PROJECTS

A dimension-adaptive sparse grid method for the Schrödinger equation

PRINCIPAL INVESTIGATOR

PROJECT RESEARCH ASSISTANT

ABSTRACT

Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, different approximations like HF, CI/CC, and DFT are used. However, these approaches more resemble simplified models than discretization procedures. Instead, a special discretization using sparse grids can be aimed at. The dimension-adaptive sparse grid method has the possibility to overcome the exponential complexity exhibited by conventional discretization procedures through the curse of dimension.
In the first application period of this project, we developed sparse grid techniques and product methods for the discretization of Schrödinger's equation with different choices of multi-level bases (real space, Fourier space) for one-dimensional one-particle states. Here we realized new and enhanced implementations of the sparse grid approach. Furthermore, we implemented a related approach based on separable product expansions.
For the second application period of this Schwerpunktprogramm, we will improve our methods to deal with three-dimensional particle spaces and apply them to atoms, molecules and unit cells of crystals with up to 6 electrons. Here, the incorporation of antisymmetry, efficient treatment of the Coulomb interactions, dimension-adaptivity and optimal preconditioning of the eigenvalue solver will be of special importance.