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large changes, every data for GW50

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Manuscript/SRGGW.bib
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@ -17104,6 +17104,17 @@
volume = {73},
year = {2006}}
@misc{Coveney_2023,
title = {A Regularized Second-Order Correlation Method from {{Green}}'s Function Theory},
author = {Coveney, Christopher J. N. and Tew, David P.},
year = {2023},
number = {arXiv:2302.13296},
eprint = {2302.13296},
eprinttype = {arxiv},
doi = {10.48550/arXiv.2302.13296},
archiveprefix = {arXiv}
}
@article{Ou_2016,
author = {Ou, Qi and Subotnik, Joseph E.},
doi = {10.1021/acs.jpca.6b03294},

349
Manuscript/SRGGW.tex
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@ -56,7 +56,7 @@
\newcommand{\ant}[1]{\textcolor{teal}{#1}}
\newcommand{\trashant}[1]{\textcolor{teal}{\sout{#1}}}
\newcommand{\ANT}[1]{\ant{(\underline{\bf ANT}: #1)}}
\DeclareMathOperator{\sgn}{sgn}
% addresses
@ -130,7 +130,7 @@ The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hyberts
These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism in $GW$-based methods.
Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022,Coveney_2023} the present work investigates the application of the SRG formalism in $GW$-based methods.
In particular, we focus here on the possibility of curing the qs$GW$ convergence issues using the SRG.
The SRG formalism has been developed independently by Wegner \cite{Wegner_1994} in the context of condensed matter systems and Glazek \& Wilson \cite{Glazek_1993,Glazek_1994} in light-front quantum field theory.
@ -268,7 +268,7 @@ The satellites causing convergence problems are the above-mentioned intruder sta
One can deal with them by introducing \textit{ad hoc} regularizers.
For example, the $\ii\eta$ term in the denominators of Eq.~\eqref{eq:GW_selfenergy}, sometimes referred to as a broadening parameter linked to the width of the quasiparticle peak, is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.
However, this $\eta$ parameter stems from a regularization of the convolution that yields the self-energy and should theoretically be set to zero. \cite{Martin_2016}
Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and, in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021,Coveney_2023} and, in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
This is one of the aims of the present work.
@ -302,7 +302,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
\end{equation}
\titou{To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.}
\ant{The flow equation can be approximately solved by introduction of an approximate form of $\boldsymbol{\eta}(s)$.}
In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
\begin{equation}
\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
@ -337,16 +337,16 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
\label{sec:srggw}
%%%%%%%%%%%%%%%%%%%%%%
\titou{By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.}
\ant{Finally, the two previous subsections will be combined by applying the SRG method to the $GW$ formalism.}
However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}
\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
(\bW^{\text{2h1p}})^\dag & \bC^{\text{2h1p}} & \bO \\
(\bW^{\text{2p1h}})^\dag & \bO & \bC^{\text{2p1h}} \\
\bF &\bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
(\bW^{\text{2h1p}})^\dag &\bC^{\text{2h1p}} & \bO \\
(\bW^{\text{2p1h}})^\dag &\bO & \bC^{\text{2p1h}} \\
\end{pmatrix},
% \begin{pmatrix}
% \bZ^{\text{1h/1p}} \\
@ -581,7 +581,7 @@ For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \
%%% FIG 2 %%%
\begin{figure*}
\includegraphics[width=0.8\linewidth]{fig1.pdf}
\includegraphics[width=0.8\linewidth]{fig2.pdf}
\caption{
Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.}
\label{fig:plot}
@ -626,10 +626,24 @@ Indeed, it has been previously mentioned that intruder states are responsible fo
Is it then possible to rely on the SRG machinery to remove discontinuities?
Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite value of $s$ is suitable to avoid intruder states in its static part.
However, performing the following bijective transformation
\titou{\begin{align}
\ant{\begin{align}
e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},
% s = t/2 - \ln 2 - \ln[\sinh(t/2)]
\end{align}}
\end{align}
on the renormalized quasiparticle equation,
\begin{multline}
F_{pq}^{(2)}(t)
= \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}
\\
\times e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] t},
\end{multline}
\begin{equation}
\begin{split}
\widetilde{\bSig}_{pq}(\omega; t)
&= \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu}}\qty[1- e^{-\qty[(\Delta_{pi}^{\nu})^2 + (\Delta_{qi}^{\nu})^2 ] t}] \\
&+ \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}\qty[1- e^{-\qty[(\Delta_{pa}^{\nu})^2 + (\Delta_{qa}^{\nu})^2 ] t}],
\end{split}
\end{equation}}
reverses the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
@ -639,8 +653,8 @@ Note that, after this transformation, the form of the regularizer is actually cl
%=================================================================%
% Reference comp det
Our set of molecules is composed by closed-shell organic compounds that correspond to the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
Our set of molecules is composed by closed-shell compounds that correspond to the 50 smallest (wrt the number of electrons) atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level (without frozen core) \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
The reference CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively.
