Generalized Sturmians Nonorthogonality

In the preceding discussion, the creation and annihilation operators always referred to a set of orthonormal spin-orbitals. We can ask how much of the formalism holds in cases where complete orthonormality can no longer be assumed, for example, in valence-bond calculations or in the generalized Sturmian method. 

The generalized Sturmian method for solving the iV-electron Schrodinger equation [11 -20] is a direct configuration interaction method, using basis functions which are antisymmetrized isoenergetic solutions of the approximate iV-electron equation 

In equation (87), Vo is as near as possible to the actual potential V, and the weighting-constants /3 are chosen in such a way as to make all of the solutions correspond to the same energy, regardless of the quantum numbers. 

In atomic calculations, equation (87) can be solved exactly if V0 is set equal to the nuclear attraction potential 

Then (as is shown in Ref. [20]) the first generalized Slater-Condon rule can be 

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