasis on the H2O molecule. A few applications of Pauling’ semiem-pirical theory of p electrons in conjugated and aromatic hydrocarbonsconclude the chapter.

9.1 THE BORN–OPPENHEIMER APPROXIMATION

This concerns the separation in molecules of the motion of the lightelectrons from the slow motion of the heavy nuclei.We want to solve the molecular wave equation

HY ¼ WY ð9:1Þ

where H is the molecular Hamiltonian (in atomic units):

H ¼Xa

� 1

2Mar2

a þXi

� 1

2r2

i þVenþVee

!þVnn

¼Xa

� 1

2Mar2

a þ He þVnn

ð9:2Þ

In the expression above, the first term is the kinetic energy operatorfor the motion of the nuclei,1 the term in parentheses is the electronicHamiltonian He and the last term is the Coulombic repulsion between thepoint-like nuclei in the molecule.Since wave equation (9.1) was too difficult to solve, Born and Oppen-

heimer (1927) suggested that, in a first approximation, the molecularwave function Y could be written as

Yðx; qÞ � Yeðx;qÞYnðqÞ ð9:3Þ

where Ye is the electronic wavefunction, an ordinary function of theelectronic coordinates x and parametric in the nuclear coordinates q. Ye

is a normalized solution of the electronic wave equation2

HeYe ¼ EeYe; hYejYei ¼ 1 ð9:4Þ

1 Ma is the mass of nucleus a in units of the electron mass.2 Which must be solved for any nuclear configuration specified by {q}.

142 VALENCE BOND THEORY AND THE CHEMICAL BOND

ConsideringYeYn as a nuclear variation function (Ye ¼ fixed), Longuet-Higgins (1961) showed that the best nuclear wavefunctionYn satisfies theeigenvalue equation

Xa

� 1

2Mar2

aþ UeðqÞ" #

YnðqÞ ¼ WYnðqÞ ð9:5Þ

where

UeðqÞ¼EeðqÞþVnn�Xa

1

2Ma

ðdxY�

er2aYe

�Pa

1

2Ma

ðdxY�

eraYe �ra

ð9:6Þ

is the potential energy operator for the motion of the heavy nuclei in theelectron cloud of the molecule. The assumption (9.3) about the molecularwavefunction is known as the first Born–Oppenheimer approximation.Consideration of just the first two terms in (9.6) gives what is known asthe second Born–Oppenheimer approximation, where the last two smallvibronic terms are omitted.3 In this second approximation, the nuclearwave equation becomes:

Xa

� 1

2Mar2

a þUeðqÞ" #

YnðqÞ ¼ WYnðqÞ ð9:7Þ

whereUe(q) isnowapurelymultiplicativepotential energy term.From(9.7)follows the possibility of defining a potential energy surface for the effectivemotion of the nuclei in the field provided by the nuclei themselves and themolecular electron charge distribution:

UeðqÞ � EeðqÞþVnn ¼ EðqÞ ð9:8Þ

Usually, we shall refer to (9.8) as the molecular energy in the Born–Oppenheimer approximation and refer to (9.7) as the Born–Oppenheimernuclear wave equation, which determines the motion of the nuclei (e.g. inmolecular vibrations) so familiar in spectroscopy.

3 These terms can be neglected, in a first approximation, since they are of the order of

1=Ma � 10�3.

THE BORN–OPPENHEIMER APPROXIMATION 143

The adiabatic approximation includes in Ue(q) the third term in (9.6),which describes the effect of the nuclear Laplacian r2

a on the electronicwavefunction Ye. Both small vibronic terms in (9.6) can be included in avariational or perturbative way in a refined calculation of the molecularenergy as a function of nuclear coordinates, and are responsible forinteresting fine structural effects in the vibrational spectroscopy of polya-tomic molecules (Jahn–Teller and Renner effects).

9.2 THE HYDROGEN MOLECULE H2

Electrons and nuclei in the H2 molecule are referred to the interatomiccoordinate system of Figure 9.1. At A and B are the two protons (chargeþ 1), a distanceR apart measured along the z-axis; at 1 and 2 are the twoelectrons (charge �1). The bottom part of the figure shows the overlap

x

1

2r12

1

r1r2

rA2 rB1

A B z

y

R

A

a(r) b(r)S

B

Figure 9.1 Interatomic coordinate system (top) and overlap S between spherical AOs(bottom)


S between two basic 1s STOs with orbital exponent c0 (¼1 for the freeatoms):

aðr1Þ ¼ c30p

!1=2

expð�c0r1Þ; bðr2Þ ¼ c30p

!1=2

expð�c0r2Þ ð9:9Þ

S ¼ hbjai ¼ ðabj1Þ ¼ expð�rÞ 1þ rþ 1

3r2

!r ¼ c0R ð9:10Þ

9.2.1 Molecular Orbital Theory

If sg and su are the normalized bonding and antibonding one-electronMOs:

sgðrÞ ¼ aðrÞþ bðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2S

p ; su ¼ bðrÞ�aðrÞffiffiffiffiffiffiffiffiffiffiffiffi2�2S

p ð9:11Þ

then the one-configuration two-electron MO wavefunction for the H2

ground state is

YðMO;1Sþg Þ ¼ jjsg�sgjj

¼ sgðr1Þsgðr2Þ 1ffiffiffi2

p aðs1Þbðs2Þ�bðs1Þaðs2Þ½ � S ¼ MS ¼ 0

ð9:12Þ

where the total spin quantum number S in (9.12) must not be confusedwith the overlap. In the molecular ground state, we allocate twoelectrons with opposite spin in the normalized spatial bonding MOsg(r).The MO energy in the Born–Oppenheimer approximation is

EðMO; 1 Sþg Þ ¼ hYðMO; 1Sþ

g ÞjHjYðMO; 1Sþg Þi

¼ sgsg h1 þ h2 þ 1

r12þ 1

R

��sgsg

* +¼ 2hsgsg

þðs2g js2

gÞþ1

R

ð9:13Þ

THE HYDROGEN MOLECULE H2 145

where

h ¼ � 1

2r2� 1

rA� 1

rBð9:14Þ

is the bare-nuclei one-electron Hamiltonian.In the hydrogenic approximation (c0 ¼ 1), we obtain for the matrix

elements

2hsgsg¼ haa þ hbb þhbaþ hab

1þ S

¼ 2EH þ ða2jVBÞþ ðb2jVAÞþ ðabjVBÞþ ðbajVAÞ1þ S

ð9:15Þ

ðs2g js2

gÞ ¼1

4ða2ja2Þþ ðb2jb2Þ� �þ 1

2ða2jb2Þþ ðabjabÞþ ða2jbaÞþ ðb2jabÞ

ð1þ SÞ2ð9:16Þ

where both one- and two-electron integrals are written in charge densitynotation:

ða2jVBÞ ¼ haj�r�1B jai; ðabjVBÞ ¼ hbj�r�1

B jai ð9:17Þ

ðabjabÞ ¼ð ð

dr1dr2½aðr2Þb�ðr2Þ�

r12aðr1Þb�ðr1Þ½ � ð9:18Þ

Because of the two-electron monocentric terms, ða2ja2Þ ¼ ðb2jb2Þ, thetwo-electron part (9.16) of the molecular energy is wrong when R!¥,since (c0 ¼ 1):

limR!¥

ðs2g js2

gÞ ¼1

2ða2ja2Þ ¼ 1

2� 5

8¼ 5

16¼ 0:3125Eh ð9:19Þ

instead of zero, as it must be, since the interaction must vanish at infiniteseparation.The interaction energy naturally decomposes into the following two

terms:

DE ¼ EðMO; 1Sþg Þ�2EH ¼ DEcb þDEexch-ovð1Sþ

g Þ ð9:20Þ


DEcb ¼ ða2jVBÞþ ðb2jVAÞþ ða2jb2Þþ 1

Rð9:21Þ

being the semiclassical Coulombic interaction energy and

DEexch-ovð1 Sþg Þ¼ ðab�Sa2jVBÞþ ðba�Sb2jVAÞ

1þ S

þ1

4ða2ja2Þþ ðb2jb2Þ� �� 1

2þ2Sþ S2

� �ða2jb2Þþ ðabjabÞþ ða2jabÞþ ðb2jbaÞ

ð1þ SÞ2

ð9:22Þthe exchange-overlap component of the interaction energy, a quantumcomponent depending on the electronic state of the molecule. The firstterm in (9.22) is the one-electron part, which has a correct behaviourwhen R!¥. In it appear the two exchange-overlap densities aðrÞbðrÞ�Sa2ðrÞ and bðrÞaðrÞ�Sb2ðrÞ (compare with Equation 4.17), having theproperty

Ðdr ½aðrÞbðrÞ�Sa2ðrÞ� ¼ 0;

Ðdr ½bðrÞaðrÞ�Sb2ðrÞ� ¼ 0 ð9:23Þ

The second term in (9.22) is the two-electron part, which is incorrectwhen R!¥, due to the presence of the one-centre integrals. Therefore,we have for the ground state MO wavefunction

limR!¥

DEðMO; 1Sþg Þ ¼ 1

2ða2ja2Þ ð9:24Þ

corresponding to the erroneous dissociation

H2ð1Sþg Þ!Hð2SÞþ 1

2H�ð1SÞ ð9:25Þ

instead of the correct one:

H2ð1Sþg Þ!2Hð2SÞ ð9:26Þ

The behaviour of the potential energy curve for the MO description ofground-state H2 is qualitatively depicted in the top part of Figure 9.2,which shows the asymptotically incorrect behaviour of the dissociation


energy. This large correlation error can be removed through CI of theground-state s2

g configuration with the doubly excited configuration s2u,

where both electrons occupy the su antibonding MO, leaving sg empty.The interconfigurational wavefunction

YðMO-CI; 1Sþg Þ ¼ N½Yðs2

g ;1Sþ

g Þþ lYðs2u;

1Sþg Þ� ð9:27Þ

with the variationally optimized mixing parameter l � �0:13 atRe ¼ 1:4a0, now correctly describes dissociation of the molecule intotwo neutral H atoms in their ground state.

9.2.2 Heitler–London Theory

Heitler and London (1927) proposed for H2 the two-configurationwavefunction

ΔΕ /10-3Εh

Σg+1

,

R/a0

R/a0

01.4

312.5 MO

Chemical Bond

HL

ΔΕΔΕ /10-6Ε h

Σu+3 ,

8

0 RScattering State

VdW Bond

Figure 9.2 MO and HL potential energy curves for 1Sþg ground state (top) and 3Sþ

uexcited state (bottom) of the H2 molecule


Y ðHL; 1Sþg Þ ¼ Nfjja�bjj�jj�abjjg

¼ aðr1Þbðr2Þþ bðr1Þaðr2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2S2

p 1ffiffiffi2

p aðs1Þbðs2Þ�bðs1Þaðs2Þ½ �

S ¼ MS ¼ 0 ð9:28Þgiving the HL energy in the Born–Oppenheimer approximation:

EðHL; 1 Sþg Þ ¼ hYðHL; 1Sþ

g Þj^HjYðHL; 1Sþg Þi

¼ abþ baffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2S2

p h1 þ h2 þ 1

r12þ 1

R

�� abþ baffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ 2S2p

* +

¼ haa þ hbb þ Sðhba þhabÞ1þ S2

þ ða2jb2Þþ ðabjabÞ1þ S2

þ 1

R

ð9:29ÞIt is seen that theHL two-electron component of themolecular energy is

much simpler than itsMOcounterpart and nowhas the correct behaviouras R!¥.In the hydrogenic approximation (c0 ¼ 1), the HL interaction energy

for 1Sþg H2 is

DEðHL; 1Sþg Þ ¼ EðHL; 1Sþ

g Þ�2EH ¼ DEcb þDEexch-ovð1Sþg Þ ð9:30Þ

where DEcb is the same as the MO expression (9.21), while

DEexch-ovð1Sþg Þ ¼ S

ðab�Sa2jVBÞþ ðba�Sb2jVAÞ1þ S2

þ ðabjabÞ�S2ða2jb2Þ1þ S2

ð9:31ÞBoth components of the interaction energy now vanish asR!¥, there-

fore describing correctly the dissociation of the ground-state H2 moleculeinto neutral ground-state H atoms (top part of Figure 9.2, HL curve).For theexcited stateof theH2molecule,wehave the tripletwavefunction

YðHL; 3Sþu Þ ¼

ka bk S ¼ 1; MS ¼ 11ffiffiffi2

p ½ka b k þ k a bk� S ¼ 1; MS ¼ 0

k a b k S ¼ 1; MS ¼ �1

8>>><>>>:

ð9:32Þ


or, by expanding the determinants:

YðHL; 3Sþu Þ¼ aðr1Þbðr2Þ�bðr1Þaðr2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�2S2p

aðs1Þaðs2Þ1ffiffiffi2

p aðs1Þbðs2Þþbðs1Þaðs2Þ½ �

bðs1Þbðs2Þ

8>>><>>>:

ð9:320Þ

The quantum mechanical exchange-overlap component of the inter-action energy for the triplet is now

DEexch-ovð3Sþu Þ ¼ �S

ðab�Sa2jVBÞþ ðba�Sb2jVAÞ1�S2

�ðabjabÞ�S2ða2jb2Þ1�S2

ð9:33Þ

and is repulsive for any value of R (a scattering state, left bottom part ofFigure 9.2). At the rather large internuclear distance of about 8a0 weobserve the formation of a rather flat van der Waals (VdW) minimum of�20� 10�6Eh, due to the prevalence of theweakLondonattractive forces(Chapter 11) over thisweak triplet repulsion.The appearance of thisweakVdWbond is depicted qualitatively in the right bottom part of Figure 9.2,which refers to large internuclear distances.4 It should be noted that,at variance with the ground state, YðMO; 3Sþ

u Þ now coincides withYðHL; 3Sþ

u Þ.

9.3 THE ORIGIN OF THE CHEMICAL BOND

The HL theory of the chemical bond in H2 can be considered as the firstapproximation including exchange in an RS perturbation expansion ofthe interatomic energy (see Chapters 10 and 11). In a VB calculation,variational optimization of the orbital exponent c0 of a simple STO basisis tantamount to including in first order a large part of the sphericaldistortion, so giving a fairly good description of the quantum exchange-overlap component of the interaction that is dominant in the bond region(Magnasco, 2007).

4 Note in the figure the change in the energy scale for chemical (top, 10�3Eh) and VdW (bottom,

10�6Eh) bonds, meaning that the latter are 1000 times smaller in their order of magnitude.


To investigate the origin of the chemical bond, optimized VB calcula-tions were done (Magnasco, 2008) for the ground states of Hþ

2 ð2Sþg Þ,

H2 ð1Sþg Þ and Heþ

2 ð2Sþu Þ, a series of simple molecules that are the

prototypes of one-, two- and three-electron s chemical bonds, and ofthe Pauli repulsion for theHe(1s)–He(1s) interaction.The bond energyDEis analysed into its polarized Coulombic DEcb and exchange-overlapDEexch-ov components, the Coulombic component being best obtainedby direct analytical calculation avoiding the round-off errors that wouldarise at the larger distances.A summary of the results5 is presented inTables 9.1–9.4. For all systems

studied, including the molecule-ions Hþ2 and Heþ

2 , the Coulombiccomponent (Table 9.2) has a charge-overlap nature rapidly decreasing

Table 9.1 DE bond energies (10�3Eh) for the ground state of simple homonucleardiatomics

R/a0 Hþ2 ð2Sþ

g Þ H2 ð1Sþg Þ Heþ

2 ð2Sþu Þ

1 59.001 6 �88.585 2 570.468 11.2 �15.921 6 �128.956 1 223.806 21.4 �55.339 8 �139.049 5 46.991 81.6 �75.319 2 �134.372 6 �39.864 21.8 �84.176 4 �122.673 4 �78.053 42 �86.506 0 �108.010 1 �89.992 03 �64.448 2 �41.699 3 �46.940 54 �37.334 5 �11.358 2 �13.819 35 �19.205 4 �2.534 91 �3.421 806 �9.080 4 �0.509 89 �0.783 36

Table 9.2 DEcb Coulombic component (10�3Eh) of the bond energies for the groundstate of simple homonuclear diatomics

R/a0 Hþ2 H2 Heþ

2

1 117.092 9 15.846 9 41.179 91.2 68.997 0 �9.928 7 �5.603 91.4 42.746 1 �19.423 1 �14.892 81.6 27.486 5 �21.831 1 �13.535 41.8 18.173 0 �21.076 3 �9.975 02 12.260 2 �18.993 9 �6.709 63 2.005 7 �7.016 22 �0.522 674 0.341 95 �1.677 58 �0.026 285 0.053 3 �0.329 08 �0.001 106 0.007 5 �0.060 06 �0.000 04

5 The optimized values of the orbital exponent c0 at the variousR are given inMagnasco (2008).

THE ORIGIN OF THE CHEMICAL BOND 151

with the internuclear distanceR as exp(�c0R) and being always repulsivefor Hþ

2 . This means that, classically, no bond can be formed betweena ground-stateH atom and a protonHþ . For the remaining systems,DEcb

is repulsive at small values and attractive at large values of R, but alwaysinsufficient to originate a sufficiently stable chemical bond. For allsystems, the greater part of the bond energy comes from the exchange-overlap contribution between spherically polarized charge distributions(Tables 9.3 and 9.4), which, in the bond region, is attractive for Hþ

2 , H2

and Heþ2 and repulsive for the He–He interaction.Hence, we conclude that, for all chemically bonded molecules, the

dominant component of the interaction in the bond region is theexchange-overlap component, which depends on the spin coupling ofthe interacting atoms; this is always attractivewhen forming the chemicalbond (Table 9.3) and is repulsivewhen bonding is forbidden (Table 9.4).So, the exchange-overlap component is at the origin of both the chemicalbond and Pauli repulsion. In an appendix given as an electronic addition

Table 9.3 DEexch-ov exchange-overlap component (10�3Eh) of the bond energies forthe ground state of simple homonuclear diatomics

R/a0 Hþ2 ð2Sþ

g Þ H2ð1Sþg Þ Heþ

2 ð2Sþu Þ

1 �58.091 3 �104.432 1 529.288 21.2 �84.918 6 �119.027 4 229.410 11.4 �98.085 9 �119.626 4 61.884 61.6 �102.805 7 �112.541 5 �26.328 81.8 �102.349 4 �101.597 1 �68.078 42 �98.766 2 �89 016 2 �83.282 43 �66.453 9 �34.683 1 �46.417 84 �37.676 4 �9.680 6 �13.793 05 �19.258 7 �2.205 8 �3.420 76 �9.087 9 �0.449 83 �0.783 32

Table 9.4 DE Pauli repulsion and its Coulombic and exchange-overlap components(10�3Eh) for the He(1s)–He(1s) interaction in the medium range of interatomicseparations

R/a0 DE DEcb DEexch-ov

2 136.616 �27.282 1 163.8982.5 42.669 3 �7.549 17 50.218 53 12.964 0 �1.930 70 14.894 73.5 3.799 79 �0.468 75 4.268 544 1.073 17 �0.109 52 1.182 694.5 0.292 81 �0 024 864 0.317 675 0.077 48 �0.005 522 0.083 00


to the paper quoted above (Magnasco, 2008), the origin of the quantummechanical exchange-overlap densities was studied in detail, showingtheir different behaviour in the case of the chemical bond in ground-stateH2 and of the Pauli repulsion between two ground-state He atoms. Thismade clear the relation between suchVB ‘exchange-overlap’ densities andthe MO density ‘interference’ terms introduced time ago by Ruedenberg(1962).The values at the minima of the potential energy calculations, not

shown in the tables, are �86:506� 10�3Eh at Re ¼ 2a0 with c0 ¼ 1:238for Hþ

2 ð2Sþg Þ, �139:079� 10�3Eh at Re ¼ 1:41a0 with c0 ¼ 1:165 for

H2 ð1Sþg Þ, �90:498� 10�3Eh at Re ¼ 2:06a0 with c0 ¼ 1:832 for

Heþ2 ð2Sþ

u Þ, while the He(1s)–He(1s) interaction is repulsive by12:964� 10�3Eh at R ¼ 3a0 with c0 ¼ 1:691. In this way, it is seen thatpractically correct Re bond lengths and about 85% of the experimentallyobserved or accurately calculated DEe bond energies (Peek, 1965; Huberand Herzberg, 1979 for H2

þ ; Wolniewicz, 1993 for H2; Cencek andRychlewski, 1995 for He2

þ ) are obtained from these optimized calcula-tions, while the calculated Pauli repulsion for He(1s)–He(1s) in themedium range at, say, R ¼ 3a0, gives over 95% of the correct repulsion(Liu andMcLean, 1973). It should be remarked that the optimized orbitalexponents c0 contractover the hydrogenic valueZ ¼ 1 forHþ

2 andH2andexpand over Z ¼ 2 for Heþ

2 and He2. This change in shapes of theoptimized AOs suggests that the guiding principle in the elementary VBtheory of the chemical bond is the maximum of the exchange-overlapenergy and not the maximum overlap, as discussed elsewhere (Magnascoand Costa, 2005).

9.4 VALENCE BOND THEORY AND THECHEMICAL BOND

This is amulti-determinant theorywhich originates from theHL theory ofthe H2 molecule just discussed above.

9.4.1 Schematization of Valence Bond Theory

Basis of (spatial) AOs) atomic spin-orbitals (ASOs)) antisymmetrizedproducts (APs), that is, Slater determinants of order equal to the numberNof the electrons in the molecule, eigenfunctions of the Sz operator witheigenvalue MS ¼ ðNa�NbÞ=2 ) VB structures, eigenstates of S

2with

VALENCE BOND THEORY AND THE CHEMICAL BOND 153

eigenvalue S(S þ 1) (defined S) ) symmetry combinations of VB struc-tures ) multi-determinant wavefunctions describing at the full-VB levelthe electronic states of the molecule.

9.4.2 Schematization of Molecular Orbital Theory

Basis of (spatial) AOs)MOs by the LCAOmethod, classified accordingto symmetry-adapted types (Chapter 12) ) single Slater determinant ofdoubly occupied MOs (for closed shells) ) MO-CI among all determi-nants belonging to the same symmetry ) multi-determinant wavefunc-tions describing at the MO-full-CI level the electronic states of themolecule.Starting from the same AO basis, full-VB andMO-full-CI methods are

entirely equivalent at the end of each process, but may be deeply differentin their early stages (as we have seen for the H2 molecule).

9.4.3 Advantages of the Valence Bond Method

1. VB structures (such as H�H, H�Hþ , HþH�, Li�H, Liþ�H�,N:N, C�:Oþ ) are related to the existence of chemical bondsin molecules, the most important ones corresponding to the rule ofthe so-called ‘perfect-pairing of bonds’ (see Figure 9.6 for H2O).

2. Principle of maximum overlap (better, minimum of exchange-over-lap bond energy)) bond stereochemistry in polyatomic molecules(Magnasco and Costa, 2005).

3. It allows for a correct description of the dissociation of the chemicalbonds (of crucial importance for studying chemical reactions).

4. It gives a sufficiently accurate description of the spin densities inradicals (see the example of the allyl radical in Section 9.6).

5. For small molecules, the ab initio formulation of the theory allowsone to account up to about 80% of the electronic correlation and toget bond distances within 0.02a0.

9.4.4 Disadvantages of the Valence Bond Method

1. Nonorthogonal basic AOs ) nonorthogonal VB structures )difficulties in the evaluation of the matrix elements of theHamiltonian.


2. The number of covalent VB structures of given total spin S increasesrapidly with the number n ¼ N=2 of bonds, according to theWigner’s (1959) formula:

fNS ¼ 2nn�S

� �� 2n

n�S�1

� �¼ ð2Sþ 1Þð2nÞ!

ðnþ Sþ1Þ!ðn�SÞ! ð9:34Þ

Table 9.5 gives a comparison between the order ofH€uckel and covalentVB secular equations for the singlet state of some polycyclic hydrocarbonswhose structure is reported in Figure 9.3. When N is increasing, there isa striking difference between the second and the last column of the table,giving the order of the respective secular equations.The situation is evenworse ifwe take into consideration ionic structures

as well. In this case, the total number of structures, covalent plus allpossible ionic, is given by Weyl’s formula (Mulder, 1966):

f ðN;m; SÞ ¼ 2Sþ 1

mþ 1

mþ 1N

2þ Sþ 1

0B@

1CA

mþ 1N

2�S

0B@

1CA ð9:35Þ

N = 6 N = 10

N = 14 N = 24

Figure 9.3 Structure of some polycyclic hydrocarbons. From top left: benzene,naphthalene, anthracene, coronene

Table 9.5 Comparison between the degree of H€uckel and covalent VB secularequations for the singlet state of some polycyclic hydrocarbons

Molecule N n ¼ N=2 f N0

Benzene 6 3 5Naphthalene 10 5 42Anthracene 14 7 429Coronene 24 12 208 012


whereN is the number of electrons,m the number of basic AOs and S thetotal spin. Recall the formula for the binomial coefficient:

nm

� �¼ n!

m!ðn�mÞ! n � m ð9:36Þ

Consider the example of the p-electron system of benzene:

(i) Single zeta (SZ) basis set

N ¼ 6; m ¼ 6; S ¼ 0 ) f ð6; 6;0Þ ¼ 175

Singlet VB structures (5 covalent þ 170 ionic).(ii) Double zeta (DZ) basis set

N ¼ 6; m ¼ 12; S ¼ 0 ) f ð6; 12; 0Þ ¼ 15 730

The total number of possible VB structures increases very rapidlywith the size of the basic AOs!

