Modern quantum chemistry introduction to advanced electronic structure theory

Một cuốn sách viết về cơ học lượng tử khá đầu đủ. Cuốn sách trình bày từ việc xây dựng vector ba chiều, ma trận, toán tử, hàm sóng và cách xây dựng hàm sóng. và các vấn đề liên quan như ứng dụng cơ học lượng tử để tính toán cấu trúc electron. Đây là một cuốn sách bổ ích cho nhà nghiên cứu lượng tử cũng như những người muốn nhập môn cơ học lượng tử

MODERN QUANTUM CHEMISTRY

Introduction to Advanced Electronic Structure Theory

DOVER PUBLICATIONS, INC

Mineola, New York

Copyright

Copyright © 1982 by Macmillan Publishing Co., Inc

Copyright © 1989 by McGraw-Hill, Inc

All rights resers 2d under Pan American and International Copyright Conventions

Published in Canada by General Publishing Company, Ltd., 30 Lesmilf Road, Don

Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanchesters,

162-164 Fulham Palace Road, London W6 9ER

Bibliographical Note

This Dover edition, first published in 1996, is an unabridged, unaltered republication of

the “First Edition, Revised,” originally published by the McGraw-Hill Publishing Com-

pany, New York, 1989 The original edition was published by the Maemillan Publishing

Company, New York, 1982

Library of Congress Cataloging-in-Publication Data

Modem quantum chemistry : introduction to advanced electronic structure theory /

Attila Szabo, Neil S Ostlund í

p cm

Previously published: Ist ed., rev New York : McGraw-Hill, c1989

Includes bibliographical references and index

11.5 Change of Basis 13

1.1.6 The Eigenvalue Problem 15

1.1.7 Functions of Matrices 2]

Orthogonal Functions, Eigenfunctions, and Operators

The Variation Method 1.3.1 The Variation Principle 31

1.3.2 The Linear Variational Problem 33

iv MODERN QUANTUM CHEMISTRY

2.3

2.4

25

2.2.3 Slater Determinants 49

2.2.4 The Hartree-Fock Approximation 53

2.2.5 The Minimal Basis H, Model 55 2.2.6 Excited Determinants 58

2.2.7 Form of the Exact Wave Function and Configuration Interaction 60

2.3.1 Minimal Basis H, Matrix Elements 64

2.3.2 Notations for One- and Two-Electron Integrals 67

2.3.3 General Rules for Matrix Elements 68

2.3.4 Derivation of the Rules for Matrix Elements 74 2.3.5 Transition from Spin Orbitals to Spatial Orbitals 87

2.3.6 Coulomb and Exchange Integrals 85 2.3.7 Pseudo-Classical Interpretation of Determinantal Energies 87

2.4.1 Creation and Annihilation Operators and Their

Anticommutation Relations 8&9 2.4.2 Second-Quantized Operators and Their Matrix Elements 95

2.5.1 Spin Operators 97 2.5.2 Restricted Determinants and Spin-Adapted Configurations 100

3.1.1 The Coulomb and Exchange Operators 112 3.1.2 The Fock Operator 114

3.2.1 Functional Vanation 1/15 3.2.2 Minimization of the Energy of a Single Determinant 117 3.2.3 The Canonical Hartree-Fock Equations 120

Interpretation of Solutions to the Hartree-Fock Equations 123 3.3.1 Orbital Energies and Koopmans’ Theorem 123

3.3.2 Brillouin’s Theorem 128 3.3.3 The Hartree-Fock Hamiltoman 130

Restricted Closed-Shell Hartree-Fock: The Roothaan Equations 131 3.4.1 Closed-Shell Hartree-Fock: Restricted Spin Orbitals 132

3.4.2 Introduction of a Basis: The Roothaan Equations 136 3.43 The Charge Density 138

3.4.4 Expression for the Fock Matrix 140 3.4.5 Orthogonalization of the Basis 142

Model Calculations on H, and HeH*

3.5.1 The 1s Minimal STO-3G Basis Set 153

3.5.2 STO-3GH, 159

3.5.3 An SCF Calculation on STO-3G HeH* 168

Polyatomic Basis Sets

3.6.1 Contracted Gaussian Functions 180 3.6.2 Minimal Basis Sets: STO-3G 184

3.6.3 Double Zeta Basis Sets: 4-31G 186 3.6.4 Polarized Basis Sets: 6-31G* and 6-31G** 189

Some Illustrative Closed-Shell Calculations

3.71 Total Energies 191

3.7.2 lonization Potentials 194

3.7.3 Equilibrium Geometries 200 3.7.4 Population Analysis and Dipole Moments 203

Unrestricted Open-Shell Hartree-Fock:

The Pople-Nesbet Equations

3.8.1 Open-Shel] Hartree Fock: Unrestricted Spin Orbitals 206 3.8.2 Introduction of a Basis: The Pople-Nesbet Equations 2/0 3.8.3 Unrestricted Density Matrices 2/2

3.8.4 Expression for the Fock Matrices 214 3.8.5 Solution of the Unrestricted SCF Equations 215 3.8.6 Illustrative Unrestricted Calculations 2/6

