small modifs

Manuscript/SRGGW.tex

@@ -303,7 +303,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
   \boldsymbol{\eta}(s) = \dv{\bU(s)}{s}	\bU^\dag(s)	= - \boldsymbol{\eta}^\dag(s).
 \end{equation}
 
-\ant{The flow equation can be approximately solved by introduction of an approximate form of $\boldsymbol{\eta}(s)$.}
+The flow equation can be approximately solved by introducing an approximate form of $\boldsymbol{\eta}(s)$.
 In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
 \begin{equation}
   \boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
@@ -338,7 +338,7 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
 \label{sec:srggw}
 %%%%%%%%%%%%%%%%%%%%%% 
 
-\ant{Finally, the two previous subsections will be combined by applying the SRG method to the $GW$ formalism.}
+Here, we combine the concepts of the two previous subsections and apply the SRG method to the $GW$ formalism.
 However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
 A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
 Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
@@ -654,7 +654,7 @@ Note that, after this transformation, the form of the regularizer is actually cl
 %=================================================================%
 
 % Reference comp det
-Our set of molecules is composed by closed-shell compounds that correspond to the 50 smallest (wrt the number of electrons) atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
+Our set of systems is composed by closed-shell compounds that correspond to the 50 smallest atoms and molecules (in terms of the number of electrons) of the $GW$100 benchmark set. \cite{vanSetten_2015}
 Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level (without frozen core) \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
 The reference CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively.
 
@@ -665,8 +665,8 @@ In all $GW$ calculations, we use the aug-cc-pVTZ cartesian basis set and self-co
 We use (restricted) HF guess orbitals and energies for all self-consistent $GW$ calculations.
 The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively.
 In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
-However, in order to perform a black-box comparison, these parameters have been fixed to these default values.
-\ant{The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculation while for (SRG-)qs$GW$ calculations the $\eta$ value has been chosen as the largest value where one successfully converges the 50 systems of the test set.}
+However, in order to perform black-box comparisons, these parameters have been fixed to these default values.
+The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations while, for the (SRG-)qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
 
 %=================================================================%
 \section{Results}
@@ -734,7 +734,6 @@ As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreas
 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 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.
-\trashant{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 approximation underestimates the reference IP.
@@ -768,6 +767,7 @@ Therefore, it seems that the effect of the TDA cannot be systematically predicte
   \includegraphics[width=\linewidth]{fig5.pdf}
   \caption{
     Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
+    \PFL{Add MAE and MSE values to each figure.}
     \label{fig:fig4}}
 \end{figure*}
 %%% %%% %%% %%%
@@ -890,7 +890,8 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
   \includegraphics[width=\linewidth]{fig7.pdf}
   \caption{
     Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
-    \label{fig:fig6}}
+   \PFL{Add MAE and MSE values to each figure.}
+   \label{fig:fig6}}
 \end{figure*}
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