working on statistical analysis

Manuscript/SRGGW.tex

@@ -703,7 +703,7 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
 
 The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
 This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
-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 be difficult to be plagued by intruder states. \cite{vanSetten_2015,Forster_2021}
+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}
 
 \begin{table}
   \centering
@@ -741,6 +741,7 @@ In addition, the MgO and O3 molecules (which are part of GW100 as well) has been
     \hline
     MSE            &  & 0.64 & 0.26 & 0.24 & 0.17 \\
     MAE           &  & 0.74 & 0.32 & 0.25 & 0.19 \\
+    SDE             &  & 0.71 & 0.39 & 0.18 & 0.15 \\
     Min             &  & -0.50 & -0.29 & -0.16 & -0.15 \\
     Max            &  & 2.35 & 1.56 & 0.56 & 0.42 \\
     \hline
@@ -758,12 +759,24 @@ In addition, the MgO and O3 molecules (which are part of GW100 as well) has been
 \end{figure*}
 %%% %%% %%% %%%
 
-Draft of outline:
+Table~\ref{tab:tab1} shows the principal IP of the 22 molecules considered in this work computed at various level of theories.
+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.
+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.
+However, there are still outliers with quite large errors, for example the IP of the dinitrogen is overestimated by \SI{1.56}{\electronvolt}.
+Self-consistency can mitigate the error of the outliers as the maximum error at the qs$GW$ level is now \SI{0.56}{\electronvolt} and the standard deviation error (SDE) is decreased from \SI{0.39}{\electronvolt} for $G_0W_0$ to \SI{0.18}{\electronvolt} for qs$GW$.
+In addition, the MSE and MAE (\SI{0.24}{\electronvolt}/\SI{0.25}{\electronvolt}) 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 $s=100$.
+For this value of the flow parameter, the MAE is converged to \SI{10d-3}{\electronvolt} (see Supplementary Material).
+The statistical descriptors corresponding to the alternative static self-energy are all improved with respect to qs$GW$.
+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.
+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}.
+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.
+
+In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
+\ANT{TO CONTINUE, waiting for s=10000}
 
-Discuss table 1 and fig4:
-black box convergence,
-mean value of s to reach plateau,
-increase of accuracy for no additional computational cost
 
 Part on approximation and correction for W:
 TDA vs RPA,