Complete nuclear permutation group

In the above we have discussed several different symmetry groups the translation group Gj, the rotation group K (spatial), the inversion group, the electron permutation group and the complete nuclear permutation... 

The complete nuclear permutation inversion (CNPI) group of the PH molecule is the direct product of the complete nuclear permutation (CNP) group A3 (see (equation A 1.4.19)) and the inversion group = E, E. This is a group of 12 elements that we call Gj2-... 

The subgroup of the complete nuclear permutation-inversion group that contains all feasible permutation-inversion operations. 

It would appear that identical particle permutation groups are not of help in providing distinguishing symmetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very useful restrictions on the way we can build up the complete molecular wavefimction from basis functions. Molecular wavefunctions are usually built up from basis functions that are products of electronic and nuclear parts. Each of these parts is further built up from products of separate uncoupled coordinate (or orbital) and spin basis functions. When we combine these separate functions, the final overall product states must conform to the permutation symmetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis functions. 

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