% GW comp det
@ -651,7 +665,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$
The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively.
In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
However, in order to perform a black-box comparison, these parameters have been fixed to these default values.
\titou{The $\eta$ value used in the convetional $G_0W_0$ and qs$GW$ calculations corresponds to the largest value where one successfully converges all systems.}
\ant{The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculation while for (SRG-)qs$GW$ calculations the $\eta$ value has been chosen as the largest value where one successfully converges the 50 systems of the test set.}
%=================================================================%
\section{Results}
@ -662,25 +676,21 @@ The results section is divided into two parts.
The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases.
Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-qs$GW$ schemes are statistically gauged over the test set of molecules described in Sec.~\ref{sec:comp_det}.
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
\label{sec:flow_param_dep}
%%%%%%%%%%%%%%%%%%%%%%
%%% FIG 2 %%%
%%% FIG 3 %%%
\begin{figure}
\includegraphics[width=\linewidth]{fig2.pdf}
\includegraphics[width=\linewidth]{fig3.pdf}
\caption{
Principal IP of the water molecule in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
\PFL{Should we have a similar figure for EAs? (maybe not water though)}
\ANT{I did the plot, let's discuss it at the next meeting}
\label{fig:fig2}}
\end{figure}
%%% %%% %%% %%%
%%% FIG 3 %%%
%%% FIG 4 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig3.pdf}
\includegraphics[width=\linewidth]{fig4.pdf}
\caption{
Principal IP of the \ce{Li2}, \ce{LiH} and \ce{BeO} in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
@ -688,6 +698,11 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
\end{figure*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
\label{sec:flow_param_dep}
%%%%%%%%%%%%%%%%%%%%%%
This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set.
Figure \ref{fig:fig2} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
The IP at the HF level (dashed black line) is too large; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019}
@ -714,14 +729,15 @@ As $s$ increases, the first states that decouple from the HOMO are the 2p1h conf
Therefore, for small $s$, only the last term of Eq.~\eqref{eq:2nd_order_IP} is partially included, resulting in a positive correction to the IP.
As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreasing and eventually goes below the $s=0$ initial value as observed in Fig.~\ref{fig:fig2}.
\ANT{I don't know if we should remove this paragraph and the TDA curves in Fig 3 and 4 or not...}
In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening (dubbed qs$GW^\TDA$ and SRG-qs$GW^\TDA$) are also considered in Fig.~\ref{fig:fig2}.
The TDA values are now underestimated the IP, unlike their RPA counterparts.
The TDA values are now underestimating the IP, unlike their RPA counterparts.
For both static self-energies, the TDA leads to a slight increase in the absolute error.
This trend is investigated in more detail in the next subsection.
\trashant{This trend is investigated in more detail in the next subsection.}
Next, we investigate the flow parameter dependence of SRG-qs$GW$ for three more challenging molecular systems.
The left panel of Fig.~\ref{fig:fig3} shows the results for the lithium dimer, \ce{Li2}, which is an interesting case because, unlike in water, the HF approximation underestimates the reference IP.
On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are too large.
On the other hand, the qs$GW$ and SRG-qs$GW$ IPs are too large.
Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig2}.
Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
@ -730,7 +746,7 @@ We now turn to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2
In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value.
However, SRG-qs$GW$ improves slightly the accuracy as compared to HF.
Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations due to the presence of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values.
Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
@ -746,216 +762,139 @@ Therefore, it seems that the effect of the TDA cannot be systematically predicte
\label{sec:SRG_vs_Sym}
%%%%%%%%%%%%%%%%%%%%%%
%%% FIG 4 %%%
%%% FIG 5 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig4.pdf}
\includegraphics[width=\linewidth]{fig5.pdf}
\caption{
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
\label{fig:fig4}}
\end{figure*}
%%% %%% %%% %%%
\begin{table*}
\caption{First ionization potential (left) and first electron attachment (right) in eV calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. The statistical descriptors are computed for the errors with respect to the reference. \ANT{Maybe change the values of SRG with the one for s=1000}}
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{l|ddddd|ddddd}
Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
\hline
\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.54 & 2.66 & 2.70 & 2.66 & 2.66 & 2.66 \\
\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 & 5.09 & 5.47 & 5.25 & 5.19 & 5.19 \\
\ce{H2} & 16.40 & 16.16 & 16.49 & 16.45 & 16.45 & 1.35 & 1.33 & 1.28 & 1.28 & 1.28 \\
\ce{Li2} & 5.25 & 4.96 & 5.38 & 5.40 & 5.37 & -0.34 & 0.08 & -0.17 & -0.18 & -0.21 \\
\ce{LiH} & 8.02 & 8.21 & 8.22 & 8.25 & 8.15 & 0.29 & -0.20 & -0.27 & -0.27 & -0.27 \\
\ce{HF} & 16.15 & 17.69 & 16.25 & 16.45 & 16.34 & 0.66 & 0.81 & 0.71 & 0.70 & 0.70 \\
\ce{Ar} & 15.60 & 16.08 & 15.71 & 15.61 & 15.63 & 2.55 & 2.97 & 2.68 & 2.64 & 2.65 \\
\ce{H2O} & 12.69 & 13.88 & 12.90 & 12.98 & 12.88 & 0.61 & 0.80 & 0.68 & 0.65 & 0.66 \\
\ce{LiF} & 11.47 & 12.91 & 11.40 & 11.75 & 11.58 & -0.35 & -0.29 & -0.33 & -0.32 & -0.33 \\
\ce{HCl} & 12.67 & 12.98 & 12.78 & 12.77 & 12.72 & 0.57 & 0.79 & 0.64 & 0.63 & 0.63 \\
\ce{BeO} & 9.95 & 10.45 & 9.74 & 10.32 & 10.18 & -2.17 & -1.80 & -2.28 & -2.10 & -2.13 \\
\ce{CO} & 13.99 & 15.11 & 14.80 & 14.34 & 14.33 & 1.57 & 1.80 & 1.66 & 1.61 & 1.62 \\
\ce{N2} & 15.54 & 16.68 & 17.10 & 15.93 & 15.91 & 2.37 & 2.20 & 2.10 & 2.10 & 2.10 \\
\ce{CH4} & 14.39 & 14.83 & 14.76 & 14.67 & 14.63 & 0.65 & 0.79 & 0.70 & 0.68 & 0.68 \\
\ce{BH3} & 13.31 & 13.59 & 13.68 & 13.62 & 13.59 & 0.09 & 0.81 & 0.46 & 0.29 & 0.30 \\
\ce{NH3} & 10.91 & 11.69 & 11.21 & 11.18 & 11.10 & 0.61 & 0.80 & 0.68 & 0.66 & 0.66 \\
\ce{BF} & 11.14 & 11.04 & 11.34 & 11.19 & 11.17 & 0.80 & 1.06 & 0.90 & 0.87 & 0.87 \\
\ce{BN} & 12.05 & 11.55 & 11.76 & 11.89 & 11.90 & -3.02 & -2.97 & -3.90 & -3.41 & -3.44 \\
\ce{SH2} & 10.39 & 10.49 & 10.51 & 10.50 & 10.45 & 0.52 & 0.76 & 0.60 & 0.58 & 0.59 \\
\ce{F2} & 15.81 & 18.15 & 16.35 & 16.27 & 16.22 & -0.32 & 1.71 & 0.53 & -0.10 & -0.07 \\
\ce{MgO} & 7.97 & 8.75 & 8.40 & 8.54 & 8.36 & -1.54 & -1.40 & -1.64 & -1.72 & -1.71 \\
\ce{O3} & 12.85 & 13.29 & 13.56 & 13.34 & 13.27 & -1.82 & -1.32 & -2.19 & -2.22 & -2.17 \\
\ce{C2H2} & & & & & & & & & & \\
\ce{NCH} & & & & & & & & & & \\
\ce{B2H6} & & & & & & & & & & \\
\ce{H2CO} & & & & & & & & & & \\
\ce{C2H4} & & & & & & & & & & \\
\ce{SiH4} & & & & & & & & & & \\
\ce{PH3} & & & & & & & & & & \\
\ce{CH4O} & & & & & & & & & & \\
\ce{H2NNH2} & & & & & & & & & & \\
\ce{HOOH} & & & & & & & & & & \\
\ce{KH} & & & & & & & & & & \\
\ce{Na2} & & & & & & & & & & \\
