Interpolative separable density fitting on adaptive real space grids

We generalize the interpolative separable density fitting (ISDF) method, used to compress the four-index electron repulsion integral (ERI) tensor, to incorporate adaptive real space grids for potentially highly localized single-particle basis functions. The ISDF method produces a decomposition of the ERI tensor by solving the Poisson equation for a collection of auxiliary densities, for which we use a fast adaptive Poisson solver. The adaptive grids are generated using a high-order accurate, black-box procedure that satisfies a user-specified error tolerance. We find that the ISDF compression efficiency for the ERI tensor with highly localized basis sets is comparable to that for smoother basis sets compatible with uniform grids, leading to a highly efficient, fully adaptive ISDF scheme. This work provides a path towards performing cubic-scaling electronic structure calculations beyond density functional theory (e.g., using the GW method) with arbitrary, possibly highly localized (e.g., all-electron and/or numerically-constructed) basis sets, and systematically controllable accuracy.