Group Theory Applied to Chemistry

ISBN-13: 9789402422443 / Twarda / 2024 / 426 str.

Arnout Jozef Ceulemans

Spis treści

1. Operations

1.1. Operations and Points

1.2. Operations and Functions

1.3. Operations and Operators

1.4. An Aide M´emoire

1.5. Problems

References

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

3.8. 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

4.7. Application: the sp3 Hybridization of Carbon

4.8. Application: the Vibrations of UF6

4.9. Application: H¨uckel Theory

4.10. Problems

References

5. What has Quantum Chemistry Got to Do with It?

5.1. The Prequantum Era

5.2. The Schr¨odinger 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

6.9. Induction Revisited: the Fibre Bundle

6.10. Application: Bonding Schemes for Polyhedra

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

7.7. 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

8.5. Plane Groups: the Graphene Lattice

8.6. Application: Nanotubes

8.7. Problems

References

Appendix A. Character Tables

A.1. Finite Point Groups

C1 and the Binary Groups: Cs,Ci,C2

The Cyclic Groups: Cn (n=3,4,5,6,7,8)

The Dihedral Groups: Dn (n=2,3,4,5,6)

The Conical Groups: Cnv (n=2,3,4,5,6)

The Cnh Groups (n=2,3,4,5,6)

The Rotation-Reflection groups: S2n (n = 2,3,4)

The Prismatic Groups: Dnh (n=2,3,4,5,6,8)

The Anti-Prismatic Groups: Dnd (n=2,3,4,5,6)

The Tetrahedral and Cubic groups

The Icosahedral Groups

The Symmetric Groups

A.2. Infinite Groups

Cylindrical Symmetry

Spherical Symmetry

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 G ↓ H

C.2. Induction H ↑ G

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

Solutions to Problems

Index

O autorze

Arnout Ceulemans is emeritus professor of theoretical chemistry at KULeuven. His research is devoted to the development and application of group theory and topology to chemistry. He has published three books on this topic. In 2013 appeared the first edition of a textbook on group theory applied to chemistry (Springer, 2013). Together with Dr. Pieter Thyssen he authored a book on continuous symmetry groups, entitled 'Shattered Symmetry, group theory from the eightfold way to the periodic table' (2017). His latest contribution is a monograph on the 'Theory of the Jahn-Teller effect, when a boson meets a fermion' (Springer 2022).

Blurb

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.