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We present an efficient, linear-scaling implementation for building the(screened) Hartree-Fock exchange (HFX) matrix for periodic systems within theframework of numerical atomic orbital (NAO) basis functions. Our implementationis based on the localized resolution of the identity approximation by whichtwo-electron Coulomb repulsion integrals can be obtained by only computingtwo-center quantities -- a feature that is highly beneficial to NAOs. Byexploiting the locality of basis functions and efficient prescreening of theintermediate three- and two-index tensors, one can achieve a linear scaling ofthe computational cost for building the HFX matrix with respect to the systemsize. Our implementation is massively parallel, thanks to a MPI/OpenMP hybridparallelization strategy for distributing the computational load and memorystorage. All these factors add together to enable highly efficient hybridfunctional calculations for large-scale periodic systems. In this work wedescribe the key algorithms and implementation details for the HFX build asimplemented in the ABACUS code package. The performance and scalability of ourimplementation with respect to the system size and the number of CPU cores aredemonstrated for selected benchmark systems up to 4096 atoms.

Efficient Hybrid Density Functional Calculations for Large Periodic Systems Using Numerical Atomic Orbitals