Localized resolution of identity for efficient Hartree-Fock exchange, based on numeric atom-centered orbitals

Methods based on an exact exchange operator (EX) are increasingly popular, but are still restricted to analytical basis functions (e.\,g. Gaussians) for medium system sizes. We here introduce a localized resolution-of-identity approach for the two-electron Coulomb operator, based on expanding single-particle basis function products separately into auxiliary atom-centered basis sets that are restricted to two centers. Our approach produces accurate results for all-electron EX, can be applied both to analytical and numeric basis functions, requires only ${\mathcal{O}}(N^2)$ intermediate storage and retains a path towards ${\mathcal{O}}(N)$ EX for large systems. We demonstrate a total-energy accuracy of $<1$\,meV/atom for systems including Alanine chains and the S22 benchmark molecule set [1], using the numeric atom-centered orbital based all-electron electronic structure code FHI-aims [2].\\[4pt] [1] P. Jure\v{c}ka \emph{et al.}, Phys. Chem. Chem. Phys.~\textbf{8}, 1985 (2006).\\[0pt] [2] V. Blum \emph{et al.}, Comput. Phys. Comm.~\textbf{180}, 2175 (2009).