3.8.7 The Dissociation Problem and its Unrestricted Solution 22/

Multiconfigurational Wave Functions and the

Structure of the Full CI Matrix

4.1.1 Intermediate Normalization and an Expression for

the Correlation Energy 237

Doubly Excited Cl Some Illustrative Calculations Natural Orbitals and the One-Particle Reduced Density Matrix

The Multiconfiguration Self-Consistent Field (MCSCF) and Generalized Valence Bond (GVB) Methods

Truncated CI and the Size-Consistency Problem

Vi MODERN QUANTUM CHEMISTRY

Coupled-Pair Theories

5.2.1 The Coupled Cluster Approximation (CCA) 287

5.2.2 The Cluster Expansion of the Wave Function 290 5.2.3 Linear CCA and the Coupled Electron Pair

Approximation (CEPA) 292

5.2.4 Some Illustrative Calculations 296

Many-Electron Theories with Single Particle Hamiltonians

5.3.1 The Relaxation Energy via Cl, IEPA, CCA, and CEPA 303

5.3.2 The Resonance Energy of Polyenes in Hiickel Theory 309

Notes

Further Reading

Chapter 6 Many-Body Perturbation Theory

6.1 Rayleigh-Schrédinger (RS) Perturbation Theory

*6.2 Diagrammatic Representation of RS Perturbation Theory

6.2.1 Diagrammatic Perturbation Theory for 2 States 327

62.2 Diagrammatic Perturbation Theory for N States 335 6.2.3 Summation of Diagrams 336

6.3 Orbital Perturbation Theory: One-Particle Perturbations

*6.4 Diagrammatic Representation of Orbital Perturbation Theory

6.5 Perturbation Expansion of the Correlation Energy

6.6 The N-Dependence of the RS Perturbation Expansion

*6.7 Diagrammatic Representation of the Perturbation

Expansion of the Correlation Energy

671 Hugenholtz Diagrams 356

672 Goldstone Diagrams 362 6.73 Summation of Diagrams 368

6.7.4 What Is the Linked Cluster Theorem? 369

6.8 Some Illustrative Calculations

TABLE OF CONTENTS Vii

Chapter 7 The One-Particle Many-Body Green’s Function 380

7.1 Green’s Functions in Single Particle Systems 381

7.2 The One-Particle Many-Body Green’s Function 387

7.2.1 The Self-Energy 389

7.2.2 The Solution of the Dyson Equation 397

7.3 Application of the Formalism to H, and HeH* 392

7.4 Perturbation Theory and the Green’s Function Method 398

Szeretettel a sziileimnek

AS

To my wonderful father and

the memory of my dear mother

NSO

PREFACE TO REVISED EDITION

This revised edition differs from its predecessor essentially in three ways First, we have included an appendix describing the important recent de-

velopments that have made the efficient generation of equilibrium geo-

metries almost routine We are fortunate that M Zerner agreed to write this since our own recent interests have been channeled in other directions Second, numerous minor but annoying errors have been corrected For

most of these we are indebted to K Ohno, T Sakai and Y Mochizuki

who detected them in the course of preparing the Japanese translation which has recently been published by Tokyo University Press Finally, we have updated the Further Reading sections of all the chapters We are extremely pleased by the many favorable comments we have received about our book and we hope that the next generation of readers will find this edition useful

ATTILA SZABO NEIL S OSTLUND

PREFACE

The aim of this graduate textbook is to present and explain, at other than a superficial level, modern ab initio approaches to the calculation of the electronic structure and properties of molecules The first three chapters contain mtroductory material culminating in a thorough discussion of the Hartree-Fock approximation The remaining four chapters describe a variety

of more sophisticated approaches, which improve upon this approximation

We have strived to make advanced ‘opics understandable to the be- ginning graduate student Our goal was to teach more than cocktail party jargon; we wanted to give insight into the nature and validity of a variety

of approximate formalisms and improve the reader’s problem-solving skills Needless to say, a book of this size cannot cover all of quantum chemistry;

it would be easy to write down a long list of important topics which we have not covered Nevertheless, we believe that anyone who has mastered the material in this book will have a solid foundation to build upon, and will

be able to appreciate most research papers and seminars dealing with electronic structure theory and its applications

The origins of this book go back to our student days when we were trying to understand a variety of quantum chemical formalisms that at first glance appeared to be extremely complicated We found that by applying such formalisms to simple, analytically tractable, model systems we not only got a feeling as to what was involved in performing actual calculations but also gained insight into the interrelationships among different approximation

schemes The models we used then, and ones we devised or learned sub-

sequently, play an important role in the pedagogical approach we adopt in this book The writing began in 1976 when we jointly taught a special topics course at Indiana University The manuscript gradually evolved in response

to the needs and reactions of students enrolled in a second-semester quantum

xi

xii MODERN QUANTUM CHEMISTRY

chemistry course we taught some five times at both Indiana University and the University of Arkansas

Animportant feature of this book is that over 150 exercises are embedded

in the body of the text These problems were designed to help the reader acquire a working knowledge of the material The level of difficulty has been kept reasonably constant by breaking up the more complicated ones into manageable parts Much of the value of this book will be missed if the exercises are ignored

