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\newcommand{\ant}[1]{\textcolor{teal}{#1}}
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\newcommand{\trashant}[1]{\textcolor{teal}{\sout{#1}}}
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\newcommand{\ANT}[1]{\ant{(\underline{\bf ANT}: #1)}}
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\newcommand{\SupInf}{\textcolor{blue}{Supporting Information}}
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\DeclareMathOperator{\sgn}{sgn}
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% addresses
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@ -656,7 +657,7 @@ Note that, after this transformation, the form of the regularizer is actually cl
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% Reference comp det
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Our set of systems is composed by closed-shell compounds that correspond to the 50 smallest atoms and molecules (in terms of the number of electrons) of the $GW$100 benchmark set. \cite{vanSetten_2015}
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We will refer to this set as $GW$50.
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Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level (without frozen core) \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
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Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3/aug-cc-pVTZ basis level \cite{Christiansen_1995b,Koch_1997} using the \textsc{cfour} program. \cite{CFOUR}
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% GW comp det
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The two qs$GW$ variants considered in this work have been implemented in an in-house program, named \textsc{quack}. \cite{QuAcK}
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@ -666,28 +667,28 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$
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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.
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In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
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However, in order to perform black-box comparisons, these parameters have been fixed to these default values.
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The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations while, for the (SRG-)qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
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The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
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These many-body perturbation theory values are compared with a $\Delta$CCSD(T) reference.
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The $\Delta$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.
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The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set.
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The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} with default parameters within the restricted and unrestricted HF formalism for the neutral and charged species, respectively.
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The numerical data associated with this study are reported in the {\SupInf}.
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%=================================================================%
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\section{Results}
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\label{sec:results}
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%=================================================================%
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The results section is divided into two parts.
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The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases.
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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}.
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%The results section is divided into two parts.
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%The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases.
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%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}.
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%%% FIG 3 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{fig3.pdf}
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\caption{
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Error [with respect to $\Delta$CCSD(T)] in the principal IP of water 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.
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Error [with respect to $\Delta$CCSD(T)] in the principal IP of water in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for SRG-qs$GW$.
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The HF and qs$GW$ values are reported as dashed lines.
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\PFL{Should we have a similar figure for EAs? (maybe not water though)}
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\ANT{I did the plot, let's discuss it at the next meeting}
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\label{fig:fig3}}
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\end{figure}
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%%% %%% %%% %%%
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@ -703,16 +704,16 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
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\subsection{Flow parameter dependence of SRG-qs$GW$}
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\label{sec:flow_param_dep}
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%%%%%%%%%%%%%%%%%%%%%%
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This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set.
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Figure \ref{fig:fig3} shows the error of various methods for the principal IP with respect to the $\Delta$CCSD(T) reference value.
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Figure \ref{fig:fig3} shows the error in the principal IP [with respect to the $\Delta$CCSD(T) reference value] as a function of the flow parameter in SRG-qs$GW$ (blue curve).
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The corresponding HF and qs$GW$ (computed with $\eta = 0.05$) values are also reported for the sake of comparison.
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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}
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The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference value.
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Figure \ref{fig:fig3} also displays the IP at the SRG-qs$GW$ level as a function of the flow parameter (blue curve).
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At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
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As $s$ grows, the IP reaches a plateau at an error that is significantly smaller than the HF starting point.
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Furthermore, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart.
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@ -738,22 +739,20 @@ As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreas
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%The TDA values are now underestimating the IP, unlike their RPA counterparts.
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%For both static self-energies, the TDA leads to a slight increase in the absolute error.
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Next, the flow parameter dependence of SRG-qs$GW$ is investigated for the IPs of two more challenging molecular systems as well as the EA of the \ce{F2} molecule.
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Next, the flow parameter dependence of SRG-qs$GW$ is investigated for the principal IPs of two additional molecular systems as well as the principal EA of the \ce{F2} molecule.
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The left panel of Fig.~\ref{fig:fig4} 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.
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Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimated as in the water molecule case.
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Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimated as in \ce{H2O}.
