Resolution-of-identity approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and<i>GW</i>with numeric atom-centered orbital basis functions

Abstract

Efficient implementations of electronic structure methods are essential for\nfirst-principles modeling of molecules and solids. We here present a\nparticularly efficient common framework for methods beyond semilocal\ndensity-functional theory, including Hartree-Fock (HF), hybrid density\nfunctionals, random-phase approximation (RPA), second-order M{\\o}ller-Plesset\nperturbation theory (MP2), and the $GW$ method. This computational framework\nallows us to use compact and accurate numeric atom-centered orbitals (popular\nin many implementations of semilocal density-functional theory) as basis\nfunctions. The essence of our framework is to employ the "resolution of\nidentity (RI)" technique to facilitate the treatment of both the two-electron\nCoulomb repulsion integrals (required in all these approaches) as well as the\nlinear density-response function (required for RPA and $GW$). This is possible\nbecause these quantities can be expressed in terms of products of\nsingle-particle basis functions, which can in turn be expanded in a set of\nauxiliary basis functions (ABFs). The construction of ABFs lies at the heart of\nthe RI technique, and here we propose a simple prescription for constructing\nthe ABFs which can be applied regardless of whether the underlying radial\nfunctions have a specific analytical shape (e.g., Gaussian) or are numerically\ntabulated. We demonstrate the accuracy of our RI implementation for Gaussian\nand NAO basis functions, as well as the convergence behavior of our NAO basis\nsets for the above-mentioned methods. Benchmark results are presented for the\nionization energies of 50 selected atoms and molecules from the G2 ion test set\nas obtained with $GW$ and MP2 self-energy methods, and the G2-I atomization\nenergies as well as the S22 molecular interaction energies as obtained with the\nRPA method.\n