Dimension-adaptive sparse grid product methods for the Schrödinger equation

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Any direct numerical solution of the electronic Schrödinger 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 exponential scaling in d to almost linear scaling in d, 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.