In the following, we briefly describe some of the highlights of the seven chapters Chapter I reviews the mathematics (mostly matrix algebra) re- quired for the rest of the book It is self-contained and suited for self-study Its presence in this book is dictated by the deficiency in mathematics of most chemistry graduate students The pedagogical strategy we use here, and in much of the book, is to begin with a simple example that illustrates most of the essential ideas and then gradually generalize the formalism to handle more complicated situations

Chapter 2 introduces the basic techniques, ideas, and notations of

quantum chemistry A preview of Hartree-Fock theory and configuration interaction is used to motivate the study of Slater determinants and the evaluation of matrix elements between such determinants A simple model system (minimal basis H,) is introduced to illustrate the development This model and its many-body generalization (N independent H, molecules) reappear in all subsequent chapters to illuminate the formalism Although not essential for the comprehension of the rest of the book, we also present here a Self-contained discussion of second quantization

Chapter 3 contains a thorough discussion of the Hartree-Fock approx- imation A unique feature of this chapter is a detailed illustration of the computational aspects of the self-consistent-field procedure for minimal basis HeH* Using the output of a simple computer program, listed in Appendix B, the reader is led iteration-by-iteration through an ab initio calculation This chapter also describes the results of Hartree-Fock calcula- tions on a standard set of simple molecules using basis sets of increasing sophistication We performed most of these calculations ourselves, and in later chapters we use these same molecules and basis sets to show how the Hartree-Fock results are altered when more sophisticated approaches are used Thus we illustrate the influence of both the quality of the one-electron basis set and the sophistication of the quantum chemical method on cal- culated results In this way we hope to give the reader a feeling for the kind

of accuracy that can be expected from a particular calculation

Chapter 4 discusses configuration interaction (CI) and is the first of the four chapters that deal with approaches incorporating electron correlation One-electron density matrices, natural orbitals the multiconfiguration self- consistent-field approximation, and the generalized valence bond method are

PREFACE Xiii

discussed from an elementary point of view The size-consistency problem associated with truncated CI is illustrated using a model consisting of

N independent hydrogen molecules This example highlights the need for

so-called many-body approaches, which do not suffer from this deficiency,

that are discussed in subsequent chapters

Chapter 5 describes the independent electron pair approximation and

a variety of more sophisticated approaches that incorporate coupling be- tween pairs Since this chapter contains some of the most advanced material

in the book, many illustrative examples are included In the second half of

the chapter, as a pedagogical device, we consider the application of many-

electron approaches to an N-electron system described by a Hamiltonian containing only single particle interactions This problem can be solved exactly in an elementary way However, by seeing how “high-powered” approaches work in such a simple context the student can gain insight into the nature of these approximations

Chapter 6 considers the perturbative approach to the calculation of the correlation energy of many-electron systems A novel pedagogical approach allows the reader to acquire a working knowledge of diagrammatic pertur-

bation theory surprisingly quickly Although the chapter is organized so

that the sections on diagrams (which are starred) can be skipped without loss of continuity, we find that the diagrammatic approach is fun to teach and is extremely well received by students

Chapter 7 contains a brief introduction to the use of the one-particle many-body Green’s function in quantum chemistry Our scope is restricted

to discussing ionization potentials and electron affinities The chapter is directed towards a reader having no knowledge of second quantization or Green’s functions, even in a simple context

This book is largely self-contained and, m principle, requires no pre- requisite other than a solid undergraduate physical chemistry course How- ever, exposure to quantum chemistry at the level of the text by I N Levine (Quantum Chemistry, Allyn and Bacon) will definitely enhance the student’s appreciation of the subject material We would normally expect the present text to be used for the second semester of a two-semester sequence on quan- tum chemistry It is also suitable for a special topics course There is probably too much material in the book to be taught in-depth in a single semester For students with average preparation, we suggest covering the first four chapters and then discussing any one of the last three, which are essentially independent Our preferred choice is Chapter 6 For an exceptionally well-

prepared class, the major fraction of the semester could be spent on the

last four chapters We have found that a course based on this text can be enriched in a number of ways For example, it is extremely helpful for students to perform their own numerical calculations using, say, the Gaussian

80 system of programs In addition, recent papers on the applications of

xiv MODERN QUANTUM CHEMISTRY

electronic structure theory can be assigned at the beginning of the course

and the students asked to give short in-class presentations on one or more

of such papers at the end-of the course

We have placed special emphasis on using a consistent notation through- out the book Since quantum chemists use a number of different notations,

it is appropriate to define the notation we have adopted Spatial molecular

orbitals (with latin mdices i, j,k ) are denoted by wy These are usually

expanded in a set of spatial (atomic) basis functions (with greek indices

Lt, v, 4, -) denoted by @ Molecular spin orbitals are denoted by y Occupied molecular orbitals are specifically labeled by a, b, c, and unoccupied (virtual) molecular orbitals are specifically labeled by r, s, t, Many- electron operators are denoted by capital script letters (for example, the Hamiltonian is #), and one-electron operators are denoted by lower case latin letters (for example, the Fock operator for electron-one is f(1)) The exact many-electron wave function 1s denoted by ®, and we use V to denote

approximate many-electron wave functions (i.e the Hartree-Fock ground

state wave function is ‘Wo, while ‘V5, is a doubly excited wave function) Exact and approximate energies are denoted by & and E, respectively All numerical quantities (energies, dipole moments, etc.) are given in atomic

to illustrate the lack of size consistency of doubly excited configuration

interaction using the model of N noninteracting H, molecules This led

to our extensive use of this model M Zerner constructively criticized the entire manuscript The following colleagues gave advice, performed cal-

culations just for the book, or assisted us in some other useful way: R Bartlett,

