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Manuscript/ufGW.tex

@@ -145,6 +145,7 @@ Its popularity in the molecular electronic structure community is rapidly growin
 
 The idea behind the $GW$ approximation is to recast the many-body problem into a set of non-linear one-body equations. The introduction of the self-energy $\Sigma$ links the non-interacting Green's function $G_0$ to its fully-interacting version $G$ via the following Dyson equation:
 \begin{equation}
+\label{eq:Dyson}
 	G = G_0 + G_0 \Sigma G
 \end{equation}
 Electron correlation is then explicitly incorporated into one-body quantities via a sequence of self-consistent steps known as Hedin's equations. \cite{Hedin_1965}
@@ -217,6 +218,12 @@ Note that we have the following important conservation rules \cite{Martin_1959,B
 \end{align}
 which physically shows that the mean-field solution of unit weight is ``scattered'' by the effect of correlation in many solutions of smaller weights.
 
+\alert{In standard $GW$ calculations in solids, \cite{Martin_2016} one assignes a quasparticle peak to the solution of the Dyson equation \eqref{eq:Dyson} that is associated with the largest value of the spectral function
+\begin{equation}
+	S(\omega) = \frac{1}{\pi} \abs{\Im G(\omega)} 
+\end{equation}
+}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Upfolding: the linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -411,6 +418,7 @@ The most common and well-established way of regularizing $\Sigma$ is via the sim
 	f_\eta(\Delta) = (\Delta \pm \ii \eta)^{-1}
 \end{equation}
 (with $\eta > 0$), \cite{vanSetten_2013,Bruneval_2016a,Martin_2016,Duchemin_2020} a strategy somehow related to the imaginary shift used in multiconfigurational perturbation theory. \cite{Forsberg_1997}
+\alert{This type of broadening is customary in solid-state calculations, hence such regularization is naturally captured in many codes. \cite{Martin_2016}}
 In practice, an empirical value of $\eta$ around \SI{100}{\milli\eV} is suggested.
 Other choices are legitimate like the regularizers considered by Head-Gordon and coworkers within orbital-optimized second-order M{\o}ller-Plesset theory, which have the specificity of being energy-dependent. \cite{Lee_2018a,Shee_2021}
 In this context, the real version of the simple energy-independent regularizer \eqref{eq:simple_reg} has been shown to damage thermochemistry performance and was abandoned. \cite{Stuck_2013,Rostam_2017}

Response_Letter/Response_Letter.tex

@@ -209,7 +209,7 @@ This is the only general method that works in solids as, rigorously, there is no
 Making this connection, especially around or before Eq.~8, would clarify the method to a readership that is not only interested in molecular systems.}
 \\
 \alert{
-We have modified the manuscript around Eq.~(8) to clarify this point.
+We have modified the manuscript below Eq.~(8) to clarify this point.
 }
 
 \item 
@@ -222,7 +222,7 @@ In addition, the term "regularized GW method" gives the impression that the GW a
 Instead, this procedure is nothing but a small broadening parameter that smooths out sudden jumps between neighboring peaks in the spectral function. }
 \\
 \alert{
-We understand the point of the reviewer but "intruder state" and "regularization" are well-defined terms in the electronic structure community which re not linked with the appearance of spurious solution.
+We understand the point of the reviewer but "intruder state" and "regularization" are well-defined terms in the electronic structure community which are not linked with the appearance of spurious solutions.
 The intruder state problem is well documented in multireference perturbation theory and comes usually from a poor choice of the active space.
 By definition, an intruder state has a similar energy than the zeroth-order wave function and should be then moved in the model space; this is exactly what is happening in the case of $GW$.
 }
@@ -232,7 +232,11 @@ By definition, an intruder state has a similar energy than the zeroth-order wave
 The authors should be aware that traditional GW calculations performed in solids often use a small broadening, and hence such a "regularization" is naturally captured by many codes - whether or not on purpose.}
 \\
 \alert{
-As detailed below (see the answers to Reviewer \#1), we have thoroughly modified and expanded this section to test the effect of the regularization function and its parameter.
+We have specified this point in Section V of the revised manuscript.
+Moreover, as detailed below (see the answers to Reviewer \#1), we have thoroughly modified and expanded this section to test the effect of the regularization function and its parameter.
+In particular, we have changed the notations regarding the various regularizers that we have studied.
+We now use $\eta$ for the traditional regularizer and $\kappa$ for Evangelista's regularizer.
+We have also included additional graphs for different values of $\eta$ and $\kappa$ which shows how the quasiparticle energies are altered by the choice of the regularizing function and the values of $\eta$ and $\kappa$.
 }
 
 \item