saving ork in revision

Manuscript/sfBSE.bib

@@ -1,13 +1,58 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2021-02-19 10:50:27 +0100 
+%% Created for Pierre-Francois Loos at 2021-02-25 09:23:35 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Chai_2008a,
+	author = {Chai, J. D. and Head-Gordon, M.},
+	date-added = {2021-02-25 09:23:14 +0100},
+	date-modified = {2021-02-25 09:23:35 +0100},
+	journal = JCP,
+	pages = {084106},
+	title = {Systematic Optimization of Long-Range Corrected Hybrid Density Functionals},
+	volume = 128,
+	year = 2008}
+
+@article{Chai_2008b,
+	author = {Chai, J. D. and Head-Gordon, M.},
+	date-added = {2021-02-25 09:21:29 +0100},
+	date-modified = {2021-02-25 09:23:17 +0100},
+	journal = PCCP,
+	pages = {6615--6620},
+	title = {Long-range Corrected Hybrid Density Functionals with Damped Atom--Atom Dispersion Corrections},
+	volume = 10,
+	year = 2008}
+
+@article{Henderson_2009,
+	author = {Henderson, T. M. and Izmaylov, A. F. and Scalmani, G. and Scuseria, G. E.},
+	date-added = {2021-02-25 09:20:49 +0100},
+	date-modified = {2021-02-25 09:21:06 +0100},
+	doi = {10.1063/1.3185673},
+	journal = {J. Chem. Phys.},
+	pages = {044108},
+	title = {Can Short-Range Hybrids Describe Long-Range-Dependent Properties?},
+	volume = {131},
+	year = {2009},
+	Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.3185673}}
+
+@article{Coulson_1949,
+	author = {Coulson, Ca and Fischer, I},
+	date-added = {2021-02-25 09:01:32 +0100},
+	date-modified = {2021-02-25 09:01:53 +0100},
+	doi = {10.1080/14786444908521726},
+	journal = {Philos. Mag.},
+	number = {303},
+	pages = {386},
+	title = {{XXXIV. Notes on the Molecular Orbital Treatment of the Hydrogen Molecule}},
+	volume = {40},
+	year = {1949},
+	Bdsk-Url-1 = {https://doi.org/10.1080/14786444908521726}}
+
 @article{Veril_2021,
 	abstract = {Abstract We describe our efforts of the past few years to create a large set of more than 500 highly accurate vertical excitation energies of various natures (π → π*, n → π*, double excitation, Rydberg, singlet, doublet, triplet, etc.) in small- and medium-sized molecules. These values have been obtained using an incremental strategy which consists in combining high-order coupled cluster and selected configuration interaction calculations using increasingly large diffuse basis sets in order to reach high accuracy. One of the key aspects of the so-called QUEST database of vertical excitations is that it does not rely on any experimental values, avoiding potential biases inherently linked to experiments and facilitating theoretical cross comparisons. Following this composite protocol, we have been able to produce theoretical best estimates (TBEs) with the aug-cc-pVTZ basis set for each of these transitions, as well as basis set corrected TBEs (i.e., near the complete basis set limit) for some of them. The TBEs/aug-cc-pVTZ have been employed to benchmark a large number of (lower-order) wave function methods such as CIS(D), ADC(2), CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, ADC(3), CC3, NEVPT2, and so on (including spin-scaled variants). In order to gather the huge amount of data produced during the QUEST project, we have created a website (https://lcpq.github.io/QUESTDB\_website) where one can easily test and compare the accuracy of a given method with respect to various variables such as the molecule size or its family, the nature of the excited states, the type of basis set, and so on. We hope that the present review will provide a useful summary of our effort so far and foster new developments around excited-state methods. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods},
 	author = {V{\'e}ril, Micka{\"e}l and Scemama, Anthony and Caffarel, Michel and Lipparini, Filippo and Boggio-Pasqua, Martial and Jacquemin, Denis and Loos, Pierre-Fran{\c c}ois},

