  In this paper, we introduce a new scheme for the efficient numerical
treatment of the electronic Schr\"odinger 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.
