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\begin{letter}%
{To the Editors of Journal of Chemical Theory and Computation,}

\opening{Dear Editors,}

\justifying
Please find attached a revised version of the manuscript entitled 
\begin{quote}
	\textit{``Spin-Conserved and Spin-Flip Optical Excitations From the Bethe-Salpeter Equation Formalism''}.
\end{quote}
We thank the reviewers for their constructive comments which, we believe, have improved the overall quality of the present manuscript.
Our detailed responses to their comments can be found below.
For convenience, changes are highlighted in red in the revised version of the manuscript. 

We look forward to hearing from you.

\closing{Sincerely, the authors.}

%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}

\begin{itemize}

	\item 
	{The authors present a new formulation of Bethe-Salpeter equations (RPAx on GW reference) for spin-flip excitations. Despite the existence of many formulations of spin-flip (SF) currently, this is the first time it has been applied to GW/BSE. All the spin-flip formulations have a similar structure, changing essentially the amount of exchange and correlation effects dug in the orbital energies and the particle-hole couplings. The authors present the theory in deep detail and perform the basic applications that allow the assessment of the performance of SF-BSE.

I recommend this manuscript for publication after the minor points addressed:}
	\\
	\alert{We thank the reviewer for recommending publication of the present manuscript.}

	\item 
	{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 following range-separated hybrid functionals: CAM-B3LYP, LC-$\omega$HPBE, and $\omega$B97X-D.
	CAM-B3LYP only has 75\% exact exchange at long range while LC-$\omega$PBE08 and $\omega$B97X-D have 100\% of HF exact exchange at longe range.
	These results have been added to the corresponding Tables and Figures.
	For the sake of clarity, in the case of \ce{H2}, we have chosen to add some of the graphs to the supporting information instead.
	In a nutshell, range-separated hybrids do not provide a clear improvement upon global hybrids and BH\&HLYP seems to be the best all-round performer.
	All these results are discussed in the revised version of the manuscript.}
	
	\item 
	{Figure 1: The similarity between SF-dBSE and SF-ADC(2)-s is more than simply the results. 
	I would say that the two formulations are equivalent, and should lead to the same results in Figure 1 if the authors would have used GW/SF-dBSE instead of G0W0/SF-dBSE.
	Could the authors add these results based on GW and comment?}
	\\
	\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 an excellent starting point in this particular case. 
	A discussion around these points have been also included in the revised version of the manuscript.}
 
	\item 
	{Figure 1: The only difference between SF-ADC(2)-s and SF-ADC(2)-x is that the energy difference in the dynamic part of the BSE equation is corrected to first order. 
	The equivalent thing in SF-BSE would be to add in Eq. 30 the direct and exchange corrections in the orbital energy difference appearing in the denominator of the second term (i.e., similar to using the diagonal part of Eq. 29a and 29c in the orbital energy difference of Eq. 30). 
	Could the authors verify that?}
	\\
	\alert{We thank the reviewer for mentioning this interesting fact. We were not aware of this.
	Actually, this is already the case in SF-dBSE; the eigenvalues differences in the denominator of the second of Eq. 30 are $GW$ quasiparticle energies.
	The $GW$ superscripts were missing in the original manuscript and they have been added. 
	We have performed SF-dBSE@$G_0W_0$ calculations replacing the $GW$ quasiparticle energies by the HF orbital energies in the denominator of Eq. (30) but it does not seem to alter much the results in the case of Be.}
	
	\item 
	{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.
	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 as illustrated by the new figure that we have included in the Supporting Information. 
	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 optimise 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 $GW$ calculations with such large basis set (cc-pVQZ) for reasons discussed elsewhere [see, for example, V\'eril et al. JCTC 14, 5220 (2018)].}
	\\

\end{itemize}

 
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