Zur Approximation elektronischer Wellenfunktionen durch anisotrope Gauß-Funktionen

The electronic Schrödinger equation is one of the most important equations in quantum mechanics and provides an explanation of the behaviour of atoms and molecules. The solutions of the eigenvalue problem - the wave functions - describe the state of the modelled quantum mechanic system and can be computed analytically in only a few simple cases. The high dimensionality of the problem is one of the difficulties in the approximation of a solution. Therefore a direct discretization is nearly impossible. We present a possible way out in this thesis. To this end we examine a substitute problem instead of the original equation. The substitute equation is constructed from an approximation of a restatement of the Schrödinger equation with the help of exponential sums. The error between the solution of the substitute problem and the original wave function can be chosen arbitrarily small by the special construction. Now we can focus on the substitute equation. We show that its solutions can be approximated theoretically by a linear combination of Gaussian functions. At the same time the effort is only marginally bigger than that of the approximation of the convolution of the wave function and a Gaussian kernel of sufficient width in this class of functions. Every convergence order is reachable in theory. The results of this thesis can be utilized for the precise analysis of the perturbed preconditioned inverse iteration which is an iteration method for the solution of eigenvalue problems. It will be shown that after a derivation of the variant of the Schrödinger equation the iteration can completely operate on Gaussian functions. The hope in doing so is to show that the singularities can be dissolved adequately by this nonlinear ansatz.

Author dc:creator

• Scholz, Stephan

Advisor dc:contributor.advisor

• Yserentant, Harry

Licence dc:rights.uri

• https://creativecommons.org/licenses/by-sa/4.0/

Language dc:language.iso

de

Identifier URI

http://dx.doi.org/10.14279/depositonce-5068

OAI identifier oai:identifier

oai:depositonce.tu-berlin.de:11303/5393