9.4.5 Construction of Valence Bond Structures

VB structures are specified by giving their parent, a Slater determinantwhere the spin-orbitals involved in the bond must have opposite spin.In brief notation, we shall represent the parent by indicating in parenth-eses only those valence orbitals that are bonded together, omitting allcore orbitals which, in a first approximation, are considered as not takingpart in the bond.TheVB structures (eigenstates of S

2belonging to the total

spin S) are then simply obtained from the parents by doing all possible spininterchanges between the bonded orbitals, with a minus sign introducedfor each interchange.

(i) LiH ð1Sþ Þ

Lið2 SÞ : 1s2Lijs Hð2SÞ : h

Covalent structure Ionic structuresLi--H

ðs�hÞLiþH� Li�Hþ

ðh�hÞ ðs�sÞ


where the parents are6

ðs�hÞ ¼ jj1sLi1�sLis�hjj; ðh�hÞ ¼ jj1sLi1�sLih�hjj; ðs�sÞ ¼ jj1sLi1�sLis�sjjð9:37Þ

the first denoting a covalent bond between Li and H, the secondthe ionic structure LiþH� (both electrons on H) and the last theionic structure Li�Hþ (both electrons on Li). The full-VB struc-ture denoting covalent–ionic resonance in LiH will be

YðLiH; 1 Sþ Þ ¼ c1c1þc2c2þc3c3

/ ½ðs�hÞ�ð�shÞ�þl1ðh�hÞþl2ðs�sÞð9:38Þ

where c1 is the (normalized) singlet covalent structure associatedwith the parent ðs�hÞ:

c1 ¼1ffiffiffi2

p ðs�hÞ�ð�shÞ� � ð9:39Þ

while the ionic parents ðh�hÞ and ðs�sÞ are already eigenstates of S2with eigenvalue S¼ 0:

c2 ¼ ðh�hÞ; c3 ¼ ðs�sÞ ð9:40Þ

The mixing coefficients in (9.38) are determined by the Ritzmethod by solving the appropriate (3�3) secular equation.Because of the strong difference in electronegativity between Liand H, it is reasonable to expect that l1 l2.

(ii) FH ð1Sþ Þ

Fð2 PÞ : 1s2F 2s2F 2pp4Fj2psF Hð2SÞ : h

Covalent structure Ionic structures

F--H

ðsF�hÞ

F�Hþ FþH�

ðsF�sFÞ ðh�hÞ

6 Parent determinants have all properties of complete Slater determinants.


where now (N¼ 10) the first parent stands for the (10� 10) Slaterdeterminant:

ðsF�hÞ ¼ jj1sF1�sF2sF2�sF2ppxF2p�pxF2ppyF2p�pyFj2psF

�hjj ð9:41Þ

the singlet covalent VB structure being

c1 ¼1ffiffiffi2

p ðsF�hÞ�ð�sFhÞ

� � ð9:42Þ

and so on.(iii) N2ð1Sþ

g Þ

NAð4 SÞ : 1s2NA2s2NA

jsAxAyA s ¼ 2ps ¼ 2pz N ¼ 7

NBð4SÞ : 1s2NB2s2NB

jsBxByB x ¼ 2ppx ¼ 2px N ¼ 7

If, in a first approximation, we neglect sp hybridization onto theN atoms, then we can describe as follows the formation of thethree N�N bonds in N2 (top row of Figure 9.4).

xx xx

σBσAA B

- -+z

+

xA xB

+

-

+

-BA z

σ bond π bondx

xx

- -C O

x 2Ox 0

C

z+ +

C O

y 0C y 2

O

(σcσoxoxoycyo)

empty doubly occC

(σcσoxcxoyoyo)

emptyO

doubly occ

+ + - -

Figure 9.4 s and px bonds inN2 (top), px and py bonds in CO from charge transferOto C (bottom)


Covalent s bond (directed along the internuclear z-axis):

ðsA�sBÞ�ð�sAsBÞ

Covalent px bond (perpendicular to the internuclear axis, directedalong x):

ðxA�xBÞ�ð�xAxBÞCovalent py bond (perpendicular to the internuclear axis, directedalong y):

ðyA�yBÞ�ð�yAyBÞ

Altogether these describe the covalent triple bond inN2 in terms ofthe parent:

ðsA�sBxA�xByA�yBÞ ð9:43Þ

The full singlet (S¼MS¼ 0) covalent VB structure of N2 is thengiven by the combination of the eight Slater determinants:

Yð1 Sþg Þ ¼ ð

ffiffiffi8

pÞ�1½ðsA�sBxA�xByA�yBÞ�ð�sAsBxA�xByA�yBÞ

�ðsA�sB�xAxByA�yBÞ�ðsA�sBxA�xB�yAyBÞþ ð�sAsB�xAxByA�yBÞþ ð�sAsBxA�xB�yAyBÞþ ðsA�sB�xAxB�yAyBÞ�ð�sAsB�xAxB�yAyBÞ�

ð9:44Þ

where orthogonality is assumed between the determinants, thefirst term (I) being a shorthand for the (normalized) complete(14� 14) Slater determinant:

ðIÞ ¼ jj1sA1�sA1sB1�sB 2sA2�sA2sB2�sBjsA�sBxA�xByA�yBjj ð9:45Þ

In (9.45), the first part on the left describes eight electrons infrozen-coreAOsand the last part the six valence electrons engagedin the triple bond.

(iv) CO ð1Sþ Þ

Cð3 PÞ : 1s2C2s2CjsCxCy0C or x0CyC N ¼ 6

Oð3PÞ : 1s2O2s2OjsOxOy2O or x2OyO N ¼ 8


TheCOmolecule is a heteropolarmolecule isoelectronic withN2.Since electron charge transfer can occur between the doublyoccupied AOs on oxygen towards the corresponding empty AOson carbon (bottom rowof Figure 9.4), themost probable structureof CO is C�:Oþ , which describes formation of an ionic triplebond as shown by the parent:

ðsC�sOxC�xOyC�yOÞ ð9:46Þ

(v) Heþ2 ð2Sþ

u Þ three-electron s bondWhile He2 is a VdWmolecule (see Chapter 11), Heþ

2 is a ratherstable diatomic molecule with De ¼ 90:78� 10�3Eh atRe ¼ 2:043a0 (Huber and Herzberg, 1979), which can be con-sidered the prototype of the three-electron s bond.With referenceto the top row of Figure 9.5, the single normalized doublet VBstructures (S ¼ MS ¼ 1

2) are

c1 ¼ N1½ða�abÞ�ðaa�bÞ� ¼ jja�abjj N1 ¼ ½6ð1�S2Þ��1=2 ð9:47Þ

c2 ¼ N2½ðb�baÞ�ðbb�aÞ� ¼ jjb�bajj N2 ¼ ½6ð1�S2Þ��1=2 ð9:48Þwhere both the second determinants in the square bracketsof either (9.47) or (9.48) are zero because of Pauli’s principle(two spin-orbitals are equal).

Figure 9.5 Three-electron s bond in Heþ2 (top) and three-electron p bonds in

ground-state O2 (bottom)


The normalized full-VB doublet wavefunction with the correctsymmetry for the 2Sþ

u ground state is

Yð2Sþu Þ¼ c1�c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ 2Sp ð9:49Þ

while that for the excited 2Sþg state7 is

Yð2Sþg Þ¼ c1þc2ffiffiffiffiffiffiffiffiffiffiffiffi

2�2Sp ð9:50Þ

Using elementary rules of determinants, it can be easily shownthat, in this case, the VB and MO wavefunctions coincide:

YðMO; 2Sþu Þ ¼ jjsg �sg sujj ¼ YðVB; 2Sþ

u Þ ð9:51Þ

YðMO; 2Sþg Þ ¼ jjsg su �sujj ¼ YðVB; 2Sþ

g Þ ð9:52Þ

Recent ab initio calculation (Magnasco, 2008) of the potentialenergy curve for ground-state Heþ2 with a minimal basis set ofoptimized 1s STOs gives De ¼ 90:49� 10�3Eh at Re ¼ 2:05a0for c0 ¼ 1:833, in excellent agreement with the spectroscopicallyobserved values given above. The excited 2Sþ

g state is alwaysrepulsive in the bond region, giving a scattering state.

(vi) O2 ð3S�g Þ three-electron p bonds

OAð3 PÞ : 1s2A2s2AjsAxAy2A or x2AyA N ¼ 8

OBð3PÞ : 1s2B2s2BjsBxBy2B or x2ByB N ¼ 8

The electronic structure of this 16-electron paramagnetic mole-cule depends entirely on the system of its p electrons (six electronsin four 2pp orbitals), while there is always one s bond between Aand B directed along the internuclear z-axis. The two possiblesinglet VB structures for the ground state would imply formationof a p bond and two parallel 2pp lone pairs containing fourelectrons, having the parents

ðxA�xB yA�yA yB�yBÞ , ðxA�xA xB�xB yA�yBÞ ð9:53Þ

7 The symmetries of ground and excited state in Heþ2 are exactly opposed to those observed

for Hþ2 .


The corresponding structures are highly improbable, however,because of the strong Pauli repulsion existing between coplanarlone pairs. The most probable VB structures are obtained bytwisting by 90 the two lone pairs (bottom row of Figure 9.5)so that Pauli repulsion is drastically reduced, leading to theparents

ðxAyA �yAxB �xByBÞ , ðxByB �yBxA �xAyAÞ ð9:54Þ

Full resonance between the corresponding two equivalent struc-tures describes now the formation of two highly stable three-electron p bonds (Wheland, 1937), yielding for the ground stateof O2 a triplet 3S�

g .

9.5 HYBRIDIZATION AND MOLECULARSTRUCTURE

9.5.1 The H2O Molecule

Let us consider now the formation of two equivalent O�H bonds in theH2O molecule making an interbond angle 2u of about 105. Hybridiza-tion, namely themixing of sp AOs onto the same nucleus, is now essentialin order to give straight bonds satisfying the principle of maximumoverlap. With reference to the top part of Figure 9.6, we see that the two2p AOs on oxygen directed along the bonds are no longer orthogonal toeach other:

hp1jp2i ¼ cos 2u ¼ �0:25882 ð9:55Þ

It is evident from (9.55) thatwe can have orthogonality only for 2u ¼ 90.We can restore orthogonality of directed orbitals on oxygen simply bymixing in some s AO, and a detailed calculation shows thatthe appropriately directed sp2 (C2v) hybrids on O are (top part ofFigure 9.6)

hy1 ¼ 0:4534sþ 0:5426zþ 0:7071y ¼ b1

hy2 ¼ 0:4534sþ 0:5426z�0:7071y ¼ b2

hy3 ¼ 0:7673s�0:6412z ¼ s

8><>: ð9:56Þ


giving 20.6% s and 79.4%p for the hybrids engaged in the two equivalentO�Hbonds and58.9%sand41.1%p for the hybrid directed awayon theback of the z-axis and forming the s lone pair.We can now consider the formation of covalent and ionic structures

involving such directed hybrids, limiting ourselves to consideration of thefour-electron problem, leaving inner shell and lone pair electrons frozen(only a first approximation). In an obvious brief notation:

Oð3 PÞ : 1s2O2s2Ox2OyOzO ) k2s2x2b1b2 Hð2SÞ : h1; h2 ð9:57ÞWeyl’s formula (N ¼ 4, m ¼ 4, S ¼ 0) gives in this case

f ð4;4; 0Þ ¼ 1

5

53

� �52

� �¼ 20 ð9:58Þ

Oy

σ

y

z

θ

b1b2

z

O

H1H2

(b1h1b2h2)

(b1h1b2b2)(b1b1b2h2)

ψ1

- -OO

HH H+ H+

ψ3 ψ4

Figure 9.6 The three sp2 hybrids of C2v symmetry (top) and the most importantcovalent and singly polar VB structures in H2O (bottom)

HYBRIDIZATION AND MOLECULAR STRUCTURE 163

so that there are altogether 20 singlet VB structures (2 covalent, 12 singlypolar, 6 doubly polar), themost important being those given in the bottompart of Figure 9.6. An approximate VB function describing correctly theobserved value of the dipole moment (m ¼ 0:73ea0) was estimated byCoulson (1961) as:

Yð1A1Þ ¼ c1c1 þ1ffiffiffi2

p ðc3 þc4Þc3þc15c15 ð9:59Þ

with c1 ¼ 0:64, c3 ¼ c4 ¼ 0:48 and c15 ¼ 0:36, where c15 is the doublypolar structurewith all four electronsonO ðb1�b1 b2 �b2Þ.Wavefunction (9.59)yields the following relative percentage weights for the VB structures:

41% c1; 23% c3; 23% c4; 13% c15 ð9:60Þ

9.5.2 Properties of Hybridization

. Hybrid AOs (see bA and bB in the bottom part of Figure 9.7) becomeunsymmetricalwith respect to their centres, acquiring what Coulson(1961) calls an atomic dipole.8

. Physically, hybridization describes polarization (distortion) of theAOs engaged in the bond. As such, it may involvemixing of occupiedand empty AOs.9

. Hybridization restores orthogonality of AOs on the same atom,allowing for interhybrid angles greater than 90. Hybrid AOs canin this way reorient themselves in an optimumway, yielding orbitalsdirected along the bonds and avoiding formation of weaker bentbonds.

. Hybridization gives in this way the AOs the appropriate directionalcharacter for forming strong covalent bonds and, therefore, is offundamental importance in stereochemistry.

. Evenwithout changes in valency (isovalent hybridization), hybridizationallows for a better disposition in space of electron lone pairs, projectingthem outside the bond region (as in the H2O molecule), which has aconsiderable effect on the electric dipole moment of the molecule.

8 A not directly observable quantity.9 TheAOsmayhavedifferent (as inH2) or identicalprincipal quantumnumbers (as inLiHorBe2,

where 2s and 2p AOs are nearly degenerate). In H2O, NH3 and CH4, 2s and 2p AOs are mixedunder theC2v,C3v andTd symmetry of the electric field provided by two, three, and four protons

respectively.


. For the carbon atom, promotion of an electron from 2s to 2pincreases its covalency from 2 to 4, giving the possibility of formingfour equivalent sp3 hybrids directed towards the vertices of a tetra-hedron,making an interhybrid angle of 109.5. The energy expendedfor promoting electrons is largely recovered by the formation of twoadditional strong single bonds.

. Hybridization increases overlap between the AOs forming the bond,yielding a stronger bond (more correctly, it increases the exchange-overlap component of the bond energy).

The interhybrid overlap between two sp hybrids bA and bB is shown inthe bottom part of Figure 9.7. Assuming positive all elementary overlapsbetween s and ps AOs, the overlap between two general sp hybridsmaking an angle 2u with respect to the bond direction z is

Sbb ¼ hbAjbBi¼ Sss cos

2vþðSss cos22uþ Spp sin22uÞsin2vþ Sss cos2u sin2v

ð9:61Þ

Figure 9.7 Overlap between two sp bond hybrids along the bond direction

HYBRIDIZATION AND MOLECULAR STRUCTURE 165

where v is the hybridization parameter for equivalent hybrids; namely:

bA ¼ sAcosvþ pAsinv; bB ¼ sBcosvþpBsinv ð9:62Þ

with

pA ¼ sAcos2uþ pAsin2u ð9:63Þ

In (9.63), sA is a 2p AO directed along the bond and pA is a 2p AOperpendicular to the bond direction.For hybrids along the bond, u ¼ 0, so that (9.61) becomes

Sbbðu ¼ 0Þ ¼ Ssscos2vþ Ssssin

2vþ Ssssin2v ð9:64Þ

For C¼C in ethylene at R ¼ 2:55a0, the elementary overlaps betweenSTOs with cs ¼ cp ¼ 1:625 are

Sss ¼ 0:4322; Sss ¼ 0:3262; Sss ¼ 0:4225 ð9:65Þ

In Table 9.6 we give the overlap between equivalent hybrids fordifferent values of the hybridization parameter v. It is seen from thetable that hybrid overlap is sensibly larger than each elementary overlap.Considering the largest overlap, Sss ¼ 0:4322, hybrid overlap is increasedby over 85.5% for sp, 75.8% for sp2 and 66.3% for sp3. These numbersshow the fantastic increase in overlap due to sp hybridization.

9.6 PAULING’S FORMULA FOR CONJUGATEDAND AROMATIC HYDROCARBONS

To obtain energy values in VB theory it is necessary to evaluate matrixelements between structures,whichmaynot be easybecause the structuresmay not be orthogonal and the usual Slater rules for orthonormaldeterminants are not valid.

Table 9.6 Overlap between sp bond hybrids along the bond direction

Hybrid spn cos w w/ Sbb

Digonal sp 1=ffiffiffi2

p45 0.8017

Trigonal sp2 1=ffiffiffi3

p54.7 0.7600

Tetrahedral sp3 1/2 60 0.7186


Pauling (1933) gave a simple pictorial formula for evaluating thematrixelements of the Hamiltonian for singlet covalent VB structures occurringin the theory of the p electrons of conjugated and aromatic hydrocarbons,which is of use in semiempirical calculations of the resonance energymuchin the same way as is H€uckel’s theory we studied in Chapter 7. Based onthe following assumptions:10

. orthonormality of the basic AOs

. consideration of singlet covalent structures only

. consideration of single interchanges between adjacent orbitals only

Pauling derived a simple formula for the matrix element of theHamiltonian between the VB structures cr and cs as

Hrs�ESrs ¼ 1

2n�iQ�Eþ

Xi;j

Kij� 1

2

Xi;j

Kij

!ð9:66Þ

where n is the number of p bonds, i the number of islands in the super-position pattern,Q ð< 0Þ the Coulomb integral andKij ð< 0Þ the integralof single exchange between the pair i and j. The first summation in roundbrackets is over all bonded orbitals in the same island, the second over allnonbonded orbitals in different islands. The superposition patterns areclosed polygons or islands, each formed by an even number of bonds,which are obtained by superposing the so-called Rumer diagrams describ-ing single covalent bonds between singly occupied AOs, as depictedin Figure 9.8 for ethylene (N ¼ 2), cyclobutadiene (N¼4), butadiene(N¼ 4), and the allyl radical (N¼ 3).The lastmolecule has anoddnumberof bonds, and is introduced just to show how Pauling’s formula can beapplied in this case.It must be remarked that in the figures the s skeleton is included just

to recall the complete chemical structure of each molecule, but Rumerdiagrams are concerned only with the p bonds (marked as double bondsin the figure). Above each superposition diagram is the correspondingcoefficient obtained from Equation 9.66.11

10 A detailed critique and partial justification of Pauling’s hypotheses, which are not very

different from those usually accepted for H€uckel’s theory, are fully discussed elsewhere(Magnasco, 2007).11 Because of molecular symmetry, all Kij integrals are equal and denoted by K.

PAULING’S FORMULA 167

As already done inH€uckel’s theory, in writing the VB secular equationsit is convenient to put

Q�E

K¼ �x ) E ¼ Qþ xK ð9:67Þ

where x is the p bond energy in units of parameter K and x > 0 meansbonding.Paralleling what we have done for H€uckel’s theory in Chapter 7, we

shall nowapply Pauling’s formula to ethylene, cyclobutadiene, butadiene,the allyl radical and benzene.

Figure 9.8 VB canonical structures and Pauling superposition patterns for ethylene,cyclobutadiene, butadiene and the allyl radical


9.6.1 Ethylene (One p-Bond, n ¼ 1)

We have n ¼ i ¼ 1, so that the (1� 1) secular equation is

Hcc�EScc ¼ Q�EþK ¼ 0

j�xþ 1j ¼ 0 ) x ¼ 1 ) E ¼ QþK

�ð9:68Þ

which is Pauling’s energy of the double bond.

9.6.2 Cyclobutadiene (n ¼ 2)

With reference to the second row of Figure 9.8, we can write for the twofully equivalent Kekul�e structures12 c1 and c2

Y ¼ c1c1 þc2c2 ð9:69Þ

The distinct matrix elements are

H11�ES11 ¼ H22�ES22 ¼ Q�Eþ2K�K ¼ Q�EþK

H12�ES12 ¼ H21�ES21 ¼ 1

2ðQ�Eþ 4KÞ ¼ 1

2ðQ�EÞþ 2K

8><>: ð9:70Þ

giving the (2� 2) secular equation

�xþ 1 � x

2þ 2

� x

2þ 2 �xþ 1

��

��¼ 0 ð9:71Þ

ð�xþ 1Þ2 ¼ � x

2þ 2

2) x2�4 ¼ 0 ) x ¼ �2 ð9:72Þ

12 Pauling VB theory distinguishes between Kekul�e structures (all p bonds equal) and Dewar

structures (containing one or more long, i.e. weaker, p bonds).


Taking the positive root, we have the following for the p energy ofcyclobutadiene:

1 Kekul�e : E ¼ QþK

2 Kekul�e : E ¼ Qþ 2K

�ð9:73Þ

In Pauling’s theory, the resonance energy is defined as the energylowering resulting from the difference between the lowest root of thesecular equation and the energy associated with the most stable singlestructure. In this case, denoting by K the Kekul�e structure:

DE ¼ Eð2KÞ�EðKÞ ¼ K < 0 ð9:74Þ

In this way, resonance between the two equivalent Kekul�e structuresstabilizes the system and the p energy decreases. This result is at variancefrom the corresponding H€uckel result, where p delocalization energy wasfound to be zero for cyclobutadiene.The calculation of the coefficients of the resonantVB functions in (9.69)

proceeds as usual from the homogeneous system and the normalizationcondition:13

ð�xþ1Þc1 þ � x

2þ 2

!c2þ 0

c21 þ c22 ¼ 1

8><>: ð9:75Þ

c2 ¼ x�1

2� x2

c1 ¼ c1 for x ¼ 2 ð9:76Þ

so that we have

c1 ¼ c2 ¼ 1ffiffiffi2

p ð9:77Þ

13 For the sake of simplicity, unless differently stated,we assumeorthogonality for the structures,

not only for the AOs. This gives no problem at this elementary level.


giving

50% c1; 50% c2 ð9:78Þ

as relative weights of the two Kekul�e structures in square cyclobutadiene,as it must be for completely resonant structures.

9.6.3 Butadiene (Open Chain, n ¼ 2)

Butadiene, the open chain with 2n ¼ 4 (two p bonds), has the canonicalstructures and the superposition patterns depicted in the third and fourthrows of Figure 9.8. c1 is the Kekul�e structure (two equivalent shortp bonds) and c2 is the Dewar structure (one long p bond). The completeVB wavefunction will be

Y ¼ c1 c1 þc2 c2 ð9:79Þ

where the coefficients must be determined by the Ritz method.The matrix elements are

H11�ES11 ¼ Q�Eþ 3

2K

H22�ES22 ¼ Q�E

H12�ES12 ¼ H21�ES21 ¼ 1

2ðQ�Eþ 3KÞ ¼ 1

2ðQ�EÞþ 3

2K

8>>>>><>>>>>:

ð9:80Þ

The (2� 2) secular equation is

�xþ 3

2� x

2þ 3

2

� x

2þ 3

2�x

��

��¼ 0 ð9:81Þ

�x �xþ 3

2

� �¼ � x

2þ 3

2

� �2

) x2�3 ¼ 0 ) x ¼ �ffiffiffi3

pð9:82Þ


Taking the positive root, we have the following for the p energy ofbutadiene:

1 Kekul�e : E ¼ Qþ 3

2K

1 Kekul�eþ1 Dewar : E ¼ Qþ ffiffiffi3

pK

8><>: ð9:83Þ

so that we obtain

DE ¼ EðKþDÞ�EðKÞ ¼ ð1:73�1:5ÞK ¼ 0:23K < 0 ð9:84Þ

for the conjugation energy.14

Calculation of the coefficients in butadiene proceeds through solutionof the system:

�xþ 3

2

!c1þ � x

2þ 3

2

!c2 þ 0

c21 þ c22 ¼ 1

8><>: ð9:85Þ

c2 ¼x� 3

232� x

2

c1 ¼ 0:362c1 for x ¼ffiffiffi3

p� 1:73 ð9:86Þ

Therefore, we obtain the following for the ‘resonance’ between Kekul�eand Dewar structures in butadiene:

Y ¼ c1ðc1þ 0:362 c2Þ ¼ 0:94 c1 þ 0:34 c2 ð9:87Þ

giving

88% c1; 12% c2 ð9:88Þ

which shows the greater importance of the Kekul�e structure versus theDewar structure.15

14 Since the structures are different, it is appropriate to speakof conjugation and not of resonanceenergy.15 Always, Dewar structures, having long bonds, are sensibly less important than Kekul�estructures, but their importance may increase with their number (e.g. naphthalene and

anthracene).


