some corrections

Manuscript/SRGGW.bib

@@ -2170,6 +2170,13 @@
 	note = {Gaussian Inc. Wallingford CT 2009},
 	title = {Gaussin~09 {R}evision {E}.01}}
 
+@misc{g16,
+author={M. J. Frisch and G. W. Trucks and H. B. Schlegel and G. E. Scuseria and M. A. Robb and J. R. Cheeseman and G. Scalmani and V. Barone and G. A. Petersson and H. Nakatsuji and X. Li and M. Caricato and A. V. Marenich and J. Bloino and B. G. Janesko and R. Gomperts and B. Mennucci and H. P. Hratchian and J. V. Ortiz and A. F. Izmaylov and J. L. Sonnenberg and D. Williams-Young and F. Ding and F. Lipparini and F. Egidi and J. Goings and B. Peng and A. Petrone and T. Henderson and D. Ranasinghe and V. G. Zakrzewski and J. Gao and N. Rega and G. Zheng and W. Liang and M. Hada and M. Ehara and K. Toyota and R. Fukuda and J. Hasegawa and M. Ishida and T. Nakajima and Y. Honda and O. Kitao and H. Nakai and T. Vreven and K. Throssell and Montgomery, {Jr.}, J. A. and J. E. Peralta and F. Ogliaro and M. J. Bearpark and J. J. Heyd and E. N. Brothers and K. N. Kudin and V. N. Staroverov and T. A. Keith and R. Kobayashi and J. Normand and K. Raghavachari and A. P. Rendell and J. C. Burant and S. S. Iyengar and J. Tomasi and M. Cossi and J. M. Millam and M. Klene and C. Adamo and R. Cammi and J. W. Ochterski and R. L. Martin and K. Morokuma and O. Farkas and J. B. Foresman and D. J. Fox},
+title={Gaussian˜16 {R}evision {C}.01},
+year={2016},
+note={Gaussian Inc. Wallingford CT}
+} 
+
 @article{QChem,
 	author = {Shao, Y. and Fusti-Molnar, L. and Jung, Y. and Kussmann, J. and Ochsenfeld, C. and Brown, S. T. and Gilbert, A. T. B. and Slipchenko, L. V. and Levchenko, S. V. and O'Neill, D. P. and Distasio Jr., R. A. and Lochan, R. C. and Wang, T. and Beran, G. J. O. and Besley, N. A. and Herbert, J. M. and Lin, C. Y. and Van Voorhis, T. and Chien, S. H. and Sodt, A. and Steele, R. P. and Rassolov, V. A. and Maslen, P. E. and Korambath, P. P. and Adamson, R. D. and Austin, B. and Baker, J. and Byrd, E. F. C. and Dachsel, H. and Doerksen, R. J. and Dreuw, A. and Dunietz, B. D. and Dutoi, A. D. and Furlani, T. R. and Gwaltney, S. R. and Heyden, A. and Hirata, S. and Hsu, C.-P. and Kedziora, G. and Khalliulin, R. Z. and Klunzinger, P. and Lee, A. M. and Lee, M. S. and Liang, W. and Lotan, I. and Nair, N. and Peters, B. and Proynov, E. I. and Pieniazek, P. A. and Rhee, Y. M. and Ritchie, J. and Rosta, E. and Sherrill, C. D. and Simmonett, A. C. and Subotnik, J. E. and Woodcock III, H. L. and Zhang, W. and Bell, A. T. and Chakraborty, A. K. and Chipman, D. M. and Keil, F. J. and Warshel, A. and Hehre, W. J. and Schaefer III, H. F. and Kong , J. and Krylov, A. I. and Gill, P. M. W. and Head-Gordon, M.},
 	date-added = {2020-12-09 22:47:45 +0100},
@@ -2180,6 +2187,20 @@
 	volume = {8},
 	year = {2006}}
 
+
+@article{CFOUR,
+  title = {Coupled-Cluster Techniques for Computational Chemistry: {{The CFOUR}} Program Package},
+  author = {Matthews, Devin A. and Cheng, Lan and Harding, Michael E. and Lipparini, Filippo and Stopkowicz, Stella and Jagau, Thomas-C. and Szalay, P{\'e}ter G. and Gauss, J{\"u}rgen and Stanton, John F.},
+  year = {2020},
+  journal = {The Journal of Chemical Physics},
+  volume = {152},
+  number = {21},
+  pages = {214108},
+  issn = {0021-9606},
+  doi = {10.1063/5.0004837}
+}
+
+
 @article{Hirata_2004,
 	author = {Hirata, S.},
 	date-added = {2020-12-09 21:02:23 +0100},

Manuscript/SRGGW.tex

@@ -547,7 +547,16 @@ For $s\to\infty$, it tends towards the following static limit
 \end{equation}
 while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
 \begin{equation}
-	\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
+  \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
+\end{equation}
+with
+\begin{equation}
+  \label{eq:SRG-GW_selfenergy}
+  \begin{split}
+    \widetilde{\bSig}_{pq}(\omega; s) 
+  &= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\ 
+  &+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
+  \end{split}
 \end{equation}
 Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
@@ -598,17 +607,17 @@ The dynamic part after the change of variable is actually closely related to the
 %=================================================================%
 
 % Reference comp det
-The geometries have been optimized without frozen core approximation at the CC3 level in the aug-cc-pVTZ basis set using the CFOUR program.
-The reference CCSD(T) ionization potential (IP) energies have been obtained using default parameters of Gaussian 16.
-This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted HF formalism.
+The geometries have been optimized without frozen core approximation at the CC3 level in the aug-cc-pVTZ basis set using the CFOUR program. \cite{CFOUR}
+The reference CCSD(T) ionization potential (IP) energies have been obtained using the default parameters of Gaussian 16. \cite{g16}
+This means that the cations use an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted HF formalism.
 
