PROJECTS

Dimension-adaptive sparse grid product methods 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, 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. Here, the sparse grid method in its adaptive version allows to overcome the exponential complexity exhibited by conventional discretization procedures and delivers a convergent numerical approach with guaranteed convergence rates. However, besides the rate, also the dependence of the complexity constants on the number of electrons plays an important role for a truly practical method. In the first and second period of this priority program, we developed and implemented our sparse grid product methods for the discretization of Schrödinger's equation with different choices of multi-level bases. Here, a Fourier basis, a wavelet basis, and a basis with almost orthonormal multiscale functions using Gaussians were employed. These approaches were then used to compute one-dimensional model systems with several particles and first small three-dimensional systems up to the lithium atom. We learned that the Gaussian multiscale basis leads to the best constants and indeed delivers the convergence rates expected in presence of the electron-electron cusp. For the third application period of this priority program, we will further improve our method, and we will apply it to several small molecules with up to 10 electrons, such as He₂, LiH, CH, Be₂, and H₂O. Here, an improved multilevel eigenvalue solver for large near-degenerate systems, an a priori matrix compression scheme to exploit a new variant of generalized Slater- Condon rules, and a generalized basis including explicit two-particle correlation factors will be of special importance.