Symmetry Properties of the Coupling Coefficients

This coefficient is known as a Clebsch-Gordan (CG) coupling coefficient and denoted by the 3F bracket (T ay a/lpfclT K). It indicates how the orbital irreps TL and Fb have to be combined to yield a product ket that transforms as Fy). The CG-coefficients can be determined by using projection operators. The results are listed in Appendix F. It is often possible to obtain these results by a simpler procedure. We illustrate this for the components of the T g two-electron state, obtained in Eq. (6.9). The z-component of this state is the only component that is totally symmetric imder the C4 splitting field. It is clear that this symmetry can be obtained only by multiplying the egc) and ) components, since these are both antisymmetric and thus will form a symmetric product. From here on we will adopt for the product functions the usual notation of small letters for the orbitals and capital letters for the coupled states. Hence  

The coupling coefficient Eg T2g T gz) is thus equal to 1. The x and y components may then immediately be obtained by applying the cyclic C3 generator. As an example for the x-component  

A direct consequence of the latter viewpoint is that the rules for complex conjugation of brackets apply  

Being expansion coefficients of SALCs, the coupling coefficients also obey two orthogonality rules. Column-wise orthonormality results from the orthonormal properties of the coupled states. 

The permutational properties of the CG-coefflcients refer to interchange of the bra and ket irreps. If Pa and Pb are not equivalent, their ordering will not affect the symmetry of the coupled state, since the factors in the direct product commute  

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