Point Group Symmetry Applications

The mathematical apparatus of group theory is a means of exploring and exploiting physical and algebraic structure in physical and chemical prob­ lems. The existence of structure in the physical processes leads to structure in the solutions. For group theory to be useful this structure need not be an exact symmetry, although as examples of exact symmetries we have that the identity of electrons leads to permutation symmetries in many-electron wave functions, the spatial structure of crystals leads to the Bloch theory of crystal eigenfunctions, and the rotational invariance of the hydrogenic Hamiltonian leads to its factorization into angular and radial parts. In the 1930's Wigner extended what is known to mathematicians as the theory of group representations and the theory of group algebras to study the coupling coefficients of angular momentum, relating various properties of the coefficients to the properties of the abstract group of rotations in 3-space. In 1949 Racah, in a paper on rare earth spectra, showed that similar coefficients occur in other situations. Immediately a number of studies of the coefficients were begun, notably by Jahn, with his applications in nuclear physics. In the years since then a large number of physicists and chemists have added to the development of a general theory of the coefficients, or have produced specialized tables for a specific application. Applications now range from high-energy physics to biology.

Contenu

1. Introduction.- 1.1. A Guide to the Use of the Text.- 1.2. Notation and Phase Conventions.- 2. Basic Concepts.- 2.1. Irrep Spaces.- 2.2. Group-Subgroup Bases.- 2.3. Coupling Coefficients.- 2.4. Character Theory.- 2.5. Complex Conjugation.- 2.6. Spin Irreps and Labeling Irreps.- 3. The jm Factors and j Symbols.- 3.1. The jm Factors.- 3.2. The j Symbols.- 3.3. Properties of jmFactors and j Symbols.- 3.4. Computational Methods.- 3.5. Phase and Multiplicity Choices.- 4. The Wigner-Eckart Theorem.- 4. 1. Basis Operators.- 4.2. The Wigner-Eckart Theorem.- 4.3. Coupled Tensors.- 4.4. The Standard Racah Tensors s of SO3.- 5. O3 and Its Subgroups.- 5.1. Direct Product Groups.- 5.2. Isomorphic Subsgroups.- 5.3. Specification of x,y,z Axes and Bases.- 6. Properties of Dihedral Groups.- 6. 1. The Structure of the Dihedral Groups.- 62. The Chain SO3 ? D? ? SO2.- 6.3. Finite Subgroups of D?.- 7. Fractional Parentage Coefficients.- 7. 1. The Parentage Concept.- 7.2. Continuous Matrix Groups.- 7.3. Complete Parentage Schemes.- 7.4. Strong-Field Parentage Schemes.- 8. Time Reversal.- 8. 1. Time of States.- 8.2 Time Reversal and Selection Rules.- 9. Applications.- 9.1. The Free-Ion Hamiltonian.- 9.2. The Zeeman Interaction.- 9.3. Ligand Fields.- 10. Programming Notes.- 11. Group Information Tables.- 11.1. The Cyclic Groups.- 11.2.The Dihedral Groups.- 11.3. The Tetrahedral Groups.- 11.4. The Octahedral Groups.- 11.5. The Icosahedzal Groups.- 12. Branching Rule Tables.- Tables.- 13. jm Factor Tables.- Tables.- 14. 3j and 6j Symbol Tables.- Tables.- 15. 9j Symbols.- Tables.- 16. Bases in Terms of Spherical Harmonics.- 16.1. Rotation Matrices in the JM Basis.- 16.2. Spherical Harmonics in Rectangular Coordinates.- Tables.- References.- Table Index.