9.6.4 The Allyl Radical (N ¼ 3)

This case (last row of Figure 9.8) is an interesting example of how to applyPauling’s rules to an odd-electron system (N¼ 3, S ¼ 1

2). We add a

phantom atom (say d), treat the system as a four-atom system and, atthe end, remove the contribution of the phantom atom from the calcula-tion. The covalent VB structures and the corresponding superpositionpatterns are then the same as those of butadiene. Hence, with reference tothe previous calculation, we can write

H11�ES11 ¼ Q�EþK� 1

2KþK ¼ Q�Eþ 1

2K

H22�ES22 ¼ Q�EþK� 1

2K� 1

2K ¼ Q�Eþ 1

2K

H12�ES12 ¼ H21�ES21 ¼ 1

2ðQ�EþKþKþKÞ ¼ 1

2ðQ�EÞþK

8>>>>>>>>>><>>>>>>>>>>:

ð9:89Þ

where we have bolded the contributions which must be removed.We then obtain the secular equation for the allyl radical:

�xþ 1

2� x

2þ 1

� x

2þ 1 �xþ 1

2

��

��¼ 0 ð9:90Þ

�xþ 1

2

� �2

¼ � x

2þ 1

2) 3

4x2� 3

4¼ 0 ) x ¼ �1 ð9:91Þ

Taking the positive root, we have the following for the p energy of theradical:

1 Kekul�e : E ¼ Qþ 1

2K

1 Kekul�eþ 1 Dewar : E ¼ QþK

8<: ð9:92Þ


so that we obtain

DE ¼ EðKþDÞ�EðKÞ ¼ 1

2K < 0 ð9:93Þ

for the conjugation energy. The conjugation energy in the allyl radical ishence larger than that of butadiene (nearly twice).Calculation of the coefficients in the allyl radical proceeds through

solution of the system

��xþ 1

2

�c1 þ � x

2þ1

!c2þ 0

c21 þ c22 ¼ 1

8>><>>: ð9:94Þ

c2 ¼ x� 12

1� x2

c1 ¼ c1 for x ¼ 1 ð9:95Þ

giving


p ð9:96Þ

if we neglect nonorthogonality between the structures.Therefore, the relative weights of the two structures c1 and c2 are

50% c1; 50% c2 ð9:97Þ

as it must be for two truly resonant structures (c1 and c2 are fullyequivalent).By writing the complete VB wavefunctions for the two resonant

structures, we can immediately obtain electron and spin density distribu-tions in the allyl radical, even without doing any effective calculationof the energy. Assuming orthonormal spin-orbitals, with reference toFigure 9.8, the complete form of the two structures in terms of theirparents is

c1 ¼1ffiffiffi2

p ða�bcÞ�ð�abcÞ� � ð9:98Þ


c2 ¼1ffiffiffi2

p ða�bcÞ�ðab�cÞ� � ð9:99Þ

The structures have the same parent but differentwavefunctions. Evenif the spin-orbitals are assumed orthonormal, the structures are non-orthogonal (they have equal the first determinant):

S12 ¼ hc1jc2i ¼1

2ð9:100Þ

Then, from

Y ¼ c1c1 þc2c2 ð9:101Þ

it is immediately obtained that


p ð9:102Þ

The apparently strange normalization of the wavefunction (9.101) isdue to the nonorthogonality of c1 and c2, as can be easily checked:

hYjYi ¼ c21 þ c22 þ 2c1c2S12 ¼ 1

3þ 1

3þ 2� 1

3� 1

2¼ 1 ð9:103Þ

Direct calculation of the a- and b-components of the one-electrondensity gives

ra1 ¼ coefficient of aa� in r1

¼ a21

2c21þ c22þ c1c2

!þ b2

1

2c21 þ

1

2c22

!þ c2 c21 þ

1

2c22 þ c1c2

!

ð9:104Þ

rb1 ¼ coefficient of bb� in r1

¼ a21

2c21

!þb2

1

2c21 þ

1

2c22þ c1c2

!þ c2

1

2c22

! ð9:105Þ


Therefore, we have

PðrÞ ¼ ra1ðrÞþ rb1ðrÞ¼ ðc21þ c22 þ c1c2Þða2 þb2 þ c2Þ ¼ a2 þ b2 þ c2

ð9:106Þ

QðrÞ ¼ ra1ðrÞ�rb1ðrÞ

¼ a2ðc21 þ c1c2Þ�b2ðc1c2Þþ c2ðc21þ c1c2Þ ¼ 2

3a2� 1

3b2 þ 2

3c2

ð9:107ÞSo, while the p electron density (9.106) is uniform (one electron onto

each carbon atom, as it must be for an alternant hydrocarbon), VB theorypredicts that an excess of a spin at the terminal atoms induces some b

spin at the central atom, in accord with experimental ESR results. This isan interesting example of electron and spin population analysis inmultideterminant wavefunctions.

9.6.5 Benzene (n ¼ 3)

The five canonical structures (2 Kekul�e þ 3 Dewar) and their distinctsuperposition patterns for the six p electrons of benzene are given inFigure 9.9. The matrix elements are

H11�ES11 ¼ H22�ES22 ¼ Q�Eþ 3

2K

H12�ES12 ¼ 1

4ðQ�EÞþ 3

2K

8>>><>>>:

ð9:108Þ

H33�ES33 ¼ H44�ES44 ¼ H55�ES55 ¼ Q�E

H34�ES34 ¼ H35�ES35 ¼ H45�ES45 ¼ 1

4ðQ�EÞþ 3

2K

8<: ð9:109Þ

H13�ES13 ¼H14�ES14 ¼ H15�ES15

¼H23�ES23 ¼ H24�ES24 ¼ H25�ES25

¼ 1

2ðQ�EÞþ 3

2K

ð9:110Þ

Wenow examine the solution of the secular equations in three differentcases.


. Resonance between the two Kekul�e structures

Y ¼ c1c1þc2c2 ð9:111Þ�xþ 3

2� x

4þ 3

2

� x

4þ 3

2�xþ 3

2

��

��¼ 0 ð9:112Þ

5x2�12x ¼ xð5x�12Þ ¼ 0 ) x ¼ 0; x ¼ 12

5ð9:113Þ

c

b

a

d f

eψ1 ψ2

ψ5ψ3 ψ4

-1/2

-1/2

+1 +1 +1

1+1+1+

1+1+2/1- +1

ψ1+ψ1 ψ1+ψ2

i 3 i 1i=3 i=1

-1/2 1+2/1-2/1-

+1 +1

+1 +1

+1 +1 +1 +10 0 0

2/1-2/1- 1+2/1- 1+1+

ψ3+ψ3 ψ3+ψ4 ψ1+ψ3

i=1i=3 i=2i=1i=3 i=2

Figure 9.9 VB canonical structures and Pauling superposition patterns for benzene


Taking the lowest root, we obtain for the ground state

E ¼ Qþ 12

5K ¼ Qþ2:4K ð9:114Þ

giving as resonance energy for the two Kekul�e structures

DE ¼ Eð2KÞ�EðKÞ ¼ 12

5� 3

2

� �K ¼ 9

10K < 0 ð9:115Þ

. Resonance between the three Dewar structures

Y ¼ c3c3 þc4c4 þc5c5 ð9:116Þ

�x � x

4þ 3

2� x

4þ 3

2

� x

4þ 3

2�x � x

4þ 3

2

� x

4þ 3

2� x

4þ 3

2�x

��

��

¼ 0 ð9:117Þ

Expanding the determinant gives the cubic equation in x:

x3 þ 2x2�4x�8 ¼ ðxþ 2Þ2ðx�2Þ ¼ 0 ) x ¼ �2 ðtwiceÞ; x ¼ 2

ð9:118Þ

The lowest root is

E ¼ Qþ 2K ð9:119Þ

giving as resonance energy between the three Dewar structures

DE ¼ Eð3DÞ�EðKÞ ¼ 2� 3

2

� �K ¼ 1

2K < 0 ð9:120Þ

The resonance between the three Dewar structures, each having a longbond, is therefore sensibly less than the resonance between the twoKekul�e structures (0.5 instead of 0.9).


. Resonance between all VB structuresThe complete VB problem, arising from the mixing of all five VBstructures, is rather tedious since it would involve solution of a fifth-order determinantal equation. We can, however, simplify the problemusing symmetry arguments, if we are only interested in the ground-stateenergy of the system, as we are. In fact, symmetry suggests that

Y ¼ ðc1 þc2ÞcK þ ðc3 þc4þc5ÞcD ¼ YKcK þYDcD ð9:121Þ

where YK and YD are the un-normalized combinations of equivalentKekul�e and Dewar structures respectively. In this way, we reduce thefull-VB problem to the solution of a simple quadratic secular equation.We have

HKK�ESKK HKD�ESKD

HKD�ESKD HDD�ESDD

�� ¼ 0 ð9:122Þ

where, from (9.108)–(9.110):

HKK�ESKK ¼ 5

2ðQ�EÞþ 6K; HDD�ESDD ¼ 9

2ðQ�EÞþ9K

HKD�ESKD ¼ 3ðQ�EÞþ 9K

8<:

ð9:123Þ

Therefore, the (2�2) secular equation is

� 5

2xþ 6 �3xþ 9

�3xþ 9 � 9

2xþ 9

��

��¼ 0 ð9:124Þ

giving upon expansion

x2þ 2x�12 ¼ 0 ) x ¼ �1þffiffiffiffiffiffi13

p; x ¼ �1�

ffiffiffiffiffiffi13

pð9:125Þ

Taking the lowest root, we obtain the strongly bonding ground state

E ¼ Qþ 2:6055K ð9:126Þ


with the resonance energy

DE ¼ Eð2Kþ 3DÞ�EðKÞ ¼ ð2:6055�1:5ÞK ¼ 1:1K ð9:127Þ

which is the largest seen so far for benzene.For the coefficients of the resonant structures, we have from the

homogeneous system

cD ¼52 x�6

�3xþ 9cK ¼ 0:4321cK for x ¼ 2:6055 ð9:128Þ

Assuming orthogonality between the structures, we obtain thenormalization factor for the Y in (9.121):

hYjYi ¼ 2c2K þ 3c2D ¼ 1 ) N ¼ ð2c2K þ 3c2DÞ�1=2 ð9:129Þ

Figure 9.10 Percentage relativeweights ofVB canonical structures in conjugated andaromatic hydrocarbons


cK ¼ 0:6243; cD ¼ 0:2710 ð9:130Þ

finally giving the following as relative weights of the covalentsinglet structures in benzene:

38:9% c1; 38:9% c2; 7:4% c3; 7:4% c4; 7:4% c5

ð9:131Þ

So, the contribution of the Dewar structures, even if individuallysmall, is quite important on the whole. The results (9.131) are insurprising agreement with recent ab initio VB calculations byCooper et al. (1986), which give for these weights 40.3% forc1,2 and 6.5% for c3–5.For the sake of comparison, the relative weights of the structures

resulting from all preceding VB calculations are collected inFigure 9.10.


10Elements of Rayleigh–

Schroedinger Perturbation

Theory

In this chapter we present a few elements of RS stationary perturbationtheory, where a stationary system acted upon by an external perturbationwill change by a small amount its energy levels and eigenstates. It consistsessentially in relating the actual eigenvalue problem to be solved to one forwhich a complete solution is exactly known, and in treating the differencebetween the two Hamiltonian operators as a small perturbation.

10.1 RAYLEIGH–SCHROEDINGER PERTURBATIONEQUATIONS UP TO THIRD ORDER

We want to solve the Schroedinger eigenvalue equation

ðH�EÞc ¼ 0 ð10:1Þfor a Hermitian decomposition of the Hamiltonian H into:

H ¼ H0 þ lH1 ð10:2Þwhere (i) l is a parameter giving the orders in perturbation theory,1 (ii) H0

is the unperturbed Hamiltonian, namely the Hamiltonian of an already

1 Orders are here carried by the suffixes or their sum.


Valerio Magnasco


solved problem (either physical or model) and (iii) H1 is the small first-order difference between H and H0, called the perturbation.We now expand both eigenvalue E and eigenfunction c into powers

of l:

E ¼ E0 þ lE1þ l2E2 þ l3E3 þ � � � ð10:3Þc ¼ c0 þ lc1 þ � � � ð10:4Þ

where the coefficients of the different powers of l are, respectively, thecorrections of the various orders to energy and wavefunction (e.g. E2 isthe second-order energy correction, c1 the first-order correction to thewavefunction, and so on). It is often useful to define corrections up to agiven order, which we write, for example, as

Eð3Þ ¼ E0þE1 þE2 þE3 ð10:5Þ

meaning that we add corrections up to the third order.By substituting the expansions into the Schroedinger Equation 10.1:

½ðH0 �E0Þþ lðH1 �E1Þ�l2E2 � l3E3� � � � �ðc0þ lc1þ l2c2 þ � � � Þ ¼ 0

ð10:6Þ

and separating orders, we obtain

l0 ðH0�E0Þc0 ¼ 0

l ðH0�E0Þc1 þðH1 �E1Þc0 ¼ 0

l2 ðH0�E0Þc2 þðH1 �E1Þc1 �E2c0 ¼ 0

� � �

8>>>><>>>>:

ð10:7Þ

which are known as RS perturbation equations of the various ordersspecified by the power of l.Because of theHermitian property of H0, bracketing Equations 10.7 on

the left by hc0j, all the first terms in the RS equations are zero, and we areleft with

l0 hc0jH0 �E0jc0i ¼ 0

l hc0jH1 �E1jc0i ¼ 0

l2 hc0jH1 �E1jc1i�E2hc0jc0i ¼ 0

� � �

8>>><>>>:

ð10:8Þ

184 RAYLEIGH–SCHROEDINGER PERTURBATION THEORY

Taking c0 normalized to 1, we obtain the RS energy corrections of thevarious orders as

l0 E0 ¼ hc0jH0jc0il E1 ¼ hc0jH1jc0il2 E2 ¼ hc0jH1 �E1jc1i ¼ �hc1jH0 �E0jc1il3 E3 ¼ hc0jH1 �E1jc2i�E2hc0jc1i ¼ hc1jH1 �E1jc1i

� � �

8>>>>>>><>>>>>>>:

ð10:9Þ

E0 and E1 are nothing but the average values of H0 and H1 respectivelyover the unperturbed functionc0,whileE2 is given as a nondiagonal term,often referred to as transition integral, connecting c0 to c1 through theoperator H1, the last expression of E2 in (10.9) showing that alwaysE2< 0. The equations above show that knowledge of c1 (the solution ofthe first-order RS differential equation) determines the energy correctionsup to third order.2 In solving the first-order RS differential equation, weimpose on c1 the orthogonality condition

hc0jc1i ¼ 0 ð10:10Þ

which follows in first order from the normalization condition on the totalwavefunction:

hcjci ¼ 1 ð10:11Þ

and the fact that we assume a normalized c0:

hc0jc0i ¼ 1 ð10:12Þ

We recall that all cn ðn 6¼ 0Þ corrections are not normalized (even ifthey are normalizable).Before proceeding any further, an explanation is due to the reader of the

symmetric form resulting for E3 in (10.9). In fact, the RS third-orderdifferential equation determining c3:

l3ðH0 �E0Þc3þðH1�E1Þc2 �E2c1 �E3c0 ¼ 0 ð10:13Þ

2 In general, cn determines E up to E2nþ 1.

RAYLEIGH–SCHROEDINGER PERTURBATION 185

gives the third-order energy correction in the form

l3E3 ¼ hc0jH1 �E1jc2i�E2hc0jc1i ð10:14Þ

It will now be shown that it is possible to shift the order from theoperator to the wavefunction, and vice versa,3 if we make repeated use ofthe RS perturbation equations given in (10.7) and take into account thefact that the operators are Hermitian. In fact, we can write

E3 ¼ hðH1�E1Þc0jc2i�E2hc0jc1i ð10:15Þ

and, using the complex conjugate of the first-order equation (the bra):

E3 ¼ �hðH0 �E0Þc1jc2i�E2hc0jc1i¼ �hc1jH0 �E0jc2i�E2hc0jc1i

ð10:16Þ

In this equation, the order has been shifted from the operator to thewavefunction. If we now make use of the second-order RS differentialequation, the last term above can be written

E3 ¼ hc1jH1 �E1jc1i�E2½hc0jc1iþ hc1jc0i� ð10:17Þ

Since the term in square brackets is identically zero, the last ofEquations 10.9 is recovered. The same can be done for E2.Finally, it must be emphasized that the leading term of the RS perturba-

tion equations (10.7), the zeroth-order equation ðH0�E0Þc0 ¼ 0,must besatisfied exactly, otherwise uncontrollable errors will affect the wholechain of equations. Furthermore, it must be observed that only energy infirst order gives an upper bound to the true energy of the ground state, sothat the energy in second order, E(2), may be below the true value.4

10.2 FIRST-ORDER THEORY

First-order RS theory is useful, for instance, in explaining the Zeemaneffect and the splitting of the multiplet structure in atoms, or in giving the

3 This is known as the Dalgarno interchange theorem.4 This is particularly true when the correct value of E2 is determined.


Coulombic component of the interaction energy between atoms andmolecules.An interesting case arises when the zeroth-order energy level E0 is

N-fold degenerate. In this case, it can be shown that the first-order energycorrections are given by the solutions of the first-order determinantalequation of degree N:

jH1�E11j ¼ 0 ð10:18ÞwhereH1 is the representativeof theperturbationH1 over theorthonormalset fc0

1;c02; . . . ;c

0Ng belonging to the degenerate eigenvalue E0. A striking

example of this first-order degenerate perturbation theory is offered by theH€uckel theory of chain hydrocarbons. In this case, the determinantalequation (10.18) is nothing but the H€uckel determinant DN:

DN ¼�x 1 0 � � �1 �x 1 � � �� 0 1 �x

��

��¼ 0 ð10:19Þ

whose solutions were studied in Chapter 7.

10.3 SECOND-ORDER THEORY

Apart from first-order theory, essential to any practical use of perturba-tion theory is the possibility of solving, at least, the first-order RSdifferential equation. This can be done exactly in some cases, as for theH atom in a uniform electric field.The problem is best understood in its physical terms if we use the simple

model described in detail elsewhere (Magnasco, 2004b). The H atom,spherical in its ground state, distorts under the action of a uniformexternal fieldF (Fx ¼ Fy ¼ 0,Fz ¼ F) acquiring an induceddipolemomentproportional to the field F:

mi ¼ aF ð10:20Þ

where a is a second-order electric property of the atom, called the(dipole) polarizability.5 We define a (Figure 10.1) in terms of the (dipole)

5 The polarizability is the physical quantity measuring the distortion of the atom (or molecule)

under the action of the field; it has the dimensions of a volume, its atomic units being a30.

SECOND-ORDER THEORY 187

transition moment mi from state c0 to c1 and its related excitation energy«i ð> 0Þ as

a ¼ 2m2i

«ið10:21Þ

In the uniform external field F, the atom acquires the potential energy

H1 ¼ �Fz ð10:22Þwhichwill be taken as the first-order perturbation. l ¼ F is nowa physicalperturbation parameter which can be acted upon from the laboratory(e.g. by swithching on/off a plane condenser).A simple Ritz variational calculation involving c0 and c1, both states

being assumed normalized and orthogonal to each other, shows that theenergy of the atom in the field is lowered by the amount

DE ¼ � jhc0jH1jc1ij2«i

¼ �F2 ðc0c1jzÞ2«i

¼ �F2 m2i

«i¼ � 1

2

2m2i

«i

!F2 ¼ �1

2aF2

ð10:23Þ

From this relation follows that a can be obtained as the negative ofthe second derivative of the energy with respect to the field F evaluatedat F ¼ 0:

a ¼ � d2DEdF2

!F¼0

ð10:24Þ

Turning to our perturbation theory, we take

H ¼ H0� Fz; H0 ¼ � 1

2r2� 1

rð10:25Þ

ψ1 = ai

Η1

∧ εi, μi

α

ψ0 = a0

Figure 10.1 Dipole transition between c0 and c1 for ground-state H


where z ¼ r cos u. Then, the first-order energy correction is zero bysymmetry:

E1 ¼ hc0j�Fzjc0i ¼ �Fhc0jzjc0i ¼ 0 ð10:26Þ

since z is odd and the spherical density {c0}2 even. In this case, the first

physically relevant correction to the energy occurs in second order ofperturbation theory. It can be shown (seeMagnasco (2007)) that the exactfirst-order dipole correction to the unperturbed wavefunction c0 is

c1 ¼ F rþ r2

2

� �c0 cos u ð10:27Þ

where

c0 ¼ 1ffiffiffip

p expð�rÞ ð10:28Þ

is the unperturbed wavefunction for the ground state of the H atom. Thestudent can verify this solution taking into account that

r2 ¼ r2r�

L2

r2; r2

r ¼d2

dr2þ 2

r

d

dr; L

2cos u ¼ 2cos u; E0 ¼ � 1

2

ð10:29Þ

The exact second-order correction to the energy is then given by

E2 ¼ hc0jH1jc1i ¼ hc0j�Fzjc1i

¼ �F2 c0 z rþ r2

2

!cos u

��c0

* +¼ �F2 9

4

ð10:30Þ

where the integral is easily evaluated in spherical coordinates (x ¼ cos u):

c0 r2 þ r3

2

!cos2u

��c0

* +¼ 2p

1

p

ð1�1

dx x2ð10

dr r2 r2þ r3

2

!expð�2rÞ

¼ 4

3

ð10

dr r4 þ r5

2

!expð�2rÞ ¼ 4

3

4� 3� 2

25þ 1

2

5� 4� 3� 2

26

0@

1A ¼ 9

4

ð10:31Þ

SECOND-ORDER THEORY 189

The exact dipole polarizability (atomic units) of ground-state H is thenobtained as

a ¼ � d2E2

dF2

!0

¼ 9

2¼ 4:5a30 ð10:32Þ

10.4 APPROXIMATE E2 CALCULATIONS:THE HYLLERAAS FUNCTIONAL

When the first-order RS differential equation cannot be solved exactly,use must be made of the second-order Hylleraas functional:

~E2½~c1� ¼ h~c1jH0�E0j~c1iþ h~c1jH1�E1jc0iþ hc0jH1�E1j~c1i ð10:33Þ

This gives the exact E2 for the exact c1, as can be easily verified by directsubstitution in (10.33). In fact:

~E2½c1� ¼ hc1jðH0�E0Þc1 þðH1 �E1Þc0iþ hc0jH1�E1jc1i¼ hc0jH1�E1jc1i ¼ E2

ð10:34Þ

since the first term on the right-hand side vanishes, because the ket isidentically zero when c1 is the exact solution of the first-order equation.For approximate c1, say ~c1, it can be shown that we obtain upper

bounds to the true E2:

~E2½~c1� ¼ ~E2 � E2 ð10:35Þ

and that the variational ~E2 is affected by a second-order error.6 In thisway, we can easily construct variational approximations to second-orderenergies and first-order wavefunctions. Much as we did for the totalenergy in Chapter 4, we introduce variational (linear or nonlinear)parameters into ~c1 andminimize theHylleraas functional ~E2 with respectto them, obtaining in this way the best variational approximation to ~E2

and ~c1.

6 The wavefunction is instead affected by a first-order (i.e. larger) error. As in all variational

approximations, energy is determined better than the wavefunction.


