Group Theory Applied to Chemistry

Cover
Theoretical Chemistry and Computational Modelling Series
Group Theory Applied to Chemistry
Copyright
Dedication
Preface
Acknowledgments
Contents
1. Operations
	1.1 Operations and Points
	1.2 Operations and Functions
	1.3 Operations and Operators
	1.4 An Aide Mémoire
	1.5 Problems
	Reference
2. Function Spaces and Matrices
	2.1 Function Spaces
	2.2 Linear Operators and Transformation Matrices
	2.3 The Adjoint Relation Between Bra's and Kets
	2.4 Unitary Matrices
	2.5 Time Reversal as an Anti-linear Operator
	2.6 Problems
	References
3. Groups
	3.1 The Symmetry of Ammonia
	3.2 The Group Structure
	3.3 Some Special Groups
	3.4 Subgroups
	3.5 Cosets
	3.6 Classes
	3.7 Overview of the Point Groups
		3.7.1 Spherical Symmetry and the Platonic Solids
		3.7.2 Cylindrical Symmetries
	3.8 Rotational Groups and Chiral Molecules
	3.9 Applications: Magnetic and Electric Fields
	3.10 Problems
	References
4. Representations
	4.1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia
	4.2 Character Theorems
	4.3 Character Tables
	4.4 Matrix Theorem
	4.5 Projection Operators
	4.6 Subduction and Induction
		4.6.1 Subduction
		4.6.2 Induction
	4.7 Application: The s p cubedsp3 Hybridization of Carbon
	4.8 Application: The Vibrations of UFSubscript 66
	4.9 Application: Hückel Theory
		4.9.1 Introduction
		4.9.2 Cyclic Polyenes
		4.9.3 Zeeman Effect in Cyclic Polyenes
		4.9.4 Polyhedral Hückel Systems of Equivalent Atoms
		4.9.5 Triphenylmethyl Radical and Hidden Symmetry
	4.10 Problems
	References
5. What has Quantum Chemistry  Got to Do with It?
	5.1 The Prequantum Era
	5.2 The Schrödinger Equation
	5.3 How to Structure a Degenerate Space
	5.4 The Molecular Symmetry Group
	5.5 Problems
	References
6. Interactions
	6.1 Overlap Integrals
	6.2 The Coupling of Representations
	6.3 Symmetry Properties of the Coupling Coefficients
	6.4 Pauli Exchange-Symmetry and Slater Determinants
	6.5 Matrix Elements and the Wigner–Eckart Theorem
	6.6 Application: The Jahn–Teller Effect
	6.7 Application: Pseudo-Jahn–Teller Interactions
	6.8 Application: Linear and Circular Dichroism
		6.8.1 Linear Dichroism
		6.8.2 Circular Dichroism
	6.9 Induction Revisited: The Fiber Bundle
	6.10 Application: Bonding Schemes for Polyhedra
		6.10.1 Edge Bonding in Trivalent Polyhedra
		6.10.2 Frontier Orbitals in Leapfrog Fullerenes
	6.11 Problems
	References
7. Spherical Symmetry and Spins
	7.1 The Spherical Symmetry Group
	7.2 Application: Crystal-Field Potentials
	7.3 Spinors and Spinor Groups
	7.4 The Coupling of Spins
	7.5 Double Groups
	7.6 Kramers Degeneracy
		7.6.1 Time-Reversal Selection Rules
	7.7 Application: Spin Hamiltonian for the Octahedral Quartet State
	7.8 Problems
	References
8. Line Groups and Plane Groups
	8.1 Translational Symmetry Along a Line
	8.2 Band Structures
	8.3 Line Groups
	8.4 Applications
		8.4.1 Line Group with a Glide Plane
		8.4.2 Line Group with a Screw Axis
	8.5 Plane Groups: The Graphene Lattice
	8.6 Application: Nanotubes
	8.7 Problems
	References
Appendix A. Character Tables
	A.1  Finite Point Groups
	A.2  Infinite Groups
Appendix B. Symmetry Breaking by Uniform Linear Electric and Magnetic Fields
	B.1  Spherical Groups
	B.2  Binary and Cylindrical Groups
Appendix C. Subduction and Induction
	C.1  Subduction upper G down arrow upper HGdownarrowH
	C.2  Induction upper H up arrow upper GHuparrowG
Appendix D. Canonical-Basis Relationships
Appendix E. Direct-Product Tables
Appendix F. Coupling Coefficients
Appendix G. Spinor Representations
	G.1  Character Tables
	G.2  Subduction
	G.3  Canonical-Basis Relationships
	G.4  Direct-Product Tables
	G.5  Coupling Coefficients
Appendix H. Solutions to Problems
References
Index