Mathematical Explorations in Quantum Chemistry

Abstract

Mathematical models, approximations, and optimisations are the foundation of the field of quantum chemistry. This thesis explores improvements in some of the theory behind computational chemistry calculations.

One of the most computationally expensive tasks of self-consistent field (SCF) energy calculations, known as the integral bottleneck, is the calculation of the O(N^4) electron repulsion integrals (ERIs) in every cycle iteration. It is desirable to reduce the total number of ERIs to reduce computational cost and allow for calculation of larger systems, and this thesis explores two methods to do so.

The first method is to eliminate integrals that would have negligible impact on the final result before they are calculated. We investigate several screening strategies that remove integrals using metrics based on the Cauchy-Schwarz inequality and the Hölder inequality by bounding the maximum integral that could be made by a given shell pair or quartet. The Cauchy-Schwarz upper bound is commonly used, but the Hölder bound, proposed many years ago, had yet to be implemented and evaluated.

The second method is to reduce sums within the integrals. In SCF calculations, shell pair densities are described by sums of Gaussian functions. By modelling each shell pair density with a sum of fewer Gaussians, the total number of integrals required is reduced. An algorithm for forming shell pair density models and analysis of some examples are presented, as well as a preliminary energy calculation using modelled shell pair densities.

Finally, we look at a different kind of mathematical model: a representation of spherical functions in a coupled basis called bipolar harmonics. We present a new form of a generalised addition theorem that reduces bipolar harmonics of high degree to a sum of simpler base harmonics, which would allow for efficient application of these in different modelling problems, including density functional development.