Approximate energy‐evaluating schemes for a system of weakly overlapping group functions

Abstract

=The energy of weakly overlapping group functions can be written as a series according to the powers of the (a -I ) matrix, where a is the molecular overlap matrix and I is the unit matrix [1,2]. This power series of the energy is studied by investigating the importance of different order terms to obtain accurate energies and to predict equilibrium bond lengths. It is found that the series is truncated advantageously at an even-order term. Approximate formulas for the first-and second-order terms are proposed in order to reduce computational work. Numerical examples are presented to illustrate the effect of these terms to the energy. The relation of the projection energy to the approximate first-and second-order terms is also discussed. It is found that, by choosing appropriate projection factors, the projection energy corrects the zeroth-order energy more efficiently than does the first-order term. The inclusion of the approximate second-order term represents a slight improvement with respect to the use of the projection energy at the expense of some extra computation. 0 1995 John WiIey & Sons, Inc.

The self-consistent nonorthogonal group function approach in reduced basis frozen-core calculations

“…Before analyzing Figure l(a)-(e>, it is worth recalling the results of [16]. In that article, a different but related problem is examined.…”

“…The valence orbitals were calculated with the orthoge nal and with the nonorthogonal formulations. In the former case, the eigenvectors of matrix (131, with X chosen as a unit matrix, were calculated and the valence energy is obtained as the sum of formulas (12) and (16). The nonorthogonal orbitals were calculated from Eq.…”

“…In this case, the leading term in the energy expression is that obtained with the neglect of orbital overlaps, and the terms arising from the overlaps can be included as corrections to the energy expression of orthogonal orbitals. The derivation of such an energy expression is discussed in detail in [16]. Here, an outline of the derivation is recapitulated and the final formula is presented.…”

“…The formula for the electronic energy is derived in [16] and, here, only a summary of the approximations and the obtained energy expressions are presented together with the formulas of the separated nuclear repulsion energy. In the third section, the orthogonal formulation, as applied in the present article, is summarized.…”

Electron group functions for the analysis of the electronic structures of molecules

“…In the more general case the working formulae become very cumbersome35, 36 such that full consideration is possible only for very small systems 37. In this case approximate schemes can be quite useful 38. Another approach for treating nonorthogonality is employing biorthogonality 39, 40.…”