Double-Tensor Operators for Configurations of Equivalent Electrons

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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 (lnv1WξS1L ‖W(κk)‖ lnv1′W′ξ′S1′L′),for fixed n, v1, v1′, W, and W′, is proportional to the set (lmv2WξS2L ‖W(κ′k)‖ lmv2′W′ξ′S2′L′),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.

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