Static correlation and electron localization in molecular dimers from the self-consistent RPA andGWapproximation

Abstract

We investigate static correlation and delocalization errors in the self-consistent $GW$ and random-phase approximation (RPA) by studying molecular dissociation of the ${\mathrm{H}}_{2}$ and LiH molecules. Although both approximations contain topologically identical diagrams, the nonlocality and frequency dependence of the $GW$ self-energy crucially influence the different energy contributions to the total energy as compared to the use of a static local potential in the RPA. The latter leads to significantly larger correlation energies, which allow for a better description of static correlation at intermediate bond distances. The substantial error found in $GW$ is further analyzed by comparing spin-restricted and spin-unrestricted calculations. At large but finite nuclear separation, their difference gives an estimate of the so-called fractional spin error normally determined only in the dissociation limit. Furthermore, a calculation of the dipole moment of the LiH molecule at dissociation reveals a large delocalization error in $GW$ making the fractional charge error comparable to the RPA. The analyses are supplemented by explicit formulas for the $GW$ Green's function and total energy of a simplified two-level model providing additional insights into the dissociation limit.