  Pair atomic density fitting (PADF) is a promising strategy to reduce the
scaling with system size of quantum chemical methods for the calculation of the
correlation energy like the direct random phase approximation (RPA) or
second-order M{\o}ller-Plesset perturbation theory (MP2). PADF can however
introduce large errors in correlation energies as the two-electron interaction
energy is not guaranteed to be bounded from below. This issue can be partially
alleviated by using very large fit sets, but this comes at the price of reduced
efficiency and having to deal with near-linear dependencies in the fit set. In
this work, we introduce an alternative methodology to overcome this problem
that preserves the intrinsically favourable scaling of PADF. We first
regularize the Fock matrix by projecting out parts of the basis set which gives
rise to orbital products that are hard to describe by PADF. We then also apply
this projector to the orbital coefficient matrix to improve the precision of
PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new
approach in a numerical atomic orbital framework using Slater Type Orbitals
(STO) and correlation consistent Gaussian type basis sets up to
quintuple-$\zeta$ quality for systems with more than 200 atoms. For the small
and medium systems in the S66 database we show the maximum deviation of
PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA
reference results to be 0.07 and 0.14 kcal/mol respectively. When the new
projector method is used, the errors only slightly increase for large molecules
and also when moderately sized fit sets are used the resulting errors are well
under control. Finally, we demonstrate the computational efficiency of our
algorithm by calculating the interaction energies of non-covalently bound
complexes with more than 1000 atoms and 20000 atomic orbitals at the
RPA@PBE/CC-pVTZ level of theory.
