(L̂²,Ŝ²)—adapted graphs

Abstract

Despite the great progress in application of group-theoretical methods to atomic spectroscopy (cf Biedenharn and Louck 1981) even such simple case like d³ configuration structure is not fully understood (Judd 1979). The problem of atomic states classification, although clear from the point of view of group theory (Wybourne 1970), is not solved satisfactorily. The celebrated methods of Racah (1949) are not well-suited for large scale computations, especially when g or higher orbitals are used (Judd 1979). The unitary calculus of Harter and Petterson (1976) has also not solved the problem; although the authors claim that they have “a perfect labeling system”, it is true only for Ŝ² eigenfunctions, where their favorite description using Weyl tablaux is sufficient, as we have seen in the previous sections. The problem is indeed that of labeling: to obtain a proper label (‘proper’ in the sense of carrying sufficient information about the construction of the labeled function to calculate arbitrary matrix elements) one should know genealogy of the function and derive a term of l^r configuration from those of l^r−1. Coefficients of fractional parentage, allowing for such a procedure, have unfortunately no closed form formulas, and the recursive relations (Redmond 1954) are complicated.