Product Symmetrization and the Pauli Exchange-Symmetry

In principle, the T g and T2g coupled two-electron states, which we obtained in Table 6.2 of the previous section, could apply to the case of the (t2 )Heg) excited states of a transition-metal ion in an octahedral ligand field, which splits the d 

The IT kCI, 2)) and T k(2, 1)) states will have exactly the same symmetries, since the factors in the direct product commute  

As a result, Pn commutes with the spatial symmetry operators, and we can symmetrize the coupled states with respect to the electron permutation. The permutation operator is the generator of the symmetric group, S2, which has only two irreps, one symmetric and one antisymmetric, corresponding, respectively, to the plus and minus combination in Eq. (6.35). 

These states have distinct permutation symmetries, and spatial symmetry operators cannot mix - - and - states. This is a very general property of multi-particle states, 

On the other hand the permutation symmetry of multi-electron wavefunctions is restricted by the Pauli principle. 

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