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MODERN QUANTUM CHEMISTRY Introduction to Advanced Electronic Structure Theory Attila Szabo and Neil S. Ostlund MODERN QUANTUM CHEMISTRY Introduction to Advanced Electronic Structure Theory ATTILA SZABO Laboratory of Chemical Physics National Institutes of Health Bethesda, Maryland NEIL S. OSTLUND Hypercube, Inc. Waterloo, Ontario SS a, So 3 & o a DOVER PUBLICATIONS, INC. Mineola, New York TABLE OF CONTENTS Preface to Revised Edition Preface Chapter 1. Mathematical Review 1.1 Linear Algebra 1.1.1 Three-Dimensional Vector Algebra 2 1.1.2 Matrices 5 1,13 Determinants 7 N-Dimensional Complex Vector Spaces 9 Change of Basis 13 The Eigenvalue Problem 15 Functions of Matrices 2] 4 AS «16 17 12 Orthogonal Functions, Eigenfunctions, and Operators 1.3 The Variation Method 1,3.1 The Variation Principle 37 1,32 The Linear Variational Problem 33 Notes Further Reading Chapter 2. Many Electron Wave Functions and Operators 2.1 The Electronic Problem 24.1 Atomic Units 47 2.1.2 The Borm-Oppenheimer Approximation 43 21.3 The Antisymmetry or Pauli Exclusion Principle 45 2.2 Orbitals, Slater Determinants, and Basis Functions 2.2.1 Spin Orbitals and Spatial Orbitals 46 2.22 Hartree Products 47 24 3 38 38 39 iv MODERN QUANTUM CHEMISTRY 23 2.4 25 22.3 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 S8 2.2.7 Form of the Exact Wave Function and Configuration Interaction 60 Operators and Matrix Elements 64 2.3.1 Minimal Basis H, Matrix Elements 64 23.2 Notations for One- and Two-Electron Integrais 67 23,3 General Rules for Matrix Elements 68 23,4 Derivation of the Rules for Matrix Elements 74 2.3.5 Transition from Spin Orbitals to Spatial Orbitals 81 23.6 Coulomb and Exchange Integrals 85 23,7 Pseudo-Classical Interpretation of Determinantal Energies 87 Second Quantization 89 24.1 Creation and Annihilation Operators and Their Anticommutation Relations 89 2.4.2 Second-Quantized Operators and Their Matrix Elements 95 Spin-Adapted Configurations 97 25.1 Spin Operators 97 25.2 Restricted Determinants and Spin-Adapted Configurations 100 2.5.3 Unrestricted Determinants 104 Notes 107 Further Reading 107 Chapter 3. The Hartree-Fock Approximation 108 31 3.2 3.3 3.4 The Hartree-Fock Equations ul 3.1.1 The Coulomb and Exchange Operators J/2 3.1.2 The Fock Operator 114 Derivation of the Hartree-Fock Equations 115 3.2.1 Functional Variation J/5 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 33.2 Brillouin's Theorem 128 3.3.3 The Hartree-Fock Hamiltonian 130 Restricted Closed-Shell Hartree-Fock: The Roothaan Equations 131 34.1 Closed-Shell Hartree-Fock : Restricted Spin Orbitals 132 3.4.2 Introduction of a Basis: The Roothaan Equations 136 3.4.3 The Charge Density 138 34.4 Expression for the Fock Matrix 140 3.4.5 Orthogonalization of the Basis 142 Vi MODERN QUANTUM CHEMISTRY Notes Further Reading Chapter 5. Pair and Coupled-Pair Theories 51 5.2 5.3 The Independent Electron Pair Approximation (IEPA) 5.1.1 Invariance under Unitary Transformations: An Example 277 5.1.2 Some Illustrative Calculations 284 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 CI, IEPA, CCA, and CEPA 303 5.32 The Resonance Energy of Polyenes in Húckel Theory 309 Notes Further Reading Chapter 6. Many-Body Perturbation Theory 61 *6.2 63 “6.4 6.5 6.6 *6.7 68 Rayleigh-Schrôdinger (RS) Perturbation Theory Diagrammatic Representation of RS Perturbation Theory 6.2.1 Diagrammatic Perturbation Theory for 2 States 327 6.22 Diagrammatic Perturbation Theory for N States 335 6.2.3 Summation of Diagrams 336 Orbital Perturbation Theory: One-Particle Perturbations Diagrammatic Representation of Orbital Perturbation Theory Perturbation Expansion of the Correlation Energy The N-Dependence of the RS Perturbation Expansion Diagrammatic Representation of the Perturbation Expansion of the Correlation Energy 67.1 Hugenholtz Diagrams 356 672 Goldstone Diagrams 362 6.7.3 Summation of Diagrams 368 6.7.4 What Is the Linked Cluster Theorem? 