Revisiting the use of explicitly correlated geminals in coupled pair calculations: new perspectives and a proposal

Abstract

A review is made of earlier work with sets of explicitly correlated Gaussian geminals in expansions of electron pair functions at perturbative and coupled-pair levels. Analysis of numerical results, in particular their trends towards nearly exact limits and dependences on various Gaussian basis set sizes, reveals their efficacy and robustness. A major problem for large-scale application of these geminals is the choice of their non-linear parameters. The approach thus far was their optimization at the second-order energy level with a novel variational functional. Although much more efficient computationally than previous functionals, the large number of these non-linear parameters makes it quite foreboding, if not impractical for larger molecules of interest. Unlike the situation for Gaussian-type orbital parameters, no pattern, uniqueness or transferability of optimized geminal parameters seems discernible. A mathematical analysis is conducted of the integro-differential equation for the pair function in first order, especially its singularities and limiting large-coordinate forms. It is argued that all mathematical properties of the full coupled-pair functions are determined by this equation. As a result, an alternative geminal basis function is proposed. Its correlation factor can be given an integral representation with a Gaussian dependence on the inter-electronic coordinate. It seems that the non-linear optimization problem can then be almost eliminated.