Inner Products of Spherical Tensor Operators: A Late Chapter of Racah Algebra

Inner products of spherical tensor operators have been used since the early eighties to define orthogonal operators. However, the basic theory and properties are largely missing in the literature. An inner product in any configuration is directly proportional to the inner product taken in the most basic configuration in which it can occur. The formula for the proportionality factor in question is presented for the first time. This allows the inner products in and between arbitrary configurations to be calculated in advance. In addition, inner products are shown to be independent of the coupling scheme used to construct the state functions. Applications such as the orthogonal operator method and projections of ab initio calculations check for the completeness of the used basis of operators and, importantly, check the matrix elements in any arbitrary configuration, as discussed and illustrated with examples. Closed formulae for the inner products of the well-known Slater and spin–orbit operators are given.