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Electron correlation in finite and extended systems is often described in aneffective single-particle framework within the $GW$ approximation. Here, we usethe statically screened second-order exchange contribution to the self-energy($G3W2$) to calculate a perturbative correction to the $GW$ self-energy. We usethis correction to calculate total correlation energies of atoms, relativeenergies, as well as charged excitations of a wide range of molecular systems.We show that the second-order correction improves correlation energies withrespect to the RPA and also improves relative energies for many, but not allconsidered systems. While the full $G3W2$ contribution does not give consistentimprovements over $GW$, taking the average of $GW$ and $GW + G3W2$ generallygives excellent results. Improvements over quasiparticle self-consistent $GW$,which we show to give very accurate charged excitations in small and mediummolecules by itself, are only minor. $G_0W_0$ quasiparticle energies evaluatedwith eigenvalue and orbitals from range-separated hybrids, however, aretremendously improved upon: The second-order corrected $G_0W_0$ outperforms allexisting $GW$ methods for the systems considered herein and also does not comewith substantially increased computational cost compared to $G_0W_0$ forsystems with up to 100 atoms.

Exploring the Statically Screened $G3W2$ Correction to the $GW$ Self-Energy: Charged Excitations and Total Energies of Finite Systems