10.5 LINEAR PSEUDOSTATES AND MOLECULARPROPERTIES

A convenient way to proceed is to apply the Ritz method to ~E2. We startfrom a convenient set of basis functions x written as the (1�N) rowvector:

x ¼ ðx1x2 . . . xNÞ ð10:36Þ

possibly orthonormal in themselves but necessarily orthogonal to c0. Weshall assume that

x�x ¼ 1; x�c0 ¼ 0 ð10:37Þ

If the xs are not orthogonal then they must be preliminarily orthogo-nalized by the Schmidt method. Then, we construct the matrices

M ¼ x�ðH0 �E0Þx ð10:38Þ

the (N�N) Hermitian matrix of the excitation energies, and

m ¼ x�ðH1c0Þ ð10:39Þ

the (N� 1) column vector of the transition moments.By expanding c1 in the finite set of the xs, we can write

c1 ¼ xC ¼XNk¼1

xkCk ð10:40Þ

~E2 ¼ C�MCþC�mþm�C ð10:41Þwhich is minimum for

d~E2

dC� ¼ MCþm ¼ 0 ) C ðbestÞ ¼ �M�1m ð10:42Þ

giving as best variational approximation to ~E2

~E2 ðbestÞ ¼ �m�M�1m ð10:43Þ

LINEAR PSEUDOSTATES AND MOLECULAR PROPERTIES 191

The Hermitian matrixM can be reduced to diagonal form by a unitarytransformation U among its basis functions x:

c ¼ xU; U�MU ¼ «; U�m ¼ mc ð10:44Þ

where « is here the (N�N) diagonal matrix of the (positive) excitationenergies:

« ¼

«1 0 � � � 0

0 «2 � � � 0

� � � � � � � � � � � �0 0 � � � «N

0BBBB@

1CCCCA ð10:45Þ

The cs are called pseudostates and give ~E2 in the form

~E2 ðbestÞ ¼ �m�c«

�1mc ¼ �XNk¼1

jhckjH1jc0ij2«k

ð10:46Þ

which is known as the sum-over-pseudostates expression. Equation 10.46has the same form as the analogous expression that would arise from thediscrete eigenstates of H0, but with definitely better convergence proper-ties, reducing the infinite summation to a sum of a finite number of termsandavoiding the needof considering the contribution fromthe continuouspart of the spectrum.For nonlinear parameters, eventually occurring in ~c1,

7 no specificexpression can be given for the best energy ~E2, which must be minimizedwith respect to these parameters by solving the necessary equations:

@~E2

@ck¼ 0; k ¼ 1;2; . . . ð10:47Þ

This can be done either analytically or numerically.At this point, we can work out a few simple examples concerning the H

atom in the external uniform dipole field (10.22), comparing the results ofour approximate calculations of the polarizability a with the exact valuegiven by (10.32). The results are collected in Table 10.1.

7 Essentially, orbital exponents.


10.5.1 Single Pseudostate

Choose as a convenient variational function satisfying all symmetryprescriptions the normalized 2p STO:

x ¼ c5

p

� �1=2

expð�crÞz; z ¼ r cos u ð10:48Þ

We have

~c1 ¼ w ¼ CFx; hc0jxi ¼ 0 ð10:49Þ

the order being carried by the linear parameterC, and omitted for brevity,while c is a nonlinear parameter.Then, optimization against the single linear variationparameter (single-

term approximation) gives

~E2 ðbestÞ ¼ �F2 m2

«ð10:50Þ

where

m ¼ hxjzjc0i ¼ ðc0xjzÞ ¼2ffiffiffic

pcþ 1

� �5

ð10:51Þ

« ¼ hxjH0 �E0jxi ¼ 1

2ðc2� cþ1Þ ð10:52Þ

Table 10.1 Pseudostate approximations to a for the ground state of the H atom

w c m=ea0 «=Eh a=a30

2pz 0.5 0.7449 0.375 2.96

2pz 0.7970 0.9684 0.4191 4.48

2pz 1-term 1 1 0.5 4.0

2pz þ 3pz 2-term 10:40820:9129

�10:4

�0:33334:1667

�

2pz þ 3pz þ 4pz 3-term 10:81530:55720:0605

8<:

0:38110:61661:7023

8<:

3:48871:00700:0043

8<:


In Equation 10.51, the first expression on the right-hand side is thetransition moment in Dirac form and the second in charge densitynotation. These two integrals are evaluated in spherical coordinates inthe Appendix.

~E2 can be further minimized with respect to the nonlinear parameter centering both m and «. It is seen that best c is obtained as a solution of thecubic equation (Magnasco, 1978)

7c3� 9c2þ 9c� 5 ¼ 0 ð10:53Þ

which has the real root c ¼ 0:7970.We then have the cases exemplified in Table 10.1.

10.5.1.1 Eigenstate of H0

In this case we put c ¼ 1=2 in expression (10.48). Evaluating the integralsby means of Equations 10.51 and 10.52 gives, in atomic units, the resultsof the first row of Table 10.1:

m ¼ 0:7449ea0; « ¼ 3

8¼ 0:375Eh; a ¼ 2:96a30 ð10:54Þ

This value ofa is only 66%of the correct value 4.5. Includinghighernpzeigenstates with c ¼ 1=n only improves this result a little. Including termsup to n ¼ 7 gives a ¼ 3:606, the asymptotic value of a ¼ 3:660 beingreached for n ¼ 30. This is only 81.3% of the exact value, the remaining18.7% coming from the contribution of the continuous part of thespectrum. These results show that the expansion in eigenstates of H0 isdisappointingly poor, the correct value being obtained only throughdifficult calculations.

10.5.1.2 Single optimized pseudostate

The best c ¼ 0:7970 is obtained in this case from the real root ofEquation 10.53. We then obtain the results of the second row ofTable 10.1:

m ¼ 0:9684; « ¼ 0:4191; a ¼ 4:48 ð10:55Þ


a is now within 99.5% of the exact result. An enormous improvementover the eigenstate result is evident. The fact that the best orbital exponentc is sensibly larger than 0.5 suggests that the excited pseudostate must berather contracted. It is then tempting to try with c ¼ 1, giving a pseu-dostate having the same decay with r as the unperturbed c0.

10.5.1.3 Single pseudostate with c ¼ 1 (Single-term approximation)

The results are those given in the third row of Table 10.1:

m ¼ 1; « ¼ 0:5; a ¼ 4 ð10:56ÞThis simple unoptimized pseudostate still gives 89%of the exact result,

suggesting that a possible improvement would be to include furtherfunctions, suitably polarized along z, by means of the Ritz method,Equation 10.46.

10.5.2 N-term Approximation

The dipole pseudospectra of H(1s) forN ¼ 1 throughN ¼ 4 are given inTable 10.2. The two-term approximation gives the exact result for thedipole polarizability a, the same being true for the three-term and thehigher N-term (N > 3) approximations. In all such cases, the dipolepolarizability of the atom is partitioned into an increasing number N ofcontributions arising from the different pseudostates:

a ¼XNi¼1

ai ð10:57Þ

Table 10.2 Dipole pseudospectra of H(1s) for N¼ 1 through N¼ 4

i ai=a30 «i/Eh

Pai

1 4.000 000� 100 5.000 000� 10�1 4.01 4.166 667� 100 4.000 000� 10�1

2 3.333 333� 10�1 1.000 000� 100 4.51 3.488 744� 100 3.810 911� 10�1

2 9.680 101� 10�1 6.165 762� 10�1

3 4.324 577� 10�2 1.702 333� 100 4.51 3.144 142� 100 3.764 643� 10�1

2 1.091 451� 100 5.171 051� 10�1

3 2.564 244� 10�1 9.014 629� 10�1

4 7.982 236� 10�3 2.604 969� 100 4.5


This increasingly fine subdivision of the exact polarizability value intodifferent pseudostate contributions is of fundamental importance for theincreasingly refined evaluation of the London dispersion coefficients fortwoH atoms interacting at long range, as we shall see in Chapter 11. Thecalculation of the two-term pseudostate approximation is fully describedelsewhere (Magnasco, 2007) and will not be pursued further here. It canonly be said that, in general, the N-term approximation will involvediagonalization of the (N�N) matrix M given by Equation 10.38, witheigenvalues giving the excitation energies «i and as eigenvectors thecorrespondingN-term pseudostates {ci}, i ¼ 1;2; . . . ;N. For a given atom(or molecule), knowledge of the so-called N-term pseudospectrumfai; «ig, i ¼ 1; 2; . . . ;N, allows for the direct calculation of the dispersioncoefficients of the interacting atoms (or molecules).Before ending this section,wenotice thatEquation10.27gives the exact

first-order solution of the RS dipole perturbation equation in un-normal-ized form. From this result we can obtain the single normalized pseudos-tate c equivalent to the exact c1 as

~c1 ¼ CFc; c ¼ N zþ r

2z

� �c0 ð10:58Þ

Easy calculation shows that

N¼ffiffiffiffiffiffi8

43

s; m¼ 9

2

ffiffiffiffiffiffi2

43

s¼ 0:9705; «¼ 18

43¼ 0:4186; a¼ 4:5 ð10:59Þ

so that the single optimized pseudostate (10.55) is seen to overestimate thetrue « by only 5�10�4Eh and underestimate the true m by 0:0021ea0,giving a dipole polarizability which differs from the exact one by just0:02a30. It must be admitted that the performance of this fully optimizedsingle function is exceptionally good, but the linearly optimized two-termapproximation does even better, giving the exact value for a (seeTable 10.1).

10.6 QUANTUM THEORY OF MAGNETICSUSCEPTIBILITIES

In this section we shall glance briefly at the use of RS perturbationtechniques in treating magnetic susceptibilities of atoms and molecules.


An introductory presentation of the subject can be found in the bookMolecular Quantum Mechanics (Atkins and Friedman, 2007), while amore specialistic presentation is given in The Theory of the Electric andMagnetic Properties of Molecules by Davies (1967). In the rest of thissection we shall make use of both atomic and cgs/emu units.The Hamiltonian for a particle of mass m and charge e in a scalar

potential V is

H0 ¼ p2

2mþV ð10:60Þ

where p is the operator for the linearmomentum (impulse). In the presenceof a magnetic perturbation due to a vector potential A, p is replaced by

p ) p� e

cA

� �ð10:61Þ

Hence, when acting on a functionY, the kinetic energy operator in thepresence of A gives

p2

2mY ¼ 1

2mp�e

cA

!� p�e

cA

!Y

¼ 1

2mp2�e

cp �A�e

cA � pþ e2

c2A2

!Y

¼ ��h22m

r2Yþ ie�h2mc

r�ðAYÞ þ ie�h2mc

A �rYþ e2

2mc2A2Y

¼ ��h22m

r2Yþ ie�h2mc

ðA �rYþYr�AÞ þ ie�h2mc

A �rYþ e2

2mc2A2Y

¼ ��h22m

r2 þ ie�hmc

A �r þ ie�h2mc

ðr�AÞ þ e2

2mc2A2

" #Y

ð10:62Þ

Therefore, we obtain the correction terms to the unperturbed Hamil-tonian (10.60) in the presence of a magnetic field:

H1 ¼ ie�h

mcA �rþ i

e�h

2mcðr �AÞ ð10:63Þ

QUANTUM THEORY OF MAGNETIC SUSCEPTIBILITIES 197

H2 ¼ e2

2mc2A2 ð10:64Þ

with (10.63) being linear and (10.64) quadratic in A.The vector potentialA originates a magnetic fieldH at point r, given by

the vector product

A ¼ 1

2H� r ¼ 1

2

i j kHx Hy Hz

x y z

�� ð10:65Þ

where the components of the field are constant. Then, the last termin (10.63) is zero:

r �A ¼ divA ¼ 0 ð10:66Þ

Next, it can be easily shown that

ðH� rÞ �r ¼ H � ðr�rÞ ð10:67Þ

so that, taking into account spin, we get for H1:

H1 ¼ �ðmLþ gebemSÞ �H ð10:68Þ

where ge � 2 is the intrinsic g-factor for the single electron (its correctvalue depends on considerations of quantum electrodynamics), be is theBohrmagneton, and mL and mS are the vector operators for the orbital andspin magnetic moments. It should be noted that the magnetic momentshave a direction opposite to that of the vectors representing orbital andspin angular momenta.For a magnetic field H uniform along z:

Hx ¼ Hy ¼ 0; Hz ¼ H; H ¼ kH ð10:69Þ

where the field strength H should not be confused with the Hamiltoniansymbol. Equation 10.65 then becomes

A ¼ 1

2H� r ¼ 1

2

i j k0 0 Hx y z

�� ¼

1

2Hð�iyþ jxÞ ð10:70Þ


giving

A2 ¼ A �A ¼ 1

4H2ðx2 þ y2Þ ð10:71Þ

Hence

H2 ¼ e2

2mc2A2 ¼ e2

8mc2H2ðx2þ y2Þ ð10:72Þ

For the energy of the system in the magnetic field we have the Taylorexpansion in powers of the magnetic field (compare with the correspond-ing expansion in the electric field):

EðHÞ ¼ E0 þ @E

@H

!0

Hþ 1

2

@2E

@H2

!0

H2 þ � � �

¼ E0�m0H� 1

2xH2 þ � � �

ð10:73Þ

where

m0 ¼ � @E

@H

� �0

ð10:74Þ

is the permanent magnetic moment (due to orbital motion) and

x ¼ � @2E

@H2

� �0

ð10:75Þ

is the magnetic susceptibility (which corresponds to the electricsusceptibility).

10.6.1 Diamagnetic Susceptibilities

Weconsiderfirst thegroundstatesof spherically symmetricatoms ðm ¼ 0Þand the singlet S ground states of diatomic molecules ðS ¼ L ¼ 0Þ.

(i) Atoms in spherical ground state

For atoms in a spherical ground state (such as hydrogen or rare gases)the first-order correctionE1 is zero (thenucleus is takenas theorigin of


the vector potential A). In fact:

Lz ¼ �i@

@wLzY‘mðWÞ ¼ mY‘mðWÞ ð10:76Þ

where

Y‘mðWÞ / expðimwÞPm‘ ðcos uÞ ð10:77Þ

is a spherical harmonic in complex form (W stands for the solid anglespecified by u and w) and

m ¼ 0;�1;�2; . . . ;�‘ ð10:78Þ

is the magnetic quantum number.Then, using RS perturbation theory up to second order in the fieldH:

E1 ¼ hc0jH1jc0i ¼ hc0j�mL �Hjc0i

¼ hc0je�h

2mcL �Hjc0i ¼

e�h

2mcHhc0jLzjc0i ¼ 0

ð10:79Þ

since m ¼ 0 for a spherical ground state;

E2 ¼ hc0jH2jc0i ¼e2

8mc2H2hc0jx2þ y2jc0i

¼ e2

12mc2H2hr2i00

ð10:80Þ

and we obtain the Langevin contribution to the molar diamagneticsusceptibility:

xL ¼ �NAe2

6mc2hr2i00 ¼ �0:792� 10�6hr2i00 < 0 ð10:81Þ

a negative contribution (when r is in au, the susceptibility is given incgs/emu).Table10.3gives the calculateddiamagnetic susceptibilities fora few

simple atoms. The value for the H atom is exact. For the two-electronatomic system, He, the hydrogenic approximation ðc ¼ Z ¼ 2Þ gives


only 68% of the accurate value ð�1:8905� 10�6 cgs=emuÞ obtainedby Pekeris (1959). The single-term SCF (optimization of the orbitalexponent c0 of a minimum STO basis) improves the result to 88%,while a five-term SCF (practically HF) gives over 99% of the Pekerisresult. Pretty good results are also obtained from the HF calculations(Strand and Bonham, 1964) on the heavier rare gases (7.4 instead ofaccurate 6.7 for Ne, 20.9 instead of 19.0 for Ar, 33.0 instead of 28.0for Kr), showing a limited effect of electronic correlation on diamag-netic susceptibility.

For the high-frequency part, expanding the second-order energy inpseudostates fckg, we have

Ehf2 ¼ hc0jH1jc1i ¼ �

Xkð 6¼ 0Þ

hc0je�h

2mcL �Hjckihckj

e�h

2mcL �Hjc0i

Ek�E0

¼ � e2�h2

4m2c2H2

Xkð 6¼ 0Þ

jhc0jLzjckij2«k

ð10:82Þwhence, referring to a mole:

xhf ¼ � @2E

@H2

� �0

¼ NAe2�h2

2m2c2

Xkð 6¼ 0Þ

hc0jLzjcki�� 2

«k> 0 ð10:83Þ

Table 10.3 Calculated diamagnetic susceptibilities for simple atoms

Atom Method hr2i00=a20 �xL=10�6 cgs=emu

H Exact 3 2.376He Hydrogenic 0.75 1.188 0

1-term SCFa 1.053 5 1.668 72-term SCFa 1.183 2 1.874 35-term SCFa 1.184 7 1.876 5Accurateb 1.193 5 1.890 5

Li HFc 15.2Be HFc 14.1Ne HFc 7.4Ar HFc 20.9Kr HFc 33.0

aClementi and Roetti, 1974.bPekeris, 1959.c Strand and Bonham, 1964.


so that the high-frequency term (paramagnetic contribution to thediamagnetic susceptibility) is positive, of a sign opposite to that of theLangevin term. So, we have for the diamagnetic susceptibility

xd ¼ xL þ xhf ð10:84Þ

a molecular property independent of T. For atoms in sphericalground states, xhf vanishes, since all transition integrals are zerobecause of the orthogonality of the excited pseudostates ck to theground state c0:

hc0jLzjcki ¼ h0jLzjki ¼ mh0jki ¼ 0 for k 6¼ 0 ð10:85Þ

(ii) Diatomic molecules in S singlet ground state

For molecules, xhf 6¼ 0 and use is made of the average susceptibility:

�xd ¼ 1

3

Xa

xLaa þ

1

3

Xa

xhfaa ¼ �xL þ �xhf ð10:86Þ

where

�xL ¼ �NAe2

6mc2hr2i00 < 0 ð10:87Þ

�xhf ¼ NAe2�h2

2m2c2

Xkð 6¼ 0Þ

hc0jLzjcki�� 2

«k> 0 ð10:88Þ

with «k > 0 the excitation energy from the ground state j0i to thepseudostate jki.

In Table 10.4 we give some values of diamagnetic susceptibilities forground-state H2 calculated with different wavefunctions (Tillieu, 1957a,1957b).We see (i) that the high-frequency contribution is sensibly smallerthan the low-fequency (Langevin) contribution, (ii) that the simple MOwavefunction exhibits an exceptionally good performance, comparingwell with the accurate James–Coolidge wavefunction result of the lastrow, (iii) that the HL (purely covalent) wavefunction shows a reasonablebehaviour, while (iv) the Weinbaum (HL plus ionic) wavefunction givesresults that are definitely too high either for xL or xhf.


In Table 10.5 we give some calculated values of diamagnetic suscept-ibilities for the singlet S ground state of a few simple diatomics. Someexperimental values are given in parentheses. The agreement with experi-ment can be judged as moderately satisfactory, even if the high-frequencycontributions calculated for the isoelectronic molecules N2 and CO aredefinitely too low.

10.6.2 Paramagnetic Susceptibilities

We now pass to consider briefly electronic states of atoms, ions andmolecules in nonsinglet spin states ðS 6¼ 0Þ, originating T-dependentparamagnetic susceptibilities in gases. In these cases, Curie’s law holds:

xp ¼ Cm

Tð10:89Þ

Table 10.4 Calculateda diamagnetic susceptibilities for ground-state H2 (units 10�6

cgs/emu)

Wavefunction xL xhf xd

MO �4.03 0.09 �3.94HL �4.22 0.11 �4.11Weinbaum �5.55 0.26 �5.29James–Coolidge �3.92 0.07 �3.85

aTillieu,1957a, 1957b.

Table 10.5 Calculated diamagnetic susceptibilities for the ground state of simplediatomics (units 10�6 cgs/emu)

Molecule xLa–c xhfc

Li2 �83.027 50.53N2 �40.800(�43.6) 23.47(30.3)F2 �64.800 46.15LiH �20.529 11.38(12.5)FH �9.588(�9.2) 0.937(0.61)LiF 16.10(15.3)CO 21.94(28.2)

a Stevens, Pitzer, and Lipscomb, 1963.b Stevens and Lipscomb, 1964a, 1964b, 1964c.cKarplus and Kolker, 1961, 1963.


where

Cm ¼ NAm2

3kð10:90Þ

is the Curie constant (referred to a mole).Adding the diamagnetic susceptibility xd, we have for the molar

magnetic susceptibility

xm ¼ xd þxp ¼ �Aþ C

Tð10:91Þ

a result similar to that existing for electric polarizabilities.For paramagnetic systems, where often electronic spin plays a funda-

mental role in determining susceptibilities, the first term in (10.91) isusually negligible with respect to xp (hence Curie’s law).A generalization of (10.89) is given by the law by Curie–Weiss:

xm ¼ Cm

T�Q) 1

xm¼ � Q

Cmþ 1

CmT ð10:92Þ

where Q ð< TÞ is called the Curie temperature. Q is a quantity character-istic of the different substances, its value marking the differencebetween paramagnetic ðQ < 0Þ and ferromagnetic ðQ > 0Þ systems.Equation 10.92 shows that the reciprocal of the magnetic susceptibilityis linear in the temperature T and can be used for the experimentaldetermination of the Curie constant (slope) and the Curie temperature(intercept).

(i) Paramagnetic susceptibilities

To investigate on the value assumed by the elementary magneticmoment m in the case of light atoms and ions, we may resort to theRussell–Saunders LS-coupling scheme in the so-called vector model(Magnasco, 2007). In this scheme, the spin vectorsL and S are coupledto a resultant vector J having a component Jz along the direction of themagnetic field. The associated ‘good’ quantumnumbers J andMJ takethe values

J ¼ Lþ S;Lþ S�1; . . . ; jL�SjMJ ¼ �J;�ðJ�1Þ; . . . ; ðJ�1Þ; J

(ð10:93Þ


The modulus (absolute value or magnitude) of the magnetic momentof the atom in state J will be

mJ ¼ �gee�h

2mc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJðJþ 1Þ

p¼ �gebe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJðJþ1Þ

pð10:94Þ

where ge is now the Land�e g-factor:

ge ¼ 1þ JðJþ 1Þþ SðSþ 1Þ�LðLþ1Þ2JðJþ 1Þ ð10:95Þ

For a single s-electron, L ¼ 0, S ¼ 12, J ¼ S ¼ 1

2, and therefore ge ¼ 2.The averagemolarmagneticmoment for an atom in the quantumstateJ for a small field will hence be

hmJi ¼ NAm2

3kTH ¼ NA

g2eb2e

3kTJðJþ1ÞH ð10:96Þ

giving for the T-dependent molar paramagnetic susceptibility xp

xp ¼ hmJiH

¼ NAg2eb

2e

3kTJðJþ1Þ ð10:97Þ

As already said, in most cases paramagnetism is given by spin onlyðge ¼ 2Þ, so that

xS ¼ hmSiH

¼ 4NAb2e

3kTSðSþ 1Þ ð10:98Þ

The effective magnetic moment for an atom (or ion) in state J in unitsof the Bohr magneton be will be

meff

be

¼ geffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJðJ þ 1Þ

pð10:99Þ

Spectroscopically, there are three cases according to the value of themultiplet width D ¼ hn with respect to the temperature T.


(a) D kT (narrow multiplets)We have

meff

be

¼ ½4JðJþ 1Þ�1=2 ð10:100Þ

This is the case of the ions of the metals of the first transitiongroup, and of the triplet ground state of O2. If L and S arequantized independently (like in Fe2þ ), then

meff

be

¼ ½4SðSþ 1ÞþLðLþ 1Þ�1=2 ð10:101Þ

(b) D kT (wide multiplets)Now, almost all particles are in the state of lowest energy J:

meff

be

¼ ge½JðJþ 1Þ�1=2; ge ¼ 3

2þ SðSþ 1Þ�LðLþ 1Þ

2JðJþ 1Þð10:102Þ

the case of the rare-earth ions.(c) D � kT (at room temperature, T ¼ 293 K)

This is the most difficult case (for example, NO, Sm3þ andEu3þ ), and it has been treated in detail by Van Vleck (1932).

