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@ -502,6 +502,16 @@ It is worth noting the close similarity of the first-order elements with the one
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\subsection{Second-order matrix elements}
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% ///////////////////////////%
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%%% FIG 1 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\linewidth]{fig1.pdf}
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\caption{
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Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.
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\label{fig:plot}}
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\end{figure*}
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%%% %%% %%% %%%
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The second-order renormalized quasiparticle equation is given by
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\begin{equation}
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\label{eq:GW_renorm}
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@ -546,16 +556,6 @@ while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends t
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Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
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This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
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%%% FIG 1 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\linewidth]{fig1.pdf}
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\caption{
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Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.
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\label{fig:plot}}
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\end{figure*}
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%%% %%% %%% %%%
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%///////////////////////////%
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\subsection{Alternative form of the static self-energy}
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% ///////////////////////////%
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@ -699,17 +699,6 @@ However, as we will see in the next subsection these are just particular molecul
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Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig3} but this is the other way around.
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Therefore, it seems that the effect of the TDA can not be systematically predicted.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Statistical analysis}
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\label{sec:SRG_vs_Sym}
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%%%%%%%%%%%%%%%%%%%%%%
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The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
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This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
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In addition, the MgO and O3 molecules (which are part of GW100 as well) has been added to the test set because they are known to possess intruder states. \cite{vanSetten_2015,Forster_2021}
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\begin{table}
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\centering
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\caption{First ionization potential 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.}
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@ -754,6 +743,11 @@ In addition, the MgO and O3 molecules (which are part of GW100 as well) has been
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\end{tabular}
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\end{table}
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Statistical analysis}
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\label{sec:SRG_vs_Sym}
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%%%%%%%%%%%%%%%%%%%%%%
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%%% FIG 4 %%%
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\begin{figure*}
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\centering
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@ -764,6 +758,10 @@ In addition, the MgO and O3 molecules (which are part of GW100 as well) has been
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\end{figure*}
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%%% %%% %%% %%%
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The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
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This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
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In addition, the MgO and O3 molecules (which are part of GW100 as well) has been added to the test set because they are known to possess intruder states. \cite{vanSetten_2015,Forster_2021}
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Table~\ref{tab:tab1} shows the principal IP of the 22 molecules considered in this work computed at various level of theories.
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As mentioned previously the HF IPs are overestimated with a mean signed error (MSE) and mean absolute error (MAE) of \SI{0.64}{\electronvolt} and \SI{0.74}{\electronvolt}, respectively.
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Performing a one-shot $G_0W_0$ calculation on top of it allows to divided by more than 2 the MSE and MAE, \SI{0.26}{\electronvolt} and \SI{0.32}{\electronvolt}, respectively.
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@ -773,23 +771,30 @@ In addition, the MSE and MAE (\SI{0.24}{\electronvolt}/\SI{0.25}{\electronvolt})
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Now turning to the new results of this manuscript, \ie the alternative self-consistent scheme SRG-qs$GW$.
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Table~\ref{tab:tab1} shows the SRG-qs$GW$ values for $s=100$.
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For this value of the flow parameter, the MAE is converged to \SI{10d-3}{\electronvolt} (see Supplementary Material).
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For this value of the flow parameter, the MAE is converged to \SI{d-3}{\electronvolt} (see Supplementary Material).
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The statistical descriptors corresponding to the alternative static self-energy are all improved with respect to qs$GW$.
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Of course these are slight improvements but this is done with no additional computational cost and can be implemented really quickly just by changing the form of the static approximation.
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The evolution of statistical descriptors with respect to the various methods considered in Table~\ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
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The decrease of the MSE and SDE correspond to a shift of the maximum of the distribution toward 0 and a shrink of the width of the distribution, respectively.
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In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
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\ANT{TO CONTINUE, waiting for s=10000}
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Indeed, up to $s=10^3$ self-consistency of the SRG-qs$GW$ scheme can be converged without any problems.
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For $s=10^4$, convergence could not be attained for the following molecules.
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On the other hand, the qs$GW$ convergence is much more erratic, the 22 molecules could be converged for $\eta=0.1$.
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However, if we decrease $\eta$ then convergence could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
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Unfortunately, the convergence of the IP is not as tight as in the SRG case because for $\eta=0.01$ the IP that could be converged can vary by $10^{-2}$ to $10^{-1}$ with respect to $\eta=0.1$.
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We will now gauge the effect of the TDA for the screening on the accuracy of the various methods considered previously.
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Part on approximation and correction for W:
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TDHF G0W0 not that bad in GW100 but bad in GW22, qsGW TDHF even worse even with SRG,
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Maybe that would be nice to add SRG G0W0 to see if it mitigates the outliers of GW20 cf Bruneval 2021,
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That would be nice to understand clearly why qsGWTDHF is worse (screening, gap, etc)
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\begin{table}
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\centering
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\caption{First ionization potential in eV calculated using \ANT{TO COMPLETE}. The statistical descriptors are computed for the errors with respect to the reference.}
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\caption{First ionization potential in eV calculated using $G_0W_{\text{TDA},0}$@HF, qs$GW_{\text{TDA}}$ and SRG-qs$GW_{\text{TDA}}$. The statistical descriptors are computed for the errors with respect to the reference.}
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\label{tab:tab1}
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-24.454162, -20.845919, -16.530702, -5.453359, -8.144555, -15.641055, -15.602545, -12.417807, -10.751003, -12.696334, -9.32538, -14.603221, -17.364737, -14.667361, -13.662675, -10.910541, -11.379826, -11.850654, -10.467744, -15.55351, -8.098392, -13.68235
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\begin{tabular}{lccc}
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\hline
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\hline
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@ -799,8 +804,8 @@ We will now gauge the effect of the TDA for the screening on the accuracy of the
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\ce{He} & 24.45 & 24.48 & 24.39 \\
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\ce{Ne} & 20.85 & 21.23 & 20.92 \\
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\ce{H2} & 16.53 & 16.46 & 16.50 \\
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\ce{Li2} & 5.45 & 5.50 & 5.46 \\
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\ce{LiH} & 8.14 & 8.17 & 8.05 \\
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\ce{Li2} & 5.45 & 5.50 & 5.46 \\
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\ce{LiH} & 8.14 & 8.17 & 8.05 \\
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\ce{HF} & 15.64 & 15.79 & 15.66 \\
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\ce{Ar} & 15.60 & 15.42 & 15.46 \\
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\ce{H2O} & 12.42 & 12.40 & 12.31 \\
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@ -829,16 +834,9 @@ We will now gauge the effect of the TDA for the screening on the accuracy of the
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\end{tabular}
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\end{table}
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Part on approximation and correction for W:
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TDHF G0W0 not that bad in GW100 but bad in GW22, qsGW TDHF even worse even with SRG,
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Maybe that would be nice to add SRG G0W0 to see if it mitigates the outliers of GW20 cf Bruneval 2021,
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That would be nice to understand clearly why qsGWTDHF is worse (screening, gap, etc)
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Part on EA:
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MgO- does not converge yet but when we have it same analysis as Table 1 and Fig 4 but for the EA
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Not sure that we need anything more, let's see once this is written
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%=================================================================%
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\section{Conclusion}
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\label{sec:conclusion}
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