Group Theory Book

Basic Mathematics
Basic Mathematical Background: Introduction     3
Definition of a Group     3
Simple Example of a Group     3
Basic Definitions     6
Rearrangement Theorem     7
Cosets     7
Conjugation and Class     9
Factor Groups     11
Group Theory and Quantum Mechanics     11
Representation Theory and Basic Theorems     15
Important Definitions     15
Matrices     16
Irreducible Representations     17
The Unitarity of Representations     19
Schur's Lemma (Part 1)     21
Schur's Lemma (Part 2)     23
Wonderful Orthogonality Theorem     25
Representations and Vector Spaces     28
Character of a Representation     29
Definition of Character     29
Characters and Class     30
Wonderful Orthogonality Theorem for Character     31
Reducible Representations     33
The Number of Irreducible Representations     35
Second Orthogonality Relation for Characters     36
Regular Representation     37
Setting up Character Tables     40
Schoenflies SymmetryNotation     44
The Hermann-Mauguin Symmetry Notation     46
Symmetry Relations and Point Group Classifications     48
Basis Functions     57
Symmetry Operations and Basis Functions     57
Basis Functions for Irreducible Representations     58
Projection Operators P[subscript kl superscript (Gamma subscript n)]     64
Derivation of an Explicit Expression for P[subscript kl superscript (Gamma subscript n)]     64
The Effect of Projection Operations on an Arbitrary Function     65
Linear Combinations of Atomic Orbitals for Three Equivalent Atoms at the Corners of an Equilateral Triangle     67
The Application of Group Theory to Quantum Mechanics     70
Introductory Application to Quantum Systems
Splitting of Atomic Orbitals in a Crystal Potential     79
Introduction     79
Characters for the Full Rotation Group     81
Cubic Crystal Field Environment for a Paramagnetic Transition Metal Ion     85
Comments on Basis Functions     90
Comments on the Form of Crystal Fields     92
Application to Selection Rules and Direct Products     97
The Electromagnetic Interaction as a Perturbation     97
Orthogonality of Basis Functions     99
Direct Product of Two Groups     100
Direct Product of Two Irreducible Representations     101
Characters for the Direct Product     103
Selection Rule Concept in Group Theoretical Terms     105
Example of Selection Rules     106
Molecular Systems
Electronic States of Molecules and Directed Valence     113
Introduction     113
General Concept of Equivalence     115
Directed Valence Bonding     117
Diatomic Molecules     118
Homonuclear Diatomic Molecules     118
Heterogeneous Diatomic Molecules     120
Electronic Orbitals for Multiatomic Molecules     124
The NH[subscript 3] Molecule     124
The CH[subscript 4] Molecule     125
The Hypothetical SH[subscript 6] Molecule     129
The Octahedral SF[subscript 6] Molecule     133
[sigma]- and [pi]-Bonds     134
Jahn-Teller Effect     141
Molecular Vibrations, Infrared, and Raman Activity     147
Molecular Vibrations: Background     147
Application of Group Theory to Molecular Vibrations     149
Finding the Vibrational Normal Modes     152
Molecular Vibrations in H[subscript 2]O     154
Overtones and Combination Modes     156
Infrared Activity     157
Raman Effect     159
Vibrations for Specific Molecules     161
The Linear Molecules     161
Vibrations of the NH[subscript 3] Molecule     166
Vibrations of the CH[subscript 4] Molecule     168
Rotational Energy Levels     170
The Rigid Rotator     170
Wigner-Eckart Theorem     172
Vibrational-Rotational Interaction     174
Application to Periodic Lattices
Space Groups in Real Space     183
Mathematical Background for Space Groups     184
Space Groups Symmetry Operations     184
Compound Space Group Operations     186
Translation Subgroup     188
Symmorphic and Nonsymmorphic Space Groups     189
Bravais Lattices and Space Groups     190
Examples of Symmorphic Space Groups     192
Cubic Space Groups and the Equivalence Transformation     194
Examples of Nonsymmorphic Space Groups     196
Two-Dimensional Space Groups     198
2D Oblique Space Groups     200
2D Rectangular Space Groups     201
2D Square Space Group     203
2D Hexagonal Space Groups      203
Line Groups     204
The Determination of Crystal Structure and Space Group     205
Determination of the Crystal Structure     206
Determination of the Space Group     206
Space Groups in Reciprocal Space and Representations     209
Reciprocal Space     210
Translation Subgroup     211
Representations for the Translation Group     211
Bloch's Theorem and the Basis of the Translational Group     212
Symmetry of k Vectors and the Group of the Wave Vector     214
Point Group Operation in r-space and k-space     214
The Group of the Wave Vector G[subscript k] and the Star of k     215
Effect of Translations and Point Group Operations on Bloch Functions     215
Space Group Representations     219
Symmorphic Group Representations     219
Nonsymmorphic Group Representations and the Multiplier Algebra     220
Characters for the Equivalence Representation     221
Common Cubic Lattices: Symmorphic Space Groups     222
The [Gamma] Point     223
Points with k [not equal] 0     224
Compatibility Relations     227