J Binkley, M Bowen, R Counts, C Dykstra, W Eaton, D Freeman,

W Goddard III S Hagstrom, A Hernandez, D Merrifield, J Neece,

I Shavitt, and R Whiteside The following students helped us clarify our presentation and eliminate errors: B Basinger G Caldwell, T Croxton,

R Farren, R Feeney M Flanagan, J Freeze V Hess K Holtzclaw, J Joens,

J Johnson, R Jones, J Kehayias, G Lipari, D Lupo, D McMullen, J Meek,

S Munchak, N Peyghambarian, R Russo, W Schinzer, D Shoup, B Stone,

E Tsang and I Waight Of all the secretaries who typed various versions

of the manuscript, K Wagner, M Mabie, and L Ferro were the most

helpful J Hawkins prepared the illustrations

ATTILA SZABO

Ne S OstLUND

CHAPTER

ONE

MATHEMATICAL

REVIEW

This chapter provides the necessary mathematical background for the rest

of the book The most important mathematical tool used m quantum chem- istry is matrix algebra We have directed this chapter towards the reader who has some familiarity with matrices but who has not used them in some time and is anxious to acquire a working knowledge of linear algebra Those with strong mathematical backgrounds can merely skim the material to acquaint

themselves with the various notations we use Our development is informal

and rigour 1s sacrificed for the sake of simplicity To help the reader develop those often neglected, but important, manipulative skills we have included carefully selected exercises within the body of the text The material cannot

be mastered without doing these simple problems

In Section 1.1 we present the elements of linear algebra by gradually generalizing the ideas encountered in three-dimensional vector algebra We consider matrices, determinants linear operators and their matrix repre- sentations, and, most importantly, how to find the eigenvalues and eigen- vectors of certain matrices We introduce the very clever notation of Dirac, which expresses our results in a concise and elegant manner This notation

is extremely useful because it allows one to manipulate matrices and derive various theorems painlessly Moreover, it highlights similarities between Imear algebra and the theory of complete sets of orthonormal functions as

will be seen in Section 1.2 Finally, in Section 1.3 we consider one of the cornerstones of quantum chemistry namely, the variation principle

2 MODERN QUANTUM CHEMISTRY

1.1 LINEAR ALGEBRA

We begin our discussion of linear algebra by reviewing three-dimensional

vector algebra The pedagogical strategy we use here, and in much of the book, 1s to start with the simplest example that iilustrates the essential ideas and then gradually generalize the formalism to handle more complicated

situations

1.1.1 Three-Dimensional Vector Algebra

A three-dimensional vector can be represented by specifying 1ts components

a,, 1=1,2,3 with respect to a set of three mutually perpendicular unit

written as a linear combination of the basis vectors However, a basis is

not unique, we could have chosen three different mutually perpendicular unit vectors, é,, 1 = 1, 2, 3 and represented G as

ä= tai + Èya; + 303 = > Fa, (12)

Given a basis, a vector is completely specified by its three components with respect to that basis Thus we can represent the vector @ by a column matrix as

a=[a,} in the basis {6} (1.3a)

or as

a =|a,]_ in the basis {é,} (1.3b)

The scalar or dot product of two vectors a and b is defined as

a@ b= a,b, + a,b, + a3b3 = > ab, (1.4)

Given a vector d we can find its component along é, by taking the

scalar product of Eq (1 1) with é, and using the orthonormality relation (1.7)

1s the unit dyadic A dyadic 1s an entity which when dotted mto a vector

gives another vector The unit dyadic gives the same vector back Equation

(1.10) is called the completeness relation for the basis {é,} since it is an alternate

form of Eg (1.1), which states that any vector @ can be written as a linear

combination of the basis vectors {é,}

We now define an operator @ as an entity which when acting on a vector

a converts it into a vector b

The operator is satd to be Jinear if for any numbers x and 3

C(xd + yb) = xOa + yOb (1.12)

A linear operator 1s completely determined if its effect on every possible vector is known Because any vector can be expressed in terms of the basis {ế,}, 1t is sufficient to know what @ does to the basis vectors Now since

Oé, 1s a vector, it can be written as a linear combination of the basis vectors

4 MODERN QUANTUM CHEMISTRY

We say that O is the matrix representation of the operator @ in the basis {é,} The matrix O completely specifies how the operator @ acts on an arbitrary vector since this vector can be expressed as a linear combination

of the basis vectors {é,} and we know what © does to each of these basis vectors

Thus if we define matrix multiplication by (1.16), then the matrix repre-

sentation of the product of two operators is just the product of their matrix representations

