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We present a new efficient way to perform hybrid density functional theory(DFT) based electronic structure calculation. The new method uses aninterpolative separable density fitting (ISDF) procedure to construct a set ofnumerical auxiliary basis vectors and a compact approximation of the matrixconsisting of products of occupied orbitals represented in a large basis setsuch as the planewave basis. Such an approximation allows us to reduce thenumber of Poisson solves from $\Or(N_{e}^2)$ to $\Or(N_{e})$ when we apply theexchange operator to occupied orbitals in an iterative method for solving theKohn-Sham equations, where $N_{e}$ is the number of electrons in the system tobe studied. We show that the ISDF procedure can be carried out in$\Or(N_{e}^3)$ operations, with a much smaller pre-constant compared to methodsused in existing approaches. When combined with the recently developedadaptively compressed exchange (ACE) operator formalism, which reduces thenumber of times the exchange operator needs to be updated, the resultingACE-ISDF method significantly reduces the computational cost \REV{associatedwith the exchange operator} by nearly two orders of magnitude compared toexisting approaches for a large silicon system with $1000$ atoms. Wedemonstrate that the ACE-ISDF method can produce accurate energies and forcesfor insulating and metallic systems, and that it is possible to obtainconverged hybrid functional calculation results for a 1000-atom bulk siliconwithin 10 minutes on 2000 computational cores. We also show that ACE-ISDF canscale to 8192 computational cores for a 4096-atom bulk silicon system. We usethe ACE-ISDF method to geometrically optimize a 1000-atom silicon system with avacancy defect using the HSE06 functional and computes its electronicstructure.

Interpolative Separable Density Fitting Decomposition for Accelerating  Hybrid Density Functional Calculations With Applications to Defects in  Silicon