The natural occupation number associated with each PNO is used to build pair-specific domains [ij] PNO . The number of ERIs needed are thereby immensely reduced. In order to provide a compact localization of the virtual space a hierarchical treatment of projected atomic orbitals, orbital-specific virtuals and pair-natural orbitals, as introduced by the group of Werner, is employed. Moreover, multipole approximations for distant pairs can be used to further reduce the scaling. There are several reasons why such a local scheme (albeit possible) is not used for the terms involving nuclei. First of all, the calculation of the nuclear-nuclear and electronicnuclear correlation is, for all the examples provided in this manuscript, a minute fraction of the computational cost as we demonstrate later on. In all cases, the computation of the electronic correlation overwhelmingly dominates the timings. Nonetheless, we significantly decreased the computational demand of the electronic-nuclear 4-center integrals by employing local approximations. The latter quantities are calculated using the nuclear fitting basis, whereby only contributions from the electrons in close proximity of the nuclei are accounted for. Although it might seem like a harsh approximation, given the local nature of the electron-proton interaction, this turns out to be a very robust approach. This point was already mentioned by Hammes-Schi↵er and coworkers.