This book presents to students of introductory physical chemistry the basic principles of symmetry and group theory, and their use in describing and predicting molecular structure and spectra. Symmetry is a crucial determinant of many chemical phenomena, and group theory is the grammar of the language of symmetry. In many cases, simple calculations suffice to explain why certain triatomic molecules are linear and others bent, or why certain transitions do not appear in molecular spectra.
In this book, the aim is understanding the ideas, and skills in application of the principles, rather than mathematical rigour. The book is intended as a supplement for students who want to follow up an interest in and recognition of the importance of group theory, and who seek a short and mathematically relatively undemanding introduction. Exercises appearing throughout the text are integrated with the presentation to give readers confidence in their assimilation of the material.
Preface. 1: The relationship between group theory and chemistry. 1.1. Introduction. 1.2. Applications of group theory. 2: Symmetry. 2.1. A bridge from geometry to arithmetic. 2.2. Classifying symmetry operations. 2.3. Full analysis of the symmetry of the water molecule: Introduction to notation. 2.4. Products of covering operations: multiplication tables. 2.5. What is a group? 3: Group theory. 3.1. Definition of a group. 3.2. Subgroups. 3.3. Examples of groups. 4: Point groups - The symmetry of groups of small molecules. 4.1. Introduction. 4.2. Axes of rotation: Cn. 4.3. Mirror planes: sigma. 4.4. Stereographic projection diagrams. 4.5. Inversion: i. 4.6. Rotary reflections, or improper rotations, Sn. 4.7. Catalogue raisonée of the common point groups: symbols, molecular examples and macroscopic examples. 5: Introduction to linear algebra. 5.1. Introduction. 5.2. Systems of coordinates. 5.3. Vectors. 5.4. Norm or length of a vector. 5.5. Angles and inner products. 5.6. Generalizations to n dimensions. 5.7. Orthogonality and normality. 5.8. Linear transformations and matrices. 5.9. Successive transformations: matrix multiplication. 5.10. The effect on a matrix of a change in coordinate system. 5.11. Orthogonal transformations. 5.12. Traces and determinants. 5.13. Matrix representation of symmetry groups. 6: Group representations and character tables. 6.1. Introduction. 6.2. Group representations. 6.3. Character tables. 6.4. Properties of character tables. 6.5. Calculations with character tables. 7: Molecular vibrations. 7.1. Introduction. 7.2. Classical description of molecular vibrations. 7.3. Eigenvalue problems. 7.4. Determination of the symmetries of the normal modes. 7.5. Use of internal coordinates. 8: Electronic structure of atoms and molecules. 8.1. The quantum-mechanical background. 8.2. Symmetry properties of wave functions. 8.3. Molecular wave functions. 8.4. Expectation values and the variation theorem. 9: Symmetry properties of molecular orbitals. 9.1. Diatomic molecules. 9.2. Triatomic molecule - Walsh diagrams. 9.3. Molecular orbitals for the bent AH2 molecule (C2v). 9.4. Molecular orbitals for the linear AH2 molecule (D8h). 9.5. Correlation of thew orbitals between bent and linear geometries. 10: Spectroscopy and selection rules. 10.1. Introduction. 10.2. The relationship between symmetry properties and the vanishing of matrix elements. 10.3. The direct-product representation. 10.4. Selection rules in spectroscopy. 11: Molecular orbital theory of planar conjugated molecules. 11.1. Introduction. 11.2. The LCAO-MO description of pyridine. 11.3. Distribution of molecular orbitals among symmetry species. 11.4.
