Recapitulation and Generalization

Computational Chemistry Using the PC, Third Edition, by Donald W. Rogers ISBN 0-471-42800-0 Copyright 2003 John Wiley Sons, Inc. 

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16) 

Every electron in a molecule has a Coulombic attraction to its own nucleus Hii = J iH idx. In addition, it has an athaction to all other nuclei in the molecule Hij = J Coulombic attraction between nuclei and the electrons normally 

These expansion coefficients ak] are the minimization parameters of a set of simultaneous equations. Imposition of all the minimization conditions (7-1) for k = 1,2. n leads to the set of n equations 

The optimization procedure is canied out to find the set of coefficients of the eigenvector that minimizes the energy. These are the best coefficients for the chosen linear combination of basis functions, best in the sense that the linear combination of arbitrarily chosen basis functions with optimized coefficients best approximates the molecular orbital (eigenvector) sought. Usually, some members of the basis set of funetions bear a eloser resemblanee to the true moleeular orbital than others. If basis function j) bears a closer resemblance to fit than does, then Uk a +i. 

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