Expansion of Ligand Set as Spherical Harmonic Functions

In the Complementary Spherical Electron Density Model [10-12], the symmetry adapted linear combinations of Oi ligand orbitals for ML compoimd can be expressed in terms of a spherical harmonic expansion  

In this fashion the linear combinations are assigned quantum numbers 1 and m which are related to those which have been defined for the S, P and D functions derived for the particle on the sphere problem. Furthermore, their nodal characteristics mimic those of the atomic wave functions of the central atom. The spherical harmonic expansion described above will provide its most accurate description of the symmetry adapted linear combinations when the polyhedral vertices are symmetry equivalent. For example, octahedral MLg has S°, Po, i and Do,2s a total of 6 symmetry adapted linear combinations. They are 

They are shown schematically in Fig. 1. This spherical expansion improves as an approximation as n increases. When there is more than one linear combination with the same symmetry then an orthogonalisation problem can arise which will be discussed in more detail below. In the following sections the utilisation of these spherical harmonic expansions are summarised for a range of common ligand co-ordination geometries. 

1 Planar MLn, Bipyramidal ML +2, Prismatic and Anti-Prismatic ML2 Complexes 

In the planar structure, MLn if the ligands are defined as lying in the xy plane, those functions that possess a nodal plane coincident with the xy plane are systematically excluded. Therefore, the ligand orbitals span the S , P i, D i,. .., L+l,. .. spherical harmonic expansions and there are total of n wave functions. For example, the L3 moiety has S°, and P i wave functions. For L4 the additional function generated is a D function i.e., it is characterised by S , P i and D s- 

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