The Matrix Elements of Tensor Operators for the Electronic Configurationsfn

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Abstract
It is shown that many simple relations exist between various reduced matrix elements of the form (fn WUSL || Uk || fnW'U'SL'), where W = (w1w2w3) and U = (u1u2) are irreducible representations of R7 and G2 respectively. This is done by using the fact that for k = 2, 4 and 6, the components Uqk of the tensor operators Uk, when multiplied by (2k + 1)1/2, transform among themselves as the irreducible representations (200) of R7 and (20) of G2. The number of linearly independent sets of matrix elements for a given W and W' is equal to the number of times the identity representation (000) of R7 occurs in the reduction of the product (w1w2w3) × (200) × (w1'w2'w3'); similarly, the number of times the identity representation (00) of G2 occurs in the reduction of the product (u1u2) × (20) × (u1'u2') determines the number of linearly independent sets of matrix elements that can exist with U = (u1u2) and U' = (u1'u2'). These two numbers, which are denoted by c(WW'(200)) and c(UU'(20)), are tabulated for all W, W', U and U' which occur in fn, and the use of the tables in calculating the splittings induced in the levels of rare earth ions by the surrounding crystal lattice is illustrated with a number of examples.

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