Thesis presented November 14, 2005

Abstract:
This thesis is devoted to the definition and the implementation of a multiresolution method to determine the fundamental state of a system composed of nuclei and electrons. In this work, we are interested in the Density Functionnal Theory (DFT), which allows to express the Hamiltonian operator with the electronic density only, by a Coulomb potential and a non-linear potential. This operator acts on orbitals, which are solutions of the so-called Kohn-Sham equations. Their resolution needs to express orbitals and density on a set of functions owing both physical and numerical properties, as explained in the second chapter. One can hardly satisfy these two properties simultaneously, that is why we are interested on orthogonal and biorthogonal wavelets basis, whose properties of interpolation are presented in the third chapter. We present in the fourth chapter tridimensionnal solvers for the Coulomb's potential, using not only the preconditioning property of wavelets, but also a multigrid algorithm. Determining this potentiel allows us to solve the self-consistent Kohn-Sham equations, by an algorithm presented in chapter five. The originality of our method consists in the construction of the stiffness matrix, combining a Galerkin formulation and a collocation scheme. We analyse the approximation properties of this method in case of linear hamiltonien, such as harmonic oscillator and hydrogen, and present convergence results of the DFT for small electrons. Finally we show how orbital compression reduces considerably the number of coefficients to keep, while preserving a good accuracy of the fundamental energy.

Keywords:
Kohn-Sham equations, Poisson's equation, wavelets, collocation method, lifting scheme, multigrid

On-line thesis.