Orbital-Dependent-Functionals within Density Functional Theory: Methodology and Applications

Density functional theory (DFT), developed in the mid 60's by Hohenberg, Kohn, and Sham, is a mathematical construction that allows to calculate the ground-state properties of matters. It is, in principle, an exact theory, which provides a bypass to solving the formidable many-body time-independent Schrodinger equation. As implied by its name, DFT considers primarily the system's charge density, without going through the full many-body wave function. Accordingly, only a relatively moderate numerical effort is required. Consequently, even large chemical systems, composed of up to tens of thousands of atoms, are computationally feasible with today's computational power. DFT has thus become the method of choice for calculating the chemical and physical properties of a great variety of materials. Nevertheless, there are still many properties that can not be calculated accurately within DFT. This is because DFT is exact only in principle. In practice, the involved equations are not exact and all the many-body quantum effects are, at least to date, approximated. The approximated term is a functional of the density, known as the exchange-correlation (XC) functional.;Orbital-dependent functionals (ODFs) are an emerging class of functionals in which the XC energy term depends explicitly on the Kohn-Sham (KS) orbitals and only implicitly on the density. In the past decade, ODFs were shown to be very promising from the functional development point of view, but a full realization of them within the KS-DFT framework had encountered several difficulties on the one hand and raised additional questions on the other. The goal of this thesis was to develop methods for correct use of such functionals within KS-DFT and to explore their special features. Both numerical and theoretical aspects of ODFs were investigated, where I mostly (but not only) focused on the exact exchange (EXX) functional as a typical ODF.;The use of ODFs within KS-DFT requires a solution of a complicated integro-differential equation, known as the optimized effective potential (OEP) equation. Finding a full stable solution of the OEP equation is not a trivial task. Worse, it was shown that common approximations that are often used within KS calculations, such as the use of basis sets (atomic or molecular) or the pseudopotential approximation of the ionic potential, make the attempt to correctly solve the OEP equation even more difficult.;In light of these difficulties, it was highly desirable to develop an OEP numerical scheme that is free of any approximations beyond the one that is inherent in the choice of the XC functional. To the best of my knowledge, before this work such computational capability was only achieved for single atoms. Consequently, the initial goal of this study was to extend this capability to diatomic systems, which already allow the study of chemical bonds.;I have thus designed and implemented a new numerical scheme for solving the KS equations for diatomic systems, together with a full solution of the OEP equation. The equations are solved on a real-space prolate spheroidal coordinate grid, such that all the system's electrons are taken into account. The OEP equation is solved via the S-iteration scheme. This newly developed software package is called DARSEC (DiAtomic Real-Space Electronic structure Calculations). It involves no approximation except for the one inherent in the XC functional. Thus it is especially suitable for examining new functionals of any kind, and ODFs in particular. It is also an ideal tool for assessing the validity of commonly used approximations, for the same reasons.;One case for which this uniqueness of DARSEC was exploited in this thesis is the examination of the validity of the pseudopotential approximation for KS gaps that are calculated with EXX OEP (xOEP). Before this study, use of the pseudopotential approximation in such calculations was called into question. I have shown that KS gaps obtained with pseudopotentials that have been constructed in a manner consistent with the exact-exchange functional agree with the all-electron results (i.e. without the pseudopotential approximation), for the cases studied. This confirmed the reliability of the pseudopotential approximation for ODFs such as EXX.;Explicit density-dependent XC functionals traditionally fail to obtain atomization-energy as well as charge-dissociation curves that are, at least qualitatively, correct for diatomic systems. On the other hand, Hartree-Fock (HF) theory encounters no such problem. Hence, an additional goal of this research was to study the performances of the EXX functional (being the DFT counterpart of HF) in describing binding energies and charge dissociations for stretched diatomic molecules. Moreover, I wanted to investigate the special features of the resulting single and local EXX KS potential, as opposed to the non-local orbital specific HF potentials. (Abstract shortened by UMI.).