One-configuration VB theory. Non-orthogonal orbitals

The above results apply in the present context because any VB-type function (7.4.1) is equivalent to a sum of Slater determinants formed by attadiing various spin products to the orbital factor Q every term corresponds to an antisymmetrized product of two determinants (one for the core and one for the valence electrons), and on extracting the common core factor the whole VB function assumes the generalized- 

To connect the approach with that used in Chapter 4 (pp. 98-99) we remember that, instead of using a space-spin antisymmetrizer as in (7.4.1), we may apply a Wigner-type operator (cf. (A3.23) etseq.) 

The above equivalence confirms that, with a spinless Hamiltonian, we may obtain the same energy expectation value either (i) by expanding (7.4.1) in terms of determinants and then using the rules in Section 3.3 or (ii) by using a linear combination of purely spatial functions (7.4.6) of appropriate symmetry. The second approach is essentially that of spin-free quantum chemistry (Matsen, 1964), which is considered in more detail in later sections. In the present case a first-principles argument will lead to the required matrix-element expressions. 

To evaluate the general matrix element ( aI H and overlap integral A I ic). it will be sufficient to start from the latter, writing 

It is convenient to introduce at this point permutation operators that act on the orbital indices rather than the electronic variables, using for example P to denote the operator that interchanges / , and (j),. The two types of operator (P, Q) will then commute because they work on different sets of indices. Moreover, two operators P, P that correspond to the same permutation may be applied simultaneously to Q without changing it  

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