\ce{HN3} & & & & & & & & & & \\
\ce{CO2} & & & & & & & & & & \\
\ce{PN} & & & & & & & & & & \\
\ce{CH2O2} & & & & & & & & & & \\
\ce{C4} & & & & & & & & & & \\
\ce{C3H6} & & & & & & & & & & \\
\ce{C2H3F} & & & & & & & & & & \\
\ce{C2H4O} & & & & & & & & & & \\
\ce{C2H6O} & & & & & & & & & & \\
\ce{C3H8} & & & & & & & & & & \\
\ce{NaCl} & & & & & & & & & & \\
\ce{P2} & & & & & & & & & & \\
\ce{F2Mg} & & & & & & & & & & \\
\ce{OCS} & & & & & & & & & & \\
\ce{SO2} & & & & & & & & & & \\
\ce{C2H3Cl} & & & & & & & & & & \\
\hline
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
\hline
MSE & & 0.64 & 0.26 & 0.24 & 0.17 & & -0.30 & -0.02 & 0.00 & 0.00 \\
MAE & & 0.74 & 0.32 & 0.25 & 0.19 & & 0.32 & 0.19 & 0.11 & 0.12 \\
SDE & & 0.71 & 0.39 & 0.18 & 0.15 & & 0.43 & 0.31 & 0.17 & 0.17 \\
Min & & -0.50 & -0.29 & -0.16 & -0.15 & & -2.03 & -0.85 & -0.22 & -0.25 \\
Max & & 2.35 & 1.56 & 0.56 & 0.42 & & 0.17 & 0.88 & 0.41 & 0.42 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in this work computed at various levels of theory.
As mentioned previously the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.64}{\eV} and a mean absolute error (MAE) of \SI{0.74}{\eV}.
Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.26}{\eV} and \SI{0.32}{\eV}, respectively.
As mentioned previously the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.56}{\eV} and a mean absolute error (MAE) of \SI{0.69}{\eV}.
Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.29}{\eV} and \SI{0.33}{\eV}, respectively.
However, there are still outliers with large errors.
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}.
Self-consistency can mitigate the error of the outliers as the maximum error at the qs$GW$ level is now \SI{0.56}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.39}{\eV} for $G_0W_0$ to \SI{0.18}{\eV} for qs$GW$.
In addition, the MSE and MAE (\SI{0.24}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
Self-consistency can mitigate the error of the outliers as the maximum error at the qs$GW$ level is now \SI{0.57}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.31}{\eV} for $G_0W_0$ to \SI{0.18}{\eV} for qs$GW$.
In addition, the MSE and MAE (\SI{0.23}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
Now turning to the new results of this manuscript, \ie the alternative self-consistent scheme SRG-qs$GW$.
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=100$.
Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative self-consistent scheme.
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=\num{e3}$.
The statistical descriptors corresponding to this alternative static self-energy are all improved with respect to qs$GW$.
In particular, the MSE and MAE are decreased by \SI{0.06}{\eV}
Of course, these are slight improvements but this is done with no additional computational cost and can be implemented really easily by changing the form of the static approximation.
The evolution of the statistical descriptors with respect to the various methods considered in Table~\ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
The decrease of the MSE and SDE correspond to a shift of the maximum toward zero and a shrink of the distribution width, respectively.
