Theory and numerics for electronic structure

Objective

Electronic structure methods allow for the simulation of molecules and materials from first principles. Among these methods, Kohn-Sham density functional theory (DFT) stands out for its balance between computational cost and accuracy. Mathematically, it is a system of coupled nonlinear partial differential equations with an interesting geometrical structure. DFT is enormously popular in applications to physics, chemistry, materials science and biology. However, it suffers from a lack of robustness, and some systems and properties remain too expensive for routine calculations. The goal of this proposal is to improve the robustness, accuracy and applicability of DFT calculations. My overall approach will be to start from a deep mathematical analysis of the relevant equations, and to use this to improve state of the art numerical methods as well as design new ones. This mathematically-based approach will result in methods that are robust by design. With collaborators in physics and chemistry, I will test these methods in challenging systems. In particular, I will focus on three areas of DFT where a mathematical approach can have a large practical impact: minimization methods for heterogeneous, non-gapped and correlated materials, methods for point defects in solids, and electron dynamics in molecules subject to strong electrical fields. The theoretical advances will push the state of mathematical and numerical analysis towards more realistic settings, addressing fundamental questions (well-posedness, asymptotic limits, locality). The numerical advances will have an outsized impact on very popular calculations. This will help pave the way towards automatic and numerically optimal calculations, with downstream applications key to tackle major 21st century challenges.