An Adaptive Multiscale Approach for Electronic Structure Methods

In this paper, we introduce a new scheme for the efficient numerical treatment of the electronic Schrödinger equation for molecules. It is based on the combination of a many-body expansion, which corresponds to the bond order dissection ANOVA approach introduced in [M. Griebel, J. Hamaekers, and F. Heber, Extraction of Quantifiable Information from Complex Systems, Springer, New York, pp. 211--235; F. Heber, Ph.D. thesis, Intitut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014], with a hierarchy of basis sets of increasing order. Here, the energy is represented as a finite sum of contributions associated to subsets of nuclei and basis sets in a telescoping sum like fashion. Under the assumption of data locality of the electronic density (nearsightedness of electronic matter), the terms of this expansion decay rapidly and higher terms may be neglected. We further extend the approach in a dimension-adaptive fashion to generate quasi-optimal approximations, i.e., a specific truncation of the hierarchical series such that the total benefit is maximized for a fixed amount of costs. This way, we are able to achieve substantial speed up factors compared to conventional first principles methods depending on the molecular system under consideration. In particular, the method can deal efficiently with molecular systems which include only a small active part that needs to be described by accurate but expensive models.