Fully self-consistent $GW$ and quasiparticle self-consistent $GW$ for molecules

Two self-consistent schemes involving Hedin's $GW$ approximation are studied for a set of sixteen different atoms and small molecules. We compare results from the fully self-consistent $GW$ approximation (SC$GW$) and the quasiparticle self-consistent $GW$ approximation (QS$GW$) within the same numerical framework. Core and valence electrons are treated on an equal footing in all the steps of the calculation. We use basis sets of localized functions to handle the space dependence of quantities and spectral functions to deal with their frequency dependence. We compare SC$GW$ and QS$GW$ on a qualitative level by comparing the computed densities of states (DOS). To judge their relative merit on a quantitative level, we compare their vertical ionization potentials (IPs) with those obtained from coupled-cluster calculations CCSD(T). Our results are futher compared with “one-shot” $G_0{W}_0$ calculations starting from Hartree-Fock solutions ($G_0{W}_0$-HF). Both self-consistent $GW$ approaches behave quite similarly. Averaging over all the studied molecules, both methods show only a small improvement (somewhat larger for SC$GW$) of the calculated IPs with respect to $G_0{W}_0$-HF results. Interestingly, SC$GW$ and QS$GW$ calculations tend to deviate in opposite directions with respect to CCSD(T) results. SC$GW$ systematically underestimates the IPs, while QS$GW$ tends to overestimate them. $G_0{W}_0$-HF produces results which are surprisingly close to QS$GW$ calculations both for the DOS and for the numerical values of the IPs.

Figure 1

Schematic representation of the cycle in the self-consistent $GW$ (SC$GW$) approach versus exact Hedin's equations. Exact equations involve the vertex function $\Gamma$ for which, unfortunately, there is not an explicit formula available. Instead, the SC$GW$ method approximates $\Gamma$ by its zeroth-order term in an expansion as function of $W$, $\Gamma (\mathit{r},{\mathit{r}}^{\prime },\omega ;{\mathit{r}}^{\prime \prime },{\omega }^{\prime })\approx \delta (\mathit{r}-{\mathit{r}}^{\prime })\delta (\mathit{r}-{\mathit{r}}^{\prime \prime })$, giving rise to Eqs. (6, 7, 8) in the text. These equations, together with Dyson's equation (1) define a self-consistent procedure to compute the interacting Green's function $G$.

Figure 2

Principle of the quasiparticle self-consistent $GW$ approximation (QS$GW$). The calculated self-energy at the $G_0{W}_0$ level in one iteration is used to define a new one-electron effective Hamiltonian. This new ${\widehat{H}}_{\mathrm{eff}}$ provides the starting point for the next $G_0{W}_0$-like iteration. The procedure is repeated until we get a stable ${\widehat{H}}_{\mathrm{eff}}$. The method is based on a heuristic mapping ${\Sigma }_{G_0{W}_0}\left(\omega \right)\to {\widehat{H}}_{\mathrm{eff}}$ as defined in Eq. (12).

Figure 3

Evolution with the iteration number of the ionization potential of methane CH${}_4$ during SC$GW$ and QS$GW$ calculations. A Hartree-Fock calculation and a density functional theory calculation using the local density approximation were used as starting points. The first iteration corresponds to a $G_0{W}_0$ calculation.

Figure 4

Ionization potential of helium (a) and methane (b) as functions of the frequency-grid resolution. The IPs are essentially independent on frequency resolution in the QS$GW$ procedures. The SC$GW$ procedure shows an almost linear dependence of the IP on $\Delta \omega$. The linear extrapolation for SC$GW$ (dotted line) is computed from two IPs calculated using frequency resolutions $0.1$ and $0.05$ eV.

Figure 5

Densities of states of methane CH${}_4$ [(a), (c), and (e)] and nitrogen N${}_2$ [(b), (d), and (f)] molecules calculated using different approximations. The frequency resolution in these calculations is $\Delta \omega =0.1$ eV, and the broadening parameter $\varepsilon =0.2$ eV.

Figure 6

Differences between vertical IPs calculated at the $G_0{W}_0$-HF, SC$GW$, and QS$GW$ levels, and those obtained from coupled-cluster calculations. (a) and (b) show calculations performed using cc-pVDZ and cc-pVTZ basis sets, respectively. The data for the IPs can be found in Table 5.