working on response letter and modification of the manuscript

Manuscript/ufGW.tex

@@ -173,7 +173,7 @@ Atomic units are used throughout.
 \section{Downfolding: The non-linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
+Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$ \alert{and assuming real values of the frequency $\omega$}, the following (non-linear) quasiparticle equation
 \begin{equation}
 \label{eq:qp_eq}
 	\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
@@ -204,7 +204,7 @@ and
 are two-electron integrals over the HF (spatial) orbitals $\MO{p}(\br)$. 
 Because one must compute all the RPA eigenvalues and eigenvectors to construct the self-energy \eqref{eq:SigC}, the computational cost is $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals, respectively, and $\Norb = \Nocc + \Nvir$ is the total number of orbitals.
 
-As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ and their corresponding weights are given by the value of the following renormalization factor
+As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ \alert{(where the index $s$ is numbering solutions)} and their corresponding weights are given by the value of the following renormalization factor
 \begin{equation}
 \label{eq:Z}
 	0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
@@ -260,6 +260,7 @@ and the corresponding coupling blocks read
 The size of this eigenvalue problem is $1 + \Nocc^2 \Nvir + \Nocc \Nvir^2 = \order*{\Norb^3}$, and it has to be solved for each orbital that one wishes to correct.
 Thus, this step scales as $\order*{\Norb^9}$ with conventional diagonalization algorithms.
 Note, however, that the blocks $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$ do not need to be recomputed for each orbital.
+\alert{Of course, this $\order*{\Norb^9}$ scheme is purely illustrative and current state-of-the-art $GW$ implementation scales as $\order*{\Norb^3}$ thanks to efficient contour deformation and density fitting techniques. \cite{Duchemin_2019,Duchemin_2020,Duchemin_2021}}
 
 It is crucial to understand that diagonalizing $\bH^{(p)}$ [see Eq.~\eqref{eq:Hp}] is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
 This can be further illustrated by expanding the secular equation associated with Eq.~\eqref{eq:Hp}
@@ -478,7 +479,7 @@ This project has received funding from the European Research Council (ERC) under
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Data availability statement}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The data that supports the findings of this study are available within the article.% and its supplementary material.
+The data that supports the findings of this study are available within the article and its supplementary material.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \bibliography{ufGW}

Response_Letter/Response_Letter.tex

@@ -84,7 +84,7 @@ Please, specify it. It may seem trivial to a regular GW practitioner, it is not
 \\
 \alert{
 The evaluation of all the quantities of interest ares performed on the real axis, i.e., for real values of $\omega$.
-This is now stated clearly in the revised version of the manuscript. 
+This is now stated clearly in the revised version of the manuscript at the very beginning of Section II.
 }
 
 \item 
@@ -102,13 +102,15 @@ We now clearly state that our $O(K^9)$ scheme is illustrative and that state-of-
 I do not think that it is clearly mentioned in the article when the notation appears first time.}
 \\
 \alert{
-We now clearly state that $s$ numbers solutions.
+We now clearly state that $s$ numbers solutions just above Eq.~(8).
 }
 
 \item 
 {I assume that in Fig.~1 authors plot the lowest solution after the diagonalization for each orbital $p$.}
 \\
 \alert{In Fig.~1 we plot the quasiparticle solution for each orbital, i.e, the solution with the largest spectral weight which is obtained using the $G_0W_0$ scheme.
+This is, in particular, mentioned in the caption of Fig.~1.
+In general, for each orbital, it exists additional (satellites) solutions with lower and higher energies than the quasiparticle solution.
 }
 
 \item 
@@ -146,6 +148,7 @@ Could authors elaborate?}
 {How the values of Fig.~4 depend on different choices of $\eta$ magnitude? This is crucial for assessing if a regularizer scheme is viable. }
 \\
 \alert{Several graphs have been added in the supporting information for different values of $\eta$ and $\kappa$.
+The discussion has been expanded accordingly.
 }
 
 \item 
@@ -153,7 +156,8 @@ Could authors elaborate?}
 Without showing such a graph it is hard to know. 
 Also would the regularizer help when the HF eigenvalues of Homo and Lumo are known to become degenerate at the stretched distance?}
 \\
-\alert{We add few graphs in the supporting where we use different values for the $\eta$ and the $\kappa$ regularizer. We can see that the value $\kappa = 1$ gives a much better result than the $\eta = 1$ one.
+\alert{We add few graphs in the supporting information where we use different values for the $\eta$ and the $\kappa$ regularizer. 
+We can see that the value $\kappa = 1$ gives a much better result than the $\eta = 1$ one.
 }
 
 \item 
@@ -164,6 +168,7 @@ How could I recognize which value is right?}
 \alert{
 Like in other regularized methods, $\eta$ is an empirical parameter that must be chosen.
 There is no definite answer but, depending on the type of properties studied, the value of $\eta$ must be chosen carefully.
+This point is mentioned in the manuscript (Section V).
 We hope to report further on this in a forthcoming paper but this requires extensive calculations. 
 }