Two-electron integrations in the quantum theory of atoms in molecules

A method to compute two-electron integrals over arbitrary regions of space is introduced and particularized to the basins appearing in the quantum theory of atoms in molecules. The procedure generalizes the conventional multipolar approach to account for overlapping densities. We show that the approach is always convergent and computationally efficient, scaling as N(4) in the worst, two-center case. Several numerical results supporting our claims are also presented.