The order in which two operators or two matrices are multiplied is

crucial In general “4 #4 Bo or AB # BA That is, two operators or

matrices do not necessarily commute For future reference, we define the

commutator of two operators or matrices as

Now that we have seen how 3 x 3 matrices naturally arise in three-dimen-

sional vector algebra and how they are multiplied, we shall generalize these

results, A set of numbers {A,,} that are in general complex and have ordered subscripts i= 1,2, ,N and j = 1,2, , M can be considered elements

of a rectangular (N x M) matrix A with N rows and M columns

Ay, Ana to Any

If N = M the matrix is square When the number of columns in the N x M matrix A is the same as the number of rows in the M x P matrix B, then A

and B can be multiplied to give a N x P matrix C

The set of M numbers {a,} i= 1,2, , M can similarly be considered

elements of a column matrix

6 MODERN QUANTUM CHEMISTRY

We now introduce some important definitions The adjoint of an N x M matrix A, denoted by At, is an M x N matrix with elements

Exercise 1.3 If A is an N x M matrix and B is a M x K matrix show

that (AB) = BtAt,

Note that Eq (1.30) is simply the complex conjugate of Eq (1.25)

We now give certain definitions and properties applicable only to square

A real unitary matrix is called orthogonal

6 A Hermitian matrix is self-adjoint, i.e.,

If U is unitary and B = UTAU, then A = UBUT

If the product C = AB of two Hermitian matrices is also Hermitian, then

A and B commute

e If A is Hermitian then A~?, if it exists, is also Hermitian

a way of ordering these numbers, and there are N! distinct permutations of

N numbers The determinant of an N x N matrix A is a number obtained

by the prescription

det(A) = |A| =|: > f= 2D (- PPA WA Any (138)

Ani °°" Awyn =1

where Z; is a permutation operator that permutes the column indices

1, 2, 3, , N and the sum runs over all N! permutations of the indices; p;

8 MODERN QUANTUM CHEMISTRY

is the number of transpositions required to restore a given permutation

ij, i,, , iy to natural order 1, 2, 3, , N Note that it is important only

whether p, is an even or odd number As an illustration we evaluate the determinant of a 2 x 2 matrix A

2 If (A); = A,6,; ix then |A] = [] Aj = 411422 - - - Ayw-

3 A single interchange of any two rows (or columns) of a determinant

changes its sign

Exercise 1.6 Using properties (1)-(5) prove that in general

6 Ifany two rows (or columns) of a determinant are equal, the value of the

determinant is zero

7 |A~!|=(|Al)"1

If AAt = 1, then |Aj(|A])* = 1

9 If UtOU = Q and U'tU = UUt = I, then |O| = |Q]

Exercise 1.7 Using Eq (1.39), note that the inverse of a 2 x 2 matrix

A obtained in Exercise |.4f can be written as

MATHEMATICAL REVIEW 9

and thus A~‘ does not exist when |A| = 0 This result holds in general for

N x N matrices Show that the equation

Ac = 0

where A is an N x N matrix and c is a column matrix with elements c,,

r= 1,2, , N can have a nontrivial solution (¢ # 0) only when |A| = 0

For a2 x 2 determinant it is easy to verify by direct calculation that

€yBy, + C2By, Aja By, Ayo

€,By, +€,By, Aga Bạc 4¿;

This result is a special case of the following property of determinants that

we shall use several times in the book

M Ayy Ayn °°" » By, +77 Ain

ANL ÁN: TC Bye `" Ayn

A similar result holds for rows

1.1.4 N-Dimensional Complex Vector Spaces

We need to generalize the ideas of three-dimensional vector algebra to an N-dimensional space in which the vectors can be complex We will use the powerful notation introduced by Dirac, which expresses our results in an exceedingly concise and simple manner In analogy to the basis {é,} in three dimensions, we consider N basis vectors denoted by the symbol |i>, i =

1, 2, , N, which are called ket vectors or simply kets We assume this basis is complete so that any ket vector |2) can be written as

¡=1

This is a simple generalization of Eq (1.1) rewritten in our new notation

10 MODERN QUANTUM CHEMISTRY

After we specify a basis, we can completely describe our vector |a> by giving its N components a,, i= 1, 2, , N with respect to the basis {|ï>} Just as

before, we arrange these numbers in a column matrix a as

and we say that a is the matrix representation of the abstract vector |a> in

the basis {|i>} Recall (Eq (1.27)) that the adjoint of the column matrix a is

the row matrix a‘

Now we introduce an abstract bra vector <a| whose matrix representation

is at The scalar product between a bra al and a ket |b> is defined as

which is the natural generalization of the scalar product defined in Eq (1.4)

