  An alternative methodology to evaluate two-electron-repulsion integrals based
on numerical approximation is proposed. Computational chemistry has branched
into two major fields with methodologies based on quantum mechanics and
classical force fields. However, there are significant shadowy areas not
covered by any of the available methods. Many relevant systems are often too
big for traditional quantum chemical methods while being chemically too complex
for classical force fields. Examples include systems in nanomedicine, studies
of metalloproteins, etc. There is an urgent need to develop fast quantum
chemical methods able to study large and complex systems. This work is a
proof-of-concept on the numerical techniques required to develop accurate and
computationally efficient algorithms for the fast calculation of
electron-repulsion integrals, one of the most significant bottlenecks in the
extension of quantum chemistry to large systems. All concepts and calculations
were developed for the three-center integral (pxApxB|pxCpxC) with all atoms
being carbon. Starting with the analytical formulae, convenient decompositions
were tested to provide smooth two-dimensional surfaces that were easily fitted.
The approximating algorithm consisted of a multilayered approach based on
multiple fittings of two-dimensional surfaces. An important aspect of the new
method is its independence on the number of contracted Gaussian primitives. The
basis set of choice was STO-6G. In future developments, larger basis sets will
be developed. This work is part of a large effort aimed at improving the
inadequacies of existing computational chemistry methods, both based on quantum
mechanics and classical force fields, in particular in describing large and
heterogeneous systems (ex. metalloproteins).
