| In this thesis, we study various density functional approximations (DFT) as applied to atoms and harmonic oscillator systems with complete subshells. First, by confining ourselves to exchange only problems, we show that the exact Kohn-Sham(KS) wavefunctions and the KS exchange potential can be calculated for any system in which the electron states are described by only two different radial wavefunctions. We present the results of the KS solutions for eight(paramagnetic) and four(spin polarized) electrons moving in a harmonic oscillator potential with a variety of spring constants and compare with the corresponding Hartree-Fock(HF) and local spin density approximation(LSD) as well as the Slater exchange results(SLA). We also study the LSD with Self-Interaction-Correction(SIC) applied(LSDSIC). It is shown that the total energy calculations for atoms could be further improved if one takes into consideration the angular dependence of the wavefunctions in the energy functional. By studying the electron-electron interaction within the same sub-shell, we are able to develop a SIC method which treats the entire intral-shell exchange exactly in the asymptotically large r region. Calculations show that it considerably improves the accuracy of the density functional maximum single particle energy eigenvalue, {dollar}varepsilonsbsp{lcub}rm max{rcub}{lcub}rm df{rcub}{dollar}, when compared to the exact HF results. In studying the Gradient Expansion Approximation(GEA), we discuss an alternative way of constructing the GEA energy functional which eliminates the singularity in the usual GEA exchange potential as r approaches infinity. With this GEA form, we are able to apply the SIC technique(the GEASIC) and find that this GEASIC can give the atomic total energies that are significantly more accurate than those given by the usual LSDSIC. Finally, by adding the correlation effects, we perform further investigations on the various SIC methods as applied to atoms. By treating exchange exactly, we find that although the total energy calculations are improved, the {dollar}varepsilonsbsp{lcub}rm max{rcub}{lcub}rm df{rcub}{dollar} for the noble gases become poorer. |
