Matrix elements between antisymmetrized products

We consider first a single determinant, written according to (3.1.5) as an antisymmetrized product 

The integrand will thus contain (iV ) products, each of the form 

The expectation value of the 1-electron part of the Hamiltonian, namely h(i), will reduce to the sum of N identical terms, since the coordinates of each electron appear synunetrically in the corresponding integrand, and by the same kind of arguments u above it is easy to show that 

The expectation value of T, g(i, ) involves slightly more subtle considerations, since contributions may arise from terms which differ by an interchange of two electrons but an essentially similar argument shows that 

Thus the expectation value of the energy (i.e. the diagonal matrix element of H) with a normalized determinant (3.3.3) is 

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The so-called Slater-Condon rules express the matrix elements of any one-electron (F) plus two-electron (G) additive operator between pairs of antisymmetrized spin-orbital products that have been arranged (by permuting spin-orbital ordering) to be in so-called maximal coincidence. Once in this order, the matrix elements between two such Slater determinants (labelled > and are summarized as follows ... 

The interaction occurs between half-filled shell states K) and IL) which possess different parities with respect to CL, ttk and ttl, respectively. This implies that necessarily the configurations are different, i.e., I A") lL), and also that the product of parities ttkttl has the value — 1. The hypothesis that Tt% is antisymmetric under time reversal is represented in equation (32) by setting thm = —1. It follows that the interaction matrix element (LlH lK) must vanish if these criteria are met. 

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