saving owkr in response letter

Response_Letter/Response_Letter.tex

@@ -46,15 +46,19 @@ In such cases, the potential energy curves are smooth within a region where one
 \\
 \alert{
 The reviewer is correct; these issues are more general and do not originate solely from the vanishing denominator.
-These multiple solutions issues stems from the fact that, in most $GW$ methods, one only consider the quasiparticle solutions at each iteration, discarding the other solutions known as satellites.
-In a fully self-consistent scheme where one takes into account all these solutions at each iteration, these issues do not appear, as we have recently discussed in Ref.~[Front. Chem. (9, 751054 (2021). ]. }
-
+For example, we have recently studied multiple solutions in another non-linear method, coupled-cluster theory [J. Chem. Theory Comput. 17, 4756 (2021)], where their origin stems from the non-linear nature of the equations, only.
+However, in the case of $GW$ at least, we have clearly observed that these multiple solution issues appear when denominators vanish.
+More precisely, in most $GW$ methods, because one only considers the quasiparticle solution at each iteration while discarding the other solutions (known as satellites, discontinuities are observed. 
+This has been studied in details in a previous study [J. Chem. Theory Comput. 14, 5220 (2018)].
+In a fully self-consistent scheme where one takes into account all these solutions at each iteration, these issues do not appear, as we have recently discussed in a recent paper [Front. Chem. (9, 751054 (2021)]. 
+The references provided by the reviewer has been added in due place and this point has been clarified in the revised version of the manuscript.}
 
 {Convergence accelerators such as DIIS, KAIN, LCIS, can be successfully used and smooth PES can be produced when one is relatively close to a local minima, one should not expect that DIIS will help with convergence where multiple close lying solutions exists and when two solutions are competing, see J. Chem. Phys. 156, 094101 (2022).}
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 \alert{
-Indeed, convergence accelerators such as DIIS can be used to ease convergence but they will not make these discontinuitius disappear as their origin is more profond.
-This particular case is discussed in Ref.~[. Chem. Theory Comput. 14, 5220 (2018)] where we have provided our implementation of DIIS within $GW$ methods (see Appendix).}
+Indeed, convergence accelerators such as DIIS can be used to ease convergence but they will not make these discontinuities disappear as their origin is more profond.
+This particular case is discussed in a recent paper [J. Chem. Theory Comput. 14, 5220 (2018)] where we have provided our implementation of DIIS within $GW$ methods (see Appendix).
+This point has been further stressed in the revised version of the manuscript.}
 
 
 {It is my understanding that the calculation on the illustrative example H2 in 6-31G basis illustrates the existence of multiple such solutions. 
@@ -63,7 +67,8 @@ Of course these issues of vanishing denominators and multiple solutions due to t
 Frankly, I find it infuriating that in the GW community many times it is not clearly spelled out which flavor of the so-called GW is used and how the equations are solved. 
 }
 \\
-\alert{We agree with the reviewer that, historically, the $GW$ literature has been unclear on how to solve these equations, but we would like to stress that our group has made a clear effort to provide all the working equations and details necessary to solve these equations in the different cases.}
+\alert{We agree with the reviewer that, historically, the literature has been unclear on how to solve the $GW$ equations, but we would like to stress that our group has made a clear effort to provide all the working equations and details necessary to solve these equations in the different cases.
+We have added additional details and references in order to guide the readers to the relevant papers when one can find all the necessary details regarding the implementation of $GW$ methods.}
 
 {Here, I will describe more specific issues that I noticed during the reading of the manuscript:}
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@@ -89,7 +94,7 @@ On the same note, it would be good if the authors mentioned that the scheme that
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 \alert{
 Thank you for pointing this out. 
-We now clearly state that $O(K^9)$ scheme is illustrative and that state-of-the-art $GW$ calculations scales as $O(K^3)$ instead of $O(K^6)$.
+We now clearly state that our $O(K^9)$ scheme is illustrative and that state-of-the-art $GW$ calculations scales as $O(K^3)$ instead of $O(K^6)$.
 }
 
 \item 
@@ -101,16 +106,19 @@ We now clearly state that $s$ numbers solutions.
 }
 
 \item 
-{I assume that in Fig.~1 authors plot the lowest solution after the diagonalization for each orbital p.}
+{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 ``normal'' $G_0W_0$ scheme.
+\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.
 }
 
