Fundamental Electron-Repulsion Integral

Most of the semiempirical MO methods currently used are based on SCF theory and differ in the approximations that are made so as to simplify the evaluation of the two-electron repulsion integrals. The approximations are then corrected for by parametrization, wherein parameters are included in the fundamental protocol to make the results match ab initio calculations on known systems. Examples of these semiempirical methods are CNDO (complete neglect of differential overlap), INDO (intermediate neglect of differential overlap), and NDDO (neglect of diatomic differential overlap). An alternative approach is to parameterize the calculations to optimize agreement with measured molecular properties, such as thermochemical, structural, or spectral data. 

Our evaluation of the Fundamental Integral cannot proceed further than (18) unless we now specify the two-electron function f(x). We are now in a position to consider some of the integral types which arise in quantum chemical calculations overlap, kinetic-energy, electron-repulsion, nuclear-attraction and anti-coulomb. 

It must be remembered that, fundamental as this phenomenon of exchange is, it appears only as a result of the necessity and the methods of approximation. James and Coolidge, by changing the system of co-ordinates and including the terms of electron repulsion, obtained the best solution of the H2 molecule. Their curve corresponds almost exactly with the experimental one, and no exchange integrals of the type of Hiu appear in their calculation. 

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

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