Reduced Matrix Elements of Tensor Operators

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Abstract
Reduced matrix elements of a tensorial product of n tensor operators in the basis of N‐particle angular‐momentum eigenstates are expanded into a sum of products of n 1‐particle reduced matrix elements and a single (N + n)‐particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N‐particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective‐operator approach is demonstrated.

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