Sparse Grids for the Electronic Schrödinger Equation: Construction and Application of Sparse Tensor Product Multiscale Many-particle Spaces with Finite-order Weights for Schrödinger's Equation

The successful numerical treatment of the electronic Schrödinger equation provides an effective predictive tool for new insights in several research areas like chemistry, biochemistry, molecular physics, material science and nanotechnology. However, any numerical solution of the electronic Schrödinger equation using conventional linear discretization schemes is not feasible due to its high dimensionality. Therefore, typically nonlinear model approximations like Hartree-Fock, coupled cluster or density functional theory are used. In this work we construct, study and apply novel sparse tensor product multiscale many-particle spaces with finite-order weights for the electronic Schödinger equation. This new variant of sparse grids combines the favorable properties of efficient Gaussian type orbitals basis sets and adaptive tensor product multiscale bases, which provide guaranteed convergence rates. In particular, the introduced formulation of the underlying particle-wise subspace splitting includes several decomposition schemes well known in different research fields. With the implementation of our approach, small atoms and molecules can be treated at very high accuracy.