Efficient hybrid density functional calculations for solids with a local basis set

The applicability of hybrid density functional approximations (DFAs) in the solid state is still largely hindered by a high computational cost in the evaluation of the exact exchange series. We review formal and computational aspects of a direct space approach to the evaluation of exact exchange in periodic systems with local, atom-centered, Gaussian-type basis functions. We discuss an efficient prescreening strategy in terms of the localized basis to either truncate the series via overlap-based criteria or partition it into distinct regions where the integrals can be computed exactly or approximately via a bipolar expansion of the Coulomb operator. The inclusion of exact exchange in hybrid DFAs largely corrects for the self-interaction error, ensures a better description of electron localization, and thus proves crucial to the characterization of defects, strongly correlated materials, band gaps in semiconductors, as well as to an effective treatment of spin-orbit and electron-phonon couplings. Three test systems are considered, which have been selected as representative of different electronic regimes: (i) bulk hematite, α-Fe₂O₃, in its insulating, antiferromagnetic configuration; (ii) semiconducting single-layer WSe₂ (with inclusion of spin-orbit coupling); and (iii) bulk EuO in its metallic ferromagnetic phase at a pressure of 20 GPa. The strategy is assessed on various electronic properties (energy, band gap, spin magnetic moment, Rashba splitting) and is documented to yield calculations with hybrid DFAs with a relative cost of just 3--6 times that of standard (semi)local DFAs.