Summary of standard results from spherical tensor algebra

Appendix 5.1. Summary of standard results from spherical tensor algebra 

In our own work on both diatomic and polyatomic molecules, we have found it valuable to have a summary of the most important results from irreducible spherical tensor algebra, particularly those relating to the evaluation of matrix elements in various angular momentum coupling schemes. We now provide a summary of those results detailed derivations are, of course, to be found in the main body of the text. 

Tp22(A2) is a tensor of rank k2 with components p2 which operates on angular momentum j2. 

The scalar product is obtained from the tensor product by putting kn = pn = 0 it is only defined for k = k2 = k (say). Then 

If angular momenta and ji couple to form /,2 and the tensor product is defined by T (/Ii) x T ( 2) = W 2(A i, feX then the reduced matrix elements 

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