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Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
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In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig3}.
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%Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
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Now turning to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4}).
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In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value.
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Now turning to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4}), we see that the qs$GW$ IP is actually worse than the fairly accurate HF value.
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However, SRG-qs$GW$ does not suffer from the same problem and improves slightly the accuracy as compared to HF.
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Finally, we also consider the evolution with respect to $s$ of an EA.
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The right panel of Fig.~\ref{fig:fig4} displays this evolution for the \ce{F2} molecule.
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Finally, we also consider the evolution with respect to $s$ of the principal EA of \ce{F2}, as displayed in Fig.~\ref{fig:fig4}.
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The HF value is largely underestimating the $\Delta$CCSD(T) reference.
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Performing a qs$GW$ calculation on top of it brings a quantitative improvement by reducing the error from \SI{-2.03}{\eV} to \SI{-0.24}{\eV}.
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The SRG-qs$GW$ EA error is smoothly increasing from the HF value at $s=0$ to an error really close to the qs$GW$ one at $s\to\infty$.
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The SRG-qs$GW$ EA error is monotonically increasing from the HF value at $s=0$ to an error really close to the qs$GW$ one at $s\to\infty$.
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% 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}
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% The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values.
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@ -880,21 +879,21 @@ The decrease of the MSE and SDE correspond to a \ant{shift of the maximum} towar
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\end{figure}
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%%% %%% %%% %%%
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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.
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In addition to this improvement of accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
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Indeed, up to $s=\num{e3}$ SRG-qs$GW$ self-consistency can be attained without any problems for the 50 compounds.
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For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50.
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This means that some intruder states were included in the static correction for this value of $s$.
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However, this is not a problem as the MAE of the test set is already well converged at $s=\num{e3}$.
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This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and $\eta=1/2s^2$.
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The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arise for $s>\num{e3}$.
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The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arises for $s>\num{e3}$.
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Therefore, for future studies using the SRG-qs$GW$ method, a default value of the flow parameter equal to $\num{5e2}$ or $\num{e3}$ seems reasonable.
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On the other hand, the qs$GW$ convergence behaviour seems to be more erratic.
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On the other hand, the qs$GW$ convergence behavior seems to be more erratic.
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At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be achieved for 13 molecules while 2 molecules were already problematic at $\eta=\num{5e-2}$ ($s=200$).
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But this convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau before these problem arise (see orange curve in Fig.~\ref{fig:fig6}).
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But these convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau before these problem arise (see the orange curve in Fig.~\ref{fig:fig6}).
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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}.
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This difference of behaviour is due to the energy (in)dependence of the regularizers.
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This difference of behavior is due to the energy (in)dependence of the regularizers.
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Indeed, the SRG regularizer first includes the terms that are important for the energy and finally adds the intruder states.
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On the other hand, the imaginary shift regularizer acts equivalently on intruder states and terms that contribute to the energy.
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@ -910,10 +909,10 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
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Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal EA energies.
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The raw results are given in Tab.~\ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig7}.
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The HF EAs are understimated in average with a MAE of \SI{0.31}{\eV} and some large outliers, for \SI{-2.03}{\eV} for \ce{F2} and \SI{1.04}{\eV} for \ce{CH2O} for example.
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The HF EAs are underestimated in average with a MAE of \SI{0.31}{\eV} and some large outliers, for \SI{-2.03}{\eV} for \ce{F2} and \SI{1.04}{\eV} for \ce{CH2O} for example.
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$G_0W_0$@HF mitigates the average error (MAE equal to \SI{0.16}{\eV}) but the minimum and maximum error values are not yet satisfactory.
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The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt}.
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The two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavours of many-body perturbation theory considered here can not decrease the maximum error with respect to their HF starting point.
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The two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavors of many-body perturbation theory considered here can not decrease the maximum error with respect to their HF starting point.
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Note that a positive EA means that the anion state is bound and therefore the methods considered here are well-suited to describe these EAs.
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On the other hand, a negative EA means that this is a resonance state.
