We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
89/2001
Wavelet approximation of correlated wavefunctions. I. Basics
Heinz-Jürgen Flad, Wolfgang Hackbusch, Dietmar Kolb and Reinhold Schneider
Abstract
We suggest an alternative approach to electronic structure calculations based on numerical methods from multiscale analysis. By this we are aiming to achieve a better description of the various length- and energy-scales inherently connected with different types of electron correlations. Taking a product ansatz for the wavefunction $\Psi =F \Phi$, where $\Phi$ corresponds to a given mean-field solution like Hartree-Fock or a linear combination of Slater determinants, we approximate the symmetric correlation factor $F$ in terms of hyperbolic wavelets. Such kind of wavelets are especially adapted to high dimensional problems and allow for local refinement in the region of the electron-electron cusp. The variational treatment of the ansatz leads to a generalized eigenvalue problem for the coefficients of the wavelet expansion of $F$. Several new numerical features arise from the calculation of the matrix elements. This includes the appearance of products of wavelets, which are not closed under multiplication. We present an approximation scheme for the accurate numerical treatment of these products. Furthermore the calculation of one- and two-electron integrals, involving the nonstandard representation of Coulomb matrix elements, is discussed in detail. No use has been made of specific analytic expressions for the wavelets, instead we employ exclusively the wavelet filter coefficients, which makes our method applicable to a wide class of different wavelet schemes. In order to illustrate the various features of the method, we present some preliminary results for the helium atom.
