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\begin{letter}%
{To the Editors of the Journal of Chemical Theory and Computation,}

\opening{Dear Editors,}

\justifying
Please find attached a revised version of the manuscript entitled 
\begin{quote}
  \textit{``A similarity renormalization group approach to Green’s function methods''}.
\end{quote}
      
We thank the reviewers for their constructive comments.
Our detailed responses to their comments can be found below.
For convenience, changes are highlighted in red in the revised version of the manuscript. 

We look forward to hearing from you.

\closing{Sincerely, the authors.}

\newpage

%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}

\begin{itemize}

\item 
{The article of Marie and Loos describes a regularized $GW$ approach inspired by the similarity renormalization group second-order perturbative analysis to the linear $GW$ eigenvalue equations. The article is well-organized and the presentation is clear. I think this article can be accepted as is. Nonetheless, I do have a few minor suggestions.}
\\
\alert{We thank the reviewer for supporting publication of the present manuscript.
}

\item 
{In Eq. (45), the authors mention a reverse approach where, if I understand correctly, the $\omega$-dependent self-energy is directly modified using the SRG regularizer. How does this approach perform on $GW$50 and compare to qs$GW$ and SRG-qs$GW$?}
\\
\alert{}

\item 
{I am a bit surprised that the SRG-qs$GW$ converges all molecules for $s = 1000$ but not for $s = 5000$. The energy cutoff window is very narrow here: $0.032$-$0.014$ Ha. Moreover, from Figs. 3, 4, and 6, the IPs are roughly converged in the order of $s = 50$ to a few $100$. I think an analysis of the denominators $\Delta^{\nu}_{pr}$ for the typical molecules would be very informative. In particular, what are the several smallest denominators at the beginning and how do they change along the self-consistency procedure?}
\\
\alert{}

\item 
{In Eq. (18), I think $H^{\text{od}}$ is generally not a square matrix and it is better to say  $H^{\text{od}}(s)^\dagger H^{\text{od}}(s)$ instead of $H^{\text{od}}(s)^2$.}
\\
\alert{Indeed, the expression suggested by the reviewer would be more precise. 
The corresponding expression in the manuscript has been updated.}

\item 
{I think the y axis (counts in each bin) should be presented in Figs. 5 and 7. Or at least the  limit of y axis should be fixed for all subplots in Fig. 5 or Fig. 7.}
\\
\alert{We added the scaling of the vertical axis of each panel of Fig.~5 and Fig.~7.}

\end{itemize}

%%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2}

\begin{itemize}

\item 
{This is an excellent manuscript, which I very much enjoyed reading. In particular, it includes a comprehensive overview of the literature in the field, which I find very valuable (ref. 119 should be updated). The final result is an expression with a slightly different regularization as before, but it works well, is well-founded, and is easy to implement. I don't see arguments against it.}
\\
\alert{We thank the reviewer for supporting publication of the present manuscript.
}

\item 
{There are two issues that may be improved:

The authors used the "dagger" symbol in eq. (21) and further, although they use real-valued spin-orbitals. In that case, also the matrices $W$ are real. It seems more consistent to either allow for complex-valued spin-orbitals (e.g. in eq. (8)) or only use the matrix transpose.}
\\
\alert{Indeed, this is inconsistent. Therefore, we have changed the definition in Eq.~(8) in order to allow for complex-valued spin-orbitals. We thank the reviewer for pointing this out.}

\item 
{I find it somewhat disturbing to see positive and negative electron affinities. The authors may wish to comment briefly on the meaning of the sign.}
\\
\alert{This point has been already discussed in the original manuscript at the very end of Section VI, see the following paragraph:}
  
  \textcolor{red}{\textit{''Note that a positive EA indicates a bounded anion state, which can be accurately described by the methods considered in this study. However, a negative EA suggests a resonance state, which is beyond the scope of the methods used in this study, including the $\Delta$CCSD(T) reference. As such, it is not advisable to assign a physical interpretation to these values. Nonetheless, it is possible to compare $GW$-based and $\Delta$CCSD(T) values in such cases, provided that the comparison is limited to a given basis set.''}}


\end{itemize}

\end{letter}
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