saving work in results before leaving

Manuscript/SRGGW.tex

@@ -1,5 +1,5 @@
 \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx}
 \usepackage[version=4]{mhchem}
 
 \usepackage[utf8]{inputenc}
@@ -367,7 +367,7 @@ As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal part
     \bO	&	\bC^{\text{2h1p}}		&	\bO	        	\\
     \bO	&	\bO				&	\bC^{\text{2p1h}}	\\
   \end{pmatrix}
-  &
+  & \\
     \bH^\text{od}(s)  &= 
   \begin{pmatrix}
     \bO			                         	&     \bV^{\text{2h1p}} &     \bV^{\text{2p1h}} \\
@@ -544,7 +544,7 @@ In fact, the dynamic part after the change of variable is closely related to the
 
 The two qs$GW$ variants considered in this work have been implemented in an in-house program.
 The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
-The geometries have been optimized at the CC3 level in the aug-cc-pvtz basis set without frozen core using the CFOUR program.
+The geometries have been optimized at the CC3 level in the aug-cc-pVTZ basis set without frozen core using the CFOUR program.
 The reference CCSD(T) IP energies have been obtained using default parameters of Gaussian 16.
 This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted formalism.
 
@@ -567,19 +567,56 @@ Then the accuracy of the IP yielded by the Sym and SRG schemes will be statistic
   \centering
   \includegraphics[width=\linewidth]{fig1.pdf}
   \caption{
-    This figure is terrible, I will clean it. I put it here just to start writing the results discussion. Maybe fig 1 with only He and Ne but also the $\eta$ evolution because possible to converge for these molecules. Then figure 2 with H2O and BeO with SRG GW and SRG GF(2).
+    Add caption
     \label{fig:fig1}}
 \end{figure*}
 %%% %%% %%% %%%
 
+This section starts by considering the Neon atom and the water molecule in the aug-cc-pVTZ cartesian basis set in Fig.~\ref{fig:fig1}.
+The HF values (orange lines) lie below the reference CCSD(T) ones, a result which is now well-understood.
+Indeed, this is due to an over(under? \ant{I will check this...}) screening of the interactions in the mean-field treatment. \cite{Lewis_2019}
+The usual Sym-qs$GW$ scheme (blue lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\electronvolt} of the reference.
+The Neon atom is a well-behaved system and could be converged without regularization parameter while for water it was set to 0.01 to help convergence.
+
+Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP energies as a function of the flow parameter.
+At $s=0$, the IPs are equal to their HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
+For $s\to\infty$ both IPs reach a plateau that are significantly better than their $s=0$ starting point.
+Even more, the values associated with these plateau are more accurate than their Sym-qs$GW$ counterparts.
+The SRG-qs$GW$ IPs do not increase smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
+
+The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
+Add sentence about $GW$ better than GF2 when the results will be here.
+The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
+We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
+The GF(2) IP admits the following perturbation expansion...
+
+
+Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
+But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.
+
+\begin{itemize}
+\item Li2 interesting because HF underestimate IP
+\item LiH interesting because Sym-qs$GW$ worst than HF
+\item BeO interesting because usually difficult to converge in qs$GW$ (cf Forster 2021)
+\end{itemize}
+
+%%% FIG 2 %%%
+\begin{figure*}
+  \centering
+  \includegraphics[width=\linewidth]{fig2.pdf}
+  \caption{
+    Add caption
+    \label{fig:fig2}}
+\end{figure*}
+%%% %%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Comparison of the Sym-qs and SRG-qs approximations}
 \label{sec:SRG_vs_Sym}
 %%%%%%%%%%%%%%%%%%%%%%
 
-The test set considered in this study is composed of the GW20 set of molecules introduced by Berkelbach \ant{check ref} and co-workers.
+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 molecules of the GW100 benchmark set.
+This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set.
 We also added the MgO and O3 molecules which are part of GW100 and are known to be difficult to converged for qs$GW$.
 In addition, we considered the Quest 1 and 2 sets which is made of small and medium size organic molecules.