%%% FIG 5 %%%
\begin{table*}
\caption{First ionization potential (left) and first electron attachment (right) in eV calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. The statistical descriptors are computed for the errors with respect to the reference.}
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{l|ddddd|ddddd}
Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} \\
\hline
\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.55 & -2.66 & -2.70 & -2.66 & -2.66 & -2.66 \\
\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 & -5.09 & -5.47 & -5.25 & -5.19 & -5.19 \\
\ce{H2} & 16.40 & 16.16 & 16.49 & 16.45 & 16.45 & -1.35 & -1.33 & -1.28 & -1.28 & -1.28 \\
\ce{Li2} & 5.25 & 4.96 & 5.38 & 5.40 & 5.37 & 0.34 & -0.08 & 0.17 & 0.18 & 0.21 \\
\ce{LiH} & 8.02 & 8.21 & 8.22 & 8.25 & 8.15 & -0.29 & 0.20 & 0.27 & 0.27 & 0.27 \\
\ce{HF} & 16.15 & 17.69 & 16.25 & 16.45 & 16.34 & -0.66 & -0.81 & -0.71 & -0.70 & -0.70 \\
\ce{Ar} & 15.60 & 16.08 & 15.72 & 15.61 & 15.63 & -2.55 & -2.97 & -2.68 & -2.64 & -2.65 \\
\ce{H2O} & 12.69 & 13.88 & 12.90 & 12.98 & 12.88 & -0.61 & -0.80 & -0.68 & -0.65 & -0.66 \\
\ce{LiF} & 11.47 & 12.91 & 11.40 & 11.75 & 11.58 & 0.35 & 0.29 & 0.33 & 0.33 & 0.33 \\
\ce{HCl} & 12.67 & 12.98 & 12.78 & 12.77 & 12.72 & -0.57 & -0.79 & -0.64 & -0.63 & -0.63 \\
\ce{BeO} & 9.95 & 10.45 & 9.74 & 10.32 & 10.18 & 2.17 & 1.80 & 2.28 & 2.10 & 2.13 \\
\ce{CO} & 13.99 & 15.11 & 14.80 & 14.34 & 14.33 & -1.57 & -1.80 & -1.66 & -1.61 & -1.62 \\
\ce{N2} & 15.54 & 16.68 & 17.10 & 15.93 & 15.91 & -2.37 & -2.20 & -2.10 & -2.10 & -2.10 \\
\ce{CH4} & 14.39 & 14.83 & 14.76 & 14.67 & 14.63 & -0.65 & -0.79 & -0.70 & -0.68 & -0.68 \\
\ce{BH3} & 13.31 & 13.59 & 13.68 & 13.62 & 13.59 & -0.09 & -0.81 & -0.46 & -0.29 & -0.30 \\
\ce{NH3} & 10.91 & 11.69 & 11.22 & 11.18 & 11.10 & -0.61 & -0.80 & -0.68 & -0.66 & -0.66 \\
\ce{BF} & 11.15 & 11.04 & 11.34 & 11.19 & 11.18 & -0.80 & -1.06 & -0.90 & -0.87 & -0.86 \\
\ce{BN} & 12.05 & 11.55 & 11.76 & 11.89 & 11.90 & 3.02 & 2.97 & 3.90 & 3.41 & 3.44 \\
\ce{SH2} & 10.39 & 10.49 & 10.51 & 10.50 & 10.45 & -0.52 & -0.76 & -0.60 & -0.58 & -0.59 \\
\ce{F2} & 15.81 & 18.15 & 16.35 & 16.27 & 16.22 & 0.32 & -1.71 & -0.53 & 0.10 & 0.07 \\
\ce{MgO} & 7.97 & 8.75 & 8.40 & 8.54 & 8.36 & 1.54 & 1.40 & 1.64 & 1.72 & 1.71 \\
\ce{O3} & 12.85 & 13.29 & 13.56 & 13.34 & 13.27 & 1.82 & 1.32 & 2.19 & 2.23 & 2.17 \\
\ce{C2H2} & 11.45 & 11.16 & 11.57 & 11.46 & 11.43 & -0.80 & -0.80 & -0.71 & -0.71 & -0.71 \\
\ce{NCH} & 13.76 & 13.50 & 13.86 & 13.75 & 13.73 & -0.53 & -0.61 & -0.52 & -0.55 & -0.54 \\
\ce{B2H6} & 12.27 & 12.84 & 12.81 & 12.67 & 12.64 & -0.52 & -0.64 & -0.56 & -0.55 & -0.55 \\
\ce{H2CO} & 10.93 & 12.09 & 11.39 & 11.33 & 11.25 & -0.60 & -0.70 & -0.61 & -0.62 & -0.62 \\
\ce{C2H4} & 10.69 & 10.26 & 10.74 & 10.70 & 10.67 & -1.90 & -0.86 & -0.75 & -0.73 & -0.74 \\
\ce{SiH4} & 12.79 & 13.23 & 13.22 & 13.15 & 13.11 & -0.53 & -0.69 & -0.59 & -0.57 & -0.58 \\
