New Approaches to Large-Scale Electronic Structure Calculations

Recent years have witnessed a growing interest of the scientific community for the use of ab initio and density functional theory methods in theoretical studies of molecules containing many atoms. However, the `scaling wall' of some of the most accurate of such methods is often an obstacle for their applicability to systems of real-life interest, e.g. in biochemistry and nanotechnologies, one bottleneck being the evaluation and storage of the two-electron repulsion integrals. We have explored the possibility to avoid the expensive evaluation of the full integral matrix by designing and implementing approximate approaches based on Cholesky decomposition techniques. The results show the general applicability of this approximation to any quantum chemical model with substantial computational savings compared to conventional implementations -- not uncommonly of 1-2 orders of magnitude. At the same time, the loss of accuracy is minimal and can be systematically reduced at the price of sustainable additional costs. A major theoretical achievement of this study has been the reformulation of the Cholesky approximation in terms of solution to a "density fitting" variational problem. This has lead to an elegant formulation of the analytic derivatives of the Cholesky vectors, a long-standing lack of this technique. Moreover, this observation has paved the way to a new generation of accurate density fitting approximations free from biases and derived fully ab initio