Kaplan / Loebl Symmetry of Many-Electron Systems

Physical Chemistry: A Series of Monographs
1. Auflage 2013
ISBN: 978-1-4831-9173-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 384 Seiten, Web PDF

Symmetry of Many-Electron Systems discusses the group-theoretical methods applied to physical and chemical problems. Group theory allows an individual to analyze qualitatively the elements of a certain system in scope. The text evaluates the characteristics of the Schrodinger equations. It is proved that some groups of continuous transformation from the Lie groups are useful in identifying conditions and in developing wavefunctions. A section of the book is devoted to the utilization of group-theoretical methods in quantal calculations on many-electron systems. The focus is on the use of group-theoretical methods to the classification and calculation of states of molecule. A chapter of the book gives a comprehensive discussion of the fractional parentage method. This application is used in atomic and nuclear spectroscopy. The method of employing coordinate wave functions is explained. The standard Young-Yamanouchi orthogonal representation is presented completely. The book will provide useful guides for physicists, chemists, engineers, students, and researchers in the field of physics.

Autoren/Hrsg.

Kaplan, I. G.

Loebl, Ernest M.

Inhaltsverzeichnis

1;Front Cover;1
2;Symmetry of Many-Electron Systems;4
3;Copyright Page;5
4;Table of Contents;6
5;Translator's Note;11
6;Preface to Russian Edition;12
7;PART I: MATHEMATICAL APPARATUS;14
7.1;CHAPTER I. Basic Concepts and Theorems of Group Theory;16
7.1.1;Part 1. Properties of Group Operations;16
7.1.1.1;1.1. Group Postulates;16
7.1.1.2;1.2. Examples of Groups;17
7.1.1.3;1.3. Isomorphism and Homomorphism;19
7.1.1.4;1.4. Subgroups and Cosets;20
7.1.1.5;1.5. Conjugate Elements. Classes;21
7.1.1.6;1.6. Invariant Subgroups. Factor Groups;22
7.1.1.7;1.7. Direct Products of Groups;23
7.1.1.8;1.8. The Semidirect Product;23
7.1.2;Part 2. Representations of Groups;24
7.1.2.1;1.9. Definition;24
7.1.2.2;1.10. Vector Spaces;25
7.1.2.3;1.11. Reducibility of Representations;28
7.1.2.4;1.12. Properties of Irreducible Representations;29
7.1.2.5;1.13. Characters;30
7.1.2.6;1.14. The Calculation of the Characters of Irreducible Representations;31
7.1.2.7;1.15. The Decomposition of a Reducible Representation;33
7.1.2.8;1.16. The Direct Product of Representations;34
7.1.2.9;1.17. Clebsch-Gordan Coefficients;37
7.1.2.10;1.18. The Regular Representation;38
7.1.2.11;1.19. The Construction of Basis Functions for Irreducible Representations;39
7.2;CHAPTER II. The Permutation Group;43
7.2.1;Part 1. General Considerations;43
7.2.1.1;2.1. Operations with Permutations;43
7.2.1.2;2.2. Classes;44
7.2.1.3;2.3. Young Diagrams and Irreducible Representations;45
7.2.2;Part 2. The Standard Young-Yamanouchi Orthogonal
Representation;47
7.2.2.1;2.4. Young Tableaux;47
7.2.2.2;2.5. Explicit Determination of the Matrices of the Standard Representation;49
7.2.2.3;2.6. The Conjugate Representation;52
7.2.2.4;2.7. The Construction of an Antisymmetric Function from the Basis Functions for Two Conjugate Representations;53
7.2.2.5;2.8. Young Operators;54
7.2.2.6;2.9. The Construction of Basis Functions for a Standard
Representation from a Product of N Orthogonal Functions;56
7.2.3;Part 3. The Nonstandard Representation;59
7.2.3.1;2.10. Definition;59
7.2.3.2;2.11. The Transformation Matrix;61
7.2.3.3;2.12. Some Generalizations;65
7.2.3.4;2.13. Young Operators in a Nonstandard Representation;66
7.3;CHAPTER III. Groups of Linear Transformations;70
7.3.1;Part 1. Continuous Groups;70
7.3.1.1;3.1. Definition. Distinctive Features of Continuous Groups;70
7.3.1.2;3.2. Examples of Linear Groups;72
7.3.1.3;3.3. Infinitesimal Operators;74
7.3.2;Part 2. The Three-Dimensional Rotation Group;76
7.3.2.1;3.4. Rotation Operators and Angular Momentum Operators;76
7.3.2.2;3.5. Irreducible Representations;77
7.3.2.3;3.6. Reduction of the Direct Product of Two Irreducible Representations;80
7.3.2.4;3.7. Reduction of t he Direct Product of k Irreducible Representations. 3n-j Symbols;82
7.3.3;Part 3. Point Groups;86
7.3.3.1;3.8. Symmetry Elements and Symmetry Operations;86
7.3.3.2;3.9. Classification of Point Groups;88
7.4;CHAPTER IV. Tensor Representations and Tensor Operators;96
7.4.1;Part 1. The Interconnection between Linear Groups
and Permutation Groups;96
7.4.1.1;4.1. Construction of a Tensor Representation;96
7.4.1.2;4.2. Reduction of a Tensor Representation into Irreducible Components;97
7.4.1.3;4.3. Formulae for t he Characters of Symmetrized Powers of Representations;101
