Efficient evaluation of exact exchange for periodic systems via concentric atomic density fitting

The evaluation of the exact [Hartree-Fock (HF)] exchange operator is a crucial ingredient for the accurate description of the electronic structure in periodic systems through ab initio and hybrid density functional approaches. An efficient formulation of periodic HF exchange in a linear combination of atomic orbitals representation presented here is based on the concentric atomic density fitting approximation, a domain-free local density fitting approach in which the product of two atomic orbitals is approximated using a linear combination of fitting basis functions centered at the same nuclei as the AOs in that product. A significant reduction in the computational cost of exact exchange is demonstrated relative to the conventional approach due to avoiding the need to evaluate four-center two-electron integrals, with sub-millihartree/atom errors in absolute HF energies and good cancellation of fitting errors in relative energies. The novel aspects of the evaluation of the Coulomb contribution to the Fock operator, such as the use of real two-center multipole expansions and spheropole-compensated unit cell densities, are also described.