The unusual names bra (for < |) and ket (for | >) were chosen because the

notation for the scalar product (¢ | >) looks like a bra-c-ket Note that

be written as a linear combination of the bra basis vectors as

The scalar product between <a and |b> now becomes

<a|b> =) at<i| J yb;

yy

For this to be identical to our definition (1.44) of the scalar product we must

have that

which is a statement of the orthonormality of the basis and is a generaliza-

tion of Eq (1.7) In summary, a ket vector la> is represented by a column

MATHEMATICAL REVIEW 11

matrix a, a bra vector <b is represented by a row matrix bt, and their scalar

product is just the matrix product of their representations

We now ask, given a ket |a) or a bra <al, how can we determine its components with respect to the basis {|i>} or {<i]}? We proceed in complete

analogy to the three-dimensional case (c.f Eq (1.8)) We multiply Eq (1.41)

by <j] on the left and Eq (1.46) by |j> on the right and obtain

<j|a» =3, Kida =3) ông = a, (1.48a)

and

<a|j> =3, 4#<i|j>= 3, dồi, = aj (1.48b)

The expression “multiplying by </| on the left” is a shorthand way of saying

“taking the scalar product with <j| Note that

<j|a = (a|73)* = <al i>* (1.49)

Using these results we can rewrite Eqs (1.41) and (1.46) as

lay = 2 lida, = 3 |i><¡|a> (1.50a)

and

which suggests we write

which is the analogue of Eq (1.10) and is a statement of the completeness of

the basis We will find that multiplying by unity and using Eq (1.51) is an extremely powerful way of deriving many relations

In analogy to Eq (1.11), we define an operator © as an entity which

when acting on a ket |a> converts it into a ket [b>

As before, the operator is completely determined if we know what it does to

the basis {|i>}:

12 MODERN QUANTUM CHEMISTRY

which provides a useful expression for the matrix elements of O It should

be noted that we can easily get the matrix representation of © by using the completeness relation (1.51) as follows

Olid = 16|i> = ), |j><7|@|› (1.55)

which upon comparison to Eq (1.53) yields

As another illustration of the use of the completeness relation and the built-in consistency and simplicity of Dirac notation, let us find the matrix repre- sentation of the operator @ = Z in terms of the matrix representations of

the operators ‹Z and # (c.f Eq (1.15))

«li = (©), = <if PAID = <i 1Al 7D

= » <¡L#|k><k|}Z|¡>

=3 (A)„(B)i,

%

We now introduce the adjoint of the operator ©, which we denote by

Ớ! If @ changes a ket |a> into the ket |b) (c.f Eq (1.52)), then its adjoint

changes the bra <al into the bra <6], ie.,

This equation is said to be the adjoint of Eq (1.52) Multiplying both sides

of Eq (1.52) by <c| on the left and multiplying both sides of Eq (1.57) by

|c> on the right, we have

Since the labels a, b, and c are arbitrary, we have shown that the matrix

representation of (“t is the adjoint of the matrix representation of @ since

<i{O"|7> = (O%), = <iJOl>* = O% (1.59)

Finally, we say that an operator is Hermitian when it is self-adjoint

MATHEMATICAL REVIEW 13

1.1.5 Change of Basis

In Subsection 1.1.1 we have seen that the choice of basis is not unique Given

two complete orthonormal bases {|i>} and {|x>} we now wish to find the

relationship between them We use latin letters i, 7, k, to specify the bras and kets in the first basis and greek letters x, 8, y to specify the bras and kets

of the second basis Thus we have

<i|j> = 6;, » i><i] = 1 (1.62a)

and

<a|B> =o — 3;|e><a| =1 (1.62b)

Since the basis {i>} is complete, we can express any ket in the basis {|z>} as

a linear combination of kets in the basis {|i>} and vice versa That is,

a> = Ia =) |A><i]a> =P DU = Y |DOa (1.63)

where we have defined the elements of a transformation matrix U as

<ija> = U,, = (U)ia (1.64)

Transforming in the opposite direction, we have

|i> = 1] = 2 |x><œ|ï> = 2 |z)U# = » |z»(U?„ (1.65)

where we have used Eq (1.49) and the definition of the adjoint matrix to show

that

<ai> = (i|ø>* = UX = (U4), (1.66)

It is important to remember that since we have defined U via Eq (1.64),

<a|i> # U,; but rather is given by Eq (1.66) We now prove that the trans- formation matrix U is unitary This is a consequence of the orthonormality

14 MODERN QUANTUM CHEMISTRY

and hence U is unitary Thus we arrive at the important result that two orthonormal bases are related by a unitary matrix via Eq (1.63) and its

inverse, Eq (1.65) As shown by Eq (1.64), the elements of the transformation

matrix U are scalar products between the two bases

Let us now consider how the matrix representations of an operator & are related in two different complete orthonormal bases The result we shall obtain plays a central role in the next subsection where we consider the eigenvalue problem Suppose O is the matrix representation of @ in the

basis {|i>}, while Q is its matrix representation in the basis {|}

Ol = VID <C> = YO, (1.68a)

Ola = 2 IB>< BlOla> = x |Ø8>O;, (1.68b)

To find the relationship between O and © we use the, by now, familiar

technique of introducing the unit operator

is not diagonal, it is always possible to find a basis {|x>} in which the matrix representation of the operator is diagonal, ie.,