We now discuss briefly a few interesting cases of paramagnetism inatoms,molecules and ions. In comparingwith experiment, it is convenientto put

NB ¼ NAe�h

2mc¼ NAbe ¼ 5:585� 103

erg

gauss molð10:103Þ

1mol of Bohr magnetons, and an effective magnetic moment meff

defined through

xm ¼ NAg2eb

2e

3kTJðJþ 1Þ ¼ g2eN

2B

3RTJðJþ 1Þ ¼ ðNBmeffÞ2

3RTð10:104Þ


where R is the gas constant, so that

m2eff ¼

3R

N2B

ðxmTÞ ¼ ½geffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJðJþ 1Þ

p�2

experiment theory

ð10:105Þ

meff ¼3R

N2B

� �1=2 ffiffiffiffiffiffiffiffiffiffixmT

p¼ 2:828

ffiffiffiffiffiffiffiffiffiffixmT

pð10:106Þ

(ii) Atoms in S states

For the H atom in its 2S ground state,L ¼ 0 and paramagnetism is dueto spin only:

L ¼ 0; J ¼ S ¼ 1

2; ge ¼ 2

mS ¼ �gebe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSðSþ 1Þp ¼ �be

ffiffiffi3

p

meff ¼jmSjbe

¼ffiffiffi3

p� 1:73

8>>>>>>>>><>>>>>>>>>:

ð10:107Þ

It has been observed that all complete atomic shells (with an electronconfiguration similar to that of rare gases) have J ¼ 0, so that the onlymagnetically active electrons are those of the valence (or optical) shell.For instance, for vapours of atomic Na, we have only a 3s-valenceelectron, so that the situation is entirely similar to that of the ground-state H atom, giving

mS ¼ �ffiffiffi3

pNB ) meff ¼

jmSjNB

¼ffiffiffi3

p� 1:73 ð10:108Þ

Experimental measurements of xmT in the temperature range900–1000K give 0.38, so that

meff ¼ 2:828ffiffiffiffiffiffiffiffiffiffixmT

p� 1:74 ð10:109Þ

which is in almost perfect agreement with the theoretical value offfiffiffi3

p.


(iii) The 3S�g ground state of the O2 molecule

We recall that

1 cm�1 ¼ 2625:5

219:47� 103kJ ¼ 315:78� 103

219:47� 103K ¼ 1:439 K ð10:110Þ

Molecular S states are practically devoided of multiplet structure,although experimentally they do have a small fine structure of theorder of 1 cm�1 or less. Hence, xp can certainly be calculated underthe assumption that the multiplet structure is small compared withkT (case a). At room temperature ðT ¼ 293 KÞ:

DEkT

¼ 1:44

293� 4:9� 10�3 ð10:111Þ

and the assumptions of Curie’s law are valid. For the triplet 3S�g

ground state of O2, L ¼ 0 and S ¼ 1, so that

xp ¼ NAb2e

3kTm2eff ¼

NAb2e

3kT4SðSþ1ÞþL2 ¼ NAb

2e

3k

8

Tð10:112Þ

Therefore:

NAb2e

3k� 0:125 ) 8

NAb2e

3k� 1 ) xp � 1

Tð10:113Þ

At T ¼ 293 K, the theory gives

xpð3SÞ � 3:41� 10�3 cgs=emu ð10:114Þ

which is almost in perfect agreement with the average experimentalvalue of 3.408 � 10�3. Original work by Curie himself shows thatCurie’s law forO2ð3SÞ is verywell satisfied over the temperature range290–720 K.


(iv) A free 3F ion without LS-coupling in a magnetic field HThemagnetic moments operators corresponding to the quantum statescharacterized by uncoupled L and S are

mL ¼ �beL; mS ¼ �2beS ð10:115Þ

Then, the potential energy of the magnetic dipole in the uniformmagnetic field H ¼ kH is

H1 ¼ �ðmLþ mSÞ �H ¼ beHðLz þ 2SzÞ ð10:116Þ

with

Lzc ¼ MLc; Szc ¼ MSc; c ¼ cðML;MSÞ ð10:117Þ

so that the Zeeman energy splitting of the ð2Lþ1Þð2Sþ 1Þ-sublevels inpresence of the field F will be ð�L � ML � L; �S � MS � SÞ

DEðML;MSÞ ¼ beHðMLþ 2MSÞ ð10:118Þ

Therearealtogetherð2Lþ 1Þð2Sþ 1Þ ¼ 7� 3 ¼ 21energylevels,sevenofwhichare stilldegenerate even inpresence of thefield (degeneracies 2,2, 3, 3, 3, 2, 2), as shown schematically in Figure 10.2.(v) An LS-coupled 3F ion in a magnetic field HIn this case:

L ¼ 3; S ¼ 1; J ¼ 4; 3; 2

ge ¼ 3

2� 5

JðJþ 1Þ

J ¼ 4; ge ¼ 5

4; J ¼ 3; ge ¼ 13

12; J ¼ 2; ge ¼ 2

3

8>>>>>>><>>>>>>>:

ð10:119Þ

There are 9þ 7þ 5 ¼ 21 levels altogether, as before, but now in thepresence of a field H any degeneracy is removed.


The operator describing spin-orbital LS-coupling is

HSO ¼ geb

2e

Z

r3L � S ¼ jðrÞL � S ð10:120Þ

where Z is the nuclear charge of the ion. But:

L � S ¼ 1

2ðJ2�L2�S2Þ ð10:121Þ

Using first-order perturbation theory, we take the expectation value

of HSO

over the ground-state wavefunction and the relative energy oflevels of given J will be

EJ ¼ A1

2JðJþ 1Þ�LðLþ 1Þ�SðSþ 1Þ½ � ð10:122Þ

whereA is a constant, characteristic of the ion.We then get the relativeenergies of the three LS-coupled states occurring for L ¼ 3 (state F)

1

1

2

2

3

3

3

2

2

1

1

− 5

− 4

− 3

− 2

− 1

0

+1

+2

+3

+4

+5

ML + 2MS

F3

H = 0 H ≠ 0

Figure 10.2 Zeeman splitting for a free 3F ion in a magnetic field H (the residualdegeneracy of each level is shown)


and S ¼ 1 (triplet):

J ¼ 4; E4 ¼ 3A

J ¼ 3; E4 ¼ �A

J ¼ 2; E4 ¼ �4A

8><>: ð10:123Þ

(vi) The Zeeman effect for the 3FJ ion in a magnetic field HThe magnetic moment operator corresponding to the quantum state Jresulting from LS-coupling is

−4

−3

−2

−1

0

+1

+2

+3

+4

−3

−2

−1

0

+1

+2

+3

−2

−1

0

+1

+2

3 A

−A

−4 A

3F

3F4

3F2

3F3

MJ

9 states

5 states

7 states

LS-coupling21-folddegenerate

Zeeman effect(H ≠ 0)

Figure 10.3 Resolution of the multiplet structure of the spin-orbit coupled 3FJ statein a magnetic field H (all levels now have different energies)


mJ ¼ �gebeJ; ge ¼ 3

2þ SðSþ 1Þ�LðLþ 1Þ

2JðJþ 1Þ ð10:124Þ

Then, the potential energy of the magnetic dipole in the uniformmagnetic field H ¼ kH is

H1 ¼ �mJ �H ¼ gebeHJzJzc ¼ MJc ð�J � MJ � JÞ

(ð10:125Þ

so that the energy of the (2J þ 1)-sublevels in presence of a field Hwill be

EMJ¼ gebeHMJ ð10:126Þ

The splitting of the Zeeman levels is linear in the strength H of themagnetic field, and is shown schematically in Figure 10.3.

APPENDIX: EVALUATION OF m AND «

For the STO (10.48), the required integrals are easily calculated inspherical coordinates.

m ¼ h2pzjzjc0i ¼ ðc02pzjzÞ ¼ffiffiffiffiffic5

p

p2pð1�1

dx x2ð10

dr r4exp½�ðcþ 1Þr�

¼ 4

3

ffiffiffiffiffic5

p 4� 3� 2

ðcþ 1Þ5 ¼ 2ffiffiffic

pcþ1

!5

ð10:127Þ

Using relations (10.29), we obtain

d

drexpð�crÞr½ � ¼ expð�crÞð1�crÞ

d2

dr2expð�crÞr½ � ¼ expð�crÞð�2cþ c2rÞ

r2r expð�crÞr½ � ¼ expð�crÞ 2

r�4cþ c2r

!

8>>>>>>>>><>>>>>>>>>:

ð10:128Þ


so that

H0 expð�crÞr cos u½ � ¼ expð�crÞ ð2c�1Þ� c2

2r

� �cos u ð10:129Þ

We then obtain for the integral:

h2pzjH0j2pzi ¼ c5

p2pð1�1

dx x2ð10

dr r2 ð2c�1Þr� c2

2r2

" #expð�2crÞ

¼ 4

3c5 ð2c�1Þ 3� 2

ð2cÞ4 �c2

2

4� 3� 2

ð2cÞ5

24

35 ¼ c2

2� c

2

ð10:130Þ

giving for the excitation energy

« ¼ h2pzjH0�E0j2pzi ¼1

2ðc2�cþ 1Þ ð10:131Þ

APPENDIX: EVALUATION OF m AND « 213

11Atomic and Molecular

Interactions

In this chapter we shall present first (Section 11.1) the elementary RSperturbation theory for the interaction between two ground-state Hatoms, taking the interatomic potential V as a first-order perturbation.The interacting atoms are assumed to be sufficiently far apart so that, inthe first approximation, we can ignore the effect of exchanging identicalelectrons between different atoms, as required by Pauli’s antisymmetryprinciple. As a consequence, the two-electronwavefunctionswill be takenin the form of simple orbital products for the two electrons, insisting onthe fact that electron 1 belongs to atomA and electron 2 to atom B, givingwhat is called theCoulombic interatomic energy,whichwill be consideredup to second order in V. The possibility of including electron exchange inthe first-order theory is examined in some detail elsewhere (Magnasco,2007), where it is seen that it is equivalent to the HL theory of Chapter 9.The expansion of the interatomic potential V into inverse powers of theinternuclear distance R, giving the so-called multipole expansion of thepotential, is then examined in Section 11.2, with particular emphasis onthe calculation of the leading term of the London attraction betweentwo H atoms. These considerations are then extended to molecules(Section 11.3), while a short discussion on the nature of the VdW andhydrogen bonds concludes the chapter.


Valerio Magnasco


11.1 THE H–H NONEXPANDED INTERACTIONSUP TO SECOND ORDER

With reference to Figure 11.1, we take as unperturbed Hamiltonian H0

the sum of the two Hamiltonians for the separate H atoms havingunperturbed wavefunctions a0(r1) and b0(r2) and energies EA

0 and EB0:

H0 ¼ HA

0 þ HB

0 ; c0 ¼ a0ðr1Þb0ðr2Þ; E0 ¼ EA0 þEB

0ð11:1Þ

The perturbation H1 is here the interatomic potential V:

V ¼ H� H0 ¼ � 1

rB1� 1

rA2þ 1

r12þ 1

Rð11:2Þ

where R is the internuclear distance measured along the z-axis. We shallrefer to (11.2) as the nonexpanded interatomic potential.If we introduce a set of excited pseudostates {ai} on A and {bj} on B

(i; j ¼ 1;2; . . . ;N), normalized and orthogonal to the respective unper-turbed functions:

haijaii ¼ hbjjbji ¼ 1haija0i ¼ ha0jaii ¼ hbjjb0i ¼ hb0jbji ¼ 0

�ð11:3Þ

x

y

zA BR(000) (00R)

1 2

r1r2

r12

rA2rB1

(x y z )111(x y z +R) 222

Figure 11.1 Interatomic reference system for the H–H interaction

216 ATOMIC AND MOLECULAR INTERACTIONS

then the RS energy corrections up to second order in V are as follows:

E0 ¼ hc0jH0jc0i ¼ ha0b0jHA

0 þ HB

0 ja0b0i ¼ EA0 þEB

0 ð11:4Þ

the unperturbed energy, the energy pertaining to the separate atoms;

Ecb1 ¼ hc0jH1jc0i ¼ ha0b0jVja0b0i¼ ða20j � r� 1

B1 Þþ ðb20j � r� 1A2 Þþ ða20jb20Þþ

1

R¼ Ees

1ð11:5Þ

the nonexpanded first-order correction, the semiclassical Coulomb inter-action of the HL theory, said the electrostatic energy; and

Ecb2 ¼ �

Xi

haib0jVja0b0ij j2«i

�Xj

ha0bjjVja0b0i�� 2

«j

�Xi

Xj

haibjjVja0b0i�� 2

«iþ «j

ð11:6Þ

the second-order Coulombic energy describing deviations from the rigidspherical atoms. The first two terms represent the induction (distortion orpolarization) energy involvingmonoexcitations (top row of Figure 11.2),

UA

ai

ai ai

a0

bj

bj bj

b0

a0 b0 a0 b0

ai

a0

bj

b0

εi εj U

B

A A BB

B polarizes A A polarizes B

εj εi

1/r12

A A BB

R√6

3

Mutual polarization of A B

εi, μi μi μj

αi

εj, μj αj

Figure 11.2 Second-order interactions between ground-state H atoms. Top row:B polarizes A (left) and A polarizes B (right). Bottom row: nonexpanded dispersion(left) and leading term of the expanded dispersion (right)

THE H–H NONEXPANDED INTERACTIONS UP TO SECOND ORDER 217

one on A (B polarizes A) and the other on B (A polarizes B), the lastterm the dispersion (interatomic electron correlation) energy involvingsimultaneous biexcitations (bottom row of Figure 11.2), one onA and theother on B.Introducing the explicit form of V, Equation 11.2, we obtain for the

transition integrals in (11.6)

haib0jVja0b0i¼ aib0j � 1

rB1� 1

rA2þ 1

r12þ 1

Rja0b0

* +

¼ ai � 1

rB1þðdr2

b0ðr2Þ½ �2r12

��a0

* +¼ ða0aijUBÞ

ð11:7Þ

where

UBðr1Þ ¼ � 1

rB1þðdr2

½b0ðr2Þ�2r12

¼ � expð� 2rB1ÞrB1

ð1þ rB1Þ ð11:8Þ

is themolecular electrostatic potential (MEP) at r1 on A due to B (nucleusand undistorted electron of B).Similarly:

ha0bjjVja0b0i¼ a0bj � 1

rB1� 1

rA2þ 1

r12þ 1

R

��a0b0

* +

¼ bj � 1

rA2þðdr1

½a0ðr1Þ�2r12

��b0

* +¼ ðb0bjjUAÞ

ð11:9Þ

with

UAðr2Þ ¼ � 1

rA2þðdr1

½a0ðr1Þ�2r12

¼ � expð� 2rA2ÞrA2

ð1þ rA2Þ ð11:10Þ

the MEP at r2 on B due to atom A in its undistorted ground state.


For the transition integral involving the biexcitations we obtain

haibjjVja0b0i¼ aibj � 1

rB1� 1

rA2þ 1

r12þ 1

R

��a0b0

* +

¼ aibj1

r12

��a0b0

* +¼ ða0aijb0bjÞ

ð11:11Þ

the interactionbetween the two transitiondensities ½a0ðr1Þa�i ðr1Þ�onAand½b0ðr2Þb�j ðr2Þ� on B.The first few orders of nonexpanded RS perturbation theory for the

H–H interaction are then

Ecb1 ¼ Ees

1 ¼ expð� 2RÞR

1þ 5

8R� 3

4R2� 1

6R3

� �ð11:12Þ

the first-order electrostatic energy, with its explicit dependence on theinternuclear distance R, showing the charge-overlap nature of the inter-action between neutral H atoms.For the second-order energy:

~E2 ¼ ~Eind;A

2 þ ~Eind;B

2 þ ~Edisp

2 ð11:13Þ

where

~Eind;A

2 ¼ �Xi

ða0aijUBÞ�� 2«i

ð11:14Þ

is the polarization energy of A by B (B distorts A from its sphericalsymmetry),

~Eind;B

2 ¼ �Xj

ðb0bjjUAÞ�� 2«j

ð11:15Þ

is the polarization energy of B by A (A distorts B from its sphericalsymmetry) and

~Edisp

2 ¼ �Xi

Xj

ða0aijb0bjÞ�� 2

«i þ «jð11:16Þ

THE H–H NONEXPANDED INTERACTIONS UP TO SECOND ORDER 219

is the nonexpanded dispersion energy due to the mutual polarization(distortion) of A and B. The tilde onE2 and its components means that weare using a variational approximation to the second-order energies.

11.2 THE H–H EXPANDED INTERACTIONSUP TO SECOND ORDER

Wenowexpand the interatomicpotentialVat long range ðR � r1; r2Þ in apower series in R� n up to R� 3:

1

rB1¼ ½x21 þ y21 þðz1 �RÞ2�� 1=2 ¼ ðR2 �2z1Rþ r21Þ� 1=2

¼ 1

R1� 2

z1R

þ r21R2

!� 1=2

� 1

R1þ z1

R� r21

2R2þ 3

8� 2

z1R

!2

þ � � �24

35

¼ 1

Rþ z1

R2þ 3z21 � r21

2R3þOðR� 4Þ

ð11:17Þ

where use was made of the Taylor expansion for x ¼ small:

ð1þ xÞ� 1=2 � 1� 1

2xþ 3

8x2 þ � � � ð11:18Þ

1

rA2¼ ½x22 þ y22 þðz2 þRÞ2�� 1=2 ¼ ðR2 þ 2z2Rþ r22Þ� 1=2

¼ 1

R1þ 2

z2R

þ r22R2

!� 1=2

� 1

R1� z2

R� r22

2R2þ 3

82z2R

!2

þ � � �24

35

¼ 1

R� z2

R2þ 3z22 � r22

2R3þOðR� 4Þ

ð11:19Þ


For the two-electron repulsion, we have

1

r12¼ ½ðx1� x2Þ2 þðy2 � y2Þ2 þðz1 � z2 �RÞ2�� 1=2

¼ ðR2� 2z1Rþ 2z2Rþ r21 þ r22� 2x1x2 � 2y1y2 � 2z1z2Þ� 1=2

¼ 1

R1� 2

z1R

þ 2z2R

þ r21R2

þ r22R2

� 2

R2ðx1x2þ y1y2þ z1z2Þ

" #� 1=2

ð11:20Þ

so that, expanding according to Taylor:

1

r12� 1

R1þ z1

R� z2

R� r21

2R2� r22

2R2þ 1

R2ðx1x2 þ y1y2 þ z1z2Þ

"

þ 3

8� 2

z1R

!2

þ 3

82z2R

!2

þ 3

8�8

z1z2R2

!þ � � �

#

¼ 1

Rþ z1 � z2

R2þ 3z21 � r21

2R3þ 3z22 � r22

2R3þ x1x2 þ y1y2 � 2z1z2

R3þOðR� 4Þ

ð11:21Þ

Adding all terms altogether with the appropriate signs, many terms docancel, finally giving:

V � 1

R3ðx1x2 þ y1y2 � 2z1z2ÞþOðR� 4Þ ð11:22Þ

which is the leading term, the dipole-dipole interaction, of the expandedform of the interatomic potentialV for neutralH atoms. It corresponds tothe classical electrostatic interaction of two point-like dipoles1 located atthe two nuclei of A and B (Coulson, 1958). Expansion (11.22) is the firstterm of what is known as the multipole expansion of the interatomicpotential in long range.

1 In atomic units.

THE H–H EXPANDED INTERACTIONS UP TO SECOND ORDER 221

Then, with such an expanded V, it is easily seen that

Ecb1 ¼ Ees

1 ¼ 0 ð11:23Þ~Eind;A

2 ¼ ~Eind;B

2 ¼ 0 ð11:24Þ

so that the only surviving term at long range is the London dispersionattraction:

~Edisp

2 ¼ � 1

R6

Xi

Xj

jhaibjjx1x2 þ y1y2 � 2z1z2ja0b0ij2«i þ «j

¼ � 6

R6

Xi

Xj

jða0aijz1Þj2jðb0bjjz2Þj2«i þ «j

ð11:25Þ

where we have taken into account the spherical symmetry of atoms A andB, giving

ða0aijx1Þ ¼ ða0aijy1Þ ¼ ða0aijz1Þ on A

ðb0bjjx2Þ ¼ ðb0bjjy2Þ ¼ ðb0bjjz2Þ on B

(ð11:26Þ

Since

aAi ¼ 2

jða0aijz1Þj2«i

¼ 2m2i

«i; aA ¼

Xi

aAi ð11:27Þ

is the ith pseudostate contribution to aA, the dipole polarizability of atomA, and

aBj ¼ 2

jðb0bjjz2Þj2«j

¼ 2m2j

«j; aB ¼

Xj

aBj ð11:28Þ

is the jth pseudostate contribution to aB, the dipole polarizability of atomB, the formula for the leading term of the expanded dispersion can bewritten in the so-called London form:

~Edisp

2 ¼ � 6

R6

Xi

Xj

m2i m

2j

«i þ «j¼ � 6

R6

1

4

Xi

Xj

2m2i

«i

!2m2

j

«j

!«i«j

«i þ «j

¼ � 6

R6

1

4

Xi

Xj

aiaj«i«j

«i þ «j¼ � 6

R6C11 ð11:29Þ


where

C11 ¼ 1

4

Xi

Xj

aiaj«i«j

«i þ «jð11:30Þ

is the dipole dispersion constant, the typical quantum mechanical part ofthe calculation of the dispersion coefficient, while 6 is a geometricalfactor2. Therefore:

C6 ¼ 6C11 ð11:31Þ

is the C6 London dispersion coefficient for the long-range interactionbetween two ground-state H atoms.In this way, the previously calculated dipole pseudospectra {ai,«i},

i ¼ 1; 2; . . . ;N, for each H atom can be used to obtain better and bettervalues for theC6 London dispersion coefficient for theH–H interaction: amolecular (two-centre) quantity, C6, can be evaluated in terms of atomic(one-centre), nonobservable, quantities, ai (a alone is useless). Thecoupling between the different components of the polarizabilities occursthrough the denominator in the London formula (11.30), so that wecannot sumover i or j to get the full, observable, 3aA oraB. An alternative,yet equivalent, formula for the dispersion constant is due to Casimir andPolder (1948) in terms of the frequency-dependent polarizabilities atimaginary frequencies of A and B:

C11 ¼ 1

2p

ð¥0

du aAðiuÞ aBðiuÞ

aAðiuÞ¼Xk

«k2mð0kÞmðk0Þ

«2k þ u2; aA ðstaticÞ ¼ aAð0Þ ¼ lim

u!0aAðiuÞ

8>>>>><>>>>>:

ð11:32Þ

where u is a real quantity.

2 Depending on the spherical symmetry of the ground-state H atoms.3 That is, measurable.

THE H–H EXPANDED INTERACTIONS UP TO SECOND ORDER 223

In this case, we must know the dependence of the frequency-dependentpolarizabilities on the real frequency u, and the coupling occurs now viathe integration over the frequencies. When the necessary data are avail-able, however, the London formula (11.30) is preferable because use ofthe Casimir–Polder formula (11.32) presents some problems in theaccurate evaluation of the integral through numerical quadrature tech-niques (Figari and Magnasco, 2003).Using the London formula and some of the pseudospectra derived in

Chapter 10, we obtain for the leading term of the H–H interaction theresults collected in Table 11.1.Table 11.1 shows that convergence is very rapid for the H–H interac-

tion. We give here the explicit calculation for N ¼ 2:

C11ð2-termÞ ¼ 1

8

25

6

!22

5þ 1

8

2

6

!2

� 1þ 1

2

25

6

2

6

2

5� 1

2

5þ 1

¼ 5� 5� 25� 2

2� 4� 36� 5þ 4

2� 4� 36þ 5� 5� 2� 5

36� �5� 7

¼ 125

4� 36þ 1

2� 36þ 50

7� 36¼ 1089

1008¼ 121

112¼ 1:080 357

so that the two-term approximation gives the dispersion constant as theratio between two not divisible integers! However, this explicit calcula-tion is no longer possible forN > 2,wherewemust resort to the numericalmethods touched upon inChapter 10.Using a nonvariational technique inmomentum space, Koga and Matsumoto (1985) gave the three-term C6

for H–H as the ratio of not divisible integers as

C6 ¼ 12 529

1928¼ 6:498443983 . . .

Table 11.1 N-term results for the dipole dispersion constant C11 and C6 Londondispersion coefficients for the H–H interaction

N C11=Eha60 C6=Eha

60 Accurate/%

1 1 6 92.32 1.080 357 6.482 1 99.73 1.083 067 6.498 4 99.994 1.083 167 6.499 00 99.9995 1.083 170 6.499 02 100


and for the four-term C6

C6 ¼ 6 313 807

971 504¼ 6:499 002 577 . . .

This last value is accurate to four decimal figures,4 while a valueaccurate to 13 decimal figures was given by Thakkar (1988):

C6 ¼ 6:499 026 705 405 840 5 . . .

The London C6 dispersion coefficient for the long-range H–H inter-action is today one of the best known ‘benchmarks’ in the Literature. Itwas calculated with an accuracy of 20 exact decimal digits by Yan et al.(1996):

C6 ¼ 6:499 026 705 405 839 313 13 . . .

and, with an accuracy of 15 decimal digits, in an independent way, byKoga and Matsumoto (1985) and by Magnasco et al. (1998):

C6 ¼ 6:499 026 705 405 839 218 . . .

It must be stressed that all such values are far beyond any possibleexperimental accuracy, being useful only for checking the accuracy ofdifferent ways of calculation!Unfortunately, the convergence rate for C6 (as well as that for a) is not

so good for two-electron systems, as we shall see shortly.

