Fully self-consistent GW calculations for molecules

We calculate single-particle excitation energies for a series of 34 molecules using fully self-consistent GW, one-shot ${\text{G}}_0{\text{W}}_0$, Hartree-Fock (HF), and hybrid density-functional theory (DFT). All calculations are performed within the projector-augmented wave method using a basis set of Wannier functions augmented by numerical atomic orbitals. The GW self-energy is calculated on the real frequency axis including its full frequency dependence and off-diagonal matrix elements. The mean absolute error of the ionization potential (IP) with respect to experiment is found to be 4.4, 2.6, 0.8, 0.4, and 0.5 eV for DFT-PBE, DFT-PBE0, HF, ${\text{G}}_0{\text{W}}_0\left[\text{HF}\right]$, and self-consistent GW, respectively. This shows that although electronic screening is weak in molecular systems, its inclusion at the GW level reduces the error in the IP by up to 50% relative to unscreened HF. In general GW overscreens the HF energies leading to underestimation of the IPs. The best IPs are obtained from one-shot ${\text{G}}_0{\text{W}}_0$ calculations based on HF since this reduces the overscreening. Finally, we find that the inclusion of core-valence exchange is important and can affect the excitation energies by as much as 1 eV.

Figure 1

(Color online) Calculated negative HOMO energy versus experimental ionization potential. Both PBE and PBE0 systematically understimates the ionization energy due to self-interaction errors while HF overestimates it slightly. The dynamical screening from the GW correlation lowers the HF energies bringing them closer to the experimental values. Numerical values are listed in Table .

Figure 2

The deviation of the calculated HOMO energy from the experimental ionization potential in GW and HF, respectively. The vertical displacement of points from the line $x=y$ gives the difference between the GW and HF energies and represents the effect of screening. Notice that the GW correction is always negative (corresponding to higher HOMO energy) and that it generally overcorrects the HF energies. Also notice that the GW correction is larger for molecules where HF presents the largest overestimation of the ionization potential.

Figure 3

(Color online) Density of states for the ${\text{NH}}_3$ molecule calculated in HF and GW, respectively. Arrows mark the level corresponding to the HOMO in the two calculations. The intersection between the line $y=\epsilon -{\epsilon }_n^{\text{HF}}$ and the real part of $⟨{\psi }_{\text{HOMO}}^0|{\Sigma }_{\text{corr}}\left(\epsilon \right)|{\psi }_{\text{HOMO}}^0⟩$ (green curve) determines the position of the GW level.

Figure 4

(Color online) Convergence of the three highest occupied levels of ${\text{H}}_2\text{O}$ obtained from GW calculations with different sizes of the augmented Wannier-function basis. SZ denotes the Wannier-function basis while, e.g., DZDP denotes the Wannier basis augmented by one extra radial function per valence state and two sets of polarization functions.

Figure 5

(Color online) Same as Fig. 4 but for CO.