Multiconfiguration VB theory. Non-orthogonal orbitals

The difficulties of dealing with non-orthogonal orbitals are once again of practice rather than principle. With the same notation as in the last section, we now need overlap and matrix-element formulae for two functions of type (7.5.1). As in (7.5.3), we start from 

In this somewhat unsatisfactory situation, there are two main ways of proceeding. In the first, every function is expressed as a linear combination of Slater determinants (by expanding each 0. in terms of elementary spin products, attaching the spatial factor Qk and anti-symmetrizing) the matrix elements then become sums of contributions from, all pairs of determinants, and these may be evaluated, even with non-orthogonal spin-orbitals, by using (3.3.11), (3.3.17) and (3.3.18). This method has been developed into a useful tool by Simonetta et al. 

The second method, which we now develop, is due mainly to Gallup and coworkers (for a review see Gallup et al., 1982), and may be regarded as another application (cf. p. 226) of the spin-free approach (Matsen, 1964) it leads to rather efficient procedures, and also involves ideas needed in Chapter 10 that are conveniently introduced at this point. 

Since the matrix elements of H between different terms in (7.6.4) vanish, through spin orthogonality, while the fs diagonal terms must be identical by symmetry (Appendix 3), the energy becomes (for a normalized i ) 

The spatial function (7.6.6) transforms under permutations like the first basis vector of the irrep with matrices V(P) but the choice p = K = l is purely conventional. 

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