11.3 MOLECULAR INTERACTIONS

The nonexpanded intermolecular potential V arises from the Coulombicinteractions between all pairs i, j of charged particles (electrons plusnuclei) in the molecules (Figure 11.3):

V ¼Xi

Xj

qiqjrij

ð11:33Þ

4 Inaccurate digits are in bold type.

MOLECULAR INTERACTIONS 225

where qi and qj are the charges of particles i (belonging to A) and j(belonging to B) interacting at the distance rij.

11.3.1 Nonexpanded Energy Corrections up to SecondOrder

If A0 and B0 are the unperturbed wavefunctions of molecules A (NA

electrons) and B (NB electrons), and Ai and Bj a pair of excited pseudos-tates describing single excitations on A and B, all fully antisymmetrizedwithin the space of A and B, we have to second order of RS perturbationtheory that

Ecb1 ¼ hA0B0 Vj jA0B0i ¼ Ees

1 ð11:34Þ

is the semiclassical electrostatic energy arising in first order from theinteractions between undistorted A and B; that

~Eind;A

2 ¼ �Xi

jhAiB0 Vj jA0B0ij2«i

¼ �Xi

jðA0AijUBÞj2«i

ð11:35Þ

is the polarization (distortion) of A by the static field of B, describedby

UB ¼ hB0 Vj jB0i ð11:36Þ

A

R

B

c.o.m. c.o.m.

ri rj

riji,qi j, qj

Figure 11.3 Interparticle distances in the intermolecular potential (c.o.m.: centre ofmass)


the MEP of B; that

~Eind;B

2 ¼ �Xj

jhA0Bj Vj jA0B0ij2«j

¼ �Xj

jðB0BjjUAÞj2«j

ð11:37Þ

is the polarization (distortion) of B by the static field of A, described by theMEP UA; and that

~Edisp

2 ¼ �Xi

Xj

hAiBjjVjA0B0i�� 2

«iþ«j¼ �

Xi

Xj

AiBj

Pi0< j0 ri0j0

�1�� A0B0

D E�� 2«iþ«j

ð11:38Þ

is the dispersion interaction, a purely electronic term arising from thedensity fluctuationsof the electrons onAandBwhichare coupled togetherthrough the intermolecular electron repulsion operator r�1

12 (1 on A, 2on B).Generalization of the previousH–H results tomolecules is possible in terms

of the charge-density operator (Longuet-Higgins, 1956) and of static andtransitionelectrondensities,PAð00jr1; r1Þ andPBð00jr2; r2Þ,PAð0ijr1; r1Þ andPBð0jjr2; r2Þ respectively on A and B. The nonexpanded dispersion energybetween molecules A and B then takes the simple integral form

~Edisp

2 ¼ �Xi

Xj

ÐÐdr1 dr2

PAð0ijr1; r1ÞPBð0jjr2; r2Þr12

��2

«i þ «jð11:39Þ

which canbe comparedwith the corresponding integral expression (11.16)found for the H–H interaction, where PAð0ijr1; r1Þ ¼ a0ðr1Þa�i ðr1Þ andPBð0jjr2; r2Þ ¼ b0ðr2Þb�j ðr2Þ.

11.3.2 Expanded Energy Corrections up to Second Order

In molecules, the interaction depends on the distance R between theircentres of mass as well as on the relative orientation of the interactingpartners, which can be specified in terms of the five independent angles5

5 These angles are simply related to the Euler angles describing the rotation of a rigid body (Brink

and Satchler, 1993).


(uA, uB, w, xA, xB) shown in Figure 11.4. The first three angles describe theorientation of the principal symmetry axes of the two molecules and thelatter two the rotation about these axes.In what follows, we shall limit ourselves mostly to consideration of the

long-range dispersion interaction between (i) two linear molecules A andB (topof Figure 11.5) and (ii) an atomA, at the origin of the intermolecularcoordinate system, and a linear molecule B, whose orientation withrespect to the z-axis is specified by the single angle u (bottom ofFigure 11.5).The linear molecule has two dipole polarizabilities, ajj, the parallel or

longitudinal component directed along the intermolecular axis, and a?,the perpendicular or transverse component perpendicular to the inter-molecular axis (McLean andYoshimine, 1967a). Themolecular isotropicpolarizability can be compared to that of atoms, and is defined as

a ¼ ajj þ2a?

3ð11:40Þ

while

Da ¼ ajj �a? ð11:41Þ

is the polarizability anisotropy, which is zero for a? ¼ ajj.The composite system of two different linear molecules hence has four

independent elementary dipole dispersion constants, which in London

x

y

z

A B

RθA θB

χA

χB

ϕ

Figure 11.4 The five angles specifying in general the relative orientation of twopolyatomic molecules


form can be written as

A ¼ 1

4

Xi

Xj

ajji a

jjj

«jji «

jjj

«jji þ «

jjj

; B ¼ 1

4

Xi

Xj

ajji a

?j

«jji «

?j

«jji þ «?j

C ¼ 1

4

Xi

Xj

a?i a

jjj

«?i «jjj

«?i þ «jjj

; D ¼ 1

4

Xi

Xj

a?i a

?j

«?i «?j

«?i þ «?j

8>>>>>><>>>>>>:

ð11:42Þ

For two identical linear molecules, there are three independent disper-sion constants, since C ¼ B.Spherical tensor expansion of the product r� 1

12 � r� 11020 (Wormer, 1975;

Magnasco and Ottonelli, 1999a) allows us to write the leading (dipole–dipole) term of the long-range dispersion interaction between two linearmolecules in the form:

~Edisp

2 ¼ �R� 6C6ðuA; uB;wÞ ð11:43Þ

x

y

z

θ

A

B

.

x

y

RθA θ B

ϕ

A B

z

Figure 11.5 Top, the three angles specifying the relative orientation of two linearmolecules. Bottom: the system atom A–linear molecule B


C6ðuA; uB;wÞ being an angle-dependent dipole dispersion coefficient,which can be expressed (Meyer, 1976) in terms of associated Legendrepolynomials on A and B as

C6ðuA; uB;wÞ ¼ C6

XLALBM

gLALBM6 PM

LAðcos uAÞPM

LBðcos uBÞ ð11:44Þ

whereLA; LB ¼ 0; 2 andM ¼ Mj j ¼ 0;1; 2. In (11.44),C6 is the isotropiccoefficient and g6 an anisotropy coefficient, defined as

gLALBM6 ¼ CLALBM

6

C6ð11:45Þ

The different components of the C6 dispersion coefficients in theLALBM scheme for (i) two different linear molecules and (ii) an atomand a linear molecule are given in Table 11.2 (Magnasco and Ottonelli,1999a) in terms of the symmetry-adapted combinations of the elementarydispersion constants (11.42). The coefficients with M=0 are not inde-pendent, but are related to that with M ¼ 0 by the relations

C2216 ¼ � 2

9C220

6 ; C2226 ¼ 1

36C220

6 ð11:46Þ

For identical molecules, C ¼ B in (11.42) and the (020) and (200)coefficients are equal.

Table 11.2 LALBM components of C6 dispersion coefficients for(i) two linear molecules and (ii) an atom and a linear molecule

LALBM (i) (ii)

00 02

3ðAþ 2Bþ 2Cþ 4DÞ 2A þ 4B

02 02

3ðA�Bþ 2C� 2DÞ 2A � 2B

20 02

3ðAþ 2B�C� 2DÞ

22 0 2(A � B � C þ D)

2 2 1 � 4

9ðA�B�CþDÞ

22 21

18ðA�B�CþDÞ


Therefore, the determination of the elementary dispersion constants(the quantum mechanical relevant part of the calculation) allows for adetailed analysis of the angle-dependent dispersion coefficients betweenmolecules.An equivalent, yet explicit, expression of the C6 angle-dependent

dispersion coefficient for the homodimer of two linear molecules as afunction of the three independent dispersion constants was derived byBriggs et al. (1971) in their attempt to determine the dispersion coeffi-cients of two H2 molecules in terms of nonlinear 1Sþ

u and 1Pu pseu-dostates:

C6 ðuA; uB;wÞ ¼ ð2Bþ 4DÞþ 3ðB�DÞðcos2 uA þ cos2 uBÞ

þ ðA� 2BþDÞðsin uAsin uBcow� 2cos uAcos uBÞ2ð11:47Þ

Since (cos u ¼ x)

hcos2 ui ¼

ð1� 1

dx x2

ð1� 1

dx

¼ 1

3; hsin2 ui ¼ 2

3; hcos2wi ¼

ð2p0

dw cos2w

ð2p0

dw

¼ 1

2

ð11:48Þ

averaging (11.47) over angles and noting that only squared terms con-tribute to the average, we obtain the following for the isotropic C6

dispersion coefficient:

hC6i¼ð2Bþ4DÞþ3ðB�DÞ 1

3þ1

3

!þðA�2BþDÞ 2

3�2

3�1

2þ4�1

3�1

3

!

¼2

3ðAþ4Bþ4DÞ¼C6 ð11:49Þ

in accord with the result of the first row of Table 11.2. Magnasco et al.(1990b)gave analternative interesting expression forC6ðuA;uB;wÞ in termsof frequency-dependent isotropic polarizabilities a(iu) and polarizability


anisotropies DaðiuÞ of the two linear molecules:

C6ðuA;uB;wÞ¼ 1

2p

ð¥0

dufaAðiuÞaBðiuÞþ½ð3cos2uB�1ÞaAðiuÞDaBðiuÞ

þð3cos2 uA�1ÞDaAðiuÞaBðiuÞ�þ½4cos2uAcos2uB�cos2uA�cos2uB�sin2 uAsin

2uB cosw

þsin2 uAsin2uBcos

2w�DaAðiuÞDaBðiuÞg ð11:50Þ

Averaging over angles, all coefficients involving polarizability aniso-tropies are zero, giving the isotropic C6 coefficient in the Casimir–Polderform (11.32).To get an illustrative numerical example, we can use the four-term

pseudospectrum of Table 11.3 for the dipole polarizabilities of the 1Sþg

ground state of the H2 molecule at R ¼ 1:4 a0, which gives for the dipolepolarizabilities of H2

ajj ¼ 6:378; a? ¼ 4:559 ð11:51Þ

results that are remarkablygood for bothpolarizabilities, being99.9%and99.6% respectively of the accurate values (ajj ¼ 6:383 and a? ¼ 4:577)obtained from the accurate 34-term pseudospectrum (Magnasco andOttonelli, 1996a).We can then calculate the four-term approximation to the three

independent elementary dispersion constants (11.42) for the homodimerH2–H2, obtaining the following numerical results:

A ¼ 2:683 99:8% of the accurate value 2:689

B ¼ C ¼ 2:018 99:3% of the accurate value 2:032

D ¼ 1:524 98:8% of the accurate value 1:542

Table 11.3 Four-termdipole pseudospectrum (atomic units) ofH2 (Sþg ) atR ¼1:4a0

i ajji «

jji a?

i «?i

1 4.567 0.473 2.852 0.4942 1.481 0.645 1.350 0.6993 0.319 0.973 0.335 1.1574 0.011 1.701 0.022 2.207


From these values, we obtain for the isotropic C6 dispersion coefficientfor H2–H2

C6 ¼ C0006 ¼ 11:23Eha

60

and for the first dipole anisotropy we have

g0200 ¼ C0206

C6¼ 0:098

which are respectively within 99.2% and þ2% of the accurate values(C6 ¼ 11:32 and g6 ¼ 0:096) given by Magnasco and Ottonelli (1996a).As a second example, illustrating a heterodimer calculation, consider

the C6 dispersion coefficient of the H–H2 system (atom–linear moleculeinteraction):

C6ðuÞ ¼ C6½1þ g6P2ðcos uÞ� ð11:52Þ

where

P2ðcos uÞ ¼ 3cos2 u� 1

2ð11:53Þ

is the Legendre polynomial of degree 2 (Chapter 3), C6 is the isotropiccoefficient and g6 is the anisotropy coefficient. From Table 11.2:

C6 ¼ C0006 ¼ 2Aþ 4B; g0206 ¼ C020

6

C6¼ A�B

Aþ 2B

so that, using the four-term pseudospectrum for the H atom (Table 10.2)and the one for the H2 molecule (Table 11.3), we obtain A ¼ 1:696 andB ¼ 1:269 and for the isotropic C6 dispersion coefficient of the H–H2

interaction we obtain

C6 ¼ 2� 1:696þ 4� 1:269 ¼ 8:468Eha60

which is within 99.6% of the accurate value C6¼ 8.502 (Magnasco,Ottonelli, 1996b). The calculated value of g6¼0.101 exceeds the correct


value (0.099) by 2%. So, these homo- and hetero-dimer calculations areaffected essentially by the same percent errors6.Tables 11.4 and 11.5 show, respectively, the convergence of N-term

dipole pseudospectra in the calculation of polarizabilities a and isotropic

Table 11.4 N-term convergence of ground-state isotropic dipole polarizabilities aða30Þ for simple atoms and molecules

N H He Hþ2

a H2b

1 4 0.694 1.485 2.5232 4.5 1.042 2.614 3.4813 1.078 2.819 3.4934 1.082 2.836 4.0785 1.135 2.837 4.45210 1.364 2.855 4.99915 1.378 2.864 5.14520 1.382 2.864 5.165Accurate 4.5c 1.383d 2.864e 5.181f

aHþ2 : Re ¼ 2a0.

bH2: Re ¼ 1:4a0.cExact value.dYan et al., 1996.eBishop and Cheung, 1978; Babb, 1994; Magnasco and Ottonelli, 1999b.fBishop et al., 1991.

6 These errors are completely removed when using best four-term reduced pseudospectra for H

and H2 obtained from a recently derived efficient interpolation technique (Figari et al., 2007).

Table 11.5 N-term convergence of isotropic C6 dispersion coefficients ðEha60Þ for the

homodimers of simple atoms and molecules

N H–H He–He Hþ2 --Hþ

2a H2–H2

b

1 6 0.695 1.449 4.7292 6.482 1.142 2.965 7.3253 6.498 1.186 3.218 7.3644 6.499 1.189 3.234 8.8255 6.499 1.242 3.234 9.64510 1.443 3.265 10.96515 1.455 3.284 11.26320 1.459 3.284 11.289Accurate 6.499c 1.461d 3.284e 11.324f

aH2þ : Re ¼ 2a0.

bH2: Re ¼ 1:4a0.cExact value.dYan et al., 1996.eBabb, 1994; Magnasco and Ottonelli, 1999b.fMagnasco and Ottonelli, 1996a.


C6 dispersion coefficients for the homodimers of some simple one-electronand two-electron atomic and molecular systems. It is seen that theconvergence rate of either a or C6 is remarkably slower for the two-electron systems.We see fromTable 11.4 that the five-term approximation gives 82% for

He, 99% for Hþ2 and 86% for H2, showing that convergence is sensibly

worst for the two-electron systems. Table 11.5 shows that the five-termapproximation, yielding practically the exact result for H–H, gives 85%forHe–He andH2–H2 and over 98% forHþ

2 --Hþ2 , confirming the similar

results for polarizabilities.

11.3.3 Other Expanded Interactions

At variance with the dispersion interaction, whose calculation at longrange requires knowledge of N-term pseudospectra of the monomermolecules, which are not observable quantities, the remaining compo-nents of the intermolecular interaction, the electrostatic and inductionenergies, are instead expressible in terms of physically observable electricproperties of the interactingmolecules, namely the permanentmomentsmand the total polarizabilities a of the individual molecules.

(a) As an example, the leading term of the electrostatic interactionbetween two dipolar HF(1Sþ ) molecules7 is given at long range bythe dipole–dipole interaction8 going as R� 3:

Ees1 ¼ m2

HF

R3ðsin uAsin uB cos w� 2 cos uAcos uBÞ ð11:54Þ

which has a minimum for the head-to-tail configuration of the twomolecules:

Ees1 ¼ � 2

m2HF

R3ð11:55Þ

for uA ¼ uB ¼ 180, leading to the formation of a collinear H-bond(H–F � � �H–F). The next quadrupole–dipole and dipole–quadrupole

7 Namely, molecules whose first nonzero permanent moment is the dipole moment.8 Equation 11.54 is just Equation 11.22 weighted in first order with the ground-state wavefunc-

tion when the resulting dipole components are expressed in spherical coordinates.


interaction terms, going as R� 4, favour instead an L configurationof the dimer, giving as a whole the structure of the homodimer withu � 60 observed by experiment and schematically represented inFigure 11.6.A similar result is obtained by the qualitative MO description of

the H-bond in (HF)2 depicted in the drawings of Figure 11.7, whichshowelectron transfer fromF lonepairs (doubly occupiedMOs) toavacant (empty MO) orbital on H. The top row shows s chargetransfer from F toH, the bottom row p charge transfer from F toH,whichmust be doubled on account of the twofold degeneracy of thep energy level (the figure shows the px MO). As a result, the dimerroughly assumes the noncollinear geometrical structure9 depicted inFigure 11.6.

(b) As a further example, the leading R� 6 term describing the polar-ization of the He atom by the HF molecule, namely

Eind2 ð6Þ ¼ � aHem

2HF

R6

3cos2uþ 1

2ð11:56Þ

H F

H

Fθ

Figure 11.6 Experimentally observed structure for (HF)2

H FH F..

H F

H

F..

Figure 11.7 MO description of the origin of the H-bond in (HF)2

9 It must be remarked that molecular beam experimentalists refer to the geometrical shape of

Figure 11.6 as a linear dimer.


is unable to discriminate between H-bonded (u ¼ 180) andanti-H-bonded (u ¼ 0) configurations, giving in both cases

Eind2 ð6Þ ¼ � 2

aHem2HF

R6ð11:57Þ

The next term inR� 7, which implies further polarization of theHeatom by themixed dipole–quadrupole moments of HF, contains acos3u term:10

Eind2 ð7Þ ¼ 3aHemHFQHF

R7cos3u ð11:58Þ

which stabilizes theH-bonded configurationHe � � �H–F (u ¼ 180),so that we can appropriately speak of formation of an H-bondbetween He and HF (Magnasco et al., 1989).

11.4 VAN DER WAALS AND HYDROGEN BONDS

From all we have seen so far, we can say that a VdW bond occurs whenthe small Pauli repulsion arising from the first-order interaction ofclosed-shell molecules at long range is overbalanced by weak attractivesecond-order induction and dispersion forces. VdW molecules are weaklybound complexeswith large-amplitudevibrational structure (Buckingham,1982).This is the caseof thedimersof the rare gasesX2 (X¼He,Ne,Ar,Kr,Xe) or the weak complexes between centrosymmetrical molecules like(H2)2 or (N2)2. Complexes between proton-donor and proton-acceptormolecules, like (HF)2 or (H2O)2, involve formation of hydrogen bonds(H-bonds), which are essentially electrostatic in nature, and lie on theborderline between VdWmolecules and ‘good’ molecules, having sensiblylarger intermolecular energies.Figure 11.8 shows the VdWpotential curve resulting for the interaction

of two ground-state He atoms at medium range. The upper curve is thefirst-order energy E1 mostly arising from repulsive (Pauli) exchange-overlap; the bottom curve is the resultant of adding the attractive energy~Edisp

2 due to London dispersion. A weak potential minimum (about33� 10�6Eh) is observed at the rather large distance of Re ¼ 5:6a0. It

10 Q is the permanent quadrupole moment of HF.

VAN DER WAALS AND HYDROGEN BONDS 237

should be remarked that, in this region of internuclear distances, bothelectrostatic and induction energies are negligible because of their charge-overlap nature. For simple 1s STOs on each He atom, with orbitalexponent c0, in fact (r ¼ c0R)

DEcb ¼ Ees1 ¼ 4c0

expð� 2rÞr

1þ 5

8r� 3

4r2 � 1

6r3

� �ð11:59Þ

~Eind

2 ¼ ~Eind;A

2 þ ~Eind;B

2 ¼ � 2Xi

jðA0AijUBÞj2«i

ð11:60Þ

and, for the neutral atom, UB decreases exponentially far from the Bnucleus.The situation is different for the long-range interaction of two ground-

state H2O molecules, whose potential energy curve is schematicallyshown in Figure 11.9. In this case, the first-order interaction E1 showsin the medium range a minimum mostly arising from the dipole–dipoleinteraction going as R� 3, and the second-order interaction simply dee-pens this minimum, strengthening the bond. As already seen in the case of(HF)2, it is appropriate in this case to speak of formation of a hydrogenbond, essentially electrostatic in origin. It is of interest to note the change

0 R/a0

5.6

ΔE /10-6 Eh

E1

E1 + E2~ disp

He2 Σg+1

Figure 11.8 Origin of the VdW bond in He2 (1Sþ

g )


in the scale factor for energy, from 10� 6Eh for He2 (VdW bond) to10� 3Eh for (H2O)2 (H-bond), even though roughly in the same region ofintermolecular distances (5.6a0 for He2 and 5.4a0 for the H2O dimer).On these grounds, some time ago, Magnasco et al. (1990a) derived a

simple electrostatic model for VdW complexes, where the angular geo-metry of the dimers was predicted in terms of just the first two observableelectric moments of the monomers. The model allowed for the successfulprediction of the most stable angular shapes of 35 VdW dimers.

11.5 THE KEESOM INTERACTION

Keesom (1921) pointed out that if two dipolarmolecules undergo thermalmotions, then they will on average assume orientations leading to attrac-tion according to

E6ðKeesomÞ ¼ � C6ðTÞR6

ð11:61Þ

C6ðTÞ being the T-dependent coefficient:

C6ðTÞ ¼ 2m2Am

2B

3kTð11:62Þ

0 R/a0

ΔE /10-3Eh

E1

E1 + E2~

(H2O)21A1

5.4

Figure 11.9 Origin of the hydrogen bond in (H2O)2 (1A1)

THE KEESOM INTERACTION 239

where T is the absolute temperature and k the Boltzmann constant. Thiscan be explained as follows.With reference to Figure 11.10, let us first give some alternative

expressions for the interaction between dipoles (Coulson, 1958):

V ¼ mA �mB

R3� 3

ðmA �RÞðmB �RÞR5

¼ mAmB

R3cosv� 3

ðmARcos uAÞðmBR cos uBÞR5

¼ mAmB

R3ðcosv� 3cos uAcos uBÞ

¼ mAmB

R3ðsin uAsin uB cos w� 2 cos uAcos uBÞ

ð11:63Þ

since, by the addition theorem (MacRobert, 1947):

cos v ¼ cos uAcos uB þ sin uA sin uB cos w ð11:64Þ

where w ¼ wA �wB is the dihedral angle between the planes specified bymA,mB andR. The last expression in (11.63) is themost convenient for us,giving the dipole interaction in terms of the spherical coordinatesR, uA, uBand w.It is convenient to put

W ¼ uA; uB;w

FðWÞ ¼ sin uA sin uB cos w� 2 cos uA cos uB

�ð11:65Þ

x

y

RθA θB

ϕA B z. ..

.

θA

θB

ω

A,µA

B, µB

R

ϕ

Figure 11.10 Different coordinate systems for two interacting dipoles


If all orientations were equally probable, then the average potentialenergy hVi, and hence the average first-order electrostatic hC3i coefficient(Magnasco et al., 1988; Magnasco, 2007), would be zero. In fact:

hViW ¼ mAmB

R3

ÐW dW FðWÞÐ

W dW¼ 0 ð11:66Þ

since

ðW

dW ¼ð2p0

dw

ðp0

duA sin uA

ðp0

duB sin uB ¼ 2pð1� 1

dxA

ð1� 1

dxB ¼ 8p

ð11:67Þ

where

xA ¼ cos uA; xB ¼ cos uB ð11:68Þ

ðW

dW FðWÞ ¼ð2p0

dw cos w

ð1� 1

dx ð1� x2Þ1=224

352

� 2

ð2p0

dw

ð1� 1

dx x

0@

1A

2

¼ 0

ð11:69Þ

The vanishing of the average potential energy for free orientations istrue for all multipoles (dipoles, quadrupoles, octupoles, hexadecapoles,etc.).The Boltzmann probability for a dipole arrangement whose potential

energy is V is instead proportional to

W / expð�V=kTÞ ð11:70Þ

We now average the quantityVexpð�V=kTÞ over all possible orienta-tions W assumed by the dipoles:

hVexpð�V=kTÞi ¼ mAmB

R3

ÐW dW FðWÞexp½aFðWÞ�Ð

W dW exp½aFðWÞ� ð11:71Þ


where we have introduced the T-dependent negative quantity

a ¼ � mAmB

R3kT<0 ð11:72Þ

We then obtain the familiar formula11

hVexpð�V=kTÞi ¼ mAmB

R3

ÐW dW FðWÞexp½aFðWÞ�Ð

W dW exp½aFðWÞ� ¼ mAmB

R3

d

dalnKðaÞ

ð11:73Þ

where

KðaÞ ¼ðW

dW exp½aFðWÞ� ð11:74Þ

is called the Keesom integral.We evaluate (11.73) for a � small (high temperatures and large

distances between the dipoles) by expanding the exponential in (11.74):

ðW

dW exp½aFðWÞ� �ðW

dW 1þ aFðWÞþ a2

2FðWÞ2 þ � � �

� �ð11:75Þ

where we have just seen that, in the expansion, the second integral12

vanishes, so that only the quadratic term contributes to the Keesomintegral.We have

ÐW dW FðWÞ2 ¼

ð2p0

dw

ðp0

duA sin uA

ðp0

duB sin uB

�ðsin2 uA sin2 uB cos2 wþ4cos2 uAcos

2 uBÞ

11 Relation (11.73) is much the same as that observed in the Debye theory of the orientation of

electric dipoles in gases, and in the Langevin (classical) or Brillouin (quantal) equations for theparamagnetic gas. In the last case, summations replace integrations over the parameter a.12 And all odd powers of it.


