A Unifying Approximation Scheme for Density Functional Theories: A Force Balance Based Approach

Abstract

First-principle methods as a way of understanding various fundamental phenomena that occur in nature is an active field of research in condensed matter physics and other related fields. There is great interest in the study of how a system or a property changes when an external perturbation is applied to it, say, by switching on a magnetic field or probing the system with a laser. Many successful theoretical developments have been made over the years to specifically treat these different situations. However using one of these theories out of its assigned setting, by construction, does not always guarantee a suitable outcome and some interesting features may not be captured. This is partly due to the approximations that are used in these methods which are geared to only specific external perturbations or properties. It is therefore of importance to have a theory that can, in a consistent way, treat these various settings and allow for the qualitative study of the changes that occur when different external stimuli (magnetic fields, lasers,...) are applied to a system. We propose here such an approach that contains all the ingredients necessary to perform such a qualitative study. In this thesis we present a unifying scheme to determine exchange correlation potentials in density and current density functional theories including vector potentials. The standard energy-based approach to determine functionals is not used here. Instead this approach relies on the equations of motion of particular current densities and is viable both for the ground state and the time-dependent setting. We aim at directly approximating the density-potential mapping thereby avoiding subtleties that arise from functional differentiability and also the costly optimized effective potential procedure of orbital-dependent energy functionals. We then show that the different density functional theories are connected through these equations of motion and demonstrate this for a local-exchange approximation. We show how these exchange-type approximations reduce to the usual local density approximation in the case of a homogeneous system. We highlight what is not captured when approximations for simple settings are used in more complex ones. In addition, these equations of motion provide a way to numerically construct density-potential mappings for different density functional theories and we show this particularly for a ground state lattice setting including the Peierl’s phase. All these show that this equation-of-motion-based approach bears many interesting advantages and provides a new path for approximations in density functional theories. Moreover, it sets a path for a more complete understanding of the properties of molecules or solids subject to different external stimuli.