Algebraic expressions for CFP of lNvLS shells

In Chapter 15, for the CFP with a detached electrons, we obtained a relationship (15.27) whose right side has the form of a vacuum average of a certain product of second-quantized operators q . To obtain algebraic formulas for CFP, it is necessary to compute this vacuum average by transposing all the annihilation operators to the right side of the creation operators. So, for N = 3, we take into account (for non-repeating terms) the explicit form of operators (15.2) and (15.5), which produce pertinent wave functions out of vacuum, and find (cf. [107]) 

Similar expressions can also be obtained at N = 4 [108]. To establish similar algebraic expressions for the repeating terms vLS only requires that the appropriate second-quantized operators (16.7), (16.8), (16.10) and (16.11) be used. For example, instead of (16.63), we have 

Using (16.16) and the above relations, we can work out algebraic expressions for SCFP, and hence for CFP, in the entire interval of the number of electrons in the shell existing for given v. Taking account of the symmetry of CFP under transposition of spin and quasispin quantum numbers further expands the number of such expressions. Formulas of this kind can be established also for larger v, but with v = 5 and above they become so unwieldy and difficult to handle that this limits their practical uses. They may be found in [108]. 

The technique used here to find algebraic expressions for CFP can also be applied to derive the recurrence relation that generalizes the well-known Redmond formula [109]. To this end, it is only required to write 

The term LjS can be chosen in an arbitrary manner, and the normalization factor is found from the normalization condition for coefficients of fractional parentage at fixed momenta L SJ and L,S. Equation (16.66) holds for repeating terms that are uniquely classified by the seniority quantum number v, but for non-repeating terms (when 5(L2S2,LS) = 0) that equation becomes the conventional Redmond formula [109]. 

---