From 16ec1904703d0eedc50ceb0e6f7911a0918fb98c Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 15 May 2023 15:34:32 +0200 Subject: [PATCH] modifs in SI --- Manuscript/si.tex | 38 ++++++++++++++++++++------------------ 1 file changed, 20 insertions(+), 18 deletions(-) diff --git a/Manuscript/si.tex b/Manuscript/si.tex index 07e5211..3f4afa3 100644 --- a/Manuscript/si.tex +++ b/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\\