19th GAMM-Seminar Leipzig on
High-dimensional problems - Numerical treatment and applications

Max-Planck-Institute for Mathematics in the Sciences
Inselstr. 22-26, D-04103 [O->]Leipzig
Phone: +49.341.9959.752, Fax: +49.341.9959.999

  19th GAMM-Seminar
January, 23th-25th, 2003
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  Abstract Heinz-Jürgen Flad, Sat, 09.00-09.50 Previous Contents Next  
  Various Notions of High-Dimensions in Quantum Chemistry
Heinz-Jürgen Flad (Max Planck Institute for Mathematics in the Sciences, Leipzig)

Solving the many-particle Schrödinger equation for N electrons resembles on a first glance to a problem in tex2html_wrap_inline19. However it is well known in quantum many-particle theory that the full problem can be decomposed into a hierarchy of lower dimensional subproblems. Starting from mean-field methods which correspond to nonlinear PDEs in tex2html_wrap_inline21, successively higher dimensional subspaces of tex2html_wrap_inline19 are considered.

From a physical point of view, these subspaces correspond to various types of electron correlations. Due to the local character of electron correlations it is possible to get data sparse representations of the wavefunction. In quantum chemistry computational approaches are usually based on atomic centered Gaussian-type basis functions. Such kind of basis sets are almost optimal for mean-field solutions, however, they have severe drawbacks for the approximation of correlated wavefunctions. We suggest an alternative approach based on a combination of sparse grids and wavelets, so called hyperbolic wavelets, to electronic structure calculations. Hyperbolic wavelets are especially adapted to higher dimensional problems and can be combined with adaptive nonlocal approximation schemes in regions of low regularity of the wavefunction.

Special attention is paid to the multi-scale character of the problem. Taking a product ansatz for the wavefunction tex2html_wrap_inline25, where tex2html_wrap_inline27 corresponds to a given mean-field solution, we approximate the correlation factor tex2html_wrap_inline29 in terms of hyperbolic wavelets. We are aiming towards a local description of electron correlations using a wavelet basis adapted to the various length and energy scales of the physical processes involved.

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