 % GW comp det
 The two qs$GW$ variants considered in this work have been implemented in an in-house program.
 The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
-All $GW$ calculations were performed without the frozen-core approximation.
+All $GW$ calculations were performed without the frozen-core approximation and in the aug-cc-pVTZ cartesian basis set.
 The DIIS space size and the maximum of iterations were set to 5 and 64, respectively.
 In practice, one could (and should) achieve convergence in some cases by adjusting these parameters or by using an alternative mixing scheme.
-However, in order to perform a black-box comparison of the methods these parameters have been fixed.
+However, in order to perform a black-box comparison of the methods these parameters have been fixed to these default values.
 
 %=================================================================%
 \section{Results}
@@ -754,13 +763,12 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
 \end{figure*}
 %%% %%% %%% %%%
 
-The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
-This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
-In addition, the MgO and O3 molecules (which are part of GW100 as well) has been added to the test set because they are known to possess intruder states. \cite{vanSetten_2015,Forster_2021}
+The test set considered in this study is based on the GW20 set introduced by Lewis and Berkelbach, \cite{Lewis_2019} which is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
+This set has been augmented with the MgO and O3 molecules (which are part of GW100 as well) because they are known to possess intruder states. \cite{vanSetten_2015,Forster_2021}
 
 Table~\ref{tab:tab1} shows the principal IP of the 22 molecules considered in this work computed at various level of theories.
-As mentioned previously the HF IPs are overestimated with a mean signed error (MSE) and mean absolute error (MAE) of \SI{0.64}{\electronvolt} and \SI{0.74}{\electronvolt}, respectively.
-Performing a one-shot $G_0W_0$ calculation on top of it allows to divided by more than 2 the MSE and MAE, \SI{0.26}{\electronvolt} and \SI{0.32}{\electronvolt}, respectively.
+As mentioned previously the HF IPs are overestimated with a mean signed error (MSE) of \SI{0.64}{\electronvolt} and a mean absolute error (MAE) of \SI{0.74}{\electronvolt}.
+Performing a one-shot $G_0W_0$ calculation on top these mean-field results allows to divided by more than two the MSE and MAE, \SI{0.26}{\electronvolt} and \SI{0.32}{\electronvolt}, respectively.
 However, there are still outliers with quite large errors, for example the IP of the dinitrogen is overestimated by \SI{1.56}{\electronvolt}.
 Self-consistency can mitigate the error of the outliers as the maximum error at the qs$GW$ level is now \SI{0.56}{\electronvolt} and the standard deviation error (SDE) is decreased from \SI{0.39}{\electronvolt} for $G_0W_0$ to \SI{0.18}{\electronvolt} for qs$GW$.
 In addition, the MSE and MAE (\SI{0.24}{\electronvolt}/\SI{0.25}{\electronvolt}) are also slightly improved with respect to $G_0W_0$@HF.
@@ -770,15 +778,16 @@ Table~\ref{tab:tab1} shows the SRG-qs$GW$ values for $s=100$.
 For this value of the flow parameter, the MAE is converged to \SI{d-3}{\electronvolt} (see Supplementary Material).
 The statistical descriptors corresponding to the alternative static self-energy are all improved with respect to qs$GW$.
 Of course these are slight improvements but this is done with no additional computational cost and can be implemented really quickly just by changing the form of the static approximation.
-The evolution of statistical descriptors with respect to the various methods considered in Table~\ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
-The decrease of the MSE and SDE correspond to a shift of the maximum of the distribution toward 0 and a shrink of the width of the distribution, respectively.
+The evolution of the statistical descriptors with respect to the various methods considered in Table~\ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
+The decrease of the MSE and SDE correspond to a shift of the maximum toward zero and a shrink of the distribution width, respectively.
 
 In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
 Indeed, up to $s=10^3$ self-consistency of the SRG-qs$GW$ scheme can be converged without any problems.
-For $s=10^4$, convergence could not be attained for the following molecules.
-On the other hand, the qs$GW$ convergence is much more erratic, the 22 molecules could be converged for $\eta=0.1$.
-However, if we decrease $\eta$ then convergence could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
-Unfortunately, the convergence of the IP is not as tight as in the SRG case because for $\eta=0.01$ the IP that could be converged can vary by $10^{-2}$ to $10^{-1}$ with respect to $\eta=0.1$.
+For $s=10^4$, convergence could not be attained for the following molecules \ANT{waiting for calculation}.
+On the other hand, the qs$GW$ convergence is much more erratic.
+The whole set considered in this work could be converged for $\eta=0.1$.
+However, if we decrease $\eta$ then self-consistency could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
+Unfortunately, the convergence of the IP is not as tight as in the SRG case because for $\eta=0.01$ the values of the IP that could be converged can vary between $10^{-3}$ and $10^{-1}$ with respect to $\eta=0.1$.
 
 We will now gauge the effect of the TDA for the screening on the accuracy of the various methods considered previously.