\ce{PH3} & 10.60 & 10.60 & 10.79 & 10.76 & 10.73 & -0.51 & -0.71 & -0.58 & -0.56 & -0.57 \\
\ce{CH4O} & 11.09 & 12.30 & 11.55 & 11.49 & 11.39 & -0.59 & -0.76 & -0.64 & -0.62 & -0.63 \\
\ce{H2NNH2} & 9.49 & 10.38 & 9.84 & 9.81 & 9.73 & -0.60 & -0.82 & -0.69 & -0.65 & -0.65 \\
\ce{HOOH} & 11.51 & 13.17 & 11.96 & 11.95 & 11.86 & -0.96 & -0.89 & -0.75 & -0.72 & -0.72 \\
\ce{KH} & 6.32 & 6.61 & 6.44 & 6.50 & 6.38 & 0.30 & 0.21 & 0.28 & 0.28 & 0.28 \\
\ce{Na2} & 4.93 & 4.53 & 4.98 & 5.03 & 5.01 & 0.36 & -0.01 & 0.26 & 0.27 & 0.30 \\
\ce{HN3} & 10.77 & 11.00 & 11.12 & 10.92 & 10.89 & -0.51 & -0.75 & -0.6 & -0.56 & -0.56 \\
\ce{CO2} & 13.80 & 14.82 & 14.24 & 14.12 & 14.06 & -0.88 & -1.22 & -0.98 & -0.95 & -0.95 \\
\ce{PN} & 11.90 & 12.00 & 12.33 & 12.12 & 12.09 & -0.02 & -0.72 & -0.03 & 0.02 & 0.00 \\
\ce{CH2O2} & 11.54 & 12.94 & 12.00 & 11.97 & 11.88 & -0.63 & -0.79 & -0.69 & -0.66 & -0.67 \\
\ce{C4} & 11.43 & 11.61 & 11.77 & 11.57 & 11.54 & 2.38 & 0.58 & 2.24 & 2.29 & 2.30 \\
\ce{C3H6} & 10.83 & 11.25 & 11.20 & 11.07 & 11.03 & -0.94 & -0.88 & -0.75 & -0.73 & -0.73 \\
\ce{C2H3F} & 10.63 & 10.48 & 10.84 & 10.73 & 10.69 & -0.65 & -0.80 & -0.69 & -0.68 & -0.68 \\
\ce{C2H4O} & 10.29 & 11.64 & 10.84 & 10.74 & 10.66 & -0.54 & -0.69 & -0.56 & -0.57 & -0.57 \\
\ce{C2H6O} & 10.82 & 12.05 & 11.37 & 11.25 & 11.15 & -0.58 & -0.78 & -0.65 & -0.62 & -0.62 \\
\ce{C3H8} & 12.13 & 12.73 & 12.61 & 12.51 & 12.46 & -0.63 & -0.83 & -0.70 & -0.67 & -0.67 \\
\ce{NaCl} & 9.10 & 9.60 & 9.20 & 9.25 & 9.16 & 0.67 & 0.56 & 0.64 & 0.64 & 0.64 \\
\ce{P2} & 10.72 & 10.05 & 10.49 & 10.43 & 10.40 & 0.43 & -0.35 & 0.47 & 0.48 & 0.47 \\
\ce{F2Mg} & 13.93 & 15.46 & 13.94 & 14.23 & 14.07 & 0.29 & -0.03 & 0.15 & 0.21 & 0.21 \\
\ce{OCS} & 11.23 & 11.44 & 11.52 & 11.37 & 11.32 & -1.43 & -1.27 & -1.03 & -0.97 & -0.98 \\
\ce{SO2} & 10.48 & 11.47 & 11.38 & 10.85 & 10.82 & 2.24 & 1.84 & 2.82 & 2.74 & 2.68 \\
\ce{C2H3Cl} & 10.17 & 10.13 & 10.39 & 10.27 & 10.24 & -0.61 & -0.79 & -0.66 & -0.65 & -0.65 \\
\hline
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} \\
\hline
MSE & & 0.56 & 0.29 & 0.23 & 0.17 & & -0.25 & 0.02 & 0.04 & 0.04 \\
MAE & & 0.69 & 0.33 & 0.25 & 0.19 & & 0.31 & 0.16 & 0.13 & 0.12 \\
SDE & & 0.68 & 0.31 & 0.18 & 0.16 & & 0.43 & 0.29 & 0.23 & 0.22 \\
Min & & -0.67 & -0.29 & -0.29 & -0.32 & & -2.03 & -0.85 & -0.22 & -0.25 \\
Max & & 2.34 & 1.56 & 0.57 & 0.42 & & 1.04 & 1.15 & 1.17 & 1.16 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%%% FIG 6 %%%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig5.pdf}
\includegraphics[width=\linewidth]{fig6.pdf}
\caption{
Temporary figure about convergence
SRG-qs$GW$ and qs$GW$ MAE of the IPs for the GW50 test set. The bottom and top axes are equivalent and related by $\eta=1/2s^2$. A different marker has been used for qs$GW$ at $\eta=0.05$ because the MAE includes only 48 molecules.