7.4.1.4;4.4. Littlewood's Theorem;104
7.4.1.5;4.5. The Reduction of U2j+1 . R3;107
7.4.2;Part 2. Irreducible Tensor Operators;110
7.4.2.1;4.6. Definition;110
7.4.2.2;4.7. The Wigner-Eckart Theorem;113
7.4.2.3;4.8. Matrix Elements of Spherical Tensors;114
8;PART II: SYMMETRY AND QUANTAL CALCULATIONS;118
8.1;CHAPTER V. Principles of the Application of Group Theory to Quantum Mechanics;120
8.1.1;5.1. The Symmetry of the Schrödinger Equation and the Classification of States;120
8.1.2;5.2. Conservation Laws;123
8.1.3;5.3. Perturbation Theory;125
8.1.4;5.4. The Variation Method;129
8.1.5;5.5. Selection Rules;132
8.2;CHAPTER VI. Classification of States;137
8.2.1;Part 1. Electrons in a Central Field;137
8.2.1.1;6.1. Equivalent Electrons. L-S Coupling;137
8.2.1.2;6.2. Additional Quantum Numbers. The Seniority Number;142
8.2.1.3;6.3. Equivalent Electrons. J-J Coupling;144
8.2.1.4;6.4. Configurations of Several Groups of Equivalent Electrons;146
8.2.2;Part 2. The Connection between Molecular Terms
and Nuclear Spin;148
8.2.2.1;6.5. The Classification of Molecular Terms and the Total Nuclear Spin;148
8.2.2.2;6.6. The Determination of the Nuclear Statistical Weights of Coordinate States;153
8.2.2.3;6.7. Statistical Weights of Rotational Levels and Molecular Spin Modifications;155
8.2.2.4;6.8. Transition from the Rotational Partition Function to an Integral over States. The Symmetry Number;161
8.2.3;Part 3. Classification of States in Approximate Quantal
Calculations;163
8.2.3.1;6.9. The Configuration Interaction Method and Partial Diagonalization of the Secular Equation;163
8.2.3.2;6.10. Determination of the Multiplets Which May Arise in
Calculations by the Valence Bond Method;166
8.2.3.3;6.11. Determination of All the Multiplets Which Arise from a Ring of Six s Orbitals with Full Configuration Interaction;174
8.3;CHAPTER VII. The Method of Coefficients of Fractional Parentage;180
8.3.1;Part 1. Equivalent Electrons;180
8.3.1.1;7.1. Definition of Coefficients of Fractional Parentage;180
8.3.1.2;7.2. Calculation of Matrix Elements of Symmetric Operators;186
8.3.2;Part 2. Configurations of Several Groups of Equivalent Electrons.
A State with Arbitrary Permutational Symmetry;190
8.3.2.1;7.3. A Single Shell;190
8.3.2.2;7.4. A Configuration of Two Shells;196
8.3.2.3;7.5. An Arbitrary Multishell Configuration;202
8.3.2.4;7.6. Formulae for the Matrix Elements of Symmetric Spin-Independent Operators;204
8.3.3;Part 3. Non-Vector-Coupled States;207
8.3.3.1;7.7. Configuration of Singly Occupied Orbitals;207
8.3.3.2;7.8. Arbitrarily Occupied Orbitals;210
8.4;CHAPTER VIII. Calculation of Electronic States of Molecular Systems;212
8.4.1;Part 1. The Hydrogen Molecule. Configuration Interaction;212
8.4.1.1;8.1. The Valence Bond Method;212
8.4.1.2;8.2. The Molecular Orbital Method;216
8.4.1.3;8.3. Orthogonal Localized Orbitals;218
8.4.2;Part 2. Calculation of the Energy Matrix for an Arbitrary
Molecular System;219
8.4.2.1;8.4. Matrix Elements of the Operators F and G;219
8.4.2.2;8.5. Expression for the Matrix Elements of the Hamiltonian;226
8.4.2.3;8.6. The Interaction of Two Subsystems In States with Definite Spins;235
8.4.3;Part 3. Symmetric Systems;241
8.4.3.1;8.7. Construction of Basis Functions in the Molecular Orbital Method for the Molecular Symmetry Group;241
8.4.3.2;8.8. Method of Calculation in the Valence Bond Approximation;246
8.4.3.3;8.9. Calculation on the H3 Molecule with Full Interaction of All Configurations of 1s Orbitals;255
8.4.4;Part 4. The Self-Consistent Field Method;259
8.4.4.1;8.10. The Hartree-Fock Equations;259
8.4.4.2;8.11. Survey of Methods of Treating Electron Correlation;267
8.4.4.3;8.12. Self-Consistent Field Equations for Spin-Degenerate States;278
8.4.4.4;8.13. Configuration of Singly Occupied Nonorthogonal Orbitals;285
8.4.4.5;8.14. The Self-Consistent Field Equations for the Method of
"Different Orbitals for Different Spins";292
9;APPENDIX 1: Character Tables for Point Groups;298
10;APPENDIX 2: Matrices of Orthogonal Irreducible Representations of the Point Groups;302
11;APPENDIX 3: Tables for the Reduction of the Representations U[.]2j+1 to the Group R3;306
12;APPENDIX 4: Character Tables for the Permutation Groups .2 to .8;309
13;APPENDIX 5: Matrices of the Orthogonal Irreducible Representations for the Permutation Groups .3 to .6;312
14;APPENDIX 6: Tables of the Matrices[.] for Values of N from 3 to 6;328
15;APPENDIX 7: Tables of the Matrices<(r'1r'2').'.'\PNac-IN\r1r2>[.] for Values of N from 3 to 6;344
16;References;366
17;Author Index;374
18;Subject Index;378
19;Physical Chemistry;384