Title: Use of multigrids to reduce the cost of performing interpolative separable density fitting

Abstract: In this article, we present an interpolative separable density fitting (ISDF) based algorithm to calculate exact exchange in periodic mean field calculations. In the past, decomposing the two-electron integrals into tensor hypercontraction (THC) form using ISDF was the most expensive step of the entire mean field calculation. Here we show that by using a multigrid-ISDF algorithm both the memory and the CPU cost of this step can be reduced. The CPU cost is brought down from cubic scaling to quadratic scaling with a low computational prefactor which reduces the cost by almost two orders of magnitude. Thus, in the new algorithm, the cost of performing ISDF is largely negligible compared to other steps. Along with the CPU cost, the memory cost of storing the factorized two-electron integrals is also reduced by a factor of up to 35. With the current algorithm, we can perform Hartree-Fock calculations on a Diamond supercell containing more than 17,000 basis functions and more than 1,500 electrons on a single node with no disk usage. For this calculation, the cost of constructing the exchange matrix is only a factor of four slower than the cost of diagonalizing the Fock matrix. Augmenting our approach with linear scaling algorithms can further speed up the calculations.