Manuscript/sfBSE.tex

@@ -598,9 +598,9 @@ However, our scheme goes beyond the plasmon-pole approximation as the spectral r
 	\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} 
 	+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} 
 	\\
-	\times \Bigg[ \frac{1}{\omega - (\eGW{s_\sigp}{} - \eGW{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} 
+	\times \Bigg[ \frac{1}{\omega - (\alert{\eGW{s_\sigp}{} - \eGW{q_\sig}{}}) - \Om{m}{\spc,\RPA} + i \eta} 
 	\\
-	+ \frac{1}{\omega - (\eGW{r_\sigp}{} - \eGW{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} \Bigg]
+	+ \frac{1}{\omega - (\alert{\eGW{r_\sigp}{} - \eGW{p_\sig}{}}) - \Om{m}{\spc,\RPA} + i \eta} \Bigg]
 \end{multline}
 The dBSE non-linear response problem is
 \begin{multline}
@@ -761,6 +761,7 @@ Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
 All the static and dynamic BSE calculations (labeled in the following as SF-BSE and SF-dBSE respectively) are performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
 The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(2)-x, respectively] as well as the SF-ADC(3) \cite{Lefrancois_2015} are performed with Q-CHEM 5.2.1. \cite{qchem4}
 Spin-flip TD-DFT calculations \cite{Shao_2003} considering the BLYP, \cite{Becke_1988,Lee_1988} B3LYP, \cite{Becke_1988,Lee_1988,Becke_1993a} and BH\&HLYP \cite{Lee_1988,Becke_1993b} functionals with contains $0\%$, $20\%$, and $50\%$ of exact exchange are labeled as SF-TD-BLYP, SF-TD-B3LYP, and SF-TD-BH\&HLYP, respectively, and are also performed with Q-CHEM 5.2.1.
+\alert{Additionally, we have performed spin-flip TD-DFT calculations considering the following range-separated hybrid (RSH) functionals: CAM-B3LYP, \cite{Yanai_2004} LC-$\omega$HPBE, \cite{Henderson_2009} and $\omega$B97X-D. \cite{Chai_2008a,Chai_2008b}}
 EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09}
 As a consistency check, we systematically perform SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
 Throughout this work, all spin-flip and spin-conserved calculations are performed with a UHF reference.
@@ -793,9 +794,13 @@ Overall, the SF-CIS and SF-BSE excitation energies are closer to FCI than the SF
 At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effects included in the SF-dBSE scheme.
 Note that the exact exchange seems to spin purified the $^3P(1s^2 2s^1 2p^1)$ state while the singlet states at the SF-BSE level are slightly more spin contaminated than their SF-CIS counterparts.
 
+\alert{Table \ref{tab:Be} and Fig.~\ref{fig:Be} also gathers results obtained at the partially self-consistent SF-(d)BSE@ev$GW$ and fully self-consistent SF-(d)BSE@qs$GW$ levels.
+The SF-(d)BSE excitation energies are quite stable with respect to the underlying $GW$ scheme which nicely illustrates that UHF eigenstates are actually an excellent starling point in this particular case.}
+
 The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
 Interestingly, SF-BSE and SF-ADC(2)-s have rather similar accuracies, except again for the $^1D$ state where SF-ADC(2)-s has clearly the edge over SF-BSE.
 Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to FCI for this simple system with significant improvements for the lowest $^3P$ state and the $^1D$ doubly-excited state.
+\alert{Although the (d)BSE and ADC(2)-s have obvious theoretical similarities, we would like to mention that they are not strictly identical as ADC(2) includes key second-order exchange contributions that are not included at the $GW$ level even in the case of more elaborate schemes like ev$GW$ and qs$GW$.}
 
 %%% TABLE I %%%
 %\begin{squeezetable}
@@ -881,7 +886,12 @@ However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ state with
 Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit.
 Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\Sigma_g^+$ doubly-excited state, resulting in an avoided crossing around $R(\ce{H-H}) = 1.6$ \AA.
 A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI} where it is shown that dynamical effects do not affect the present conclusions.
-
+\alert{One would also notice a little ``kink'' in the potential energy curves of the $\text{B}\,{}^1\Sigma_u^+$ and $\text{E}\,{}^1\Sigma_g^+$ states around $R(\ce{H-H}) = 1.2~\AA$ computed at the (d)BSE@{\GOWO} level. 
+This unfortunate feature is due to the appearance of the symmetry-broken UHF solution and the lack of self-consistent in {\GOWO}.	
+Indeed, $R = 1.2~\AA$ corresponds to the location of the well-known Coulson-Fischer point. \cite{Coulson_1949}
+Note that, as mentioned earlier, all the calculations are performed with a UHF reference even the ones based on a closed-shell singlet reference.
+If one relies solely on the restricted HF solution, this kink disappears and one obtains smooth potential energy curves.} 
+	
 The right side of Fig.~\ref{fig:H2} shows the amount of spin contamination as a function of the bond length for SF-CIS (top), SF-TD-BH\&HLYP (center), and SF-BSE (bottom).
 Overall, one can see that $\expval{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level.
 For all spin-flip methods, the $\text{E}$ state is strongly spin contaminated as expected, while the $\expval{\hS^2}$ values associated with the $\text{F}$ state 
@@ -973,7 +983,7 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
 			Method					&	$1\,{}^3B_{1g}$	&	$1\,{}^1B_{1g}$	&	$2\,{}^1A_{g}$	\\
 			\hline
 			SF-TD-B3LYP\fnm[1] 		&	$1.750$	&	$2.260$	&	$4.094$	\\
-			SF-TD-CAM-B3LYP \fnm[1] & $1.790$ & $2.379$ & $4.238$ \\
+			\alert{SF-TD-CAM-B3LYP} &	$1.790$ &	$2.379$ &	$4.238$ \\
 			SF-TD-BH\&HLYP\fnm[1] 	&	$1.583$	&	$2.813$	&	$4.528$	\\
 			SF-CIS\fnm[2]			&	$1.521$	&	$3.836$	&	$5.499$	\\
 			EOM-SF-CCSD\fnm[3]		&	$1.654$	&	$3.416$	&	$4.360$	\\
@@ -1006,7 +1016,7 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
 			Method					&	$1\,{}^3A_{2g}$	&	$2\,{}^1A_{1g}$	&	$1\,{}^1B_{2g}$	\\
 			\hline
 			SF-TD-B3LYP\fnm[1]		&	$-0.020$	&	$0.547$	&	$0.486$	\\
-			SF-TD-CAM-B3LYP \fnm[1] & $0.012$  & $0.677$ &  $0.595$ \\
+			\alert{SF-TD-CAM-B3LYP} &	$0.012$		&	$0.677$ &	$0.595$ \\
 			SF-TD-BH\&HLYP\fnm[1]	&	$0.048$		&	$1.465$	&	$1.282$	\\
 			SF-CIS\fnm[2]			&	$0.317$		&	$3.125$	&	$2.650$	\\
 			EOM-SF-CCSD\fnm[3]		&	$0.369$		&	$1.824$	&	$2.143$	\\

