Front Cover 1
Symmetry of Many-Electron Systems 4
Copyright Page 5
Table of Contents 6
Translator's Note 11
Preface to Russian Edition 12
PART I: MATHEMATICAL APPARATUS 14
CHAPTER I. Basic Concepts and Theorems of Group Theory 16
Part 1. Properties of Group Operations 16
1.1. Group Postulates 16
1.2. Examples of Groups 17
1.3. Isomorphism and Homomorphism 19
1.4. Subgroups and Cosets 20
1.5. Conjugate Elements. Classes 21
1.6. Invariant Subgroups. Factor Groups 22
1.7. Direct Products of Groups 23
1.8. The Semidirect Product 23
Part 2. Representations of Groups 24
1.9. Definition 24
1.10. Vector Spaces 25
1.11. Reducibility of Representations 28
1.12. Properties of Irreducible Representations 29
1.13. Characters 30
1.14. The Calculation of the Characters of Irreducible Representations 31
1.15. The Decomposition of a Reducible Representation 33
1.16. The Direct Product of Representations 34
1.17. Clebsch-Gordan Coefficients 37
1.18. The Regular Representation 38
1.19. The Construction of Basis Functions for Irreducible Representations 39
CHAPTER II. The Permutation Group 43
Part 1. General Considerations 43
2.1. Operations with Permutations 43
2.2. Classes 44
2.3. Young Diagrams and Irreducible Representations 45
Part 2. The Standard Young-Yamanouchi Orthogonal 
47 
2.4. Young Tableaux 47
2.5. Explicit Determination of the Matrices of the Standard Representation 49
2.6. The Conjugate Representation 52
2.7. The Construction of an Antisymmetric Function from the Basis Functions for Two Conjugate Representations 53
2.8. Young Operators 54
2.9. The Construction of Basis Functions for a Standard 
56 
Part 3. The Nonstandard Representation 59
2.10. Definition 59
2.11. The Transformation Matrix 61
2.12. Some Generalizations 65
2.13. Young Operators in a Nonstandard Representation 66
CHAPTER III. Groups of Linear Transformations 70
Part 1. Continuous Groups 70
3.1. Definition. Distinctive Features of Continuous Groups 70
3.2. Examples of Linear Groups 72
3.3. Infinitesimal Operators 74
Part 2. The Three-Dimensional Rotation Group 76
3.4. Rotation Operators and Angular Momentum Operators 76
3.5. Irreducible Representations 77
3.6. Reduction of the Direct Product of Two Irreducible Representations 80
3.7. Reduction of t he Direct Product of k Irreducible Representations. 3n-j Symbols 82
Part 3. Point Groups 86
3.8. Symmetry Elements and Symmetry Operations 86
3.9. Classification of Point Groups 88
CHAPTER IV. Tensor Representations and Tensor Operators 96
Part 1. The Interconnection between Linear Groups 
96 
4.1. Construction of a Tensor Representation 96
4.2. Reduction of a Tensor Representation into Irreducible Components 97
4.3. Formulae for t he Characters of Symmetrized Powers of Representations 101
4.4. Littlewood's Theorem 104
4.5. The Reduction of U2j+1 . R3 107
Part 2. Irreducible Tensor Operators 110
4.6. Definition 110
4.7. The Wigner-Eckart Theorem 113
4.8. Matrix Elements of Spherical Tensors 114
PART II: SYMMETRY AND QUANTAL CALCULATIONS 118
CHAPTER V. Principles of the Application of Group Theory to Quantum Mechanics 120
5.1. The Symmetry of the Schrödinger Equation and the Classification of States 120
5.2. Conservation Laws 123
5.3. Perturbation Theory 125
5.4. The Variation Method 129
5.5. Selection Rules 132
CHAPTER VI. Classification of States 137
Part 1. Electrons in a Central Field 137
6.1. Equivalent Electrons. L-S Coupling 137
6.2. Additional Quantum Numbers. The Seniority Number 142
6.3. Equivalent Electrons. J-J Coupling 144
6.4. Configurations of Several Groups of Equivalent Electrons 146
Part 2. The Connection between Molecular Terms 
148 
6.5. The Classification of Molecular Terms and the Total Nuclear Spin 148
6.6. The Determination of the Nuclear Statistical Weights of Coordinate States 153
6.7. Statistical Weights of Rotational Levels and Molecular Spin Modifications 155
6.8. Transition from the Rotational Partition Function to an Integral over States. The Symmetry Number 161
Part 3. Classification of States in Approximate Quantal 
163 
6.9. The Configuration Interaction Method and Partial Diagonalization of the Secular Equation 163
6.10. Determination of the Multiplets Which May Arise in 
166 
6.11. Determination of All the Multiplets Which Arise from a Ring of Six s Orbitals with Full Configuration Interaction 174
CHAPTER VII. The Method of Coefficients of Fractional Parentage 180
Part 1. Equivalent Electrons 180
7.1. Definition of Coefficients of Fractional Parentage 180
7.2. Calculation of Matrix Elements of Symmetric Operators 186
Part 2. Configurations of Several Groups of Equivalent Electrons. 
190 
7.3. A Single Shell 190
7.4. A Configuration of Two Shells 196
7.5. An Arbitrary Multishell Configuration 202
7.6. Formulae for the Matrix Elements of Symmetric Spin-Independent Operators 204
Part 3. Non-Vector-Coupled States 207
7.7. Configuration of Singly Occupied Orbitals 207
7.8. Arbitrarily Occupied Orbitals 210
CHAPTER VIII. Calculation of Electronic States of Molecular Systems 212
Part 1. The Hydrogen Molecule. Configuration Interaction 212
8.1. The Valence Bond Method 212
8.2. The Molecular Orbital Method 216
8.3. Orthogonal Localized Orbitals 218
Part 2. Calculation of the Energy Matrix for an Arbitrary 
219 
8.4. Matrix Elements of the Operators F and G 219
8.5. Expression for the Matrix Elements of the Hamiltonian 226
8.6. The Interaction of Two Subsystems In States with Definite Spins 235
Part 3. Symmetric Systems 241
8.7. Construction of Basis Functions in the Molecular Orbital Method for the Molecular Symmetry Group 241
8.8. Method of Calculation in the Valence Bond Approximation 246
8.9. Calculation on the H3 Molecule with Full Interaction of All Configurations of 1s Orbitals 255
Part 4. The Self-Consistent Field Method 259
8.10. The Hartree-Fock Equations 259
8.11. Survey of Methods of Treating Electron Correlation 267
8.12. Self-Consistent Field Equations for Spin-Degenerate States 278
8.13. Configuration of Singly Occupied Nonorthogonal Orbitals 285
8.14. The Self-Consistent Field Equations for the Method of 
292 
APPENDIX 1: Character Tables for Point Groups 298
APPENDIX 2: Matrices of Orthogonal Irreducible Representations of the Point Groups 302
APPENDIX 3: Tables for the Reduction of the Representations U[.]2j+1 to the Group R3 306
APPENDIX 4: Character Tables for the Permutation Groups .2 to .8 309
APPENDIX 5: Matrices of the Orthogonal Irreducible Representations for the Permutation Groups .3 to .6 312
APPENDIX 6: Tables of the Matrices< r'."/PNab-1N " 1N/r>
APPENDIX 7: Tables of the Matrices< (r'1r'2').'.'/PNac-IN/r1r2>
References 366
Author Index 374
Subject Index 378
Physical Chemistry 384