Overlap integral, calculation definition

Fock matrix elements are actually calculated. The EHM Fock matrix elements are calculated from well-defined physical quantities (ionization energies) with the aid of well-defined mathematical functions (overlap integrals), and so are closely related to ionization energies and have definite quantitative values. 

It is in the definition of the integrals or the lack of it, that most m.o. methods differ. The overlap integrals are easy to compute and were, indeed, tabulated for Slater-type atomic orbitals a long time ago (ref. 79). In some simple calculations, namely extended Huckel-type calculations a rough estimation of Hmn is as follows (see also page 79). 

As explained in the preceding section, the bond index and the valency have definite physical meanings only when basis functions are orthonormalized. MOPAC satisfies this assumption(6), but general MO methods, including DV-Xa, don t satisfy it. Fortunately basis functions can be orthonormalized after MO calculation without any loss of accuracy, provided that all overlap integrals are known. 

It is important to remember, however, that each basis function has the definite form of (3.22S) and that, while the functions approximate Slater functions, there is no approximation being made, other than the Hartree-Fock approximation, once the basis functions are chosen. The next step in the SCF calculation involves the evaluation of all integrals over the tosis set i.e., S v. W"". and the two-electron integrals (/iv 2ff). All these integrals can be evaluate using formulas developed in Appendix A. Consider the overlap integral,... 

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