Excitation spectra of aromatic molecules within a real-spaceGW-BSE formalism: Role of self-consistency and vertex corrections

We present first-principles calculations on the vertical ionization potentials (IPs), electron affinities (EAs), and singlet excitation energies on an aromatic-molecule test set (benzene, thiophene, 1,2,5-thiadiazole, naphthalene, benzothiazole, and tetrathiafulvalene) within the $GW$ and Bethe-Salpeter equation (BSE) formalisms. Our computational framework, which employs a real-space basis for ground-state and a transition-space basis for excited-state calculations, is well suited for high-accuracy calculations on molecules, as we show by comparing against ${G}_{0}{W}_{0}$ calculations within a plane-wave-basis formalism. We then generalize our framework to test variants of the $GW$ approximation that include a local density approximation (LDA)--derived vertex function (${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{LDA}}$) and quasiparticle-self-consistent (QS) iterations. We find that ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{LDA}}$ and quasiparticle self-consistency shift IPs and EAs by roughly the same magnitude, but with opposite sign for IPs and the same sign for EAs. ${G}_{0}{W}_{0}$ and $\mathrm{QS}GW{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{LDA}}$ are more accurate for IPs, while ${G}_{0}{W}_{0}{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{LDA}}$ and $\mathrm{QS}GW$ are best for EAs. For optical excitations, we find that perturbative $GW$-BSE underestimates the singlet excitation energy, while self-consistent $GW$-BSE results in good agreement with previous best-estimate values for both valence and Rydberg excitations. Finally, our work suggests that a hybrid approach, in which ${G}_{0}{W}_{0}$ energies are used for occupied orbitals and ${G}_{0}{W}_{0}{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{LDA}}$ for unoccupied orbitals, also yields optical excitation energies in good agreement with experiment but at a smaller computational cost.