Irreducible tensors in the space of complex configurations

So far we have considered only one shell of equivalent electrons, but the mathematical techniques discussed can be translated fairly simply to the case of complex atomic configurations. With LS coupling the wave functions of ni/ n/ 2...nulNu configuration are normally constructed by the vectorial coupling of orbital and spin momenta of all the shells  

Individual operator terms in these sums are given by (14.15) and (14.16), and here we have introduced additional subscripts of the operators to indicate the space of shells in which they are defined. 

The irreducible components of the operators of orbital and spin angular momenta of the n lNlri2lNl configuration have the form 

When dealing with complex configurations, it is necessary, apart from one-shell tensors, to consider the irreducible tensorial products of creation 

In the group-theoretical treatment of mixed configurations, we may, in analogy with the case of one shell, introduce the basis tensors for two shells of equivalent electrons [101] 

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