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\begin{letter}%
{To the Editors of the Journal of Chemical Physics,}

\opening{Dear Editors,}

\justifying
Please find attached a revised version of the manuscript entitled 
\begin{quote}
	\textit{``Reference Energies for Cyclobutadiene: Automerization and Excited States''}.
\end{quote}
We thank the reviewers for their constructive comments and to support publication 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. 
In addition, we have taken into account the non-scientific changes requested by the editorial team.
We look forward to hearing from you.

\closing{Sincerely, the authors.}

\newpage

%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}
 
{This article presents a survey of spin-flip TD-DFT, spin-flip ADC, multireference (CASSCF and MRPT), and equation-of-motion coupled cluster methods as applied to the automerization and vertical excitation energies of cyclobutadiene (CBD). As the smallest example of anti-aromaticity (and one of the smallest and most interesting exemplars of strong PJT distortion), CBD is an illuminating and challenging test case for these methods. (EOM-)CCSDTQ values, with a “pyramidal” basis set extrapolation scheme are used as the newly-proposed theoretical best estimates, and limited selected full CI (CIPSI) calculations confirm their excellent accuracy. The authors reach some interesting and useful conclusions concerning the tested methods.
}
\\
\alert{
Thank you for supporting publication of the present manuscript.
As detailed below, we have taken into account the comments and suggestions of the reviewers that we believe have overall improved the quality of the present paper.}



{This work could be published as-is in JPC, but some suggestions for ways in which the manuscript could be improved follow:}
\\
\alert{
}

\begin{enumerate}

\item 
{The authors opt not to test SF-EOM-CC methods. 
A justification or rationalization would be helpful. 
Also, is it expected that these methods would improve on SF-ADC and/or EOM-CC?}
\\
\alert{The authors thanks the reviewer for this comment. 
Results for SF-EOM-CCSD, SF-EOM-CCSD(dT) and SF-EOM-CCSD(fT) have been added in the manuscript (and in the supporting information) and are discussed in the text.}

\item 
{The issue of reference symmetry frame is very important at the D4h geometry. 
The correlated calculation (and often the reference SCF calculation) are performed in a D2h subgroup, of which there are two distinct possibilities: one with the C2’ axes running through the carbon atoms and one with the C2’ axes bisecting them. 
It seems the former has been employed. 
The latter actually could potentially provide a faster convergence to the A1g state since it exhibits strong mixing between the two major determinants via T2 even at the CCSD level. 
However, this same property leads to a distinct inability to properly access the B1g ground state via a single excitation in EOM-CC. 
Some illuminating comments on this issue would be welcome.}
\\
\alert{}

\item{The authors note a significant improvement in the MRPT results as the active space is enlarged. 
However, it seems to me that the most appropriate active space (for the D4h geometry at least) is in fact (2e,2o) [i.e. $Eg^2$ at D4h]. 
Within this space, the CI coefficients become fixed at D4h, leading to an “exact” SCF reference, at least in terms of static correlation. 
Perhaps the major problem with the MRPT results is not active space insufficiency, then, but intruder states? 
Can the authors perform MRCI+Q  or MRAQCC calculations for comparison?}
\\
\alert{}

\item 
{It seems that extrapolated CCSDTQ/aQZ values are available for the automerization barrier. 
Why are the aTZ numbers used as the TBE instead?}
\\
\alert{For the sake of consistency with the excitation energies and comparison, we have defined all the TBEs of the manuscript at the aug-cc-pVTZ level.
We believe that aug-cc-pVTZ is an adequate basis in order to get accurate values for the automerization barrier and the vertical excitation energies.
Defining the TBE at the aug-cc-pVQZ level would make comparison with other methods quite expensive (and sometimes undoable for some of the most expensive methods.}

