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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\begin{document}
\title{\large A similarity renormalization group approach to Green's function methods}
%\title{Excited states, symmetry breaking, and unphysical solutions in CASSCF theory}
%\title{Characterising state-specific CASSCF theory: Excited states, symmetry breaking, and unphysical solutions}
%\title{Exploring the CASSCF energy landscape: Excited states, symmetry breaking, and unphysical solutions}
\author{Antoine \surname{Marie} \textsuperscript{*} }
\email{amarie@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract}
\end{abstract}
\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{SRG-$G_0W_0$, ev$GW$ and SRG-ev$GW$ statistics}
\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*}[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$GW$.
All calculations are performed with the aug-cc-pVTZ basis.
\label{fig:supporting}}
\end{figure*}
%%% %%% %%% %%%
%%% FIG 2 %%%
\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$GW$.
All calculations are performed with the aug-cc-pVTZ basis.
\label{fig:supporting}}
\end{figure*}
%%% %%% %%% %%%
%%% 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$GW$.
The statistical descriptors associated with the errors with respect to the reference values are also reported.
All calculations are performed with the aug-cc-pVTZ basis.}
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{ldddddddddd}
& \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{($s=\num{50}$)} & \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\\
\ce{H2} & 16.40 & 16.49 & 16.49 & 16.52 & 16.51 & -1.35 & -1.28 & -1.28 & -1.28 & -1.28 \\
\ce{Li2} & 5.25 & 5.38 & 5.38 & 5.44 & 5.42 & 0.34 & 0.17 & 0.17 & 0.16 & 0.17 \\
\ce{LiH} & 8.02 & 8.22 & 8.22 & 8.26 & 8.23 & -0.29 & 0.27 & 0.27 & 0.27 & 0.27 \\
\ce{HF} & 16.15 & 16.25 & 16.25 & 16.10 & 16.09 & -0.66 & -0.71 & -0.71 & -0.71 & -0.71 \\
\ce{Ar} & 15.60 & 15.72 & 15.72 & 15.67 & 15.66 & -2.55 & -2.68 & -2.68 & -2.67 & -2.67 \\
\ce{H2O} & 12.69 & 12.90 & 12.90 & 12.80 & 12.79 & -0.61 & -0.68 & -0.68 & -0.68 & -0.68 \\
\ce{LiF} & 11.47 & 11.40 & 11.40 & 11.20 & 11.18 & 0.35 & 0.33 & 0.33 & 0.33 & 0.33 \\
\ce{HCl} & 12.67 & 12.78 & 12.78 & 12.76 & 12.75 & -0.57 & -0.64 & -0.64 & -0.64 & -0.64 \\
\ce{BeO} & 9.95 & 9.74 & 9.74 & 9.64 & 9.61 & 2.17 & 2.28 & 2.28 & 2.30 & 2.31 \\
\ce{CO} & 13.99 & 14.80 & 14.80 & 14.77 & 14.76 & -1.57 & -1.66 & -1.66 & -1.65 & -1.65 \\
\ce{N2} & 15.54 & 17.10 & 17.10 & 17.10 & 17.09 & -2.37 & -2.10 & -2.10 & -2.10 & -2.10 \\
\ce{CH4} & 14.39 & 14.76 & 14.76 & 14.76 & 14.75 & -0.65 & -0.70 & -0.70 & -0.69 & -0.69 \\
\ce{BH3} & 13.31 & 13.68 & 13.68 & 13.70 & 13.69 & -0.09 & -0.46 & -0.46 & -0.46 & -0.45 \\
\ce{NH3} & 10.91 & 11.22 & 11.22 & 11.19 & 11.17 & -0.61 & -0.68 & -0.68 & -0.68 & -0.68 \\
\ce{BF} & 11.15 & 11.34 & 11.34 & 11.37 & 11.36 & -0.80 & -0.90 & -0.90 & -0.90 & -0.90 \\
\ce{BN} & 12.05 & 11.76 & 11.76 & 11.78 & 11.76 & 3.02 & 3.90 & 3.90 & 3.95 & 3.95 \\
\ce{SH2} & 10.39 & 10.51 & 10.51 & 10.51 & 10.50 & -0.52 & -0.60 & -0.60 & -0.60 & -0.60 \\
\ce{F2} & 15.81 & 16.35 & 16.35 & 16.15 & 16.14 & 0.32 & -0.53 & -0.53 & -0.47 & -0.47 \\