\label{fig:fig5}}
\end{figure}
%%% %%% %%% %%%
The difference in
In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=\num{e3}$ self-consistency can be attained without any problems for the 50 compounds.
For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50, meaning that some intruder states were included in the static correction for this value of $s$.
However, this is not a problem as the MAE of the test set is already converged for this value of $s$.
This is illustrated by the blue curve of Fig.~\ref{fig:fig5} which shows the evolution of the MAE with respect to $s$ and $\eta=1/2s^2$.
The plateau of the MAE is reached far before the convergence problem arrives.
In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=10^3$ self-consistency can be attained without any problems (mean and max number of iterations = n for s=100).
For $s=10^4$, convergence could not be attained for 12 molecules out of 22, meaning that some intruder states were included in the static correction for this value of $s$.
This is illustrated in the case of \ce{H2O} in the upper panel Fig.
However, this is not a problem as the MAE is already well converged before the intruder states are added to the SRG-qs$GW$ static self-energy (lower panel).
On the other hand, the qs$GW$ convergence behaviour seems to be more erratic.
At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be achieved for 13 molecules while 2 molecules were already problematic at $\eta=\num{5e-2}$ ($s=200$).
But this convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau when these problem arise.
For example, out of the 37 molecules that could be converged for $\eta=\num{e-2}$ the variation of the IP with respect to $\eta=\num{5e-2}$ can go up to \SI{0.1}{\eV}.
On the other hand, the qs$GW$ convergence with respect to $\eta$ is more difficult to evaluate.
The whole set considered in this work could be converged for $\eta=0.1$.
However, as soon as we decrease $\eta$ self-consistency could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
Unfortunately, the convergence of the IP is not as tight as in the SRG case.
The values of the IP that could be converged for $\eta=0.01$ can vary between $10^{-3}$ and $10^{-1}$ with respect to $\eta=0.1$.
This difference of behaviour is due to the energy (in)dependence of the regularizers.
Indeed, the SRG regularizer first includes the terms that are important for the energy and finally adds the intruder states.
On the other hand, the imaginary shift regularizer acts equivalently on intruder states and terms that contribute to the energy.
% \begin{table}
% \caption{First ionization potential in eV calculated using $G_0W_0^{\text{TDA}}$@HF, qs$GW^{\text{TDA}}$ and SRG-qs$GW^{\text{TDA}}$. The statistical descriptors are computed for the errors with respect to the reference.}
% \label{tab:tab1}
% \begin{ruledtabular}
% \begin{tabular}{lddd}
% Mol. & \mcc{$G_0W_0^{\text{TDA}}$@HF} & \mcc{qs$GW^{\text{TDA}}$} & \mcc{SRG-qs$GW^{\text{TDA}}$} \\
% & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{5e-2}$} & \mcc{$s=\num{e2}$} \\
% \hline
% \ce{He} & 24.45 & 24.48 & 24.39 \\
% \ce{Ne} & 20.85 & 21.23 & 20.92 \\
% \ce{H2} & 16.53 & 16.46 & 16.50 \\
% \ce{Li2} & 5.45 & 5.50 & 5.46 \\
% \ce{LiH} & 8.14 & 8.17 & 8.05 \\
% \ce{HF} & 15.64 & 15.79 & 15.66 \\
% \ce{Ar} & 15.60 & 15.42 & 15.46 \\
% \ce{H2O} & 12.42 & 12.40 & 12.31 \\
