Wavelets and Fourier analysis in the study of a quantum chemistry problem.

Summary We present in this work two mathematical techniques for the study of the Hartree-Fock equations, the Fourier analysis and the wavelet analysis, trying to show their advantages as well as their drawbacks. The Fourier analysis, known for a long time but little used by chemists, leads to an impulse representation and allows to simplify the writing of the equations whose resolution becomes possible for small chemical structures. In particular, the equations related to diatomic molecules, H2 and HeH+, could be solved thanks to an iterative method defined in the impulse space obtained by Fourier transform. The analytical derivation of the first iteration is described in detail, and the improvements, both qualitative and quantitative, brought to the initial wave function by this first iteration are analyzed and commented. Wavelet analysis has never been applied in quantum chemistry. Initially developed in signal processing, it finds here a new field of application. After a brief reminder of the general foundations of wavelet theory, the representation of the Hartree-Fock equations in a mixed position-impulse space is obtained thanks to a continuous wavelet transform. An interpretation of this representation is presented, and an iterative method of resolution is proposed. The improvements brought by a first analytical iteration are also analyzed and commented. Another type of wavelet has also been used: ortho-normal wavelets which lead to a fully numerical treatment of the Hartree-Fock equations. The matrix representation provided by the BCR algorithm (Beylkin, Coifman, Rokhlin) is used in a resolution method based on an iterative process whose convergence as well as the problems related to the discretization of the data are studied more particularly.