Variation of Energy-Nonorthogonal Basis Set

In an overlapping basis of spinorbitals, the Hamiltonian can be put down in the following form [cf. Eq. (13.25)]  

It is essential now to use this form of the second quantized Hamiltonian which is expressed over the true fermion operators obeying the anticommutation relations of Eqs. (13.13). The use of Eqs. (13.28) or (13.33) for the Hamiltonian would complicate the following treatment since the appearance of the overlap matrix in the commutation rules would destroy the purely algebraic character of the creation/annihilation operators. The proper anticommutation rules permit us to consider and as abstract operators creating and annihilating electrons, respectively. Accordingly, taking the variation of Eq. (14.8) we get  

Substitution of 5H into the formal Hellmann-Feynman theorem of Eq. (14.4) gives the variation of the electronic energy as the expectation value  

In order to bring this formula into a more transparent form, two problems have 

Second, the variation of the integrals in Eq. (14.10) should be evaluated by taking into account the S transformation in their hra-indices. This is easier to do in matrix notations. For the one-electron integrals we get  

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