% \ce{LiF} & 10.75 & 11.02 & 10.85 \\
% \ce{HCl} & 12.70 & 12.65 & 12.59 \\
% \ce{BeO} & 9.33 & 10.21 & 10.05 \\
% \ce{CO} & 14.60 & 13.82 & 13.84 \\
% \ce{N2} & 17.36 & 15.15 & 15.21 \\
% \ce{CH4} & 14.67 & 14.50 & 14.47 \\
% \ce{BH3} & 13.66 & 13.57 & 13.54 \\
% \ce{NH3} & 10.91 & 10.75 & 10.68 \\
% \ce{BF} & 11.38 & 11.11 & 11.12 \\
% \ce{BN} & 11.85 & 12.05 & 12.04 \\
% \ce{SH2} & 10.47 & 10.44 & 10.38 \\
% \ce{F2} & 15.55 & 15.23 & 15.22 \\
% \ce{MgO} & 8.10 & 7.76 & 7.58 \\
% \ce{O3} & 13.68 & 12.22 & 12.22 \\
% \hline
% MSE & 0.07 & -0.12 & -0.18 \\
% MAE & 0.37 & 0.22 & 0.25 \\
% SDE & 0.55 & 0.26 & 0.27 \\
% Min & -0.72 & -0.63 & -0.63 \\
% Max & 1.82 & 0.26 & 0.22 \\
% \end{tabular}
% \end{ruledtabular}
% \end{table}
% \begin{table}
% \caption{First electron attachment in eV calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. The statistical descriptors are computed for the errors with respect to the reference.}
% \label{tab:tab2}
% \begin{ruledtabular}
% \begin{tabular}{lddddd}
% Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
% & & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
% \hline
% \ce{He} & 2.66 & 2.70 & 2.66 & 2.66 & 2.66 \\
% \ce{Ne} & 5.09 & 5.47 & 5.25 & 5.19 & 5.19 \\
% \ce{H2} & 1.35 & 1.33 & 1.28 & 1.28 & 1.28 \\
% \ce{Li2} & -0.34 & 0.08 & -0.17 & -0.18 & -0.21 \\
% \ce{LiH} & 0.29 & -0.20 & -0.27 & -0.27 & -0.27 \\
% \ce{HF} & 0.66 & 0.81 & 0.71 & 0.70 & 0.70 \\
% \ce{Ar} & 2.55 & 2.97 & 2.68 & 2.64 & 2.65 \\
% \ce{H2O} & 0.61 & 0.80 & 0.68 & 0.65 & 0.66 \\
% \ce{LiF} & -0.35 & -0.29 & -0.33 & -0.32 & -0.33 \\
% \ce{HCl} & 0.57 & 0.79 & 0.64 & 0.63 & 0.63 \\
% \ce{BeO} & -2.17 & -1.80 & -2.28 & -2.10 & -2.13 \\
% \ce{CO} & 1.57 & 1.80 & 1.66 & 1.61 & 1.62 \\
% \ce{N2} & 2.37 & 2.20 & 2.10 & 2.10 & 2.10 \\
% \ce{CH4} & 0.65 & 0.79 & 0.70 & 0.68 & 0.68 \\
% \ce{BH3} & 0.09 & 0.81 & 0.46 & 0.29 & 0.30 \\
% \ce{NH3} & 0.61 & 0.80 & 0.68 & 0.66 & 0.66 \\
% \ce{BF} & 0.80 & 1.06 & 0.90 & 0.87 & 0.87 \\
% \ce{BN} & -3.02 & -2.97 & -3.90 & -3.41 & -3.44 \\
% \ce{SH2} & 0.52 & 0.76 & 0.60 & 0.58 & 0.59 \\
% \ce{F2} & -0.32 & 1.71 & 0.53 & -0.10 & -0.07 \\
% \ce{MgO} & -1.54 & -1.40 & -1.64 & -1.72 & -1.71 \\
% \ce{O3} & -1.82 & -1.32 & -2.19 & -2.22 & -2.17 \\
% \hline
% MSE & & -0.30 & -0.02 & 0.00 & 0.00 \\
% MAE & & 0.32 & 0.19 & 0.11 & 0.12 \\
% SDE & & 0.43 & 0.31 & 0.17 & 0.17 \\
% Min & & -2.03 & -0.85 & -0.22 & -0.25 \\
% Max & & 0.17 & 0.88 & 0.41 & 0.42 \\
% \end{tabular}
% \end{ruledtabular}
% \end{table}
%%% FIG 6 %%%
%%% FIG 7 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig6.pdf}
\includegraphics[width=\linewidth]{fig7.pdf}
\caption{
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
\label{fig:fig6}}
\end{figure*}
%%% %%% %%% %%%
Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal electron attachement (EA) energies.
The raw results are given in Tab.~\ref{tab:tab2} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
The raw results are given in Tab.~\ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
The HF EA are understimated in averaged with some large outliers while $G_0W_0$@HF mitigates the average error there are still large outliers.
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt} and the error of the outliers is reduced with respect to $G_0W_0$@HF.

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