Abstract
Abstract A major difference between the problems of single-electronic systems and that of the much more complicated many-electronic systems arises from the electron-electron interactions Uee. In the later case, we have to deal with 3N-dimensional wave functions, since each electron contains three spatial dimensions. The magnitude of the number of degrees of freedom for the wave functions was a major bottleneck for serious numerical calculations on these systems. Instead of dealing directory with 3N-dimensional wave functions, the density functional theory (DFT) recasts this problem in terms of the 3N dimensional electronic density distribution function n(r). The most important theoretic justification for DFT was provided by the work of Hohenberg and Kohn (HK) . Hohenberg and Kohn proved that the external potential v(r) of a many-electronic-system can be readily deduced from the electronicdensity of the ground state n(r) Further progress of the Density Functional Theory (DFT) was furnished by the Local-Density Approximation(LDA) proposed by Kohn and Sham. With the Local-Density Approximation(LDA), the Density Functional Theory(DFT) became apractical and an efficient tool for studying complicatedelectronic systems through ab inito calculations. The Local-Density Approximation(LDA) may be generalized by considering the contributions from distinctspin states, (σ =↑, ↓), as distinct contributions, this leads to the local spin density (LSD) approximation. In this thesis we follow the interpolation formulas by Gunnarsson, Lundqvist and Wilkin for the exchange and correlation functional of the LSD. The hydrogen atom was used as a simple electronic system to check the applicability of the electron exchange and correlation energy approximation in the LDA and LSD approaches. The goal of this thesis is to apply python to the atomic system calculation. As a first step for complicate atomic system we start with the hydrogen atom. The asymptotic behavior of the solution at r → 0 and r → ∞ were used for stable numerical solution. of the atomic Schrぴodinger equations.