Local Exchange Potentials in Density Functional Theory

DFT is a method that deals eciently with the ground state any-electron problem. It replaces the solution of the many-electron Schrodinger's equation with an equation to determine the electronic density alone. In the Kohn-Sham (KS) scheme, this density is obtained as the ground state density of a ctitious system of non-interacting electrons. The aim is to determine the local potential for these electrons so that their density equals the interacting density of the physical system. This potential is the sum of the electron-nuclear attraction, the Hartree repulsion from the density and nally the exchange and correlation potential. The central approximation in DFT is the functional form of the exchange-correlation potential. The most basic approximate functionals are explicit functions of the electron density. More sophisticated approximations are orbital dependent functionals or hybrids of density and orbital dependent functionals. In this work we present the implementation of some accurate local exchange potentials, the exact exchange (EXX) potential, the local Fock exchange (LFX) potential and an approximation to EXX, the common energy denominator approximation (CEDA) potential. The EXX potential minimises the Hartree-Fock (HF) total energy and is calculated using perturbation theory and the Hylleraas variational method, improving upon previous implementations. Optimising a local potential that adopts the HF density as its own ground state density, gives the LFX potential, which is simple to calculate and physically equivalent to the EXX potential. Both the EXX and LFX methods are extended to be applicable to metallic systems. The implemented potentials are used to calculate the electronic band structures for semiconductors, insulators, antiferromagnetic insulators and metals. For the semiconducting, insulating and metallic systems studied, the LFX method gives very similar results to EXX. In the systems characterised by stronger correlations, we observe a small disparity between the two exchange methods. When compared to experiment, the results are surprisingly accurate, given the complete neglect of correlation in these calculations. This is remarkable for the strongly correlated systems and also for the simple metals, given the well-known qualitative failure of Hartree-Fock for metals. The fundamental gap of a system is the sum of the KS eigenvalue gap and a correction known as the derivative discontinuity. The exact derivative discontinuity for a system is derived from ensemble density functional theory, thus allowing the full calculation of fundamental band gaps. Approximate forms of the discontinuity for the local density approximation (LDA), generalised gradient approximations (GGA), EXX and LFX are also derived and implemented. Contrary to the accepted wisdom, that the derivative discontinuity for local approximations (LDA/GGA) vanishes, calculated LDA and GGA fundamental band gaps give a much improved result over the corresponding Kohn-Sham band gaps, with accuracy comparable to EXX and LFX KS band gaps. Finally the derivative discontinuity using exact exchange and an orbital dependent correlation functional was also derived but not implemented.