  Electron correlation in finite and extended systems is often described in an
effective single-particle framework within the $GW$ approximation. Here, we use
the statically screened second-order exchange contribution to the self-energy
($G3W2$) to calculate a perturbative correction to the $GW$ self-energy. We use
this correction to calculate total correlation energies of atoms, relative
energies, as well as charged excitations of a wide range of molecular systems.
We show that the second-order correction improves correlation energies with
respect to the RPA and also improves relative energies for many, but not all
considered systems. While the full $G3W2$ contribution does not give consistent
improvements over $GW$, taking the average of $GW$ and $GW + G3W2$ generally
gives excellent results. Improvements over quasiparticle self-consistent $GW$,
which we show to give very accurate charged excitations in small and medium
molecules by itself, are only minor. $G_0W_0$ quasiparticle energies evaluated
with eigenvalue and orbitals from range-separated hybrids, however, are
tremendously improved upon: The second-order corrected $G_0W_0$ outperforms all
existing $GW$ methods for the systems considered herein and also does not come
with substantially increased computational cost compared to $G_0W_0$ for
systems with up to 100 atoms.
