Double-Tensor Operators for Configurations of Equivalent Electrons

Abstract

To every irreducible representation W of the rotation group in 2l + 1 dimensions that is used to classify states of the electronic configurations lⁿ, there correspond two couples (v, S), where v and S stand for the seniority number and total spin, respectively. Determinantal product states are introduced to examine this correspondence in detail. It is shown that for two double tensors W^(κk) and W^(κ′k), the set of reduced matrix elements $${\mathrm{(l}}^n{v}_1{\mathrm{W\xi S}}_1{\mathrm{L\hspace{0.17em}\Vert W}}^{\mathrm{(\kappa k)}}{\mathrm{\Vert \hspace{0.17em}l}}^n{v}_1^{\prime }{W}^{\prime }{\xi }^{\prime }{S}_1^{\prime }{L}^{\prime }\mathrm{),}$$ for fixed n, v₁, v₁′, W, and W′, is proportional to the set $${\mathrm{(l}}^m{v}_2{\mathrm{W\xi S}}_2{\mathrm{L\hspace{0.17em}\Vert W}}^{{\mathrm{(\kappa }}^{\prime }\mathrm{k)}}{\mathrm{\Vert \hspace{0.17em}l}}^m{v}_2^{\prime }{W}^{\prime }{\xi }^{\prime }{S}_2^{\prime }{L}^{\prime }\mathrm{),}$$ where ξ and ξ′ are additional labels that may be required to define the states uniquely, provided (a) the two couples (v₁, S₁) and (v₂, S₂) are distinct, (b) the two couples (v₁′, S₁′) and (v₂′, S₂′) are distinct, and (c) the sum κ + κ′ + k is odd. The amplitudes of the double tensors are chosen so that the constant of proportionality is equal to the ratio of two 3‐j symbols, multiplied by a phase factor. An explicit expression for this factor is given for f electrons, and a number of applications are discussed.