The overlap concept of Clebsch-Gordan coefficients and Racah lemma constants: The formalism of orthonormal sets of operators in ligand field theory

Abstract

By virtue of the isomorphic relationship between simple tensor products | l 1 t 1〉| l 2 t 2〉, formed from othonormal bases of real functions, and the corresponding operators S ̂ l 1t 1l 2t 2 = |l 1t 1〉〈l 2t 2| obtains analogous overlap expressions (scalar products) for functions and operators when using the trace definition of operator overlap. Defining R 3-irreducible tensorial quantities [| l 1{⊗} l 2|] l 3 t 3 (for functions) and [| l 1{ ⊗ } l 2|] l 3 t 3 (for operators) by the same expansion formulae, one makes Clebsch-Gordan coefficients expressible as function overlaps of the type 〈 l 1 t 1|〉 l 2 t 2|[| l 1{ ⊗ | l 2{] l 3 t 3 and as operator overlaps of the type 〈 S ̂ l 1t 1l 2t 2 |[|l 1{ ⊗ }l 2|] l 3 t 3 〉 . Similarly, Racah lemma constants (isoscalar factors) obtain overlap interpretations. The formalism of orthonormal sets of operators, which includes a symmetry adaptation to the symmetric group P 2, allows on the one hand the strong-field and weak-field schemes of the parameterized l q model ( q = 2) and on the other hand the ligand field and crystal field parameterization schemes of a ligand model-field to become related by permutationally symmetry-adapted Racah lemma matrices. The formalism, which allows a generalization of ligand field theory as the orthonormal operators model, appears to have wider applications.