In the next subsection we consider the problem of diagonalizing Hermitian

matrices by unitary transformations

Exercise 1.8 Show that the trace of a matrix is invariant under a unitary

transformation, 1e if

Q=UtOU

then show that trQ = trO

MATHEMATICAL REVIEW 15

1.1.6 The Eigenvalue Problem

When an operator (@ acts on a vector |x> the resulting vector Is in general

distinct from |x If Ga) is simply a constant times |e), ice.,

Ole> = œ„|&> (1.72)

then we say that |) is an eigenvector of the operator (© with an eigenualue œ„

Without loss of generality we can choose the eigenvectors to be normalized

In this book we are interested in the eigenvectors and eigenvalues of Hermi- tian operators (Ot = ©) They have the following properties

1 The eigenvalues of a Hermitian operator are real This follows imme-

diately from Eq (1.61), which states that

KofOloe> = <ø|@”|ab = <a|@|z>* (1.24)

Multiplying the eigenvalue relation Eq (1.72) by <a| and substituting into (1.74) we have

which is the required result

2 The eigenvectors of a Hermitian operator are orthogonal To prove this consider

16 MODERN QUANTUM CHEMISTRY

We now show that degenerate eigenvectors can always be chosen to be orthogonal We first note that any linear combination of degenerate cigen-

vectors is also an eigenvector with the same eigenvalue, i.e.,

6(x|I> + y|2>)= xé|1› + yứ|23 = œ(x|l) + y2) (1.79)

There are many ways we can find two linear combinations of |1> and |2), which are orthogonal One such procedure is called Schmidt orthogonali- zation We assume that |1> and |2) are normalized and let (1]2> = S # 0

We choose |I> = |1> so that {I{ I> = 1 We set |II’> = |1> + cl2) and choose

cso that <I| I’) = 0 = | + cS Finally we normalize |IJ’> to obtain

The eigenvalue problem we wish to solve can be posed as follows Given the matrix representation, O, of a Hermitian operator € in the orthonormal

basis {|i), i= 1,2, ,.N} we wish to find the orthonormal basis {|«),

a= 1,2, ,N} in which the matrix representation, Q, of € is diagonal, i 1, = ©,5,5 In other words, we wish to diagonalize the matrix O We have seen in the last subsection, that the two representations of the operator

© are related by a unitary transformation (c.f Eq (1.70a))

MATHEMATICAL REVIEW 17

There exist numerous, efficient algorithms for diagonalizing Hermitian

matrices.’ For our purposes, computer programs based on such algorithms

can be regarded as “black boxes,” which when given O determine U and a

In order to make contact with the discussion of the eigenvalue problem found in most elementary quantum chemistry texts we now consider a computationally inefficient procedure that is based on finding the roots of

the secular determinant

The eigenvalue problem posed above can be reformulated as follows

Given an N x N Hermitian matrix O, we wish to find all distinct column

vectors c (the eigenvectors of O) and corresponding numbers œ (the eigen- values of O) such that

O Once we have found the eigenvalues, we can find the corresponding eigen- vectors by substituting each w, into Eq (1.84) and solving the resulting equations for c* In this way, c* can be found to within a multiplicative constant, which is finally determined by requiring c* to be normalized

In order to establish the connection with our previous development, let

us now construct a matrix U defined as U,, = cj, ie.,

= (c'e? +++ c%) (1.89)

18 MODERN QUANTUM CHEMISTRY

Thus the eth column of U is just the column matrix c* Then using (1.87)

it can be shown that

Oy

Since U,,, = c?, the orthonormality relation (1.88) is equivalent to

>» URU ig = YU ai(Wip = Sag (1.91)

which in matrix notation is

Finally, multiplying both sides of Eq (1.90) by Ut and using Eq (1.92) we have

which is identical to Eq (1.83) Thus Eq (1.89) gives the relationship between

the unitary transformation (U), which diagonalizes the matrix O, and the

MATHEMATICAL REVIEW 19

This quadratic equation has two solutions

@, = 4011 + O22 — (O22 — O11)" + 401 2021)"7] (1.96a)

2 = [0,1 + O22 + ((O22 — O11)? + 404202,)'"'7] — (196B)

which are the eigenvalues of the matrix O To find the eigenvector corre- sponding to a given eigenvalue, say «, we substitute «w, into Eq (1.94) to obtain

is later determined by the normalization, one can set c, = 1 and ce, =c

in Eq (1.94) If this is done, Eq (1.94) becomes

01, + O,.c =O

Ox, + Ó;;c= cc

20 MODERN QUANTUM CHEMISTRY

After eliminating c, find the two roots of the resulting equation and show that they are the same as those given in Eq (1.96) This technique, which we shall use numerous times in the book for finding the lowest eigenvalue of a matrix, is basically the secular determinant approach without determinants Thus one can use it to find the lowest eigenvalue of certain N x N matrices without having to evaluate an N x N determinant