 \item 
-{In Fig.~2, the thick and thin solid lines are hardly visible. A better way of displaying is necessary. 
+{In Fig.~2, the thick and thin solid lines are hardly visible. 
+A better way of displaying is necessary. 
 Otherwise it all is a bit incomprehensible.}
 \\
-\alert{Visibility of the different solid lines has been improved. The authors thank the reviewer for this valuable comment.
+\alert{Visibility of the different solid lines has been improved. 
+We thank the reviewer for this valuable comment.
+Figure 2 illustrates one of the key results of our work and must be as clear as possible.
 }
 
 \item 
@@ -118,23 +126,26 @@ Otherwise it all is a bit incomprehensible.}
 It seems to me that the difference between the values of qp energies before and post shifting are of the same order of magnitude for both regularizers. 
 Could authors elaborate what they see differently and what my untrained eyes could not see?}
 \\
-\alert{We had additionnal graphs for different values of $\eta$, i.e., the traditional shift, and $\kappa$, i.e., the Evangelista shift.
+\alert{We agree that this part was incomplete and, therefore, we have made extensive efforts in order to improve it.
+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 
 {In Fig.~4, again I do not understand author's conclusions. 
 I understand that the regularization seems to introduce only 10 meV correction to the Homo and Lumo which is good. 
-However, I cannot understand how the authors can claim that the correction introduced to the p=3 and 4 is viable. 
+However, I cannot understand how the authors can claim that the correction introduced to the $p=3$ and 4 is viable. 
 These qp energies are different by 2-3 eV. How are the smooth curves advantageous if the results are so incorrect? 
 Could authors elaborate?}
 \\
-\alert{Indeed the HOMO and LUMO orbitals do not show discontinuities along the dissociation coordinate so no need for a correction. Thus, it is an important feature that the regularization introduces only a small correction for these orbitals. It is also true that the regularization introduces a correction of few eVs for the LUMO+1 (p=3) and LUMO+2 (p=4) orbitals but we have to note that the quasiparticle solutions of Eq.~2 for these orbitals appear at the poles of the self-energy. So the regularized self-energy has to do a large correction which leads to large error on the quasiparticle energies. Moreover, it is also essential to notice that we talk here about the $G_0W_0$ scheme but in case of a partially self-consistent scheme then the use of regularization seems critical.
+\alert{Indeed the HOMO and LUMO orbitals do not show discontinuities along the dissociation coordinate so no need for a correction. Thus, it is an important feature that the regularization introduces only a small correction for these orbitals. It is also true that the regularization introduces a correction of few eVs for the LUMO+1 ($p=3$) and LUMO+2 ($p=4$) orbitals but we have to note that the quasiparticle solutions of Eq.~2 for these orbitals appear at the poles of the self-energy. So the regularized self-energy has to do a large correction which leads to large error on the quasiparticle energies. Moreover, it is also essential to notice that we talk here about the $G_0W_0$ scheme but in case of a partially self-consistent scheme then the use of regularization seems critical.
 }
 
 \item 
 {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 for different values of $\eta$ and $\kappa$.
+\alert{Several graphs have been added in the supporting information for different values of $\eta$ and $\kappa$.
 }
 
 \item 
@@ -151,6 +162,9 @@ It is reasonable to expect that the value of $\eta$ is system dependent.
 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.
+We hope to report further on this in a forthcoming paper but this requires extensive calculations. 
 }
 
 \item 
@@ -173,6 +187,7 @@ Can this paper be published as a rapid communication?
 Yes, but a plot showing that the total energy is relatively insensitive as a function of regularizer $\eta$ would make the argument more convincing to broaden the applicability of the partially self-consistent Green's function on the real axis.}
 \\
 \alert{
+As stated above, the regularizer section has been improved and we are happy if our manuscript is published as a Regular Article (as recommended by the editor).
 }
 
 \end{enumerate}
@@ -182,7 +197,8 @@ Yes, but a plot showing that the total energy is relatively insensitive as a fun
  
 {The article by Monino and Loos presents a way to avoid numerical difficulties arising from GW calculations in moderately correlated electronic systems, where a small perturbation in a particular parameter of the problem, such as the interatomic distance, leads to a sudden jump in the main quasiparticle peak, as computed within the GW approximation. The article is written clearly and includes new, original insights that could be useful for an audience of specialists interested in electronic-structure calculations. I support the article for publication, but ask the authors to address my points below: }
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-\alert{
+\alert{Thank you for these positive comments and for supporting publication of our manuscript. 
+Below, we address the points raised by the reviewer.
 }
 
 \begin{enumerate}