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@ -931,7 +930,7 @@ The problems caused by intruder states in many-body perturbation theory are mult
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The central equation of the SRG formalism is the flow equation and needs to be solved numerically in the general case.
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Yet, it can still be solved analytically for low perturbation order.
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Doing so in the (upfolded) $GW$ context yields analytical expressions for second-order renormalized Fock matrices elements and two-electrons screened integrals.
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Doing so in the (upfolded) $GW$ context yields analytical expressions for second-order renormalized Fock matrix elements and two-electrons screened integrals.
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These renormalized quantities lead to a renormalized $GW$ quasiparticle equation, referred to as SRG-$GW$, which is the main equation of this work.
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The usual approximate solving schemes of the quasiparticle equation can be applied to SRG-$GW$ as well.
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In particular, isolating the static part of this equation yields an alternative Hermitian static and intruder-state-free self-energy that can be used for qs$GW$ calculation.
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@ -940,19 +939,19 @@ In addition to this new static form, we also explained how to use the SRG-$GW$ t
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This gave a first-principle rationale for the SRG-inspired regularizer introduced in Ref.~\onlinecite{Monino_2022}.
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The flow parameter dependence of the SRG-qs$GW$ IPs has been studied for a few test cases.
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It has been shown that the IPs gradually evolves from the HF starting point at $s=0$ to a plateau value for $s\to\infty$ that is much closer to the CCSD(T) reference than the HF initial value.
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For small values of the flow parameter the SRG-qs$GW$ IPs are actually worst than their starting point.
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Therefore, in practice one should always use a value of $s$ as large as possible.
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This is similar to qs$GW$ calculations where one need to use the smallest possible $\eta$.
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It has been shown that the IPs gradually evolve from the HF starting point at $s=0$ to a plateau value for $s\to\infty$ that is much closer to the CCSD(T) reference than the HF initial value.
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For small values of the flow parameter, the SRG-qs$GW$ IPs are actually worst than their starting point.
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Therefore, in practice, one should always use a value of $s$ as large as possible.
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This is similar to qs$GW$ calculations where one needs to use the smallest possible $\eta$.
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The second stage of this study was to statistically gauge the accuracy of the SRG-qs$GW$ IP for a test set of 50 atoms and molecules (referred to as $GW$50).
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It has been shown that in average SRG-qs$GW$ is slightly better than its traditional qs$GW$ counterpart for principal ionization energies.
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Note that while the accuracy improvements are quite small, it comes with no additional computational cost and its really fast to implement as one only needs to change the expression of the static self-energy.
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In addition, it has been shown that the SRG-qs$GW$ can be converged in a much more black-box fashion than the traditional qs$GW$ thanks to its intruder-state free nature.
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It has been shown that on average SRG-qs$GW$ is slightly better than its traditional qs$GW$ counterpart for principal ionization energies.
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Note that while the accuracy improvements are quite small, it comes with no additional computational cost, and its really fast to implement as one only needs to change the expression of the static self-energy.
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In addition, it has been shown that the SRG-qs$GW$ can be converged in a much more black-box fashion than the traditional qs$GW$ thanks to its intruder-state-free nature.
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Finally, the EAs have been investigated as well.
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It has been found that the performance of the two qs$GW$ flavours for the EAs of the $GW$50 set are quite similar.
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However, there is a caveat because most of the anions of the $GW$50 set are actually resonance states and their associated physcis can not be accurately described by the methods considered in this study.
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It has been found that the performances of the two qs$GW$ flavours for the EAs of the $GW$50 set are quite similar.
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However, there is a caveat because most of the anions of the $GW$50 set are actually resonance states and their associated physics can not be accurately described by the methods considered in this study.
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A test set of molecules with bound anions with an accompanying benchmark of accurate reference values would certainly be valuable to the many-body perturbation theory community.
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -962,9 +961,9 @@ This project has received funding from the European Research Council (ERC) under
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Data availability statement}
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%\section*{Data availability statement}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The data that supports the findings of this study are available within the article.% and its supplementary material.
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%The data that supports the findings of this study are available within the article and its supplementary material.
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% \appendix
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