\ce{MgO} & 7.97 & 8.40 & 8.40 & 8.34 & 8.28 & 1.54 & 1.64 & 1.64 & 1.65 & 1.66 \\
\ce{O3} & 12.85 & 13.56 & 13.56 & 13.53 & 13.51 & 1.82 & 2.19 & 2.19 & 2.25 & 2.25 \\
\ce{C2H2} & 11.45 & 11.57 & 11.57 & 11.60 & 11.59 & -0.80 & -0.71 & -0.71 & -0.71 & -0.71 \\
\ce{HCN} & 13.76 & 13.86 & 13.86 & 13.87 & 13.86 & -0.53 & -0.52 & -0.52 & -0.52 & -0.52 \\
\ce{B2H6} & 12.27 & 12.81 & 12.81 & 12.81 & 12.80 & -0.52 & -0.56 & -0.56 & -0.56 & -0.56 \\
\ce{CH2O} & 10.93 & 11.39 & 11.39 & 11.34 & 11.32 & -0.60 & -0.61 & -0.61 & -0.60 & -0.60 \\
\ce{C2H4} & 10.69 & 10.74 & 10.74 & 10.78 & 10.77 & -1.90 & -0.75 & -0.75 & -0.74 & -0.74 \\
\ce{SiH4} & 12.79 & 13.22 & 13.22 & 13.23 & 13.22 & -0.53 & -0.59 & -0.59 & -0.59 & -0.59 \\
\ce{PH3} & 10.60 & 10.79 & 10.79 & 10.82 & 10.81 & -0.51 & -0.58 & -0.58 & -0.58 & -0.58 \\
\ce{CH4O} & 11.09 & 11.55 & 11.55 & 11.48 & 11.47 & -0.59 & -0.64 & -0.64 & -0.64 & -0.64 \\
\ce{H2NNH2} & 9.49 & 9.84 & 9.84 & 9.80 & 9.79 & -0.60 & -0.69 & -0.69 & -0.68 & -0.68 \\
\ce{HOOH} & 11.51 & 11.96 & 11.96 & 11.85 & 11.83 & -0.96 & -0.75 & -0.75 & -0.75 & -0.75 \\
\ce{KH} & 6.32 & 6.44 & 6.44 & 6.48 & 6.42 & 0.30 & 0.28 & 0.28 & 0.28 & 0.28 \\
\ce{Na2} & 4.93 & 4.98 & 4.98 & 5.03 & 5.02 & 0.36 & 0.26 & 0.26 & 0.24 & 0.26 \\
\ce{HN3} & 10.77 & 11.12 & 11.12 & 11.11 & 11.10 & -0.51 & -0.6 & -0.6 & -0.59 & -0.59 \\
\ce{CO2} & 13.80 & 14.24 & 14.24 & 14.16 & 14.15 & -0.88 & -0.98 & -0.98 & -0.97 & -0.97 \\
\ce{PN} & 11.90 & 12.33 & 12.33 & 12.34 & 12.33 & -0.02 & -0.03 & -0.03 & 0.01 & 0.01 \\
\ce{CH2O2} & 11.54 & 12.00 & 12.00 & 11.90 & 11.88 & -0.63 & -0.69 & -0.69 & -0.68 & -0.68 \\
\ce{C4} & 11.43 & 11.77 & 11.77 & 11.77 & 11.76 & 2.38 & 2.24 & 2.24 & 2.34 & 2.35 \\
\ce{C3H6} & 10.83 & 11.20 & 11.20 & 11.20 & 11.19 & -0.94 & -0.75 & -0.75 & -0.75 & -0.75 \\
\ce{C2H3F} & 10.63 & 10.84 & 10.84 & 10.85 & 10.84 & -0.65 & -0.69 & -0.69 & -0.68 & -0.68 \\
\ce{C2H4O} & 10.29 & 10.84 & 10.84 & 10.76 & 10.75 & -0.54 & -0.56 & -0.56 & -0.56 & -0.56 \\
\ce{C2H6O} & 10.82 & 11.37 & 11.37 & 11.31 & 11.30 & -0.58 & -0.65 & -0.65 & -0.64 & -0.64 \\
\ce{C3H8} & 12.13 & 12.61 & 12.61 & 12.60 & 12.59 & -0.63 & -0.70 & -0.70 & -0.70 & -0.70 \\
\ce{NaCl} & 9.10 & 9.20 & 9.20 & 9.16 & 9.13 & 0.67 & 0.64 & 0.64 & 0.64 & 0.64 \\
\ce{P2} & 10.72 & 10.49 & 10.49 & 10.52 & 10.51 & 0.43 & 0.47 & 0.47 & 0.53 & 0.54 \\
\ce{MgF2} & 13.93 & 13.94 & 13.94 & 13.74 & 13.72 & 0.29 & 0.15 & 0.15 & 0.16 & 0.16 \\
\ce{OCS} & 11.23 & 11.52 & 11.52 & 11.50 & 11.49 & -1.43 & -1.03 & -1.03 & -1.02 & -1.01 \\
\ce{SO2} & 10.48 & 11.38 & 11.38 & 11.34 & 11.33 & 2.24 & 2.82 & 2.82 & 2.87 & 2.88 \\
\ce{C2H3Cl} & 10.17 & 10.39 & 10.39 & 10.39 & 10.38 & -0.61 & -0.66 & -0.66 & -0.65 & -0.65 \\
\hline
MSE & & 0.29 & 0.29 & 0.26 & 0.25 & & 0.02 & 0.02 & 0.03 & 0.00 \\
MAE & & 0.33 & 0.33 & 0.32 & 0.31 & & 0.16 & 0.16 & 0.16 & 0.17 \\
RMSE & & 0.43 & 0.43 & 0.41 & 0.40 & & 0.28 & 0.28 & 0.29 & 0.28 \\
SDE & & 0.31 & 0.31 & 0.31 & 0.32 & & 0.29 & 0.29 & 0.29 & 0.29 \\
Min & & -0.29 & -0.29 & -0.31 & -0.34 & & -0.85 & -0.85 & -0.79 & -0.82 \\
Max & & 1.56 & 1.56 & 1.56 & 1.55 & & 1.15 & 1.15 & 1.16 & 1.14 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{document}
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