Ensemble density functional theory on a lattice

Submitted to the Faculty of Science Wits University May 29, 2018Density Functional Theory (DFT) is an elegant reformulation of quantum mechanics in which the density distribution is the variable that formally contains all the information about a system. It was placed on a formally sound theoretical footing by Hohenberg and Kohn [1] in 1964 and an implementation for determining the ground state density and energy was proposed by Kohn and Sham the following year [2]. Despite more than fty years since Hohenberg and Kohn showed that the density can be used as the controlling variable, there is no known exact way to implement DFT. Nevertheless, DFT has been successfully applied using approximations and has become the standard approach for investigating structural properties of solids and molecules. In this project we examine properties of DFT functionals for a nite single band Hubbard chain. The advantage of using a Hubbard model is that for short chains exact solutions can be found numerically and for a uniform in nite chain an analytic solution is available. The exact solutions can be used as a reference for approximate implementations of DFT. We explore DFT on a lattice in an ensemble formulation which allows a formal implementation of DFT for fractional particle numbers. We show that even for a simple uniform density approximation the resulting functional derivatives have a spatially independent discontinuity as a function of particle numbers at integer particle number, as the required by the exact formalism. An approximate exact implementation of Kohn-Sham DFT with the neglect of the DFT correlation energy can be implemented exactly and results show that it can compare very well with the exact solution, but that the success of the approximation is not consistent under all circumstances. Finally we show that it is possible to achieve the original goal of Kohn-Sham Density Functional Theory which was to nd the ground state density and energy of an interacting system while all calculations are performed for a ctitious independent particle model. We introduce a mapping of the ground state wavefunction basis function expansion coe cients of a single band Kohn-Sham Hubbard model onto the coe cients of the interacting Hubbard model and derive a set of exact self-consistent equations that can be solved within an ctitious Kohn-Sham framework to nd the interacting ground state density and energy.MT 201