saving work

Fig/flow.tex

@@ -6,6 +6,7 @@
 \begin{tikzpicture}[]
 
 	% frame
+	\draw[-,thick] (0,0) -- (4,0) node[right,sloped,below,yshift=-0.25cm]{flow};
 	\draw[-,thick] (0,0) node[anchor=north west]{} -- (5.5,0);
 	\draw[-,dash pattern=on 20pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 20pt,thick] (5,0) -- (7,0);
 	\draw[->,thick] (6.5,0) -- (8,0) node[anchor=west]{$s = \Lambda^{-2}$};

Manuscript/SRGGW.tex

@@ -81,12 +81,12 @@ Despite this, self-consistent versions still pose challenges in terms of converg
 A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem.
 In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods.
 The SRG formalism enables us to derive, from first principles, the expression of a new, naturally hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations.
-The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations and slightly improves the overall accuracy.
-%\bigskip
-%\begin{center}
-%	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
-%\end{center}
-%\bigskip
+The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.
+\bigskip
+\begin{center}
+	\boxed{\includegraphics[width=0.5\linewidth]{flow}}
+\end{center}
+\bigskip
 \end{abstract}
 
 \maketitle
@@ -551,7 +551,7 @@ which can be solved by simple integration along with the initial condition $\bF^
   \centering
   \includegraphics[width=\linewidth]{flow}
   \caption{
-    Schematic evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan). \ANT{Maybe we should replace dynamic by full?}
+    Schematic evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan). 
     \label{fig:flow}}
 \end{figure}
 %%% %%% %%% %%%
@@ -600,7 +600,7 @@ This yields a $s$-dependent static self-energy which matrix elements read
    \\
    \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ].
 \end{multline}
-Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
+Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is straightforward to implement in existing code and is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
 Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator.
 Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms.
 
@@ -686,8 +686,8 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
 
 This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set.
 Figure \ref{fig:fig2} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
-The IP at the HF level (dashed black line) is overestimated; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019}
-The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference.
+The IP at the HF level (dashed black line) is too large; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019}
+The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference value.
 
 Figure \ref{fig:fig2} also displays the IP at the SRG-qs$GW$ level as a function of the flow parameter (blue curve).
 At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
@@ -711,28 +711,30 @@ Therefore, for small $s$, only the last term of Eq.~\eqref{eq:2nd_order_IP} is p
 As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreasing and eventually goes below the $s=0$ initial value as observed in Fig.~\ref{fig:fig2}.
 
 In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening (dubbed qs$GW^\TDA$ and SRG-qs$GW^\TDA$) are also considered in Fig.~\ref{fig:fig2}.
-The TDA IPs are now underestimated, unlike their RPA counterparts.
+The TDA values are now underestimated the IP, unlike their RPA counterparts.
 For both static self-energies, the TDA leads to a slight increase in the absolute error.
 This trend is investigated in more detail in the next subsection.
 
 Next, we investigate the flow parameter dependence of SRG-qs$GW$ for three more challenging molecular systems.
-The left panel of Fig.~\ref{fig:fig3} shows the results for the lithium dimer, \ce{Li2}, which is an interesting case because, unlike in water, the HF IP underestimates the reference IP.
+The left panel of Fig.~\ref{fig:fig3} 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.
 On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are too large. 
 Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
 In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig2}.
-Both TDA results are worse than their RPA counterparts but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
+Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
 
-We now turn to the lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2}).
+We now turn to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2}).
 In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value.
 However, SRG-qs$GW$ 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 because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
-The SRG-qs$GW$ could be converged without any problem even for large values of $s$.
+The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values.
 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 the TDA one.
-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.
-Therefore, it seems that the effect of the TDA can not be systematically predicted.
+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.
+Also, SRG-qs$GW^\TDA$ is better than qs$GW^\TDA$ in the three cases of Fig.~\ref{fig:fig3}.
+However, it is not a general rule.
+Therefore, it seems that the effect of the TDA cannot be systematically predicted.
 
 \begin{table*}
   \caption{First ionization potential (left) and first electron attachment (right) 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. \ANT{Maybe change the values of SRG with the one for s=1000}}