Cover	1
	Main title	5
	Copyright page	6
	Contents	7
	Preface	13
	1 Introduction	17
	1.1 Symmetries in Solid-State Physics and Photonics	20
	1.2 A Basic Example: Symmetries of a Square	22
	Part One Basics of Group Theory	25
	2 Symmetry Operations and Transformations of Fields	27
	2.1 Rotations and Translations	27
	2.1.1 Rotation Matrices	29
	2.1.2 Euler Angles	32
	2.1.3 Euler–Rodrigues Parameters and Quaternions	34
	2.1.4 Translations and General Transformations	39
	2.2 Transformation of Fields	41
	2.2.1 Transformation of Scalar Fields and Angular Momentum	42
	2.2.2 Transformation of Vector Fields and Total Angular Momentum	43
	2.2.3 Spinors	44
	3 Basics Abstract Group Theory	49
	3.1 Basic Definitions	49
	3.1.1 Isomorphism and Homomorphism	54
	3.2 Structure of Groups	55
	3.2.1 Classes	56
	3.2.2 Cosets and Normal Divisors	58
	3.3 Quotient Groups	62
	3.4 Product Groups	64
	4 Discrete Symmetry Groups in Solid-State Physics and Photonics	67
	4.1 Point Groups	68
	4.1.1 Notation of Symmetry Elements	68
	4.1.2 Classification of Point Groups	72
	4.2 Space Groups	75
	4.2.1 Lattices, Translation Group	75
	4.2.2 Symmorphic and Nonsymmorphic Space Groups	78
	4.2.3 Site Symmetry, Wyckoff Positions, and Wigner–Seitz Cell	81
	4.3 Color Groups and Magnetic Groups	85
	4.3.1 Magnetic Point Groups	85
	4.3.2 Magnetic Lattices	88
	4.3.3 Magnetic Space Groups	89
	4.4 Noncrystallographic Groups, Buckyballs, and Nanotubes	91
	4.4.1 Structure and Group Theory of Nanotubes	91
	4.4.2 Buckminsterfullerene C60	95
	5 Representation Theory	99
	5.1 Definition of Matrix Representations	100
	5.2 Reducible and Irreducible Representations	104
	5.2.1 The Orthogonality Theorem for Irreducible Representations	106
	5.3 Characters and Character Tables	110
	5.3.1 The Orthogonality Theorem for Characters	112
	5.3.2 Character Tables	114
	5.3.3 Notations of Irreducible Representations	114
	5.3.4 Decomposition of Reducible Representations	118
	5.4 Projection Operators and Basis Functions of Representations	121
	5.5 Direct Product Representations	128
	5.6 Wigner–Eckart Theorem	136
	5.7 Induced Representations	139
	6 Symmetry and Representation Theory in k-Space	149
	6.1 The Cyclic Born–von Kármán Boundary Condition and the Bloch Wave	149
	6.2 The Reciprocal Lattice	152
	6.3 The Brillouin Zone and the Group of the Wave Vector k	153
	6.4 Irreducible Representations of Symmorphic Space Groups	158
	6.5 Irreducible Representations of Nonsymmorphic Space Groups	159
	Part Two Applications in Electronic Structure Theory	165
	7 Solution of the Schrödinger Equation	167
	7.1 The Schrödinger Equation	167
	7.2 The Group of the Schrödinger Equation	169
	7.3 Degeneracy of Energy States	170
	7.4 Time-Independent Perturbation Theory	173
	7.4.1 General Formalism	175
	7.4.2 Crystal Field Expansion	176
	7.4.3 Crystal Field Operators	180
	7.5 Transition Probabilities and Selection Rules	185
	8 Generalization to Include the Spin	193
	8.1 The Pauli Equation	193
	8.2 Homomorphism between SU(2) and SO(3)	194
	8.3 Transformation of the Spin–Orbit Coupling Operator	196
	8.4 The Group of the Pauli Equation and Double Groups	199
	8.5 Irreducible Representations of Double Groups	202
	8.6 Splitting of Degeneracies by Spin–Orbit Coupling	205
	8.7 Time-Reversal Symmetry	209
	8.7.1 The Reality of Representations	209
	8.7.2 Spin-Independent Theory	210
	8.7.3 Spin-Dependent Theory	212
	9 Electronic Structure Calculations	213
	9.1 Solution of the Schrödinger Equation for a Crystal	213
	9.2 Symmetry Properties of Energy Bands	214
	9.2.1 Degeneracy and Symmetry of Energy Bands	216
