Systematic routes to improved approximations in Kohn–Sham theory

Abstract

The Kohn–Sham (KS) formulation of density-functional theory (DFT) has become the pre-eminent method for modelling electrons in matter. In many calculations, KS theory offers an unrivalled balance between accuracy and speed. However, the most widely-used approximations are known to be inadequate in certain applications, such as strongly-correlated systems. Furthermore, it is not straightforward to systematically converge to the correct result in these cases; this is in contrast to wave-function methods, which lend themselves more naturally to systematic improvements. In this thesis, we develop methods to help understand and improve systematic failings of common approximations for the exchange and correlation functional in KS theory.

One major theme of this thesis is the use of wave-function theories to develop accurate reference KS potentials in DFT. We consider an alternative derivation of the KS potential based on the minimization of a wave-function expression, which establishes a link between DFT and wave-function theories. We use this tool to develop perturbative expansions of the KS potential: one such expansion yields a novel KS potential which is expected to have exact exchange and accurate correlation character. Continuing this theme, we explore a method to obtain the KS potential corresponding to a given density. We focus on the role of the screening density in this method, a concept which also helps our understanding of the pervasive self-interaction error in DFT.

The other major theme of this thesis is the development and application of implicit density functionals. We explore how this class of functionals can be used to develop a new formalism for open-shell systems in KS theory. Implicit density functionals in KS theory require the optimized effective potential framework, whose implementation in finite basis set codes has proven problematic in the past. We develop an implementation which is both simple to apply and formally avoids these mathematical difficulties.