update figs and modification discussion fig4

Manuscript/SRGGW.tex

@@ -738,7 +738,7 @@ As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreas
 %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.
 
-Next, the flow parameter dependence of SRG-qs$GW$ is investigated for three more challenging molecular systems.
+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.
 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.
 Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimated as in the water molecule case. 
 Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
@@ -749,10 +749,16 @@ Now turning to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4
 In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value.
 However, SRG-qs$GW$ does not suffer from the same problem and 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 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.
-The convergence properties of both schemes will be systematically investigated in the next subsection.
-Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
+Finally, we also consider the evolution with respect to $s$ of an EA.
+The right panel of Fig.~\ref{fig:fig4} displays this evolution for the \ce{F2} molecule.
+The HF value is largely underestimating the $\Delta$CCSD(T) reference.
+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}.
+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 $s\to\infty$.
+
+% 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.
+% The convergence properties of both schemes will be systematically investigated in the next subsection.
+% Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
 
 %Note that, for \ce{LiH} and \ce{BeO}, the TDA actually improves the accuracy compared to the RPA-based qs$GW$ schemes.
 %However, as we shall see in Sec.~\ref{sec:SRG_vs_Sym}, these are special cases, and, on average, the RPA polarizability performs better than its TDA version.