| Preface | V |
| 1 | Introduction | 1 |
| 1-1 | The Nature of the Problem | 1 |
| 1-2 | The Role of Symmetry | 3 |
| 2 | Abstract Group Theory | 6 |
| 2-1 | Definitions and Nomenclature | 6 |
| 2-2 | Illustrative Examples | 7 |
| 2-3 | Rearrangement Theorem | 8 |
| 2-4 | Cyclic Groups | 9 |
| 2-5 | Subgroups and Cosets | 9 |
| 2-6 | Example Groups of Finite Order | 10 |
| 2-7 | Conjugate Elements and Class Structure | 12 |
| 2-8 | Normal Divisors and Factor Groups | 13 |
| 2-9 | Class Multiplication | 15 |
| Exercises | 16 |
| References | 17 |
| 3 | Theory of Group Representations | 18 |
| 3-1 | Definitions | 18 |
| 3-2 | Proof of the Orthogonality Theorem | 20 |
| 3-3 | The Character of a Representation | 25 |
| 3-4 | Construction of Character Tables | 28 |
| 3-5 | Decomposition of Reducible Representations | 29 |
| 3-6 | Application of Representation Theory in Quantum Mechanics | 31 |
| 3-7 | Illustrative Representations of Abelian Groups | 37 |
| 3-8 | Basis Functions for Irreducible Representations | 39 |
| 3-9 | Direct-product Groups | 43 |
| 3-10 | Direct-product Representations within a Group | 46 |
| Exercises | 47 |
| References | 48 |
| 4 | Physical Applications of Group Theory | 50 |
| 4-1 | Crystal-symmetry Operators | 51 |
| 4-2 | The Crystallographic Point Groups | 54 |
| 4-3 | Irreducible Representations of the Point Groups | 62 |
| 4-4 | Elementary Representations of the Three-dimensional Rotation Group | 65 |
| 4-5 | Crystal-field Splitting of Atomic Energy Levels | 67 |
| 4-6 | Intermediate Crystal-field-splitting Case | 69 |
| 4-7 | Weak-crystal-field Case and Crystal Double Groups | 75 |
| 4-8 | Introduction of Spin Effects in the Medium-field Case | 78 |
| 4-9 | Group-theoretical Matrix-element Theorems | 80 |
| 4-10 | Selection Rules and Parity | 82 |
| 4-11 | Directed Valence | 87 |
| 4-12 | Application of Group Theory to Directed Valence | 89 |
| Exercises | 92 |
| References | 93 |
| 5 | Full Rotation Group and Angular Momentum | 94 |
| 5-1 | Rotational Transformation Properties and Angular Momentum | 94 |
| 5-2 | Continuous Groups | 98 |
| 5-3 | Representation of Rotations through Eulerian Angles | 101 |
| 5-4 | Homomorphism with the Unitary Group | 103 |
| 5-5 | Representations of the Unitary Group | 106 |
| 5-6 | Representation of the Rotation Group by Representations of the Unitary Group | 109 |
| 5-7 | Application of the Rotation-representation Matrices | 111 |
| 5-8 | Vector Model for Addition of Angular Momenta | 115 |
| 5-9 | The Wigner or Clebsch-Gordan Coefficients | 117 |
| 5-10 | Notation, Tabulations, and Symmetry Properties of the Wigner Coefficients | 121 |
| 5-11 | Tensor Operators | 124 |
| 5-12 | The Wigner-Eckart Theorem | 131 |
| 5-13 | The Racah Coefficients | 133 |
| 5-14 | Application of Racah Coefficients | 137 |
| 5-15 | The Rotation-Inversion Group | 139 |
| 5-16 | Time-reversal Symmetry | 141 |
| 5-17 | More General Invariances | 147 |
| Exercises | 151 |
| References | 153 |
| 6 | Quantum Mechanics of Atoms | 154 |
| 6-1 | Review of Elementary Atomic Structure and Nomenclature | 155 |
| 6-2 | The Hamiltonian | 157 |
| 6-3 | Approximate Eigenfunctions | 157 |
| 6-4 | Calculation of Matrix Elements between Determinantal Wavefunctions | 162 |
| 6-5 | Hartree-Fock Method | 167 |
| 6-6 | Calculation of L-S-term Energies | 170 |
| 6-7 | Evaluation of Matrix Elements of the Energy | 173 |
| 6-8 | Eigenfunctions and Angular-momentum Operations | 178 |
| 6-9 | Calculation of Fine Structure | 181 |
| 6-10 | Zeeman Effect | 188 |
| 6-11 | Magnetic Hyperfine Structure | 193 |
| 6-12 | Electric Hyperfine Structure | 201 |
| Exercises | 206 |
| References | 208 |
| 7 | Molecular Quantum Mechanics | 210 |
| 7-1 | Born-Oppenheimer Approximation | 210 |
| 7-2 | Simple Electronic Eigenfunctions | 213 |
| 7-3 | Irreducible Representations for Linear Molecules | 216 |
| 7-4 | The Hydrogen Molecule | 219 |
| 7-5 | Molecular Orbitals | 220 |
| 7-6 | Heitler-London Method | 223 |
| 7-7 | Orthogonal Atomic Orbitals | 226 |
| 7-8 | Group Theory and Molecular Orbitals | 227 |
| 7-9 | Selection Rules for Electronic Transitions | 233 |
| 7-10 | Vibration of Diatomic Molecules | 234 |
| 7-11 | Normal Modes in Polyatomic Molecules | 238 |
| 7-12 | Group Theory and Normal Modes | 242 |
| 7-13 | Selection Rules for Vibrational Transitions | 248 |
| 7-14 | Molecular Rotation | 250 |
| 7-15 | Effect of Nuclear Statistics on Molecular Rotation | 252 |
| 7-16 | Asymmetric Rotor | 255 |
| 7-17 | Vibration-Rotation Interaction | 257 |
| 7-18 | Rotation-Electronic Coupling | 260 |
| Exercises | 264 |
| References | 266 |
| 8 | Solid-state Theory | 267 |
| 8-1 | Symmetry Properties in Solids | 267 |
| 8-2 | The Reciprocal Lattice and Brillouin Zones | 270 |
| 8-3 | Form of Energy-band Wavefunctions | 275 |
| 8-4 | Crystal Symmetry and the Group of the k Vector | 279 |
| 8-5 | Pictorial Consideration of Eigenfunctions | 281 |
| 8-6 | Formal Consideration of Degeneracy and Compatibility | 284 |
| 8-7 | Group Theory and the Plane-wave Approximation | 290 |
| 8-8 | Connection between Tight- and Loose-binding Approximations | 293 |
| 8-9 | Spin-orbit Coupling in Band Theory | 295 |
| 8-10 | Time Reversal in Band Theory | 297 |
| 8-11 | Magnetic Crystal Groups | 299 |
| 8-12 | Symmetries of Magnetic Structures | 303 |
| 8-13 | The Landau Theory of Second-order Phase Transitions | 309 |
| 8-14 | Irreducible Representations of Magnetic Groups | 311 |
| Exercises | 313 |
| References | 315 |
| Appendix | |
| A | Review of Vectors, Vector Spaces, and Matrices | 317 |
| B | Character Tables for Point-symmetry Groups | 323 |
| C | Tables of c[superscript k] and a[superscript k] Coefficients for s, p, and d Electrons | 331 |
| Index | 335 |
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