369 Some Illustrative Calculations Notes Further Reading 269 269 2n 22 297 318 319 320 322 327 338 350 354 356 370 378 379 TABLE OF CONTENTS vii Chapter 7. The One-Particle Many-Body Green's Function 7.1 Green's Functions in Single Particle Systems 7.2 The One-Particle MinBody Green's Function 7.2.1 The Self-Energy 722 e Sola eia Djs Eus 394 7.3 Application of the Formalism to H; and HeH* 7.4 Perturbation Theory and the Green's Function Method 7.5 Some Illustrative Calculations Notes Further Reading Appendix A. Integral Evaluation with 1s Primitive Gaussians Appendix B. Two-Electron Self-Consistent-Field Program Appendix C. Analytic Derivative Methods and Geometry Optimization by M.C. Zerner Appendix D. Molecular Integrais for H, as a Function of Bond Length Index 380 381 387 392 398 405 410 417 437 459 461 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 Nei $. OSTLUND xii MODERN QUANTUM CHEMISTRY chemistry course we taught some five times at both Indiana University and the University of Arkansas. An important feature ofthis 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 | 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. À 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 ii 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 à 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, in 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 1. 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 CHAPTER ONE MATHEMATICAL REVIEW This chapter provides the necessary mathematical background for the rest ofthe book. The most important mathematical tool used in 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 is 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 linear 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, is to start with the simplest example that illustrates 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 its components ap i=1,2,3 with respect to a set of three mutually perpendicular unit vectors (2,) as à=êa,+8a+êa; = a, (1.1) t The vectors 2, are said to form a basis, and are called basis vectors. The basis is complete in the sense that any three-dimensional vector can be 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, i = 1, 2, 3 and represented à as à=êay + a) + Edy =D Eai (1.2) 7 Given a basis, a vector is completely specified by its three components with respect to that basis. Thus we can represent the vector à by a column matrix as a a=[a;| inthe basis (8) (1.3a) as oras A a'=[0;) inthebasis (6) (1,36) ds The scalar or dot product of two vectors à and b is defined as à b=aby+ab,+ab;=Sab, (1.4) ] Note that àâ=al+ai+ai=|a? (1.5) is simply the square of the length (/á)) of the vector à. Let us evaluate the scalar product à - b using Eq. (1.1) 16=L Lê ad; (1.6) 4 MODERN QUANTUM CHEMISTRY We say that O is the matrix representation of the operator O in the basis (ê). The matrix O completely specifies how the operator O acts on an arbitrary vector since this vector can be expressed as a linear combination of the basis vectors (2,) and we know what & does to each of these basis vectors. Exercise 1.1 a) Show that 0,,=8,:0%, b) If Cd=b show that b= E Ouas WA and B are the matrix representations of the operators aí and 4, the matrix representation of the operator ', which is the product of .” and (E = 8), can be found as follows: Cã = L &Cy = AB, = z By = z 2AuBy (115) so that Cu=D 4uBy (1.16) which is the definition of matrix multiplication and hence C=AB (117) 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 JB + 8 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 [7,8] = 43 - As (1.184) [A,B]=AB-BA (1.18b) and their anticommutator as (4, B)= 48 + Ad (1.194) (A,BJ=AB + BA (1.19) MATHEMATICAL REVIEW S Exercise 1.2 Calculate [A, B] and (A, Bj when 110 o | A=|1 2 2 B=[|-1 0 0 O 2-1 1 0 1 1.1.2 Matrices 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 (4,;; that are in general complex and have ordered subscripts i=1,2,...,Nandj=1,2,...,M can be considered elements ofa rectangular (N x M) matrix A with N rows and M columns An A ct Am Am(da das voo Mau (1.20) Avi Ama Co Any MN = M the matrix is square. When the number of columns in the N x M matrix À is the same as the number of rows in the M x P matrix B, then À and B can be multiplied to givea N x P matrix C C=AB (1.21) where the elements of C are given by the matrix multiplication rule i=1,...0,N j=1 Pp (1.22) M Cy= D A4uBy us The set of M numbers (a;) i = 1,2,...,M can similarly be considered elements of a column matrix a a» (1.23) Note that for an N x M matrix A aa=b (1.24) where b is a column matrix with N elements M = Aja d=1,2...,N (1.25) J=1 