\end{enumerate}

%%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2}
 
{This is a useful addition to the literature, presenting extensive benchmarks on a popular system, cyclobutadiene or CB. I recommend it for the publication once the following issues are addressed.}
\\
\alert{Thank you for these positive comments and for supporting publication of our manuscript. 
Below, we address the points raised by Reviewer \#2.
}

\begin{enumerate}

\item 
{The results for EOM-SF-CCSD and EOM-SF-CCSD(fT/dT) must be included in the paper and in the
analysis/discussion of the results. Why to exclude the best-performing SF methods? Since this paper aspires to be a comprehensive benchmark on CB, I believe it is absolutely essential.  Moreover, some of these results are already available (e.g., Ref. 105 has the results for excitation energies obtained in the same basis -- aug-cc-pVTZ that is used in the paper). Even if one needs to redo the calculations, they are very quick and can be done on a laptop in a few minutes.}
\\
\alert{As mentioned in the response to Reviewer \#1, results for SF-EOM-CCSD, SF-EOM-CCSD(dT) and SF-EOM-CCSD(fT) have been added in the manuscript (and in the supporting information) and are discussed in the text.}

\item 
{The comparison of SF-ADC with EOM-SF-CCSD will be illuminating for the readers. 
For example, EOM-SF is more robust wrt reference spin-contamination compared to SF-ADC because of the CC ansatz. 
There could be other interesting differences to discuss.}
\\
\alert{See previous point.}

\item 
{I also recommend to include EOM-DEA-CCSD results -- this is another extension of EOM-CCSD, which can treat diradicals. It does not suffer from spin-contamination. The method is available in Q-Chem. See here for theory description and examples: J. Chem. Phys. 154, 114115 (2021). EOM-DIP is another method, which can deal wit this type of electronic structure, but it has difficulties with diffuse basis sets (e.g., J. Chem. Phys. 135, 084109 (2011)) -- so I am not asking to add the DIP numbers, but mentioning it would be appropriate.}
\\
\alert{Adding values from the literature? Outside the scope of the present paper?}

\item 
{The analysis would benefit greatly if the authors provide Head-Gordon's indices, which can be used to compare wave-functions computed by different methods in a meaningful way, as illustrated here:J. Chem. Theo. Comp. 14, 638 (2018). }
\\
\alert{T2: I have to check this paper... The authors thanks the referee for this valuable comment. Unfortunately, in order to obtain the Head-Gordon's indices for the different spin-flip methods used we would have to do all the calculations or at least for the aug-cc-pVTZ basis which will take too much time and resources. We have mentioned these indices in the text and we will definitely use them in future works.}

\item 
{CAS-based methods are multi-reference (and also able to treat multi-configutional wfns). 
EOM-SF and EOM-EE are single-reference methods that are able to describe multi-configurational wfns. 
Please correct the section names and discussion appropriately.}
\\
\alert{We have modified the section names and discussion accordingly to the reviewer's suggestion.}

\item 
{Abstract and introduction:
please insert the word 'standard' before 'time-dependent density-functional theory (TD-DFT) or equation-of-motion ... are notoriously known to struggle in such situations.' 
The SF and DEA/DIP variants of these methods do not struggle.
}
\\
\alert{This has been corrected.}

\item
{Intro: "Of course, single-reference methods are naturally unable to describe such situations." 
This is incorrect -- see above (EOM-SF/DIP/DEA are single reference methods capable of describing multi-configurational wfns). 
Adding the word 'standard' might help. 
Below: "and remain tortuous for state-of-the-art methods ..." -- again, need to correct, e.g., consider 'remains challenging for standard hierarchy of EOM-CC methods that are using ground-state Hartree-Fock reference'.}
\\
\alert{We have performed these two corrections.}

\item{First page, last paragraph -- replace multi-configurational by multi-reference, as per above.}
\\
\alert{This has been corrected.}

\item{Last paragraph of intro -- here you can introduce the idea of single-reference approach to multi-reference wfns and describe SF/DEA/DIP methods.  
Please do not call SF 'cheaper' -- this does not make sense (the cost depends on correlation treatment). 
SF is more robust and more effective at each correlation level. 
It is also more black-box, as it does not require active-space selection. 
Also, EOM-SF can systematically converge to the exact FCI answer once sufficiently high excitations are included. 
'Obviously, spin-flip methods have their own flaws, especially spin' -- why 'obviously'? 
I suggest to reword, e.g., 'One drawback of SF methods is ...'}
\\
\alert{The reviewer is right. We have followed the suggestion of the reviewer and performed the required modifications.}

\item{Section IIC -- please rename.}
\\
\alert{Done.}

\end{enumerate}

\end{letter}
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