Now let us solve the 2 x 2 eigenvalue problem by directly finding the

orthogonal matrix U that diagonalizes the symmetric matrix O, ie

The requirement that

UjyUii Ð+U¿¿zU¿i U¡¿U¡; + U¿¿;U¿; 0 1

places three constraints (two diagonal and one off-diagonal) on the four elements of the matrix U Therefore U can be completely specified by only one parameter Since

cos 8 sin 8 /cos Ø sin #\ _ cos? #+sin? 6 0 1

UƯ'OU= cos0 sin@\/0O,, O42 cos 6 sin 8

snØ_ —cos0/\OÓ,; O22/\sin@ —cosé

O,, cos? + O,, sin? @ 4(0,, — O22) sin 2@ — 0, cos20

_ + 0,2 sin26

~ 14(0,,; — O22) sin26 — 0, cos26 O,, sin? @ + O22 cos? @

— O,, sin26

is diagonal This can be done if we choose @ such that

3(O1, — O22) sin 26 — O,, cos26 =0

This has the solution

2 On x O22

MATHEMATICAL REVIEW 21

Thus the two eigenvalues of O are

@, = 0,, cos* 6) + O22 sin? Oy + O,2 sin26, (1.106a)

and

W2 = O,, sin? Oy + 02 cos? Oy — O12 sin26, (1.106b)

Upon comparison of Eqs (1.104) and (1.89), we fñnd the two eigenvectors

It should be mentioned that the Jacobi method for diagonalizing N x N

matrices is a generalization of the above procedure The basic idea of this method is to eliminate iteratively the off-diagonal elements of a matrix by repeated applications of orthogonal transformations, such as the ones we have considered here

1.1.7 Functions of Matrices

Given a Hermitian matrix A, we can define a function of A i.e f(A), in much the same way we define functions f(x) of a simple variable x For example, the square root of a matrix A, which we denote by A", is simply that matrix

which when multiplied by itself gives A, ie.,

22 MODERN QUANTUM CHEMISTRY

or In general

n=0

After these definitions, we are still faced with the problem of calculating A‘/?

or exp (A) If A is a diagonal matrix

Thus to calculate any function of a Hermitian matrix A, we first diagonalize

A to obtain a, the diagonal matrix containing all the eigenvalues of A We then calculate f(a), which is easy because a is diagonal Finally we “undr-

agonalize” f(a) using (1.113b) to obtain (1.115) For example, we can find

the square root of a matrix A as

A12 = Ua1/2U†

Since

A12A12? = Ua!⁄2UTUa!⁄2UT = Ual/2al Ut = UaUt =A

If the above procedure were to yield a result for f(A) that was infinite, then f(A) does not exist For example, if we try to calculate the inverse of a matrix

A that has a zero eigenvalue (say a; = 0), then f(a;) = 1/a,; = co and so AT] does not exist As Exercise 1.12(a) shows, the determinant of a matrix is just the product of its eigenvalues Thus if one of the eigenvalues of A is zero, det(A) is zero and the above argument shows that A~? does not exist This same result was obtained in a different way in Exercise 1.7

24 MODERN QUANTUM CHEMISTRY

If we want x as a function of @ we need to invert the matrix for each value

of w However, if we diagonalize A, we can write

}L/(a+ bì + fla—b)] aL fla+b)— fla- ni)

IE flat b)— fla— by] 3 [flat b) + fla—b)]

We know from the theory of Fourier series that it 1s possible to represent

a sufficiently well-behaved function f(x) on some interval as an infinite

linear combination of sines and cosines with coefficients that depend on the function Thus any such function can be represented by specifying these coefficients This seems very similar to the idea of expanding a vector in terms of a set of basis vectors The purpose of this section is to explore this similarity We consider an infinite set of functions {Wi i= 1,2,. } that

Let us suppose that any function a(x) can be expressed as a linear com-

bination of the set of functions {y;}

In other words, the basis {y,(x)} 1s complete Given a function a(x) we can

determme its components a; with respect to the basis {y,;} by multiplying

Eg (1 117) by w#(x) and integrating over x, smce

fax Ứ#{x)a(x) = » fax /⁄?O2,()4 = » biG, = Gj (1.118)

Substitutmg this result for the coefficients mto the original expansion (1 117),

we have

atx) = [dx B Waly | a(x’) (1.119)

The quantity in square brackets is a function of x and x, and has the unusual

property that when multiplied by a(x’) and integrated over all x one obtains

a(x) An entity with this property 1s called the Dirac delta function d(x — x)

26 MODERN QUANTUM CHEMISTRY

Figure 1.1 Successive approximations to the Dirac delta function 6(x)

when multiplied by a function a(x) and mtegrated over any interval con- taming x = 0, it “plucks” out the value of the function at x = 0 (see Eq (1 123)) The Dirac delta function 6(x) can be thought of as the limit of a

sequence of functions that simultaneously become more and more peaked

about x = 0 and narrower and narrower such that the area is always unity

For example, one representation of 6(x), which is shown in Fig 1.1, is

theory of complete orthonormal functions can be regarded as a generaliza-

tion of ordinary linear algebra To make the analogy explicit, it is convenient

to mtroduce the shorthand notation

or more generally

a(x) = a> a*(x) = Ca| (1.127b)

and define the scalar product of two functions as