Group Theory Applied to Chemistry

2. Auflage 2024
ISBN: 978-94-024-2244-3
Verlag: Springer Netherlands

Buch, Englisch, 337 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 711 g

Reihe: Theoretical Chemistry and Computational Modelling

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.

Zielgruppe

Graduate

Autoren/Hrsg.

Ceulemans, Arnout Jozef

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Naturwissenschaften Chemie Physikalische Chemie Quantenchemie, Theoretische Chemie

Inhaltsverzeichnis

1. Operations1.1. Operations and Points1.2. Operations and Functions1.3. Operations and Operators1.4. An Aide M´emoire1.5. ProblemsReferences
2. Function Spaces and Matrices2.1. Function Spaces2.2. Linear Operators and Transformation Matrices2.3. The Adjoint Relation between Bra’s and Kets2.4. Unitary Matrices2.5. Time Reversal as an Anti-linear Operator2.6. ProblemsReferences
3. Groups3.1. The Symmetry of Ammonia3.2. The Group Structure3.3. Some Special Groups3.4. Subgroups3.5. Cosets3.6. Classes3.7. Overview of the Point Groups3.8. Rotational Groups and Chiral Molecules3.9. Applications: Magnetic and Electric Fields3.10. ProblemsReferences
4. Representations4.1. Symmetry-adapted Linear Combinations of Hydrogen Orbitals in Ammonia4.2. Character Theorems4.3. Character Tables4.4. Matrix Theorem4.5. Projection Operators4.6. Subduction and Induction4.7. Application: the sp3 Hybridization of Carbon4.8. Application: the Vibrations of UF64.9. Application: H¨uckel Theory4.10. ProblemsReferences
5. What has Quantum Chemistry Got to Do with It?5.1. The Prequantum Era5.2. The Schr¨odinger Equation5.3. How to Structure a Degenerate Space5.4. The Molecular Symmetry Group5.5. ProblemsReferences
6. Interactions6.1. Overlap Integrals6.2. The Coupling of Representations6.3. Symmetry Properties of the Coupling Coefficients6.4. Pauli Exchange-Symmetry and Slater Determinants6.5. Matrix Elements and the Wigner-Eckart Theorem6.6. Application: the Jahn-Teller Effect6.7. Application: Pseudo-Jahn-Teller Interactions6.8. Application: Linear and Circular Dichroism6.9. Induction Revisited: the Fibre Bundle6.10. Application: Bonding Schemes for Polyhedra6.11. ProblemsReferences
7. Spherical Symmetry and Spins7.1. The Spherical Symmetry group7.2. Application: Crystal-field Potentials7.3. Spinors and Spinor Groups7.4. The Coupling of Spins7.5. Double Groups7.6. Kramers Degeneracy7.7. Application: Spin Hamiltonian for the Octahedral Quartet State7.8. ProblemsReferences
8. Line Groups and Plane Groups8.1. Translational Symmetry along a Line8.2. Band Structures8.3. Line Groups8.4. Applications8.5. Plane Groups: the Graphene Lattice8.6. Application: Nanotubes8.7. ProblemsReferences
Appendix A. Character Tables

A.1. Finite Point GroupsC1 and the Binary Groups: Cs,Ci,C2The 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 groupsThe Icosahedral GroupsThe Symmetric Groups
A.2. Infinite GroupsCylindrical SymmetrySpherical Symmetry
Appendix B. Symmetry Breaking by Uniform Linear Electric and Magnetic Fields
B.1. Spherical groupsB.2. Binary and Cylindrical GroupsAppendix C. Subduction and InductionC.1. Subduction G ? HC.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 TablesG.2. SubductionG.3. Canonical-Basis RelationshipsG.4. Direct-Product tablesG.5. Coupling Coefficients
Solutions to Problems
Index

Über Autor(innen)

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).