modifs in SI

Manuscript/si.tex

@@ -1,6 +1,5 @@
 %\documentclass[aps,prb,reprint,showkeys,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[version=4]{mhchem}
 \usepackage[utf8]{inputenc}
@@ -130,8 +129,25 @@
 \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 %%% 
-\begin{figure*}
+\begin{figure*}[h]
   \includegraphics[width=\linewidth]{supporting}
   \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$GW$.
@@ -140,7 +156,7 @@
 \end{figure*}
 %%% %%% %%% %%%
 %%% FIG 2 %%% 
-\begin{figure*}
+\begin{figure*}[h]
   \includegraphics[width=\linewidth]{supporting2}
   \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$GW$.
@@ -148,20 +164,6 @@
     \label{fig:supporting}}
 \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 %%%
 \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$GW$. 
@@ -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} \\
     \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$GW$} \\
-    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
     \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\\