Selection rules quasispin

The s.o.c. operator is a one-electron operator which is even under time reversal, and non-totally symmetric in spin and orbit space. The trace of the spin-orbit coupling matrix for the t2g-shell thus vanishes. As a result the s.o.c. operator is found to transform as the MK = 0 component of a pure quasi-spin triplet (Cf. Eq. 26). Application of the selection rule in Eq. 28 shows that allowed matrix elements must involve a change of one unit in quasi-spin character, i.e. AQ = 1. Since 4S and 2D are both quasi-spin singlets while 2P is a quasispin triplet, s.o.c. interactions will be as follows ... 

Hence in his 1984 paper [7], when Ceulemans referred to the parity of a half-filled shell state with respect to a linear particle-hole conjugation operator he was actually referring to the quasispin character of the state, see equation (34). For a half-filled shell state Q is always integral, hence (— l)e = ttq = 1. This is just the result derived by Ceulemans in 1984 [7] with no knowledge whatsoever of the concept of quasispin. The governing equation from which Ceulemans 84 selection rules stem, expressed in terms of half-filled shell states, may be labelled with the quasispin scheme proposed by Ceulemans in 1994 [10] ... 

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