¼ð2p0

dw cos2 w

ð1� 1

dxA ð1� x2AÞð1� 1

dxB ð1� x2BÞ

þ 4

ð2p0

dw

ð1� 1

dxA x2A

ð1� 1

dxB x2B ¼ 8p

2

3ð11:76Þ

so that

ðW

dWa2

2FðWÞ2 ¼ 8p

a2

3ð11:77Þ

Then:

ðW

dW 1þ a2

2FðWÞ2

� �¼ 8p 1þ a2

3

� �ð11:78Þ

d

daln 8p 1þ a2

3

!" #¼ d

daln8pþ ln 1þ a2

3

!" #

¼ 1

1þ a2

3

2

3a � 2

3a

ð11:79Þ

for a � small. Hence, we obtain the final result for the average attractionenergy between the dipoles:

hVexpð�V=kTÞi � mAmB

R3

2

3a ¼ � 2

3kT

m2Am

2B

R6ð11:80Þ

This is known as the Keesom or dipole orientation energy (Equa-tions 11.61 and 11.62). This term depends on R� 6, but is temperaturedependent and decreases in importance with increasing T.It is of interest to compare the relative importance of all attractive

contributions to the intermolecular energy in the VdW region. For atomsand centrosymmetrical molecules, induction is zero, so that the only


contribution comes from attractive dispersion. For dipolar molecules,induction is usually negligible with respect to dispersion except forðLiHÞ2. The electrostatic energy is not zero when its thermal average istaken. The corresponding Keesom attractive energies (11.80) are, hence,the isotropic electrostatic contributions to the interaction energy and aretemperature dependent. A comparison between isotropic C6 coefficientsfor some homodimers at T ¼ 293 K is given in Table 11.6. It is seen thatKeesom C6ðTÞ is negligible compared with dispersion and inductioncoefficients for the homodimers of CO, NO, N2O, while for (NH3)2,(HF)2 and (H2O)2 the Keesom dipole orientation forces become increas-ingly dominant at room temperature, so they cannot be neglected inassessing collective gas properties such as the equation of state for realgases and virial coefficients.Magnasco et al. (2006) recently extended Keesom’s calculations up to

the R� 10 term, showing that deviations of the Keesom approximationfrom the full series expansion are less important than consideration ofthe higher order terms in the R� 2n expansion of the intermolecularpotential.Battezzati and Magnasco (2004) also gave an asymptotic evaluation

of the Keesom integral (11.74) for |a|� large,13 obtaining theformula

KðaÞ ffi 4p3

expð� 2aÞa2

1� 2

3a

� �ð11:81Þ

Table 11.6 Comparison between isotropic C6 coefficients ðEha60Þ for some homo-

dimers of atoms and molecules in the gas phase at T ¼ 293K

Atom–atom Dispersion Molecule–molecule Dispersion Induction Keesom

He2 1.46 (H2)2 12.1 0 0Ne2 6.35 (N2)2 73.4 0 0H2 6.50 (CO)2 81.4 0.05 0.002Ar2 64.9 (NO)2 69.8 0.08 0.009Kr2 129 (N2O)2 184.9 0.19 0.017Be2 213 (NH3)2 89.1 9.82 81.3Xe2 268 (H2O)2 45.4 10.4 204Mg2 686 (HF)2 19.0 6.3 227Li2 1450 (LiH)2 125 299 8436

13 This is the case of low temperatures and small distances between the dipoles, aswell as the case

of (LiH)2, where the interacting dipoles are very large.


which was confirmed by a recent independent calculation by Abbott(2007). The expansion in inverse powers of |a| now gives the pairpotential energy at low temperatures as

hVexpð�V=kTÞi ffi � 2mAmB

R3þ 2kTþ 2

3

R3

mAmB

ðkTÞ2 ð11:82Þ


12Symmetry

In what follows we give a short outline of the importance of molecularsymmetry in quantum mechanics and how group theoretical techniquesmay be of help in assessing symmetry-adapted functions for the factor-ization of secular equations either in MO or VB calculations.

12.1 MOLECULAR SYMMETRY

Symmetry is a property that can be used for simplifying the study of aphysical system, even without doing any effective calculation. Mostsimple molecules have a symmetry depending on the geometrical config-uration of their nuclei, and which can be classified according to themolecular point group to which they belong, for example, for the first-row hydrides, linear HF (C¥v), bent H2O (C2v), pyramidal NH3 (C3v),tetrahedral CH4 (Td).Of the greatest importance is the symmetry of any function for use in

atomic or molecular calculations, since the fundamental theorem ofsymmetry states that thematrix element of any totally symmetric operatorO (the identity I, or any model Hamiltonian such as total H, Fock F, orH€uckel H) between functions having different symmetries is identicallyzero:

hxmjOjxni ¼ 0; O ¼ I; H; F;H; n=m ð12:1Þ


Valerio Magnasco


Functions having definite symmetry properties can be obtainedby letting suitable projection operators (projectors) act upon a functionhaving no specific symmetry, as the following two examples show.First, it is known from elementary mathematics that a function f(x) can

be classified as even or odd with respect to the interchange x ) �x (forinstance, cos x or sin x). An arbitrary function without any symmetry canalways be expressed as a linear combination of an even and an oddfunction as

f ðxÞ ¼ 1

2f ðxÞþ f ð�xÞ½ � þ 1

2f ðxÞ�f ð�xÞ½ � ¼ 1

2gðxÞþ 1

2uðxÞ ð12:2Þ

where we use g(x) (from German gerade) for the even function and u(x)(German, ungerade) for the odd function. The operation (12.2) can beaptly called the resolution of the function f(x) into its components havingdefinite symmetry properties, and it is immediately evident that anyintegral like

ð¥�¥

dx gðxÞuðxÞ ¼ 0 ð12:3Þ

is identically zero.1

Second, there is the split-shell description of the atomic or moleculartwo-electron problem. We have seen in Chapters 4 and 9 that an accep-table two-electron wavefunction for the ground states of He (1S) or H2

(1Sþg ) is given by

Y ¼ aðr1Þbðr2Þþbðr1Þaðr2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2S2

p 1ffiffiffi2

p aðs1Þbðs2Þ�bðs1Þaðs2Þ½ � ð12:4Þ

where a ¼ 1s, b ¼ 1s0 forHe (1S) (the Eckart wavefunction), and a ¼ 1sA,b ¼ 1sB for H2 (

1Sþg ) (the HLwavefunction). Both wavefunctions are the

product of a symmetrical space part by an antisymmetrical spin part. Thesimple product a(r1)b(r2) has no symmetry properties and must be acted

1 Changing x into �x the integral is changed into itself with a minus sign.

248 SYMMETRY

upon by the appropriate symmetrizing operator Ps:

Ps ¼ 1

2ðIþ PÞ; P

a ¼ 1

2ðI�PÞ ð12:5Þ

where P is the operator that interchanges r1 and r2. It is easily seen that Ps

and Pahave the typical properties of projection operators (idempotency,

mutual exclusivity, completeness2):

PsPs ¼ P

s; P

aPa ¼ P

a; P

sPa ¼ P

aPs ¼ 0; P

s þ Pa ¼ I ð12:6Þ

A symmetry operation R (reflection across a symmetry plane, positiveor negative rotation3 about a symmetry axis, roto-reflection, inversionabout a centre of symmetry, etc.) is that operation that interchangesidentical nuclei, and can be defined in either of two equivalent ways:

Active representation, where we interchange physical pointsleaving unaltered the coordinate frame ð12:7Þ

or

Passive representation, where we act upon axesand leave unaltered points ð12:8Þ

Even if conceptually the passive representation is to be preferred, theactive representation is often more easily visualizable. We shall alwaysrefer to a symmetry operation as a change of coordinate axes.Symmetry operations are described by linear operators R4 having

as representatives in a given orthonormal basis x orthogonal matricesD(R)¼R, constructed as follows:

Rx ¼ xDxðRÞ ð12:9Þ

x�Rx ¼ ðx�xÞDxðRÞ ¼ DxðRÞ ð12:10Þ

2 In mathematics, also called the resolution of the identity.3 Positive rotations are always assumed anticlockwise and negative rotations clockwise.4 The operators R commute with the Hamiltonian operator H, namely are constants of the

motion.

MOLECULAR SYMMETRY 249

If a basis x is changed into a basis x0 by the transformation U, then therepresentativeDx0 ðRÞ of R in the new basis is related to the representativeDxðRÞ in the old basis by what is called a similarity transformation:

x0 ¼ xU ) Dx0 ðRÞ ¼ U�1DxðRÞU ð12:11Þ

Now, let Rqbe the coordinate transformationof the spacepointqunderthe symmetry operation R:

q0 ¼ Rq; ð12:12Þ

and

f 0ðqÞ ¼ Rf ðqÞ ð12:13Þ

be the transformation of the function f(q) under the operation R. Atransformation of the coordinate axes does not alter the value of thefunction at every point of space:

f 0ðq0Þ ¼ f ðqÞ ð12:14Þ

Namely:

Rf ðRqÞ ¼ f ðqÞ ð12:15Þ

If this equation is applied to the point R�1q, then we obtain the basicrelation

Rf ðqÞ ¼ f ðR�1qÞ ð12:16Þ

Namely, the function transformed under a symmetry operation is equal tothe function obtained by subjecting its argument to the inversetransformation.As an example, with reference to Figure 12.1, consider the rotationCþ

a

of the functions (px py) whose angular part is defined as

px / sin u cos w; py / sin u sin w ð12:17Þ

250 SYMMETRY

We obtain the transformations

w0 ¼ Cþa w ¼ w�a; C�

aw ¼ wþa ð12:18Þ

Cþa pxðwÞ ¼ pxðC�

a wÞ ¼ pxðwþaÞ ¼ sin u cosðwþaÞ¼ sin uðcos w cos a�sin w sin aÞ ¼ px cos a�py sin a

ð12:19Þ

Cþa pyðwÞ ¼ pyðC�

awÞ ¼ pyðwþaÞ ¼ sin u sinðwþaÞ¼ sin uðsin w cos aþ cos w sin aÞ ¼ px sin aþ py cos a

ð12:20Þ

which can be written in matrix form as5

Cþa ðpx pyÞ ¼ ðCþ

a pxCþa pyÞ ¼ ðpx pyÞ cos a sin a

�sin a cos a

� �¼ ðpx pyÞDpðCþ

a Þ

ð12:21Þ

5 The corresponding matrix representative for reflection across the plane specified by sa is

cos 2a sin 2asin 2a �cos 2a

� �.

qf (q )

qf(q)

′′ ′

0 0x

x

y

y

′

′

αϕϕ

Cα+

Figure 12.1 The function transformed under the positive rotationCþa is equal to the

function whose argument is transformed under the negative rotation C�a

MOLECULAR SYMMETRY 251

Care must be taken in noting that the same operation R may have adifferent effectwhen acting on adifferentbasis. The transformationunderCþ

a of the d-functions in the xy-plane:

dx2--y2 / sin2 u cos 2w; dxy / sin2 u sin 2w ð12:22Þ

gives in fact

Cþa ðdx2--y2dxyÞ¼ðCþ

a dx2�y2Cþa dxyÞ ¼ ðdx2�y2dxyÞ cos2a sin2a

�sin2a cos2a

� �¼ðdx2--y2dxyÞDdðCþ

a Þð12:23Þ

Therefore, it is always necessary to specify the basis which the matrixrepresentative refers to. These and other matrix representatives ofdifferent symmetry operations may be found elsewhere (Magnasco,2007).As a last point on symmetry, it must be recalled that

If RS ¼ T in coordinate space; then RS ¼ T in function space

andDðRÞDðSÞ ¼ DðTÞ in matrix space ð12:24Þ

12.2 GROUP THEORETICAL METHODS

Let us now briefly introduce the axioms defining the concept of agroup.An abstract groupGfG1;G2; . . . ;Ghg of order h is given by a closed set

of h elements satisfying the following properties:

(i) There is a composition law (usually, but not necessarily, themultiplication law) such that, for Gr and Gs belonging to G,GrGs ¼ Gt still belongs to G (we then say that a group is a setclosed with respect to symbolic multiplication).

(ii) The composition law is associative: ðGrGsÞGt ¼ GrðGsGtÞ.(iii) There is an identity (or neutral) element Gm, such that GrGm ¼

GmGr ¼ Gr.(iv) Each element has an inverse G�1

r , such that GrG�1r ¼ G�1

r Gr ¼Gm.

252 SYMMETRY

(v) In general, the commutative law does not hold for the symbolicmultiplication; namely: GrGs =GsGr. If GrGs ¼ GsGr, then thegroup is said to be Abelian (a commutative group).

Almost all textbooks on group theory contain tables of molecular pointgroups, their relation to the chemical structure of some polyatomicmolecules being given for instance in Eyring et al. (1944). We give belowtwo examples of molecular point groups and their multiplication tables(Tables 12.1–12.4): the Abelian group C2v of the H2O molecule andthe groupC3v ofNH3.Apoint groupof orderhhash symmetry operationscharacterizing it. So, C2v is a group of order 4, while C3v is a group oforder 6.Tables 12.1 and 12.3 are called character tables of the point groups. In

the first column of both tables, below the denomination of the pointgroup, are the symbols denoting the symmetry-defined types or, in the

Table 12.2 Themultiplication table of the point groupC2v

C2v I C2 sv s0v

I I C2 sv s0v

C2 C2 I s0v sv

sv sv s0v I C2

s0v s0

v sv C2 I

Table 12.1 The molecular point group C2v

C2v I C2 sv s0v

A1 1 1 1 1A2 1 1 �1 �1B1 1 �1 �1 1B2 1 �1 1 �1

Table 12.3 The molecular point group C3v

C3v I Cþ3 C�

3 s1 s2 s3

A1 1 1 1 1 1 1A2 1 1 1 �1 �1 �1

E1 00 1

� ��c s�s �c

� ��c �ss �c

� �1 00 �1

� ��c �s�s c

� ��c ss c

� �

GROUP THEORETICAL METHODS 253

language of group theory, the irreducible representations (in short, irreps)of that group. For each irrep, symmetry operations are given in the formofmatrices like (12.21) or (12.23), whose order equals the dimensionality ofthe irrep.6 The characters are simply the trace of suchmatrices, and are theonly numbers given in textbooks. While C2v has only one-dimensionalirreps, the higher symmetryC3v group has two one-dimensional irreps (A1

and A2) and one two-dimensional irrep (E).7

Tables 12.2 and12.4 give themultiplication tables of the twogroups. Inconstructing a multiplication table, we recall that Rk ¼ RiRj is the resultof the intersection of the column headed by Ri and the row headed by Rj,and that the operation on the right must be done first. It is seen that eachsymmetry operation occurs once in each row.We now give a few further definitions and fundamental theorems.

12.2.1 Isomorphism

Two groups G and G0 of the same order h are isomorphic if (i) there is aone-to-one correspondence between each element Gr of G and G0

r ofG0ðr ¼ 1; 2; . . . ; hÞ and (ii) the symbolic multiplication rule is preserved,namely, ifGrGs ¼ Gt inG () G0

rG0s ¼ G0

t inG0. If only (ii) is true then

the groups are said to be homomorphic.

12.2.2 Conjugation and Classes

Any two elementsA andB of a groupG are said to be conjugate if they arerelated by a similarity transformation with one other element X of the

Table 12.4 The multiplication table of the point group C3v

C3v I Cþ3 C�

3 s1 s2 s3

I I Cþ3 C�

3 s1 s2 s3

Cþ3 Cþ

3 C�3 I s2 s3 s1

C�3 C�

3 I Cþ3 s3 s1 s2

s1 s1 s3 s2 I C�3 Cþ

3s2 s2 s1 s3 Cþ

3 I C�3

s3 s3 s2 s1 C�3 Cþ

3 I

6 In C3v, c ¼ 1=2, s ¼ ffiffiffi3

p=2.

7 Cubic groups have also three-dimensional irreps (T).

254 SYMMETRY

group, namely: A ¼ X�1BX. The set of all conjugate elements defines aclass. Conjugate operations are always of the same type (rotations withrotations, reflections with reflections, etc.). The number of classes equalsthe number of irreps.

12.2.3 Representations and Characters

Let GfG1;G2; . . . ;Ghg be a group of h elements and fDðG1Þ;DðG2Þ;. . . ;DðGhÞg a group of matrices isomorphic to G. We then say that thegroupofmatrices gives a representation (German,darstellung, henceD) ofthe abstract group. If we have a representation of a group in the form of agroupofmatrices, thenwealsohave an infinite numberof representations.In fact, we can always subject all matrices of a given representation to asimilarity transformation, thereby obtaining a new representation, and soon, the multiplication rule being preserved during the similarity transfor-mation. If, byapplyinga similarity transformationwithaunitarymatrixUtoarepresentationofagroupG intheformofagroupofmatrices,weobtaina new representation whose matrices have a block-diagonal form, we saythat the representation has been reduced.The set of functions that are needed to find a (generally, reducible)

representation G forms a basis for the representation. The functionsforming a basis for the irreducible representations (irreps) of a symmetrygroup are said to be symmetry-adapted functions, and transform in thesimplest and characteristic way under the symmetry operations of thegroup. It is of basic importance in quantum chemistry to find suchsymmetry-adapted functions, starting from a given basis set through useof suitable projection operators, as we shall see.As already said before, the character is the trace of the matrix repre-

sentative of the symmetry operator R and is denoted by x(R). Thecharacters have the following properties:

1. They are invariant under any transformation of the basis.2. They are the same for all symmetry operations belonging to the same

class.3. The condition for two representations to be equivalent is that they

have the same characters.

12.2.4 Three Theorems on Irreducible Representations

1. The necessary and sufficient condition that a representation G beirreducible is that the sum over all operations R of the group of the

GROUP THEORETICAL METHODS 255

squares of the moduli of the characters be equal to the order h of thegroup: X

R

x�ðRÞxðRÞ ¼XR

jxðRÞj2 ¼ h ð12:25Þ

2. Given any two irreducible representations G i and Gj of a group, wehave the orthogonality theorem for the characters:

XR

xiðRÞ�xjðRÞ ¼ hdij ð12:26Þ

3. This theorem is a particular case of the more general orthogonalitytheorem for the components of the representative matrices of the helements of the group:

XR

DiðRÞ�mnDjðRÞm0n0 ¼

h

‘idijdmm0dnn0 ð12:27Þ

where ‘i is the dimensionality of the ith irrep.

12.2.5 Number of Irreps in a Reducible Representation

The number of times aj a given irrep Gj occurs in the reducible representa-tion G follows from the orthogonality theorem and is given by

aj ¼ 1

h

XR

xjðRÞ�xGðRÞ ð12:28Þ

12.2.6 Construction of Symmetry-adapted Functions

Symmetry-adapted functions transforming as the l-component of the jthirreducible representation Gj are obtained by use of the projector

Pj

ll ¼‘jh

XR

DjðRÞ�llR ð12:29Þ

256 SYMMETRY

which requires the complete knowledge of matrices Dj(R) for eachoperation R of the group. This projector is needed in the case of multi-dimensional irreps, but for one-dimensional irreps it is sufficient to use thesimpler projector involving the characters

Pj /

XR

xjðRÞ�R ð12:30Þ

12.3 ILLUSTRATIVE EXAMPLES

Use of symmetry-adapted functions is particularly useful for predictingwhich matrix elements of an operator are zero (selection rules, mostly inspectroscopy), even without doing any effective calculation, and in thefactorization of the secular equations arising either in MO, CI, or VBcalculations. We shall illustrate below these factorizations for theminimal basis set H€uckel calculations for the H2O (C2v) and NH3

(C3v) molecules.

12.3.1 Use of Symmetry in Ground-state H2O (1A1)

The H2Omolecule is chosen to lie in the yz-plane, with the O atom at theorigin of a right-handed coordinate system having z as binary symmetryaxis (Figure 12.2). The nuclear symmetry implies two symmetry planes (yz

y

z

σv

x

H2

O

. .

.

H1

σv′

C2

Figure 12.2 Elements of C2v symmetry in H2O

ILLUSTRATIVE EXAMPLES 257

and zx), their intersection determining the twofold symmetry axisC2. Themolecular point group is hence C2v, whose character table was given inTable 12.1. Theminimal set for theMOcalculation is given, in an obviousbrief notation, by the row matrix of the m ¼ 7 STOs:

x ¼ ðk s z x y h1 h2Þ ð12:31Þ

which must be combined linearly to give seven MOs, the first n ¼ 5being doubly occupied by electrons with opposite spin, so accomodat-ing the N ¼ 2n ¼ 10 electrons of the molecule in its totally symmetricsinglet 1A1 ground state. The LCAO coefficients of the Roothaanequations are obtained by the Ritz method through the iterative solu-tion of a (7� 7) secular equation, but for our illustrative purposes it willbe sufficient to consider the simple H€uckel approximation in the samebasis. The construction of the symmetry-adapted basis for the calcula-tion can be done at once by simple inspection, since it is immediatelyevident that the functions belonging to the different irreps of the pointgroup C2v are

k; s; z;hz ¼ 1ffiffiffi2

p ðh1 þ h2Þ ) A1 ð12:32Þ

x ) B1 ð12:33Þ

y; hy ¼ 1ffiffiffi2

p ðh1�h2Þ ) B2 ð12:34Þ

Since functions of different symmetry cannot mix under the totallysymmetric H€uckel operator H, the (7� 7) H€uckel representative H overthe minimum set will be factorized into three blocks of different symme-tries: A1 (4�4), B1 (1� 1) and B2 (2� 2).Turning to the more formal group theoretical techniques, we give in

Table 12.5 the dimensions of the representative matrices for thedifferent operations R in the reducible representation G in the originalbasis, and those of the irreducible representations Gj corresponding tothe symmetry-adapted functions of the last column. The latter are easilyobtained by letting the projector (12.30) act on the original AObasis (12.31). It is left to the reader to verify in this case all the

258 SYMMETRY

properties and theorems introduced in the formal group theory of thepreceding section.Just to be clear, we give below the construction of the matrix repre-

sentative D(C2) for the reducible representation G in the original ba-sis (12.31). We must first construct the transformation table of the basisfunctions (12.31) under the symmetry operations of C2v. Using the activetransformation, we obtain Table 12.6.Following the recipe (12.9), we then immediately obtain

DðC2Þ ¼

1 � � � � � �� 1 � � � � �� 1 � � � �� 1 � � �� 1 � �� 1

� � � � � 1 �

0BBBBBBBBBBB@

1CCCCCCCCCCCA; trDðC2Þ ¼ 1 ð12:35Þ

Table 12.5 Reducible representation G and symmetry-adapted AOs for H2O

C2v I C2 sv s0v Symmetry basis

A1 1 1 1 1 k; s; z; hz ¼ 1ffiffi2

p ðh1 þ h2ÞA2 1 1 �1 �1

B1 1 �1 �1 1 x

B2 1 �1 1 �1 y; hy ¼ 1ffiffi2

p ðh1�h2ÞG 7 1 5 3 GA1 ¼ 4; GB1 ¼ 1; GB2 ¼ 2

Table 12.6 Transformation table of the AO basis forH2O under the operations of C2v

Rx I C2 sv s0v

k k k k ks s s s sz z z z zx x �x �x xy y �y y �yh1 h1 h2 h1 h2h2 h2 h1 h2 h1


as reported in the last row of Table 12.5. We can proceed similarly forD(I), DðsvÞ, and Dðs0

vÞ. Using the standard techniques of the matrixeigenvalue problem, it is left as an easy exercise for the reader to verifythat the lower off-diagonal block of the representative matrices D(C2)and Dðs0

vÞ in the original basis set (12.31) are diagonalized in thesymmetry-adapted basis (hzhy), namely by transformation with theunitary matrix

U ¼

1ffiffiffi2

p 1ffiffiffi2

p

1ffiffiffi2

p � 1ffiffiffi2

p

0BBBBB@

1CCCCCA ð12:36Þ

12.3.2 Use of Symmetry in Ground-state NH3 (1A1)

The three hydrogen atoms of theNH3molecule are chosen to lie in the xy-plane, numbered anticlockwise with H1 on the positive x-axis, the originof the right-handed coordinate system being taken at their intersectionand theNatomalong the symmetry axiswith z positive (Figure 12.3). Thenuclear symmetry now implies three symmetry planes at 120� (s1,s2,s3),their intersection determining the threefold symmetry axis C3. The

σ3

x

z

σ1

σ2

y

H2

H3

H1

0

N

C3

C3+

_

Figure 12.3 Elements of C3v symmetry in NH3

260 SYMMETRY

molecular point group is hence C3v, whose character table was given inTable 12.3. The minimal basis set for the MO calculation is given, in anobvious brief notation, by the row matrix of the m ¼ 8 STOs:

x ¼ ðk s z x y h1 h2 h3Þ ð12:37Þ

which must be combined linearly to give eight MOs, the first n ¼ 5 beingdoubly occupied by electrons with opposite spin, so accommodating theN ¼ 2n ¼ 10 electrons of themolecule in its totally symmetric singlet 1A1

ground state.Even in this case, the constructionof the symmetry-adapted basis for the

calculation can be done at once by simple inspection, since it is imme-diately evident that the functions belonging to the different irreps of thepoint group C3v are

k; s; z; hz ¼ 1ffiffiffi3

p ðh1 þ h2 þ h3Þ ) A1 ð12:38Þ

x; hx ¼ 1ffiffiffi6

p ð2h1�h2�h3Þ


p ðh2�h3Þ) E

8>>>><>>>>:

ð12:39Þ

the last being a doubly degenerate irrep whose basic vectors transform as(x y).Turning to the group theoretical techniques, following what

was done before for H2O, Table 12.7 gives the dimensions of therepresentative matrices for the different operations R in the reduciblerepresentation G in the original basis and those of the irreduciblerepresentations Gj corresponding to the symmetry-adapted functionsof the last column. The latter are obtained by letting the full projec-tor (12.29) act on the original AO basis (12.37). Table 12.8 is thetransformation table of the original AO basis (recall that in the tablec ¼ 1=2, s ¼ ffiffiffi

3p

=2).In the case of NH3, because of the presence of the doubly degenerate

irrep E, the simple projector (12.30) based on characters would yieldnonorthogonal linearly dependent symmetry functions, which shouldthen be Schmidt orthogonalized to give a linearly independent set. It is


convenient, therefore, to use at once the full projector (12.29), as we shallshow in detail below.

