Reduced coefficients subcoefficients of fractional parentage

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46) 

Specifically, this relationship holds for the electron creation operator a(jfj which, by (15.49), is a component v = 1/2 of the triple tensor 0. If we take into consideration that the submatrix element of the creation operator is expressed in terms of the CFP from (15.21), we have 

We refer to the last factor in this expression as the reduced coefficient (or subcoefficient) of fractional parentage (SCFP) [91]. It follows from (16.16) that these coefficients can be expressed fairly simply in terms of CFP for the term with N = v that occur for the first time  

In Chapters 9 and 15 we derived the set of equations (9.9), (9.10), (15.22)-(15.24) for the sums of products of CFP. Similar sets can be derived for the sums of products of SCFP. With this in view we consider the submatrix elements of the triple tensors VF(°K) (15.52). On the one hand, they can be expressed, using (5.16), in terms of the submatrix elements of the 

On the other hand, the submatrix elements of tensors WiKkK) at certain values of their ranks have simple algebraic expressions determined by the quantum numbers of lN configuration. So, when K = k = k = 0, by (16.19), we obtain, using (15.54), 

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