Genealogical coupling of operators

From this point of view, then, the construction of CSFs from determinants (2.6.1) may be viewed as the construction of a spin tensor operator from strings of creation operators 

in the construction of a spin tensor operator, we employ (2.6.33) for creation operators and (2.6.34) for annihilation operators, assuming that the number and relative ordering of the creation and annihilation operators are fixed. 

As an example of the genealogical scheme for tensor operators, we consider the construction of a one-electron excitation operator of triplet symmetry S = 1 and M = 0. Because of the requirement of zero spin projection, only two strings will contribute  

We recall that the spin projection of the annihilation operators and is — and respectively. Evaluating the coupling coefficients, we arrive at the operator 

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