A1 : PA1k ¼ k P

A1s¼ s P

A1z¼ z

PA1x ¼ 1

6ðx�cx�sy�cxþ syþx�cx�sy�cxþ syÞ ¼ 0

PA1y ¼ 1

6ðyþ sx�cy�sx�cy�y�sxþ cyþ sxþ cyÞ ¼ 0

PA1h1 ¼ 1

6ð2h1þ2h2þ2h3Þ ¼ 1

3ðh1þh2þh3Þ ) 1ffiffiffi

3p ðh1þh2þh3Þ

Table 12.8 Transformation table of the AO basis for NH3 under the operationsof C3v

Rx I Cþ3 C�

3 s1 s2 s3

k k k k k k ks s s s s s sz z z z z z zx x �cxþ�sy �cxþ sy x �cxþ�sy �cxþ syy y sxþ�cy �sxþ�cy �y �sxþ cy sxþ cyh1 h1 h3 h2 h1 h3 h2h2 h2 h1 h3 h3 h2 h1h3 h3 h2 h1 h2 h1 h3

Table 12.7 Reducible representation G and symmetry-adapted AOs for NH3

C3v I Cþ3 C�

3 s1 s2 s3 Symmetrybasis

A1 1 1 1 1 1 1 k; s; z; hz ¼ 1ffiffi3

p ðh1 þ h2 þ h3Þ

A2 1 1 1 �1 �1 �1

E1 00 1

� ��c s�s �c

� ��c �ss �c

� �1 00 �1

� ��c �s�s c

� ��c ss c

� � x; hx ¼ 1ffiffiffi6

p ð2h1�h2�h3Þ


p ðh1�h2Þ

8>>>><>>>>:

G 8 2 2 4 4 4 GA1 ¼4; GEx ¼2; GEy ¼2

262 SYMMETRY

A2 : PA2k¼ 1

6ðkþkþk�k�k�kÞ ¼ 0 P

A2s¼ P

A2z¼ 0

PA2x¼ 1

6ðx�cx�sy�cxþ sy�xþ cxþ syþ cx�syÞ ¼ 0

PA2y¼ 1

6ðyþ sx�cy�sx�cyþyþ sx�cy�sx�cyÞ ¼ 0

PA2h1 ¼ 1

6ðh1þh3þh2�h1�h3�h2Þ ¼ 0

E : PE

xxk¼ 2

6ðk�ck�ckþk�ck�ckÞ ¼ 0 P

E

xxs¼ PE

xxz¼ 0

PE

xxx¼ 2

6x�cð�cx�syÞ�cð�cxþ syÞþx�cð�cx�syÞ�cð�cxþ syÞ½ �

¼ 1

3ð2xþ c2xþ c2xþ c2xþ c2xÞ ¼ x

8>>><>>>:

PE

xxy¼2

6y�cðsx�cyÞ�cð�sx�cyÞ�y�cð�sxþ cyÞ�cðsxþ cyÞ½ �

¼ 1

3ðy�csxþ c2yþ csxþ c2y�yþ csx�c2y�csx�c2yÞ ¼ 0

8>>><>>>:

PE

xxh1 ¼2

6ðh1�ch3�ch2þh1�ch3�ch2Þ ¼ 1

3ð2h1�h2�h3Þ

) 1ffiffiffi6

p ð2h1�h2�h3Þ

8>>><>>>:

PE

yyk¼ 2

6ðk�ck�ck�kþ ckþckÞ ¼ 0 P

E

yys¼ PE

yyz¼ 0

PE

yyx¼ 2

6x�cð�cx�syÞ�cð�cxþ syÞ�xþ cð�cx�syÞþcð�cxþ syÞ½ �

¼ 1

3ðxþ c2xþ csyþ c2x�csy�x�c2x�csy�c2xþ csyÞ ¼ 0

8>>><>>>:


PE

yyy¼2

6y�cðsx�cyÞ�cð�sx�cyÞþyþcð�sxþ cyÞþcðsxþ cyÞ½ �

¼ 1

3ðy�csxþ c2yþ csxþc2yþy�csxþc2yþcsxþ c2yÞ ¼ y

8>>><>>>:

PE

yyh1 ¼2

6ðh1�ch3�ch2�h1þ ch3þ ch2Þ ¼ 0

PE

yyh2 ¼2

6ðh2�ch1�ch3�h3þch2þ ch1Þ ¼ 1

3

3

2h2�3

2h3

!

) 1ffiffiffi2

p ðh2�h3Þ

8>>>><>>>>:

The reducible representation G of Table 12.7 splits into the followingirreps:

G ¼ 4A1 þ 2E ð12:40Þ

in accord with the result of Equation 12.28.Therefore, the (8� 8) secular equation in the original AO basis fac-

torizes into a (4� 4) block of symmetry A1 and two (2� 2) blocks ofsymmetry E, belonging to the symmetries Ex and Ey (Figure 12.4) whichare mutually orthogonal and not interacting.We end by recalling that MOs resulting from all LCAO methods are

classified according to the irreducible representations to which they

8 x 8 4 x 4

A1

Ex

Ey

2 x 2

2 x 2

= 0

0

0

Figure 12.4 Factorization under C3v of the minimum basis set MO secular equationfor NH3

264 SYMMETRY

belong, using lower case letters preceded by a principal quantum numbern specifying the order of the orbital energies (Roothaan, 1951b). In thisway, the electron configurations of ground and excited states ofmoleculesare given much in the same way as those familiar for atoms. In our case oftheN¼ 10-electronmolecules H2O andNH3, the electron configurationsof their totally symmetric singlet 1A1 ground states are given by

H2O ðC2v;1A1Þ : 1a21 2a

21 1b

22 3a

21 1b

21 ð12:41Þ

NH3 ðC3v;1A1Þ : 1a21 2a

21 1e

4 3a21 ð12:42Þ


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REFERENCES 273

Author Index

Abbott, P.C., 245

Abramowitz, M., 40, 47

Aitken, A.C., 21, 23

Atkin, R.H., 37

Atkins, P.W., 197

Babb, J.F., 234

Babb, J.F., see Yan, Z.C., 225, 234

Bacon, G.E., 92

Ballinger, R.A., 111

Bartlett, R.J., 136

Battezzati, M., 244

Battezzati, M., seeMagnasco, V., 244

Bishop, D.M., 234

Bonham, R.A., see Strand, T.G., 201

Born, M., 1, 142

Boys, S.F., 49

Briggs, M.P., 231

Brink, D.M., 44, 227

Buckingham, A.D., 237

Cade, P.E., 111, 112

Casimir, H.B.G., 223

Cencek, W., 153

Cheung, L.M., see Bishop, D.M., 234

Clementi, E., 111, 201

Cooper, D.L., 181

Corongiu, G., see Clementi, E., 111

Costa, C., see Figari, G., 234

Costa, C., see Magnasco, V., 153,

154, 225, 231, 237, 239, 241, 244

Coulson, C.A., 164, 221, 240

Cybulski, S.M., see Bishop,

D.M., 234

Dalgarno, A., see Yan, Z.C., 225, 234

Das, G., see Wahl, A.C., 135

Davies, D.W., 197

Dirac, P.A.M., 82, 86

Dixon, R.N., 77

Drake, G.W.F., see Yan, Z.C.,

225, 234

Dunning, T.H., Jr., 110

Dunning, T.H., Jr., see Woon,

D.E., 110

Eckart, C., 64

Ermler, W.C., see Rosenberg,

B.J., 136

Eyring, H., 35, 42, 253

Figari, G., 224, 234

Figari, G., see Magnasco, V., 225,

231, 237, 239, 241

Methods of Molecular Quantum Mechanices: An Introduction to Electronic Molecular Structure

Valerio Magnasco


Fock, V., 99

Frankowski, K., 135

Friedman, R.S., see Atkins, P.W., 197

Gerratt, J., see Cooper, D.L., 181

Hall, G.G., 100, 103

Hartree, D.R., 13, 99

Heisenberg, W., 17

Heitler, W., 148

Herzberg, G., see Huber, K.P.,

153, 160

Hobson, E.W., 40

Hohenberg, P., 138

Hohn, F.E., 21

Huber, K.P., 153, 160

Huo, W.H., see Cade, P.E., 111, 112

H€uckel, E., 100

Ince, E.L., 38

Karplus, M., 203

Kato, T., 134

Keesom, W.H., 239

Kettle, S.F.A., see Murrell, J.N., 118

Kimball, G.E., see Eyring, H., 35,

42, 253

Klopper, W., 136

Klopper, W., see Noga, J., 138

Koga, T., 224, 225

Kohn, W., 138

Kohn, W., see Hohenberg, P., 138

Kolker, H.J., see Karplus, M., 203

Kutzelnigg, W., 134

Kutzelnigg, W., see Klopper, W., 136

Kutzelnigg, W., see Noga, J., 137

Laaksonen, L., see Sundholm, D.,

111

Lazzeretti, P., 109, 111

Lennard-Jones, J.E., 102, 114, 130, 131

Lipscomb, W.N., see Stevens,

R.M., 203

Liu, B., 153

London, F., see Heitler, W., 148

Longuet-Higgins, H.C., 143, 227

L€owdin, P.O., 82

MacDonald, J.K.L., 67

MacRobert, T.M., 44, 240

Magnasco, V., 30, 54, 58, 63, 68, 69,

70, 95, 100, 150, 151, 153, 154,

161, 167, 187, 189, 194, 196, 204,

215, 225, 229, 230, 231, 232, 233,

234, 237, 239, 241, 244, 252

Magnasco, V., see Battezzati, M., 244

Magnasco,V., see Figari,G., 224, 234

Margenau, H., 2, 6, 21

Matsumoto, S., seeKoga,T., 224, 225

McLean, A.D., 112, 228

McLean, A.D., see Liu, B., 153

McWeeny, R., 49, 85, 89, 91, 97,

130, 131, 137

Meyer, W., 230

Mohr, P.J., 13

Møller, C., 136

Moore, C.E., 69, 70

Mulder, J.J.C., 155

Mulliken, R.S., 95

Murphy, G.M., see Margenau, H., 21

Murrell, J.N., 118

Murrell, J.N., see Briggs, M.P., 231

Noga, J., 137, 138

Noga, J., see Tunega, D., 109

Oppenheimer, J.R., see Born,M., 142

Ottonelli, M., see Magnasco V., 225,

229, 230, 232, 233, 234

Pauli, W., 75

Pauling, L., 167

Peek, J.M., 153

Pekeris, C.L., 64, 201

Pekeris, C.L., see Frankowski, K., 135

Pelloni, S., see Lazzeretti, P., 111

Pipin, J., see Bishop, D.M., 234

Pitzer, R.M., see Stevens, R.M., 203

276 AUTHOR INDEX

Plesset, M.S., see Møller, C., 136

Polder, D., see Casimir, H.B.G., 223

Purvis III, G.D., see Bartlett,

R.J., 136

Pyykk€o, P., see Sundholm, D., 111

Raimondi, M., see Cooper, D.L., 181

Rapallo, A., see Magnasco, V., 244

Ritz, W., 64

Roetti, C., see Clementi, E., 111, 201

Roos, B., 134

Roothaan, C.C.J., 71, 100, 103, 265

Rosenberg, B.J., 109, 136

Ruedenberg, K., 153

Rui, M., see Figari, G., 234

Rui, M., see Magnasco, V., 225

Rutherford, D.E., 7

Rychlewski, J., see Cencek, W., 153

Salahub, D.R., see Briggs, M.P., 231

Satchler, G.R., see Brink, D.M.,

44, 227

Schroedinger, E., 17, 19

Sham, L.J., see Kohn, W., 138

Shavitt, I., see Bartlett, R.J., 136

Shavitt, I., see Rosenberg, B.J.,

109, 136

Slater, J.C., 46, 86, 97

Stegun, I.A., see Abramowitz, M.,

40, 47

Stevens, R.M., 203

Strand, T.G., 201

Sundholm, D., 111

Taylor, B.N., see Mohr, P.J., 13

Tedder, J.M., see Murrell, J.N., 118

Thakkar, A.J., 225

Tillieu, J., 202, 203

Troup, G., 19

Tunega, P., 109

Valiron, P., see Noga, J., 138

Van Duijneveldt, F.B., see Van

Duijneveldt-Van de Rijdt,

J.G.C.M., 108

Van Duijneveldt-Van de Rijdt,

J.G.C.M., 108

Van Vleck, J.H., 206

Wahl, A.C., 135

Walter, J., see Eyring, H., 35, 42, 253

Wheland, G.W., 162

Wigner, E.P., 82, 155

Wolniewicz, L., 153

Woon, D.E., 110

Wormer, P.E.S., 229

Yan, Z.C., 225, 234

Yoshimine, M., see McLean 112, 228

Zanasi, R., see Lazzeretti, P., 109

Zener, C., 46

AUTHOR INDEX 277

Subject Index

angular momentum

orbital, 8, 9, 11

spin, 76, 78–82, 198

approximation methods

Ritz method, 64–71, 77, 113, 133,

157, 171, 191, 195, 258

RS perturbative method, 183

variation method, 53–71

atomic integrals, 71–73

atomic interactions

exchange-overlap density, 147, 153

exchange-overlap energy, 60, 147,

149–153

expanded, 220–223

non-expanded, 216–220

atomic orbitals

Gaussians (GTOs), 31, 32, 49–51,

56, 58, 108–112, 137

hydrogenic (HAOs), 31–46, 69

Slater (STOs), 31, 46–48

atomic units, 13, 14, 57, 61, 111, 142,

190, 194, 232

band theory of solids

(a model for), 129–131

bond

chemical, 141, 150–153

hydrogen, 237–239

Van der Waals, 237–239

Born-Oppenheimer

(approximation), 141–144

Brillouin (theorem), 136, 137

commutation (relations), 9, 76, 78

Condon-Shortley (phase), 44

conjugation (energy), 120, 172, 174

coordinates

Cartesian, 5, 7, 9

spherical, 10, 32, 63, 70, 71, 189,

194, 212, 240

spheroidal, 9, 10, 12, 71

correlation

energy, 99, 100, 133, 136, 218

error, 141, 148

Dalgarno (interchange theorem), 186

degeneracy, 5, 43, 68, 69, 77,

209–211, 236

delocalization (energy), 118, 120,

124, 125, 129, 170

density

electron, 92, 94, 96, 118, 128,

175, 176

exchange-overlap, 147, 153

Methods of Molecular Quantum Mechanices: An Introduction to Electronic Molecular Structure

Valerio Magnasco


density (Continued )

functional theory (DFT), 133,

138–139

matrices, 89, 90

one-electron, 90–96, 175

population analysis, 94, 95, 106, 176

spin, 92, 117, 118, 120, 128,

174, 176

transition, 219, 227

two-electron, 90, 96

determinants

elementary properties, 21, 23

parents (VB theory), 156–162

Slater, 82, 86–89, 97, 153, 154,

156, 158, 159

Dirac

formula, 79–82

notation, 2

eigenvalue equation, 5, 16, 32, 41, 75,

77, 82

electric properties

dipole moment, 111, 112, 136,

164, 235

of atoms, 187

of molecules, 235

polarizability, 187, 190, 192

quadrupole moment, 237

electrostatic potentials, 63, 71, 218,

226, 227

error

basis set, 108

correlation, 141, 148

truncation, 6, 108, 112

Fock-Dirac (density matrix), 100

functions

basis, 4, 105, 108–110

expansion theorem, 6

linear independence, 3

orthonormal set, 3

regular (Q-space), 1

scalar product, 2

Schmidt orthogonalization, 3

gradient

Cartesian, 5, 7, 10

spherical, 11

spheroidal, 12

group theory

axioms, 252

characters, 253–255

conjugation and classes, 254–255

examples, 257–264

factorization of secular

equations, 247, 257, 258, 264

irreducible representations

(irreps), 253–255

isomorphism, 254

orthogonality theorems, 256

projectors, 256–257

reducible representations

(G), 255–256

symmetry-adapted

functions, 256–257, 258, 261

Hartree-Fock (HF)

Coulombic potential, 102, 104

electronic energy, 102

equations, 103

exchange potential, 103, 104

fundamental invariant, 100–102

HF/2D, 111, 112

theory for closed shells, 100–104

H-H interactions

dispersion coefficients, 223–225

expanded, 220

non-expanded, 216

HOMO-LUMO, 106, 107

H€uckel theory

allyl radical, 115–118

alternant hydrocarbons, 118

benzene, 126–129

butadiene, 119–120

cyclobutadiene, 120–124

ethylene, 114–115

hexatriene, 124–125

model of band theory of

solids, 129–131

280 SUBJECT INDEX

hybridization

equivalent hybrids, 166

non-equivalent hybrids in

H2O, 162–163

properties of, 164

hydrogenic system

angular eigenfunctions, 40–41

energy levels, 42

Hamiltonian, 14, 42

hydrogenic orbitals (HAOs), 45

quantum numbers, 32, 35, 37, 39,

41–42

radial density, 43

radial eigenfunctions, 35–36

solution of the angular equation, 37

solution of the radial equation, 33

variational approximations, 57–61

hydrogen molecule

bond energy, 151, 153

CI theory, 148

correlation error, 148

Coulombic energy, 147, 151

exchange-overlap energy, 147,

149–153

Heitler-London theory, 148–150

MO theory, 145–148

singlet ground state 1Sgþ , 148–149

triplet excited state 3Suþ , 149–150

independent particle model

(IPM), 99, 102, 118

Laplacian

Cartesian, 5, 8, 10

radial, 11, 32

spherical, 11

spheroidal, 12

magnetic susceptibilities

diamagnetic, 199–203

paramagnetic, 203–212

quantum theory of, 196–212

matrices

definition, 21

density, 89, 96

diagonalization, 27, 65, 66,

192, 196

eigenvalue problem, 25–30

elementary properties, 21, 22

representatives, 7, 251, 252, 255

secular equation, 26–28

special, 24–25

transformation of, 7, 26–28

molecular interactions

anisotropy of, 230–233

Casimir-Polder formula, 223, 224

dispersion coefficients, 223–225,

230, 231, 233–235

dispersion constants, 223, 224,

228–233

electrostatic model, 239

expanded, 227–237

isotropic coefficients, 230,

233, 234

Keesom coefficients, 239, 244

Keesom interaction, 239–245

London formula, 222–224

molecular electrostatic potential

(MEP), 226, 227

non-expanded, 226–227

perturbation theory of, 226–245

MO theory

bases for molecular

calculations, 108–112

correlation error in H2, 141, 148

electron configurations, 88, 89,

133, 265

Fock operator, 103–105

Hartree-Fock 2D, 111, 112

H€uckel theory, 112–129

LCAO approximation, 105, 108

Roothaan equations, 105–108, 258

SCF method, 100, 104

schematization of, 154

operators

definition, 4

Hermitian, 5

SUBJECT INDEX 281

operators (Continued )

integral, 101, 103

linear, 4

projection, 30, 101, 249, 256–257,

262–264

optimization

CI coefficients, 133, 135, 148

linear parameters (Ritz), 64–67

MC-SCF method, 135

non-linear parameters, 57–64, 109

SCF method, 100, 104

second-order energies, 190–196

origin of the chemical bond, 150–153

overlap integral, 93

Pauli

equations for spin, 78

principle, 86, 87, 99, 141

repulsion, 95, 151–153, 162, 237

spin theory, 75–83

polarizabilities

dynamic (FDPs), 137, 223, 231,

232

H atom, 187, 190, 193

H2 molecule, 232

static, 222, 228, 234

Post-Hartree-Fock methods

CC-R12, 137–138

CI, 133–135

cusp condition, 134

density functional theory

(DFT), 138–139

MC-SCF, 135

Møller-Plesset (MP), 135–136

MP2-R12, 136–137

projectors, 30, 101, 249, 256–257,

262–264

pseudospectra, 195, 223, 224, 232

pseudostates, 191–196

quantum mechanics

basic postulates, 12–16

mathematics for, 2–12

observables, 12, 18

physical principles, 2, 17

Schroedinger equation, 5, 16

state function, 15–18

theory of magnetic susceptibilities,

196–212

uncertainty relations, 17

Rayleigh

functional, 53

ratio, 53, 65

variational principles, 53–56

Rayleigh-Schroedinger (RS)

Dalgarno interchange

theorem, 186

Hylleraas functional, 190

perturbative theory, 183–213

RS energies, 185

RS equations, 184

variational approximations, 190–192

resonance (energy), 167, 170, 172,

178, 180

singularities, 38, 134

Slater

determinants, 86–89

orbitals (STOs), 46–48

spin

Dirac formula, 79–83

ESR (origin), 77

N-electron, 79–83

NMR (origin), 78

one-electron, 78–79

Pauli equations, 78, 79

Pauli theory, 75–78

two-electron, 79–81

Wigner formula, 82, 83

Zeeman splitting, 77

symmetry

active and passive

representation, 249

fundamental theorem, 247

operations, 249–252

representatives, 249, 251, 252,

255, 256, 258–262

282 SUBJECT INDEX

similarity transformations, 250,

254, 255

transformation in function

space, 250–252

transformation in point space, 250

upper bounds

second-order energy, 190

total energy, 54

VB theory

advantages of, 154

allyl radical, 173–176

benzene, 176–181

butadiene, 171–172

canonical structures, 168, 177, 180

conjugated hydrocarbons, 166–181

construction of VB structures,

156–164

cyclobutadiene, 169–171

Dewar structures, 171, 172, 178,

179, 181

disadvantages of, 154–156

elementary theory, 153–164

ethylene, 169

H2O molecule, 162–164

Kekul�e structures, 169–173,

176–179

parent determinants, 156–162

Pauling formula, 167–181

resonance energy, 167, 170, 172,

178, 180

Rumer diagrams, 167

schematization of, 153–154

Weyl formula, 155, 163

Wigner formula, 155

Wigner formula, 82, 83, 155

SUBJECT INDEX 283

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