Response_Letter/Response_Letter.tex

@@ -40,7 +40,8 @@ I recommend this manuscript for publication after the minor points addressed:}
 	{Figure 1/3: these show quite a relevant assessment of the performance of different SF methods. However, I think that the comparison with SF-TDDFT is unfair. None of the DFT exchange functionals is long-range corrected, whereas all other methods have the exact long-range exchange. Could the authors add the data for a long-range corrected functional?}
 	\\
 	\alert{Following the excellent advice of Reviewer \#1, we have added data for the long-range corrected CAM-B3LYP functional.
-	We believe that it is a choice with the other functionals (BLYP, B3LYP, and BH\&HLYP) already included in the manuscript.
+	We believe that it is a consistent choice which fits nicely with the other functionals (BLYP, B3LYP, and BH\&HLYP) already included in the manuscript.
+	The CAM-B3LYP results have been added to the corresponding Tables and Figures.
 	As one can see...}
 	
 	\item 
@@ -48,7 +49,7 @@ I recommend this manuscript for publication after the minor points addressed:}
 	\\
 	\alert{The reviewer is right to mention similarities between the SF-dBSE and SF-ADC(2)-s schemes.
 	However, they are not strictly identical as ADC(2) includes second-order exchange diagrams which are not present in SF-dBSE@$GW$, even in the case of more elaborate schemes like ev$GW$ and qs$GW$.
-	To illustrate this and accordingly to the reviewer's suggestion, we have added the partially self-consistent SF-dBSE@ev$GW$ results as well as the fully self-consistent SF-dBSE@qs$GW$ results. As one can see, in the case of Be, there is not much differences between these schemes and the original SF-dBSE@$G_0W_0$ which nicely illustrates that HF eigenstates are actually are an excellent starling point in this particular case. 
+	To illustrate this and accordingly to the reviewer's suggestion, we have added the partially self-consistent SF-dBSE@ev$GW$ results as well as the fully self-consistent SF-dBSE@qs$GW$ results. As one can see, in the case of Be, there is not much differences between these schemes and the original SF-dBSE@$G_0W_0$ which nicely illustrates that HF eigenstates are actually an excellent starling point in this particular case. 
 	A discussion around these points have been also included in the revised version of the manuscript.}
  
 	\item 
@@ -63,15 +64,20 @@ I recommend this manuscript for publication after the minor points addressed:}
 	{Figure 2: Could the authors discuss the kink in G0W0/SF-BSE and G0W0/SF-dBSE (in supporting) appearing at around 1.2 Angstroms between $1\Sigma_g^+$ and $1\Sigma_u^+$. It is really puzzling. Is it due to the lack of self consistency in the G0W0 approximation? What does GW/SF-BSE gives in this case?}
 	\\
 	\alert{The kink in the SF-BSE@$G_0W_0$ and SF-dBSE/$G_0W_0$ curves for \ce{H2} are due to the appearance of the symmetry-broken UHF solution.
-	Indeed, $R = 1.2 \AA$ corresponds to the location of the well-known Coulson-Fischer point.
+	Indeed, $R = 1.2~\AA$ corresponds to the location of the well-known Coulson-Fischer point.
 	Note that, as mentioned in our manuscript, all the calculations are performed with a UHF reference (even the ones based on a closed-shell singlet reference).
-	Of course, if one relies solely on the RHF solution, this kink disappears. it would be, nonetheless, inconsistent with the rest of the paper.
+	Of course, if one relies solely on the RHF solution, this kink disappears (see figure below). it would be, nonetheless, inconsistent with the rest of the paper.
 	The appearance of this kink is now discussed in the revised version of the manuscript.
 	At the ev$GW$ level, this kink would certainly still exist as one does not self-consistently optimised the orbitals in this case.
 	However, it would likely disappear at the qs$GW$ level but it remains to be confirmed (work is currently being done in this direction).
 	Unfortunately, it is extremely tedious to converge (partially) self-consistent $GW$ calculation with such large basis set (cc-pVQZ) for reasons discussed elsewhere [see, for example, V\'eril et al. JCTC 14, 5220 (2018)].}
-	
+	\\
+	\begin{center}
+		\includegraphics[width=0.5\textwidth]{SF-BSE-RHF}
+	\end{center}
+
 \end{itemize}
+
  
 \end{letter}
 \end{document}