Matrix representation Wigner-Eckart theorem

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

Powerful though the generalized Wigner-Eckart theorem is, we are no further forward in the calculation of the matrix elements of the e, if the relevant CG coefficients are not known. No tables for the CG coefficients for SO(7) or G2 were available in 1949 and even today we have only a very limited knowledge of them. One way to proceed is to invert the problem and calculate the matrix elements of the e, by other means we can then deduce sets of CG coefficients that may be used in the future when similar representations of SO(7) and G2 appear. Equation (1) of Racah (1949) specifies the basic mechanism for calculating matrix elements step by step along the f shell. In a slightly different notation it runs... 

The matrix element in eq 1 represents a relativistic spin-dependent correction. Since the conventional scalar Hamiltonian commutes with and one significant use of symmetry is to apply the Wigner-Eckart theorem (19). Secondly, commutes with a set of rotation and reflection operators, applied to both orbital and spin variables, forming the double group (21) G, just as the scalar Hamiltonian commutes with a set of space rotations and reflections forming group G. The eigenvectors of the matrix representation of H in die... 

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