Date: Feb 22, 2019, 14:00
Candidate: Guilherme de Souza Tavares de Morais
Advisor: Rogério Custodio
Abstract:
A grid-based integral variational numerical method was developed to solve the Schrödinger equation. This method combines techniques of numerical differentiation and integration, as well as imposing Schmidt orthonormalization to calculate fundamental and excited states of quantum systems. The solution of the radial equation for multielectronic atoms was performed with a generalized exponential spacing referred to as q-exponential. This method is very simple and reduces difficulties encountered in solving the conventional Hartree-Fock equations. The evaluation of numerical wave functions with different number of points and different values of the q-parameter yielded energies with a crossing point near the numerical Hartree-Fock results. Such behavior allowed to obtain an accurate total energy using its derivative with respect to the q-parameter. For the helium atom an energy of -2.86167977 u.a. was achieved, which differs from the Hartree-Fock energy found in the literature by 2 ∙ 10−7 u.a.. The methodology was also applied to confined atoms. For the confined helium in a spherical box of radius 1 u.a. the electronic energy was 1.06120 u.a., which differs by 2.10-5 from the Hartree-Fock energy from the literature. For other confinement radius this difference may be even smaller. This work focused on the application of the method to determine the electronic energies of the fundamental and some excited states of free (1st to 4th) and confined (2nd period) atoms. The method along with the exponential mesh-grid showed excellent performance in obtaining exact results under modest calculation conditions. The results suggest that the procedure described is a consequence of numerical error, which provides energies below the variational limit and, consequently, an alternative way to achieve accurate results. Well-behaved results suggest that this is not a random or accidental effect.
