Quasispin for a shell of equivalent electrons

The operator of total quasispin angular momentum of the shell can be obtained by the vectorial coupling of quasispin momenta of all the pairing states. For a shell of equivalent electrons, instead of (15.35) we have... 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

But this phase factor can be selected so that the signs of the CFP for almost filled shells are the same as in the quasispin method. It is worth recalling here that finite transformations generated by quasispin operators define the passage to quasiparticles. In much the same way, in the quasispin space of a shell of equivalent electrons the unitary transformations... 

Coupling vectorially the quasispin operators of each shell Q = Qi +Q2 and assuming Mq = —J[ 2(/i + I2 + 1) — N, where N — iVj + N2, we generalize the quasispin concept to cover the case of complex electronic configurations. Then we can define the total quasispin quantum number for any configuration. For two shells of equivalent electrons we have, in such a case, the following wave function ... 

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