Galerkin method using optimized wavelet-Gaussian mixed bases for electronic structure calculations in quantum chemistry

This thesis aims to be a contribution to numerical methods for ab initio molecular simulation, and more specifically for electronic structure calculations by means of the Schrödingerequation or formalisms such as the Hartree-Fock theory or the Density Functional Theory. It puts forward a strategy to build mixed wavelet-Gaussian bases for the Galerkinapproximation, combining the respective advantages of these two types of bases in orderto better capture the cusps of the wave function.Numerous software programs are currently available to the chemists in this field (VASP,Gaussian, ABINIT... ) and differ from each other by various methodological choices,notably that of the basis functions used for expressing atomic orbitals. As a newcomer tothis market, the massively parallel BigDFT code has opted for a basis of wavelets. Dueto performance considerations, the number of multiresolution levels has been limited andtherefore users cannot benefit from the full potential of wavelets. The question is thus howto improve the accuracy of all-electron calculations in the neighborhood of the cusp-typesingularities of the solution, without excessively increasing the complexity of BigDFT.The answer we propose is to enrich the scaling function basis (low level of resolutionof the wavelet basis) by Gaussian functions centered on each nucleus position. The maindifficulty in constructing such a mixed basis lies in the optimal determination of the numberof Gaussians required and their standard deviations, so that these additional Gaussiansare compatible in the best possible way with the existing basis within the constraint of anerror threshold given in advance. We advocate the conjunction of an a posteriori estimateon the diminution of the energy level and a greedy algorithm, which results in a quasi-optimal incremental sequence of additional Gaussians. This idea is directly inspired bythe techniques of reduced bases.We develop the theoretical foundations of this strategy on two 1-D linear models thatare simplified versions of the Schrödinger equation for one electron in an infinite domainor a periodic domain. These prototype models are investigated in depth in the firstpart. The definition of the a posteriori estimate as a residual dual norm, as well as theimplementation of the greedy philosophy into various concrete algorithms, are presented inthe second part, along with extensive numerical results. These algorithms allow for moreand more saving of CPU time and become more and more empirical, in the sense thatthey rely more and more on the intuitions with which chemists are familiar. In particular,the last proposed algorithm partly assumes the validity of the atom/molecule transfer andis somehow reminiscent of atomic orbitals bases.