Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Chemistry

Supervisor

Prof. Viktor N. Staroverov

Abstract

Practical Kohn–Sham density-functional calculations require approximations to the exchange-correlation energy functional, EXC[ρ], or the exchange-correlation potential, vXC(r), defined as the functional derivative of EXC[ρ] with respect to the electron density, ρ. This thesis focuses on the following problems: (i) development of approximate exchange-correlation potentials by modelling the exchange-correlation charge distribution; (ii) accurate approximation of functional derivatives of orbital-dependent functionals; (iii) generation of exchange-correlation potentials from many-electron wavefunctions; (iv) analysis of accurate exchange-correlation potentials in atoms and molecules.

The advantage of modelling the exchange-correlation potential through the exchange-correlation charge distribution, qXC(r), is that it produces potentials with correct asymptotic behavior. We present an important caveat for attempts to enforce Coulombic asymptotics of vXC(r) by normalizing the exchange-correlation charge distribution. We also formulate integrability conditions that a model qXC(r) must satisfy in order for the corresponding vXC(r) to be a functional derivative of some density functional.

Functional derivatives of orbital-dependent functionals cannot be derived in closed form and have to be evaluated numerically using the optimized effective potential (OEP) method. We propose a way to avoid the OEP equation in finite-basis-set Kohn–Sham calculations employing orbital-dependent functionals. To this end, we develop a hierarchy of approximations to the functional derivative of a given orbital-dependent exchange-correlation functional. The highest level in the hierarchy is practically indistinguishable from the true OEP and is obtained from the requirement that the Kohn–Sham and the generalized Kohn–Sham densities be equal. By imposing the same requirement on the Kohn–Sham and wavefunction densities we devise and implement a method for calculating vXC(r) from a given electronic wavefunction. Our method is free from numerical limitations and basis-set artifacts of conventional schemes that fit the effective potential vXC(r) to a given ground-state electron density. In the remainder of the thesis, we apply our techniques to elucidate the mechanism of formation of the step structure of the exact vXC(r) and analyze exchange-correlation potentials derived from restricted Hartree–Fock wavefunctions of stretched diatomic molecules.

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