Advances in the understanding of Kohn-Sham DFT via the optimised effective potential method

Kohn-Sham (KS) density functional theory (DFT) has paved its way to becoming the most widely used method for performing electronic structure calculations. Its major success relies heavily on the underlying approximations that are employed to describe the exchange-correlation (xc) energy functional; hence understanding these approximations proves to be of vital importance. The main goal of this thesis is to explore and develop a deeper understanding of approximations made within DFT; with a focus on systematically improving existing (semi-)local density functional approximations (DFAs). To do so, we build upon the existing constrained minimisation method, which requires the optimised effective potential (OEP) scheme, improving its implementation and removing one of its major computational bottlenecks. This thesis also addresses a long-standing question in the field as to why the KS equations of spin-DFT do not reduce to those of DFT in the limit of zero applied magnetic field. A new OEP scheme is derived to construct DFT approximations that yield near spin-DFT accuracy and correct for a systematic error in the exchange energy for open-shell systems. This work is then extended to ensemble systems of varying electron number, where it is shown that (semi-)local approximations can yield non-zero xc derivative discontinuities; an exotic, non-analytic feature of the exact KS potential. Building on these new OEP formulations, a novel new method for decomposing the molecular screening density into screening densities localised on individual atoms is introduced. This method is shown to yield the predicted but elusive steps in the xc potential as a diatomic dissociates; a very exciting result given that these steps cannot be captured at all from any DFT so far, let alone a (semi-)local DFA