Overlap Matrices and the Neglect of Some Permutations

The (1 T) in Eq. (2) arises from the normalization of the wave function for H2. In this section we will investigate the extent to which it might be permissible to ignore the permutations of some order and higher when normalizing a VB function for n electrons. We shall do this for a standard tableau function, where we have an expression for the wave function of any multiplicity. 

Therefore, consider a standard tableaux function with orbitals i, 2, , un, where they need not all be different, of course, 

The orbitals are assumed real, normalized, but not necessarily orthogonal. The overlaps are symbolized by Sij = S = ( ,-)- It is shown elsewhere[36] that the normalization constant for such a standard tableaux function can be written as the integral of a functional determinant, 

Consider an N x N symmetric matrix S that has principal diagonal elements all equal to one.  

Theorem 1 A simple transformation of the characteristic polynomial of such a matrix will present it in a form where the contribution from each order of permutation to the value of its determinant is displayed as an elementary symmetric function of the eigenvalues of S — I. 

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