Racah notation

The Racah notation for rare gas excited states neatly summarizes all of these pertinent angular momenta in the term symbol nllK]j. As two examples of Ne multiplets, we consider the 3s[X]j and the np K]j states. The former are the lowest-lying (2p) (3s) excited states with L = 0 and S = Since one always has Lg = I, Sc = i, and or f in the rare-gas atoms, the possible K values are from Eq. 9.14... 

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... 

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