Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW

Manuscript/SRGGW.tex

@@ -207,7 +207,7 @@ with
 \end{subequations}
 and where
 \begin{equation}
-	\braket{pq}{rs} = \iint \frac{\textcolor{red}{\SO{p}^*(\bx_1) \SO{q}^*(\bx_2)}\SO{r}(\bx_1) \SO{s}(\bx_2)  }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2
+	\braket{pq}{rs} = \iint \frac{\SO{p}(\bx_1) \SO{q}(\bx_2)\SO{r}(\bx_1) \SO{s}(\bx_2)  }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2
 \end{equation}
 are bare two-electron integrals in the spin-orbital basis.
 
@@ -620,11 +620,13 @@ Performing a bijective transformation of the form,
   e^{- \Delta s} &= 1-e^{-\Delta t},
 \end{align}
 on the renormalized quasiparticle equation \eqref{eq:GW_renorm} reverses the situation and makes it possible to choose $t$ such that there is no intruder states in the dynamic part, hence removing discontinuities.
-
-\textcolor{red}{The intruder-state free dynamic part makes it possible to define a SRG-$G_0W_0$ and SRG-ev$GW$.
-The main manuscript focus on SRG-qs$GW$ but the performance of SRG-$G_0W_0$ and SRG-ev$GW$ are discussed in the {\SupInf} for the sake of completeness.}
 Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
 
+\textcolor{red}{The intruder-state-free dynamic part of the self-energy makes it possible to define SRG-$G_0W_0$ and SRG-ev$GW$ schemes.
+Although the manuscript focuses on SRG-qs$GW$, the performance of SRG-$G_0W_0$ and SRG-ev$GW$ are discussed in the {\SupInf} for the sake of completeness.
+In a nutshell, the SRG regularization improves the overall convergence properties of SRG-ev$GW$ without altering its performance.
+Likewise, the statistical indicators for $G_0W_0$ and SRG-$G_0W_0$ are extremely close.}
+
 %=================================================================%
 \section{Computational details}
 \label{sec:comp_det}

Response_Letter/Response_Letter.tex

@@ -43,13 +43,13 @@ We look forward to hearing from you.
 \\
 \alert{This reversed approach allows one to remove the intruder states from the dynamic part rather than the static part.
   Therefore, this should be used for SRG-$G_0W_0$ and SRG-ev$GW$ calculations.
-  Following the reviewer's suggestion and for the sake of completeness, the performance of these two methods are now discussed in addition to SRG-qs$GW$.
+  Following the reviewer's suggestion and for the sake of completeness, the performance of these two methods is now discussed in addition to SRG-qs$GW$.
 Yet, we decided to include this discussion in the supporting information in order not to bring confusion to the take-home message of the manuscript which is the new SRG-qs$GW$ static form.}
 
 \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{We thank the reviewer for this suggestion. Unfortunately, we have not been able to extract useful information out of the enormous number of denominators. Indeed, the number of small denominators is large and their associated screened integrals can be very different as well, hence it is difficult to explain quantitatively why the convergence is not reached for a given value of the energy cutoff. Note that one could maybe converge the calculation for $s=5000$ by changing the DIIS parameters for example.}
+\alert{We thank the reviewer for this suggestion. Unfortunately, we have not been able to extract useful information out of the enormous number of denominators. Indeed, the number of small denominators is large and their associated screened integrals can be very different as well, hence it is difficult to explain quantitatively why the convergence is not reached for a given value of the energy cutoff. Note that it is possible one would be able to converge the calculations for $s=5000$ by changing, for example, the DIIS parameters.}
 
 \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 $H^{\text{od}}(s)^2$.}
@@ -80,12 +80,13 @@ The corresponding expression in the manuscript has been updated.}
 
 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.}
+%\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.}
+\alert{Although we agree with the reviewer, we would prefer to stick with the more general "dagger" notation. Allowing complex-valued orbitals would modify equations in several places in a non-trivial way. This would definitely alter the readability and accessibility of the paper which we would like to avoid.}
 
 \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:}
+\alert{This point has been already discussed in the original manuscript at the very end of Section VI:}
   
   \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.''}}