Introduction to Group Theory with Applications

Preface

Acknowledgments

Chapter 1 Symmetry Operations


1-1 Introduction


1-2 Point Symmetry Operations


1-3 The Stereographic Projection


1-4 The 32 Crystallographic Point Groups


1-5 Related Considerations


1-6 Space Group Example

Notes

Problems

Chapter 2 Group Concepts


2-1 Introduction


2-2 Definition of a Group


2-3 Symmetry Operations Form a Group


2-4 Related Group Concepts


2-5 Isomorphism and Homomorphism


2-6 Special Kinds of Groups


2-7 More Involved Group Concepts (including a Factor Group of a Space Group)

Appendix to Chapter 2

Notes

Problems

Chapter 3 Matrix Representations of Finite Groups


3-1 Introduction


3-2 Representations


3-3 Irreducible Representations


3-4 Representations of a Factor Group

Appendix to Chapter 3

Notes

Problems

Chapter 4 Characters of Matrix Representations of Finite Groups


4-1 Properties of Characters of Irreducible Representations


4-2 Character Tables


4-3 Reduction of a Reducible Representation


4-4 Basis Functions


4-5 Examples—Neumann Principle


4-6 Atomic Positions


4-7 The Hamiltonian

Appendix to Chapter 4

Notes

Problems

Chapter 5 Vibrations of Molecules and Crystals


5-1 3N Degrees of Freedom


5-2 General Considerations


5-3 Number and Type of Normal Modes for Molecules


5-4 Internal Coordinates


5-5 Crystals


5-6 Eigenvectors and Symmetry Adapted Vectors


5-7 Projection Operators


5-8 Projection Operators Applied to Normal Coordinates

Notes

Problems

Chapter 6 Normal Modes (Direct Product and Selection Rules)


6-1 Direct Product of Irreducible Representations


6-2 Vibrational Wave Function


6-3 Selection Rules—Infrared and Raman


6-4 Molecular Approximations (Site Symmetry and Davydov Splitting)

Notes

Problems

Chapter 7 Quantum Mechanics


7-1 Atomic Wave Functions


7-2 Transformation of Functions


7-3 Eigenfunctions as Basis Functions


7-4 Proper Rotations and Angular Momentum


7-5 Perturbations


7-6 Matrix Elements (Selection Rules)


7-7 General Secular Equation Problem

Appendix to Chapter 7

Notes

Problems

Chapter 8 Crystal Field Theory (and Atomic Physics)


8-1 Rotations in Terms of Euler Angles


8-2 Representations of the Full Rotation Group


8-3 Reduction of Symmetry


8-4 Energy Level Diagrams (Correlation Diagrams)


8-5 Crystal Double Groups


8-6 Correlation Diagrams including Double Groups


8-7 Other Crystal Field Effects

Appendix to Chapter 8

Notes

Problems

Chapter 9 Hybrid Functions


9-1 Introduction


9-2 Simple Hybrid Functions and Bonding


9-3 Tetrahedral Hybridization


9-4 Other Hybrid Functions


9-5 π-Hybrid Functions


9-6 Comment on Hybrid Orbitals (Slater Determinant)

Notes

Problems

Chapter 10 Molecular Orbital Theory


10-1 Hydrogen Molecular Ion


10-2 Simple MO Theory


10-3 Transition Metal Complexes


10-4 LCAO-MO of π-Electrons in Conjugated Hydrocarbons


10-5 Woodward-Hoffman Rules

Notes

Problems

Chapter 11 Symmetry of Crystal Lattices


11-1 The Real Affine Group


11-2 Space Group


11-3 Translational Lattice


11-4 International Tables for X-Ray Crystallography, International Notation, etc.


11-5 Magnetic Groups (Color Groups)

Notes

Problems

Chapter 12 Band Theory of Solids


12-1 Translational Symmetry


12-2 Symmorphic Space Groups


12-3 Nonsymmorphic Space Groups


12-4 Spin-Orbit Effects on Bands


12-5 Time Reversal Symmetry

Notes

Problems

Chapter 13 The Full Rotation Group


13-1 The Homomorphism between the Special Unitary Group in Two Dimensions, SU(2), and the Three-Dimension Rotation Group


13-2 Irreducible Representations of SU(2)


13-3 Wigner Coefficients


13-4 Irreducible Tensor Operators


13-5 The Wigner-Eckart Theorem


13-6 Survey of 3j and Racah Coefficients

Notes

Problems

Appendix 1 Crystal Systems

Appendix 2 The 32 Point Groups

Appendix 3 Character Tables

Appendix 4 Space Groups

Appendix 5 Matrices, Vector Spaces, and Linear Operators

Appendix 6 Direct Product Tables

Appendix 7 Correlation Tables

Appendix 8 Spherical Harmonics

Appendix 9 Tanable-Sugano Diagrams

Appendix 10 Double Group Character Tables

Bibliography

Index