Site Symmetry in Crystals: Theory and Applications

1. Introduction.- 2. Finite Groups and Their Representations.- 2.1 Elements of Group Theory.- 2.1.1 Groups. Generators and Generating Relations. Subgroups. Cosets. Invariant Subgroups. The Factor Group.- 2.1.2 Conjugate Elements and Classes. Factorization of Groups.- 2.1.3 Homomorphism and Isomorphism of Groups.- 2.2 Elements of Group Representation Theory.- 2.2.1 Representations of a Group. Equivalent, Reducible and Irreducible Representations. Orthogonality Relations. Representation Characters.- 2.2.2 Decomposition of Representations. Complex Conjugate Representations.- 2.3 Generation of Representations.- 2.3.1 Direct Product of Representations.- 2.3.2 Subduction of Representations.- 2.3.3 Induction of Representations.- 2.3.4 Little Group Method of Irreducible Representation Generation.- 3. Symmetry Groups and Their Representations.- 3.1 The Euclidean Group and Its Subgroups.- 3.1.1 Translation Group.- 3.1.2 Rotation Group.- 3.1.3 Inversion Group.- 3.1.4 Full Orthogonal Group.- 3.1.5 Euclidean Group.- 3.2 Point Symmetry Groups.- 3.2.1 Symmetry Elements of Molecules and Crystallographic Point Groups.- 3.2.2 Site Symmetry Subgroups of Point Groups.- 3.3 Space Groups.- 3.3.1 Symmetry of a Model of an Infinite Crystal. Symmorphic and Nonsymmorphic Space Groups.- 3.3.2 Symmetry of a Cyclic Model of a Crystal.- 3.4 Site Symmetry in Space Groups.- 3.4.1 Crystallographic Orbits. Wyckoff Positions.- 3.4.2 Oriented Site Symmetry Groups. Choice of Origin.- 3.4.3 Crystal Structure Types. Crystals with Space Group D4h14.- 3.5 Symmetry Operations in Quantum Mechanics.- 3.5.1 Symmetry Group of a Quantum Mechanical System.- 3.5.2 Wigner's Theorem.- 3.5.3 Time-Reversal Symmetry.- 3.6 Irreducible Representations of Rotation and Full Orthogonal Groups.- 3.7 Representations of Point Groups.- 3.8 Representations of Space Groups.- 3.8.1 Irreducible Representations of the Translation Group. The Brillouin Zone.- 3.8.2 Stars of Wave Vectors. Little Group. Full Representations of Space Groups.- 3.8.3 Small Representations of a Little Group. Projective Representations of Point Groups.- 3.8.4 Double-Valued Representations of Space Groups.- 3.8.5 Dependence of the Labeling of the Irreducible Representations of a Space Group on the Setting.- 3.8.6 Example: Irreducible Representations of Space Group D4h14. Compatibility Tables.- 4. Site Symmetry and Induced Representations of Symmetry Groups.- 4.1 Induced Representations of Point Groups. Correlation Tables.- 4.2 Induced Representations of Space Groups.- 4.2.1 Induction from Site Symmetry Subgroups of Space Groups.- 4.2.2 Induced Representations in the k-Basis. Band Representations.- 4.2.3 Simple and Composite Induced Representations.- 4.3 Double-Valued Induced Representations.- 4.4 Generation of the Simple Induced Representations of the Space Group D4h14.- 4.5 The Twenty-Four Most Common Space Groups: Crystal Structures and Tables of Simple Induced Representations.- 4.5.1 Tables of Simple Induced Representations and Their Use.- 4.5.2 Space Groups and Crystal Structures with Cubic Lattices.- 4.5.3 Space Groups and Crystal Structures with Hexagonal and Trigonal Lattices.- 4.5.4 Space Groups and Crystal Structures with Tetragonal Lattices.- 4.5.5 Space Groups and Crystal Structures with Orthorhombic Lattices.- 4.5.6 Space Group Setting and Simple Induced Representations for Monoclinic Space Groups.- 5. Application of Induced Representations in the Electron Theory of Molecules and Crystals.- 5.1 Adiabatic and One-Electron Approximations.- 5.1.1 Space Symmetry of the One-Electron Approximation Hamiltonian.- 5.2 Induced Representations in the Electron Theory of Molecules.- 5.2.1 Canonical, Localized and Hybridized Molecular Orbitals.- 5.2.2 Localized Two-Center Bonds and Hybridized Orbitals in AB4 and AB3 Molecules.- 5.2.3 Multicentered Bonds in the 1,6-C2B4H6 Molecule.- 5.2.4 Canonical and Localized Orbitals in the MnO4- Molecular Ion.- 5.2.5 Localized Orbitals in the Tetrahedral Bi4 Molecule.- 5.3 One-Electron Approximation for Crystals.- 5.3.1 Crystalline Orbitals. Degenerate and Nondegenerate Energy Bands.