	9.2.2 Compatibility Relations and Crossing of Bands	217
	9.3 Symmetry-Adapted Functions	219
	9.3.1 Symmetry-Adapted Plane Waves	219
	9.3.2 Localized Orbitals	221
	9.4 Construction of Tight-Binding Hamiltonians	226
	9.4.1 Hamiltonians in Two-Center Form	228
	9.4.2 Hamiltonians in Three-Center Form	232
	9.4.3 Inclusion of Spin–Orbit Interaction	240
	9.4.4 Tight-Binding Hamiltonians from ab initio Calculations	241
	9.5 Hamiltonians Based on Plane Waves	243
	9.6 Electronic Energy Bands and Irreducible Representations	246
	9.7 Examples and Applications	252
	9.7.1 Calculation of Fermi Surfaces	252
	9.7.2 Electronic Structure of Carbon Nanotubes	254
	9.7.3 Tight-binding Real-Space Calculations	256
	9.7.4 Spin–Orbit Coupling in Semiconductors	261
	9.7.5 Tight-Binding Models for Oxides	263
	Part Three Applications in Photonics	267
	10 Solution of Maxwell's Equations	269
	10.1 Maxwell's Equations and the Master Equation for Photonic Crystals	270
	10.1.1 The Master Equation	270
	10.1.2 One- and Two-Dimensional Problems	272
	10.2 Group of the Master Equation	273
	10.3 Master Equation as an Eigenvalue Problem	275
	10.4 Models of the Permittivity	276
	10.4.1 Reduced Structure Factors	280
	10.4.2 Convergence of the Plane Wave Expansion	282
	11 Two-Dimensional Photonic Crystals	285
	11.1 Photonic Band Structure and Symmetrized Plane Waves	286
	11.1.1 Empty Lattice Band Structure and Symmetrized Plane Waves	286
	11.1.2 Photonic Band Structures: A First Example	289
	11.2 Group Theoretical Classification of Photonic Band Structures	292
	11.3 Supercells and Symmetry of Defect Modes	295
	11.4 Uncoupled Bands	299
	12 Three-Dimensional Photonic Crystals	303
	12.1 Empty Lattice Bands and Compatibility Relations	303
	12.2 An example: Dielectric Spheres in Air	307
	12.3 Symmetry-Adapted Vector Spherical Waves	309
	Part Four Other Applications	315
	13 Group Theory of Vibrational Problems	317
	13.1 Vibrations of Molecules	317
	13.1.1 Permutation, Displacement, and Vector Representation	318
	13.1.2 Vibrational Modes of Molecules	321
	13.1.3 Infrared and Raman Activity	323
	13.2 Lattice Vibrations	326
	13.2.1 Direct Calculation of the Dynamical Matrix	328
	13.2.2 Dynamical Matrix from Tight-Binding Models	330
	13.2.3 Analysis of Zone Center Modes	331
	14 Landau Theory of Phase Transitions of the Second Kind	335
	14.1 Introduction to Landau's Theory of Phase Transitions	336
	14.2 Basics of the Group Theoretical Formulation	340
	14.3 Examples with GTPack Commands	342
	14.3.1 Invariant Polynomials	342
	14.3.2 Landau and LifshitzCriterion	343
	Appendix A Spherical Harmonics	347
	A.1 Complex Spherical Harmonics	348
	A.1.1 Definition of Complex Spherical Harmonics	348
	A.1.2 Cartesian Spherical Harmonics	348
	A.1.3 Transformation Behavior of Complex Spherical Harmonics	349
	A.2 Tesseral Harmonics	350
	A.2.1 Definition of Tesseral Harmonics	350
	A.2.2 Cartesian Tesseral Harmonics	351
	A.2.3 Transformation Behavior of Tesseral Harmonics	352
	Appendix B Remarks on Databases	353
	B.1 Electronic Structure Databases	353
	B.1.1 Tight-Binding Calculations	353
	B.1.2 Pseudopotential Calculations	354
	B.1.3 Radial Integrals for Crystal Field Parameters	355
	B.2 Molecular Databases	355
	B.3 Database of Structures	355
	Appendix C Use of MPB together with GTPack	357
	C.1 Calculation of Band Structure and Density of States	357
	C.2 Calculation of Eigenmodes	358
	C.3 Comparison of Calculations with MPB and Mathematica	359
	Appendix D Technical Remarks on GTPack	361
	D.1 Structure of GTPack	361
	D.2 Installation of GTPack	362
	References	365
	Index	375
	EULA	382