Abstract

Density Functional Theory - Part 2

Richard Martin
University of Illinois at Urbana-Champaign
Physics

At the most fundamental level, properties of materials are determined by the fundamental quantum equations for the electrons.

Density functional theory (DFT) is a theory of the many-body problem of interacting electrons. Incorporating effects of interactions is essential in any approach that aims to provide quantitatively accurate predictions. Yet all practical implementations of DFT are based upon the Kohn-Sham approach that involves non-interacting electrons. Instead of the many-body problem (which is insoluble, except in special cases) the Kohn-Sham equations are soluble for large complex systems using modern computational methods. The ingenious idea of Kohn and Sham was to include the effects of correlation among the electrons as a functional of the density – the exchange correlation functional. The approach is in principle exact, but the real power of the idea is that is provides a way to approximate the effects of exchange and correlation using approximations to the functional. This lecture provides insights into the nature of the functionals used in the Kohn-Sham approach – basic ideas involved in the functionals, rationales for the success of widely-used forms, and caution that all current DFT methods involve approximations. The application of the specific functionals must be tested and used in appropriate regimes. There are definite failures and inaccuracies that must be appreciated to avoid errors and unphysical results. Examples are the famous “band gap problem” and cases of strongly-correlated electrons. In addition, there is a growing body of research that utilize many-body methods that are not written as explicit functionals of the density; the new developments are making possible increased reliability and accuracy for band gaps, optical spectra, and other properties.