Clebsch‐Gordan coefficients for chains of groups of interest in quantum chemistry

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Abstract
The algebra of the representation of the special unitary group SU(2), the universal covering of the proper rotation group SO(3), is studied in a nonstandard basis. We are using a basis adapted to a chain of type SU(2) ⊃ …︁ ⊃ G″ ⊃ G′ ⊃ G. The introduction of such a chain enables us to label, at least partially, the elements of the irreducible tensorial sets under SU(2) with irreducible representations of G, G″ G″, …. We are thus led to introduce the restriction SU(2) → …︁ → G″ → G′ → G in the Wigner‐Racah algebra of the group SU(2). The physical interest of this machinery lies in the fact that the double group of any point symmetry group belongs, up to an isomorphism, to the considered chain. The formalism described in this paper thus appears to be useful in molecular and solid‐state calculations. It is particularly efficient in the fields of vibrational‐rotational and electronic spectroscopy of molecules. In Appendix A the master formulae, principally the Wigner‐Eckart‐Racah theorem, for the Wigner‐Racah algebra of a chain of compact topological groups (discrete or continuous) are briefly discussed. Lastly, a programme for computing Clebsch‐Gordan coefficients for a chain SU(2) ⊃ …︁ ⊃ G″ ⊃ G′ ⊃ G and numerical results for chains isomorphic to SU(2) ⊃ O′ ⊃ D′4 ⊃ D′2 are described in Appendix B.

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