Assessing the G₀W₀Γ₀⁽¹⁾ Approach: Beyond G₀W₀ with Hedin's Full Second-Order Self-Energy Contribution

We present and benchmark a self-energy approach for quasiparticle energy calculations that goes beyond Hedin's GW approximation by adding the full second-order self-energy (FSOS-W) contribution. The FSOS-W diagram involves two screened Coulomb interaction (W) lines, and adding the FSOS-W to the GW self-energy can be interpreted as first-order vertex correction to GW (GWΓ⁽¹⁾). Our FSOS-W implementation is based on the resolution-of-identity technique and exhibits better than O(N⁵) scaling with system size for small- to medium-sized molecules. We then present one-shot GWΓ⁽¹⁾ (G₀W₀Γ₀⁽¹⁾) benchmarks for the GW100 test set and a set of 24 acceptor molecules. For semilocal or hybrid density functional theory starting points, G₀W₀Γ₀⁽¹⁾ systematically outperforms G₀W₀ for the first vertical ionization potentials and electron affinities of both test sets. Finally, we demonstrate that a static FSOS-W self-energy significantly underestimates the quasiparticle energies.