A07 – A new sparse grid method for the electronic Schrödinger equation
Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, several model approximations are successfully used in computational chemistry, which, however, more resemble simplified models than discretization procedures and lack a sound theory on their convergence rates. The principal aim of this project is the development of efficient numerical methods with guaranteed convergence rates. Here, besides the rates, also the dependence of the complexity constants on the number of electrons and the size-consistency plays an important role for a truly practical method.
In this project, on the one hand, we aim at a highly accurate sparse tensor product based method for small systems. Here, we will study the ANOVA smoothing effects in configuration interaction (CI) methods and develop new appropriate error estimators, which allow for dimension-adaptivity and space-adaptivity. On the other hand, we aim at the generalization of the dimension-adaptive sparse grid approach by means of Fulde’s cumulant expansion. This allows us to obtain new, size-consistent methods based on ANOVA-like decompositions which opens the possibility to also efficiently treat larger systems. Thus, the electronic Schrödinger equation may become tractable for a wide class of molecular systems with locality properties of the electrons.