modifs in SI

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Pierre-Francois Loos 2023-05-15 15:34:32 +02:00
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%\documentclass[aps,prb,reprint,showkeys,superscriptaddress]{revtex4-1} %\documentclass[aps,prb,reprint,showkeys,superscriptaddress]{revtex4-1}
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{subcaption}
\usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,enumitem} \usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,enumitem}
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -130,8 +129,25 @@
\label{app:appendixA} \label{app:appendixA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 self-energy and quasiparticle equation used for the SRG-$G_0W_0$ and SRG-ev$GW$ calculations are reported below:
\begin{equation}
\epsilon_p^{\text{HF}} + \Sigma^\text{SRG-$GW$}_{pp}(\omega) - \omega = 0,
\end{equation}
with
\begin{equation}
\Sigma^\text{SRG-$GW$}_{pp}(\omega) = \sum_{i\nu} \frac{(W_{pi}^{\nu})^2 }{\omega - \epsilon_i + \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_i + \Omega_{\nu})^2 s} + \sum_{a\nu} \frac{(W_{pa}^{\nu})^2}{\omega - \epsilon_a - \Omega_{\nu}}e^{-2(\epsilon_p - \epsilon_a - \Omega_{\nu})^2 s},
\end{equation}
Therefore, the SRG-$G_0W_0$ values are obtained by solving once these equations (one-shot procedure) without linearization, while the SRG-ev$GW$ results correspond to solutions of these equations where self-consistency on the $\epsilon_p$'s has been reached.
One observe in Table \ref{tab:tab1} that the $G_0W_0$ and SRG-$G_0W_0$ values are the same for all systems (up to $\num{e-2}$\si{\electronvolt}).
Figure \ref{fig:supporting} shows that ev$GW$ provides a slight improvement over $G_0W_0$, while ev$GW$ and SRG-ev$GW$ perform similarly.
One interesting fact is that the convergence of SRG-ev$GW$ deteriorates faster than for SRG-qs$GW$ with respect to $s$.
We suspect that it is due to the absence of the off-diagonal terms.
%%% FIG 1 %%% %%% FIG 1 %%%
\begin{figure*} \begin{figure*}[h]
\includegraphics[width=\linewidth]{supporting} \includegraphics[width=\linewidth]{supporting}
\caption{ \caption{
Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal IP of the $GW$50 test set calculated using $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$. Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal IP of the $GW$50 test set calculated using $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$.
@ -140,7 +156,7 @@
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% FIG 2 %%% %%% FIG 2 %%%
\begin{figure*} \begin{figure*}[h]
\includegraphics[width=\linewidth]{supporting2} \includegraphics[width=\linewidth]{supporting2}
\caption{ \caption{
Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal EA of the $GW$50 test set calculated using $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$. Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal EA of the $GW$50 test set calculated using $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$.
@ -148,20 +164,6 @@
\label{fig:supporting}} \label{fig:supporting}}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%% %%% %%%
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 quasiparticle equation used for the $G_0W_0$ and ev$GW$ calculations is reported here
\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,
\end{equation}
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.
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.
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.
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{Principal IP and EA (in eV) of the $GW$50 test set calculated using $\Delta$CCSD(T) (reference), $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$. \caption{Principal IP and EA (in eV) of the $GW$50 test set calculated using $\Delta$CCSD(T) (reference), $G_0W_0$@HF, SRG-$G_0W_0$@HF, ev$GW$, and SRG-ev$GW$.
@ -173,7 +175,7 @@ This is probably due to the absence of the off-diagonal terms.
& \mc{5}{c}{Principal IP} & \mc{5}{c}{Principal EA} \\ & \mc{5}{c}{Principal IP} & \mc{5}{c}{Principal EA} \\
\cline{2-6} \cline{7-11} \cline{2-6} \cline{7-11}
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{$G_0W_0$@HF} & \mcc{SRG-$G_0W_0$@HF} & \mcc{ev$GW$} & \mcc{SRG-ev$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{$G_0W_0$@HF} & \mcc{SRG-$G_0W_0$@HF} & \mcc{ev$GW$} & \mcc{SRG-ev$GW$} \\ & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{$G_0W_0$@HF} & \mcc{SRG-$G_0W_0$@HF} & \mcc{ev$GW$} & \mcc{SRG-ev$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{$G_0W_0$@HF} & \mcc{SRG-$G_0W_0$@HF} & \mcc{ev$GW$} & \mcc{SRG-ev$GW$} \\
Mol. & \mcc{(Ref.)} & \mcc{($\eta=\num{e-3}$)} & \mcc{($s=\num{e3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{e3}$)} & \mcc{(Ref.)} & & \mcc{($\eta=\num{e-3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{e3}$)} \\ Mol. & \mcc{(Ref.)} & \mcc{($\eta=\num{e-3}$)} & \mcc{($s=\num{e3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{50}$)} & \mcc{(Ref.)} & \mcc{($\eta=\num{e-3}$)} & \mcc{($s=\num{e3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{50}$)} \\
\hline \hline
\ce{He} & 24.54 & 24.59 & 24.59 & 24.58 & 24.57 & -2.66 & -2.66 & -2.66 & -2.66 & -2.66 \\ \ce{He} & 24.54 & 24.59 & 24.59 & 24.58 & 24.57 & -2.66 & -2.66 & -2.66 & -2.66 & -2.66 \\
\ce{Ne} & 21.47 & 21.46 & 21.46 & 21.30 & 21.29 & -5.09 & -5.25 & -5.25 & -5.24 & -5.24\\ \ce{Ne} & 21.47 & 21.46 & 21.46 & 21.30 & 21.29 & -5.09 & -5.25 & -5.25 & -5.24 & -5.24\\