The Diamond Structure: Nonsymmorphic Space Group     230
Factor Group and the [Gamma] Point     231
Points with k [not equal] 0     232
Finding Character Tables for all Groups of the Wave Vector     235
Electron and Phonon Dispersion Relation
Applications to Lattice Vibrations     241
Introduction     241
Lattice Modes and Molecular Vibrations     244
Zone Center Phonon Modes     246
The NaCl Structure     246
The Perovskite Structure     247
Phonons in the Nonsymmorphic Diamond Lattice     250
Phonons in the Zinc Blende Structure     252
Lattice Modes Away from k = 0     253
Phonons in NaCl at the X Point k = ([pi]/a)(100)     254
Phonons in BaTiO[subscript 3] at the X Point     256
Phonons at the K Point in Two-Dimensional Graphite     258
Phonons in Te and [alpha]-Quartz Nonsymmorphic Structures     262
Phonons in Tellurium     262
Phonons in the [alpha]-Quartz Structure     268
Effect of Axial Stress on Phonons     272
Electronic Energy Levels in a Cubic Crystals     279
Introduction     279
Plane Wave Solutions at k = 0     282
Symmetrized Plane Solution Waves along the [Delta]-Axis     286
Plane Wave Solutions at the X Point      288
Effect of Glide Planes and Screw Axes     294
Energy Band Models Based on Symmetry     305
Introduction     305
k [middle dot] p Perturbation Theory     307
Example of k [middle dot] p Perturbation Theory for a Nondegenerate [characters not reproducible] Band     308
Two Band Model: Degenerate First-Order Perturbation Theory     311
Degenerate second-order k [middle dot] p Perturbation Theory     316
Nondegenerate k [middle dot] p Perturbation Theory at a [Delta] Point     324
Use of k [middle dot] p Perturbation Theory to Interpret Optical Experiments     326
Application of Group Theory to Valley-Orbit Interactions in Semiconductors     327
Background     328
Impurities in Multivalley Semiconductors     330
The Valley-Orbit Interaction     331
Spin-Orbit Interaction in Solids and Double Groups     337
Introduction     337
Crystal Double Groups     341
Double Group Properties     343
Crystal Field Splitting Including Spin-Orbit Coupling     349
Basis Functions for Double Group Representations     353
Some Explicit Basis Functions     355
Basis Functions for Other [Gamma subscript 8 superscript +] States      358
Comments on Double Group Character Tables     359
Plane Wave Basis Functions for Double Group Representations     360
Group of the Wave Vector for Nonsymmorphic Double Groups     362
Application of Double Groups to Energy Bands with Spin     367
Introduction     367
E(k) for the Empty Lattice Including Spin-Orbit Interaction     368
The k [middle dot] p Perturbation with Spin-Orbit Interaction     369
E(k) for a Nondegenerate Band Including Spin-Orbit Interaction     372
E(k) for Degenerate Bands Including Spin-Orbit Interaction     374
Effective g-Factor     378
Fourier Expansion of Energy Bands: Slater-Koster Method     389
Contributions at d = 0     396
Contributions at d = 1     396
Contributions at d = 2     397
Summing Contributions through d = 2     397
Other Degenerate Levels     397
Other Symmetries
Time Reversal Symmetry     403
The Time Reversal Operator     403
Properties of the Time Reversal Operator     404
The Effect of T on E(k), Neglecting Spin     407
The Effect of T on E(k), Including the Spin-Orbit Interaction     411
Magnetic Groups      416
Introduction     418
Types of Elements     418
Types of Magnetic Point Groups     419
Properties of the 58 Magnetic Point Groups {lcub}A[subscript i], M[subscript k]{rcub}     419
Examples of Magnetic Structures     423
Effect of Symmetry on the Spin Hamiltonian for the 32 Ordinary Point Groups     426
Permutation Groups and Many-Electron States     431
Introduction     432
Classes and Irreducible Representations of Permutation Groups     434
Basis Functions of Permutation Groups     437
Pauli Principle in Atomic Spectra     440
Two-Electron States     440
Three-Electron States     443
Four-Electron States     448
Five-Electron States     451
General Comments on Many-Electron States     451
Symmetry Properties of Tensors     455
Introduction     455
Independent Components of Tensors Under Permutation Group Symmetry     458
Independent Components of Tensors: Point Symmetry Groups     462
Independent Components of Tensors Under Full Rotational Symmetry     463
Tensors in Nonlinear Optics     463
Cubic Symmetry: O[subscript h]     464
Tetrahedral Symmetry: T[subscript d]     466
Hexagonal Symmetry: D[subscript 6h]     466
Elastic Modulus Tensor     467
Full Rotational Symmetry: 3D Isotropy     469
Icosahedral Symmetry     472
Cubic Symmetry     472
Other Symmetry Groups     474
Point Group Character Tables     479
Two-Dimensional Space Groups     489
Tables for 3D Space Groups     499
Real Space     499
Reciprocal Space     503
Tables for Double Groups     521
Group Theory Aspects of Carbon Nanotubes     533
Nanotube Geometry and the (n, m) Indices     534
Lattice Vectors in Real Space     534
Lattice Vectors in Reciprocal Space     535
Compound Operations and Tube Helicity     536
Character Tables for Carbon Nanotubes     538
Permutation Group Character Tables     543
References     549
Index     553