## Seminar in Numerical Analysis: Jan Hamaekers (Fraunhofer SCAI)

**An Adaptive Multiscale Approach for Electronic Structure Methods**

In this talk, we introduce a new scheme for the efficient numerical treatment of the electronic SchrÃ¶dinger equation for molecules. It is based on the combination of a many-body expansion, which corresponds to the so-called bond order dissection Anova approach, 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. Finally, we discuss to apply such a multi-level many-body decomposition in the context of machine learning for many-body systems.

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