working on fig1

Manuscript/SRGGW.tex

@@ -232,8 +232,9 @@ One obvious drawback of the one-shot scheme mentioned above is its starting poin
 Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
 As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot $GW$ quasiparticle energies. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
 Alternatively, one may solve this set of quasiparticle equations self-consistently leading to the ev$GW$ scheme.
-\titou{The solutions $\epsilon_p$  are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equations are solved for $\omega$ again.}
-This procedure is iterated until convergence on the quasiparticle energies is reached.
+\ant{To do so the quasiparticle energies are used to define a new RPA problem leading to updated two-electron screened integrals.
+ Then the diagonal elements of the self-energy are updated as well and Eq.~\eqref{eq:G0W0} is solved again to obtain new quasiparticle energies.}
+This procedure is then iterated until convergence on the quasiparticle energies is reached.
 However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible. 
 Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}
 
@@ -265,7 +266,6 @@ One can deal with them by introducing \textit{ad hoc} regularizers.
 Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
 Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
 This is the central aim of the present work.
-\PFL{SRG is energy-dependent.}
 
 %%%%%%%%%%%%%%%%%%%%%%
 \section{The similarity renormalization group}
@@ -530,6 +530,16 @@ Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see E
 \titou{Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.}
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
+%%% FIG 1 %%% 
+\begin{figure*}
+  \centering
+  \includegraphics[width=\linewidth]{fig1.pdf}
+  \caption{
+    Add caption
+    \label{fig:fig1}}
+\end{figure*}
+%%% %%% %%% %%%
+
 %///////////////////////////%
 \subsection{Alternative form of the static self-energy}
 % ///////////////////////////%
@@ -543,14 +553,21 @@ This yields a $s$-dependent static self-energy which matrix elements read
   \Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}  \\ \times \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
 \end{multline}
 Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by symmetrization.
+Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
 Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
 
 It is well-known that in traditional qs$GW$ calculation increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.
 Similarly, in SRG-qs$GW$ one might need to decrease the value of $s$ to ensure convergence.
 The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
 Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
+
+It is instructive to plot both regularizing functions, this is done in Fig.~\ant{???}.
+The surfaces correspond to a value of the regularizing parameter value of 1.
+The SRG surface is much smoother than its qs counterpart.
+In fact the SRG regularization has less work to do because for $\eta=0$ there is a single singularity at $x=y=0$.
+On the other hand the function $f_{\text{qs}}(x,y;0)$ is singular on two entire axis, $x=0$ and $y=0$.
+The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$.
 The convergence properties and the accuracy of both static approximations will be quantitatively gauged in the results section.
-In addition, 
 
 To conclude this section, the case of discontinuities will be briefly discussed.
 Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
@@ -598,14 +615,24 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
 %%% FIG 1 %%%
 \begin{figure}
   \centering
-  \includegraphics[width=\linewidth]{fig1.pdf}
+  \includegraphics[width=\linewidth]{fig2.pdf}
   \caption{
     Principal IP of the water molecule in the aug-cc-pVTZ cartesian basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
     Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
-    \label{fig:fig1}}
+    \label{fig:fig2}}
 \end{figure}
 %%% %%% %%% %%%
 
+%%% FIG 2 %%% 
+\begin{figure*}
+  \centering
+  \includegraphics[width=\linewidth]{fig3.pdf}
+  \caption{
+    Add caption
+    \label{fig:fig3}}
+\end{figure*}
+%%% %%% %%% %%%
+
 This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
 Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
 The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
@@ -657,15 +684,7 @@ However, as we will see in the next subsection these are just particular molecul
 Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around.
 Therefore, it seems that the effect of the TDA can not be systematically predicted.
 
-%%% FIG 2 %%% 
-\begin{figure*}
-  \centering
-  \includegraphics[width=\linewidth]{fig2.pdf}
-  \caption{
-    Add caption
-    \label{fig:fig2}}
-\end{figure*}
-%%% %%% %%% %%%
+
 
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Statistical analysis}