Excitation spectra of aromatic molecules within a real-space $GW$-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 (${\mathrm{\Gamma }}_{\mathrm{LDA}}$) and quasiparticle-self-consistent (QS) iterations. We find that ${\mathrm{\Gamma }}_{\mathrm{LDA}}$ 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 $\mathrm{QS}GW{\mathrm{\Gamma }}_{\mathrm{LDA}}$ are more accurate for IPs, while $G_0{W}_0{\mathrm{\Gamma }}_{\mathrm{LDA}}$ 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 $G_0{W}_0{\mathrm{\Gamma }}_{\mathrm{LDA}}$ for unoccupied orbitals, also yields optical excitation energies in good agreement with experiment but at a smaller computational cost.

Figure 1

Error of the IPs of all molecules predicted via $G_0{W}_0$, relative to experiment [72, 73, 74, 75, 76, 77]. Black circles are computed using RGWBS (real-space framework) and orange squares are computed using BerkeleyGW (plane-wave framework).

Figure 2

$GW$ self-energy corrections to Kohn-Sham DFT eigenvalues for benzene (BNZ), naphthalene (NPH), thiophene (THP), 1,2,5-thiadiazole (THD), benzothiazole (BZT), and TTF. The shifts for orbitals with $\pi$ character are in black, and shifts for all other orbitals ($\sigma$ or lone pair character) are shown in cyan. The “boat” form of TTF is not aromatic and its self-energy corrections (dark blue) cannot be partitioned into two groups as in the other molecules.

Figure 3

Shift of quasiparticle energies from predictions at $G_0{W}_0$, for $GW$ variants including self-consistency and vertex corrections.

Figure 4

Error for the first IP (top) and the mean absolute error of orbitals with IPs up to 15 eV (bottom), relative to experiment for each molecule [72, 73, 74, 75, 76, 77].

Figure 5

The absorption cross section for each of the molecules convoluted with a Gaussian broadening of 0.1 eV, as predicted by various levels of theory. The ${\text{TDLDA}}_{\mathrm{big}}$ calculations are performed in simulation cells with radii of 20 bohrs, while ${\text{TDLDA}}_{\mathrm{small}}$ simulation cells have radii that are either 12 bohrs (for benzene, thiophene, and 1,2,5-thiadiazole) or 14 bohrs (for naphthalene, benzothiazole, and TTF). $GW$-BSE calculations are performed in the smaller simulation cells.

Figure 6

The top panel compares $GW$-BSE calculations for valence (localized) excitations, taking the mixed $GW$-BSE predictions as reference. The bottom panel again uses the mixed $GW$-BSE results as reference and illustrates the deviations from the best available theoretical values in the literature [80, 81, 82, 83].

Figure 7

The deviation of various first-principles methods relative to previous best available theoretical values (labeled as reference) for the vertical valence (left) and Rydberg (right) excitation energies [80, 81, 82, 83].