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 . 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 , successively
higher dimensional subspaces of 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 ,
where corresponds to a given mean-field solution,
we approximate the correlation factor 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.