Valence Bond Formalism Using an Effective Hamiltonian

By using the expression of the exact Hamiltonian in Equation 3.20, the selfenergy of the ionic terms and the off-diagonal Hamiltonian matrix elements are readily obtained 

By using these matrix elements and the calculated overlaps between determinants, the accurate VB wave function of H2 (Eq. 3.4) can be variationally determined by 3 x 3 nonorthogonal Cl. 

4 VALENCE BOND FORMALISM USING AN EFFECTIVE HAMILTONIAN 

The use of the exact Hamiltonian for calculating matrix elements between VB determinants leads, in the general case, to complicated expressions involving numerous bielectronic integrals, owing to the 1/r,-,- terms. Thus, for practical qualitative or semiquantitative applications, one uses an effective molecular Hamiltonian in which the nuclear repulsion and the 1/r,-, terms are only implicitly taken into account, in an averaged manner. Then, one defines a Hamiltonian made of a sum of independent monoelectronic Hamiltonians, much as in simple MO theory  

It is apparent that the above matrix element is made of a sum of four terms, which are calculated independently (also consult Appendix 3.A.1). The calculation of each of these terms, for example, the first one (HI)), is quite analogous to the calculation of the overlap in Equation 3.17, except that the first monoelectronic overlap S in each product is replaced by a monoelectronic Hamiltonian term  

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