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Manuscript/ufGW.tex

@@ -422,7 +422,7 @@ 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{Not that this type of broadening is customary in solid-state calculations, hence such regularization is naturally captured in many codes. \cite{Martin_2016}}
+\alert{Note that 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 \alert{(MP2)}, 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}
@@ -450,7 +450,7 @@ Let us now discuss the SRG-based energy-dependent regularizer provided in Eq.~\e
 For $\alert{\kappa} = \SI{10}{\hartree}$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves).
 For $\alert{\kappa} = \SI{0.1}{\hartree}$, we have the opposite scenario where $\alert{\kappa}$ is too small and some irregularities remain (green curves).
 We have found that $\alert{\kappa} = \SI{1.0}{\hartree}$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions. 
-Moreover, this value performs well in all scenarios that we have encountered.
+Moreover, \alert{although the optimal $\kappa$ is obviously system-dependent}, this value performs well in all scenarios that we have encountered.
 However, it can be certainly refined for specific applications. 
 \alert{For example, in the case of regularized MP2 theory (where one relies on a similar energy-dependent regularizer), a value of $\kappa = 1.1$ have been found to be optimal for noncovalent interactions and transition metal thermochemistry. \cite{Shee_2021}}
 
@@ -460,12 +460,13 @@ Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brou
 \alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ [and the simple regularizer given in Eq.~\eqref{eq:simple_reg}] are reported as {\SupMat}, where one clearly sees that the larger the value of $\kappa$, the larger the difference between regularized and non-regularizer quasiparticle energies.}
 
 As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis.
-In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol detailed in Ref.~\onlinecite{Loos_2020e}.
+In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol as detailed in Ref.~\onlinecite{Loos_2020e}.
 These results are compared to high-level coupled-cluster \alert{(CC)} calculations extracted from the same work: \alert{CC with singles and doubles (CCSD) \cite{Purvis_1982} and the non-perturbative third-order approximate CC method (CC3). \cite{Christiansen_1995b}}
 As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve computed with $\alert{\kappa} = 1$) allows to smooth it out without significantly altering the overall accuracy.
 Moreover, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface (gray curve). 
 \alert{For the sake of completeness, a similar graph for $\kappa = 10$ is reported as {\SupMat}.
-Interestingly, for this rather large value of $\kappa$, the smooth BSE@{\GOWO}@HF and BSE@ev$GW$@HF curves are superposed, and of very similar quality as CCSD.}
+Interestingly, for this rather large value of $\kappa$, the smooth BSE@{\GOWO}@HF and BSE@ev$GW$@HF curves are superposed, and of very similar quality as CCSD.
+It is therefore clear that a smaller value of $\kappa$ is more suitable.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Concluding remarks}

Response_Letter/Response_Letter.tex

@@ -59,7 +59,7 @@ The references provided by the reviewer has been added in due place and this poi
 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 point was already stressed in our original manuscript.
 This particular case is further 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 clearified in the revised version of the manuscript where we have also added the reference provided by Reviewer \#1.}
+This point has been clarified in the revised version of the manuscript where we have also added the reference provided by Reviewer \#1.}
 
 
 {It is my understanding that the calculation on the illustrative example H2 in 6-31G basis illustrates the existence of multiple such solutions. 
@@ -133,6 +133,7 @@ Could authors elaborate what they see differently and what my untrained eyes cou
 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 $\kappa$.
+The discussion has been modified accordingly and is now much more complete.
 }
 
 \item 
@@ -142,14 +143,19 @@ However, I cannot understand how the authors can claim that the correction intro
 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 of a correction in these cases. 
+Thus, it is an important feature that the regularization introduces only a small correction for these orbitals, as shown in Fig.~4.
+Additional graphs have been added as supplementary material where one shows that if one considers a larger value of $\kappa$, the energies are much more altered.
+As mentioned by the reviewer, 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 must emphasize that the quasiparticle solutions of Eq.~(2) for these orbitals appear at the poles of the self-energy. 
+Therefore, the regularization has to do its job which leads to large deviation of the quasiparticle energies; this is the only way to get rid of these discontinuities.
+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 is 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 information for different values of $\eta$ and $\kappa$.
-The discussion has been expanded accordingly.
+The discussion has been expanded accordingly (see also the previous point).
 }
 
 \item 
@@ -157,8 +163,9 @@ The discussion has been expanded accordingly.
 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 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.
+\alert{We have added several graphs in the supporting information where we use different values for $\eta$ and $\kappa$. 
+The discussion in Section V has been expanded to discuss these new results.
+As one can see, the value $\kappa = 1$ gives much better results than $\eta = 1$.
 }
 
 \item 
@@ -167,9 +174,9 @@ 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.
+Like in other regularized methods, $\eta$ is an empirical parameter.
 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).
+This point is mentioned in the manuscript (Section V) where we explicitly mention that the optimal $\kappa$ value is system-dependent. 
 We hope to report further on this in a forthcoming paper but this requires extensive calculations. 
 }
 
@@ -182,6 +189,11 @@ If they do, then I find it highly unusual.
 Or are these competing solutions that could be removed by passing the starting point between neighboring geometries without a regularizer?}
 \\
 \alert{
+We understand the point of the review.
+However, this strategy would not work in the present context as the various solutions are not continuously linked, and one cannot follow indefinitely a given solution  as its weight eventually becomes extremely small.
+Some extra "bumpiness" may have been introduced by the second-order interpolation of the curves between each pair of successive points.
+We have removed this interpolation for all the curves and, as the reviewer shall see, the curves remain very "bumpy".
+To answer the reviewer's question, yes the denominator already vanishes at equilibrium and this is far from being unusual in $GW$ methods [see J. Chem. Theory Comput. 14, 5220 (2018)].
 }
 
 
@@ -194,7 +206,7 @@ Yes, but a plot showing that the total energy is relatively insensitive as a fun
 \\
 \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).
-}
+From a more personal point of view, we believe that an efficient regularization scheme of $GW$ methods might significantly improve their applicability in quantum chemistry.}
 
 \end{enumerate}
 
@@ -243,6 +255,7 @@ Moreover, as detailed below (see the answers to Reviewer \#1), we have thoroughl
 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 $\kappa$.
+The discussion has been also expanded.
 }
 
 \item