- 5.3.2 Equivalent Hamiltonians for the Same Crystal Structures.- 5.3.3 k?p Perturbation Method in the Energy Band Theory.- 5.3.4 Zero-Slope Points of Energy Bands.- 5.3.5 Energy Bands in the Neighborhood of Degeneracy Points.- 5.3.6 Additional Degeneracy of Energy Bands Due to the Reality of the Hamiltonian.- 5.3.7 Density of States of an Energy Band.- 5.4 Induced Representations and the Theory of Chemical Bonding in Crystals.- 5.4.1 Energy Band States and Localized Functions.- 5.4.2 Localized Orbitals and Atomic States in Crystals.- 5.4.3 Hybridized Orbitals in Crystals.- 5.4.4 Crystals with Space Group Oh7.- 5.4.5 Crystals with Space Group Oh5.- 5.4.6 Crystals with Space Group D4h14.- 5.4.7 One-Electron States in High-Tc Superconductors.- 5.5 Energy Bands and Localized States.- 5.5.1 Localized Orbitals and Parameters of an Energy Band.- 5.5.2 Generation of Localized Functions in Crystals.- 5.5.3 Interpolation Scheme Using Localized Functions.- 5.6 Localized Orbitals in Molecular Models of Crystals.- 5.6.1 Cluster Model of Perfect Crystals.- 5.6.2 Cluster and Crystal Localized Orbitals.- 5.6.3 Energy Bands of AgBr from Cluster Calculations of [Ag14Br13]+.- 5.6.4 Cyclic Model as a Molecular Model of Crystals.- 5.6.5 Localized Orbitals in the Cyclic Model.- 6. Induced Representations in the Theory of Imperfect Crystals.- 6.1 Point Defects in Crystals.- 6.1.1 Single Defect Model.- 6.1.2 Cluster Model of Imperfect Crystals.- 6.1.3 Cyclic Model of Imperfect Crystals.- 6.1.4 Band Model of Imperfect Crystals.- 6.1.5 Localized Orbitals in the Band Model of Point Defects.- 6.2 Diperiodic Space Groups. Surface Electron States.- 6.2.1 Diperiodic (Layer) Space Groups.- 6.2.2 Site Symmetry in Layer Groups.- 6.2.3 Irreducible Representations of Diperiodic Groups.- 6.2.4 Induced Representations of Diperiodic Groups.- 6.2.5 Use of Translational Symmetry in the Comparison of Bulk and Surface Crystalline States.- 7. Application of Induced Representations of Space Groups to Second Order Phase Transitions.- 7.1 Symmetry Rules in the Landau Theory of Second Order Phase Transitions.- 7.2 Tensor Fields in Crystals and Induced Representations of Space Groups. Tensor Fields for Space Group D4h14.- 7.3 Vibrational Field Representation and Phase Transitions in High-Temperature Superconductors.- 8. Induced Representations of Space Groups in Phonon Spectroscopy of Crystals.- 8.1 Phonon Symmetry Analysis.- 8.2 Infrared and Raman Spectra Selection Rules.- 8.3 Phonon Symmetry and Optical Spectra Selection Rules in Semiconductor Superlattices.- 8.3.1 (GaAs)m(AlAs)n Superlattices.- 8.3.2 (Si)m(Ge)n Superlattices.- 8.3.3 Experimental Applications.- 8.4 Phonon Symmetry in High-Temperature Superconductors.- 8.5 Phonon Symmetry in Diperiodic Systems.- 9. Site Symmetry in Magnetic Crystals and Induced Corepresentations.- 9.1 Shubnikov Space Groups of Symmetry of Magnetic Crystals.- 9.2 Site Symmetry in Magnetic Crystals.- 9.3 Corepresentations of Shubnikov Space Groups.- 9.4 Induced Corepresentations of Magnetic Space Groups.- 9.5 Corepresentations of the Space Groups of Antiferromagnetic La2CuO4.- 10. Site Symmetry in Permutation - Inversion Symmetry Groups of Nonrigid Crystals.- 10.1 Symmetry Groups of Nonrigid Crystals.- 10.1.1 Labeling of Nuclei. Sampling of Coordinate Systems.- 10.1.2 Description of Permutation - Inversion Symmetry Elements.- 10.1.3 Coordinate Transformations Induced by Permutation - Inversion Symmetry Elements.- 10.1.4 Site Symmetry Group of a Rotating Molecule in a Nonrigid Crystal.- 10.1.5 Permutation - Inversion Group of a Nonrigid Sodium Nitrate Crystal.- 10.2 Irreducible Representations of a Nonrigid Crystal Symmetry Group.- 10.2.1 Generation of Irreducible Representations.- 10.2.2 Irreducible Representations of a Site Symmetry Group.- 10.2.3 Classification of States.- 10.3 Generalized Symmetry of High-Temperature Phase of Fullerite C60.- 10.3.1 Permutation - Inversion Symmetry Group of Fullerite C60 in the High-Temperature Phase.- 10.3.2 Irreducible Representations of the Groups [n] and Pc.- 10.3.3 Classification of States of Nonrigid Fullerite C60.- References.