PROJECTS

A dimension-adaptive sparse grid method for the Schrödinger equation

PRINCIPAL INVESTIGATOR

PROJECT RESEARCH ASSISTANT

ABSTRACT

Any direct numerical solution of the electronic Schr\'f6dinger \par equation 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.In this project, we propose to use a sparse grid method for the direct discretization of Schrödinger's equation. the conventional sparse grid technique allows to reduce the complexity of a d-dimensional problem from O(Nd) to O(N(logN)d-1), provided that certain smoothness assumptions are fulfilled. It uses a multi-level basis to represent one-particle states and employs a certain determinant-product approach to represent many-particle states, which takes anti-symmetry (Pauli priciple) into account. A certain truncation of the corresponding multi-level series expansion directly results in a cost-optimal discretization of the total electronic space. Here, a dimension-adaptive procedure allows to detect correlations between one-particle states. This new approach gives the perspective to reduce the computational complexity of a N-electron problem to that of a one-electron problem. For different choices of multi-level bases (real space, Fourier space) for the one-particle state, we will implement the resulting dimension-adaptive sparse grid approaches and compare their properties for Schrödinger's equation. Furthermore, this code will later be parallelized and implemented on distributed memory processors.