Symmetry-adapted linear combinations derivation

In the derivation of molecular orbitals we started with individual orbitals and created symmetry-adapted linear combinations by solving a secular... 

In previous discussion, we constructed the symmetry-adapted linear combinations of the hydrogen Is orbitals for AHM molecules. In this section, we consider AL molecules, where L is a ligand capable of both a and n bonding. For such systems, the procedure to derive the linear combinations of ligand orbitals can become fairly complicated. As an illustration, we take an AL4 molecule with Id symmetry as an example. 

Molecules such as NHj and HjO etc. are described in terms of an inequivalent hybridization scheme based on sp in valence bond theory. The construction of hybridized orbitals in such molecules is different from that developed above. The tetrahedral molecule XAY3 (3) provides a useful starting point. Since the hyl is distinguished from hy2, hy3 and hy4, the symmetry-adapted linear combinations of these hybrids cannot be generated in terms of the spherical harmonic expansion in Eq. (1). But they can be derived as follows ... 

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... 

To simplify the derivation of the singlet and triplet energy, we first construct spin-symmetry adapted CSFs by forming linear combination of the above-listed determinants with unpaired electrons ... 

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