Group Theory Applied to Chemistry

The second edition of this textbook provides a more elaborate explanation of several important group-theoretical concepts in quantum chemistry, such as: the bra-ket conjugation relation, the connection between point groups and isometries, the practical use of subduction tables, the eigenvalues of Cayley graphs, and the symmetry of Slater determinants. A new chapter introduces the application of line and plane groups to the properties of nanostructured low-dimensional molecular systems. In addition, several extra study problems are inserted to illustrate group theory at work in molecular science. The book is of great interest to advanced undergraduate and graduate students, enabling them to put the tools of group theory into practice when studying chemical problems of their own research. More experienced researchers will find in this book useful leads to the mathematical aspects of their subject. 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 B Symmetry Breaking by Uniform Linear Electric and Magnetic Fields B.1 Spherical Groups B.2 Binary and Cylindrical Groups C Subduction and Induction C.1 Subduction upper G down arrow upper HGdownarrowH C.2 Induction upper H up arrow upper GHuparrowG D Canonical-Basis Relationships E Direct-Product Tables F Coupling Coefficients G Spinor Representations G.1 Character Tables G.2 Subduction G.3 Canonical-Basis Relationships G.4 Direct-Product Tables G.5 Coupling Coefficients H Solutions to Problems References Index