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Antoine Marie 2023-05-15 14:41:44 +02:00
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\end{figure*} \end{figure*}
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In this section, the values of the two others SRG-based methods derived in the main manuscript are reported along with their corresponding histogram plot of the errors . In this section, the values obtained with the two alternative SRG-based methods derived in the main manuscript, SRG-$G_0W_0$ and SRG-ev$GW$, are reported along with their corresponding histogram plot of the errors.
For the sake of completeness, the SRG-regularized quasi-particle equation used for the $G_0W_0$ and ev$GW$ calculations is reported here For the sake of completeness, the SRG-regularized quasiparticle equation used for the $G_0W_0$ and ev$GW$ calculations is reported here
\begin{equation} \begin{equation}
\epsilon_p^{\text{HF}} + \sum_{i\nu} \frac{W_{pi}^{\nu} W_{pi}^{\nu} }{\omega - \epsilon_i + \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_i + \Omega_{\nu})^2 s} + \sum_{a\nu} \frac{W_{pa}^{\nu} W_{pa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_a - \Omega_{\nu})^2 s} - \omega = 0, \epsilon_p^{\text{HF}} + \sum_{i\nu} \frac{W_{pi}^{\nu} W_{pi}^{\nu} }{\omega - \epsilon_i + \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_i + \Omega_{\nu})^2 s} + \sum_{a\nu} \frac{W_{pa}^{\nu} W_{pa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_a - \Omega_{\nu})^2 s} - \omega = 0,
\end{equation} \end{equation}
Therefore, the SRG-$G_0W_0$ values correspond to one-shot solutions of these equations (without linearization). Therefore, the SRG-$G_0W_0$ values correspond to one-shot solutions of these equations (without linearization).
While the SRG-ev$GW$ results correspond to iterative solutions of these equations where self-consistency on the $\epsilon_p$ has been reached. While the SRG-ev$GW$ results correspond to iterative solutions of these equations where self-consistency on the $\epsilon_p$ has been reached.
Table \ref{tab:tab1} shows that the SRG-$G_0W_0$ values are the same as the $G_0W_0$ ones for all systems (up to $\num{e-2}$\si{\electronvolt}). One can observe in Tab.~\ref{tab:tab1} that the SRG-$G_0W_0$ values are the same as the $G_0W_0$ ones for all systems (up to $\num{e-2}$\si{\electronvolt}).
Figure \ref{fig:supporting} shows that ev$GW$ offers a slight improvement over $G_0W_0$ while SRG-ev$GW$ performs similarly to its traditional ev$GW$ counterpart. Figure \ref{fig:supporting} shows that ev$GW$ offers a slight improvement over $G_0W_0$ while SRG-ev$GW$ performs similarly to its traditional ev$GW$ counterpart.
One interesting fact to note is that the convergence of SRG-ev$GW$ deteriorates faster than for SRG-qs$GW$. One interesting fact to note is that the convergence of SRG-ev$GW$ deteriorates faster than for SRG-qs$GW$ with respect to $s$.
This is probably due to the absence of the off-diagonal terms. This is probably due to the absence of the off-diagonal terms.
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