{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.
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.}
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 now discussed in the text.}
{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.}
Indeed, at the $D_{4h}$$T_1$ optimized geometry, we have used the conventional standard orientation where two $C_2$ axes run through the carbon atoms.
In this conventional orientation, the singlet ground state $1^1B_{1g}$ remains $1^1B_{1g}$ in the $D_{2h}$ point group and the singlet excited state $1^1A_{1g}$ becomes $1^1A_g$ in the $D_{2h}$ point group.
As pointed out by the reviewer, upon rotation of the molecular framework by 45 degrees in the ($xy$) plane, the two $C_2$ axes bisect the carbon-carbon bonds.
This induces a different orbital picture. The correlation between the orbitals and states in the new molecular framework are illustrated in the figure below at the CASSCF(4,4) level.
Because of the different orbital picture (the frontier orbitals are localized on two carbon atoms in the standard orientation and on four carbon atoms in the other orientation), the new CI coefficients resulting from this rotation also bring a different wave function representation.
Whereas the $1^1B_{1g}$ ground state is described in a one-electron-excitation picture in the standard orientation (the $1^1A_{1g}$ excited state involves a double excitation), the corresponding $1^1B_{1g}$ ground state in the new orientation involves a two-electron-excitation picture (the $1^1A_{1g}$ excited state also involves a double excitation).
Of course, these two representations are perfectly equivalent at the CASSCF level which describes single and double excitations on the same footing.
As mentioned in our manuscript in section II.B., for the $D_{4h}$ arrangement, we have considered the lowest closed-shell singlet state $1^1A_{g}$ as reference.
Because this state has a substantial double-excitation character, we expect a significant error at the CCSD level.
The $1^1B_{1g}$ ground state is obtained as a singly excited state from that reference, while the $1^1B_{2g}$ excited state should also be a mixture involving a double excitation.
In the other (non-standard) orientation, the lowest $^1A_g$ state correlates with the $1^1B_{1g}$ ground state, which in this orientation has a strong double-excitation character.
Then, the $1^1 A_{1g}$ excited state has also a strong double-excitation character, while the $1^1B_{2g}$ excited state is obtained by one-electron excitation.
Thus, whatever the orientation of the molecule, we will face the same problem for the reference state.
Note that in the case of the SF formalism, these three singlet states should all be described correctly if one takes the $1^3A_{2g}$ state as the reference high spin state, whatever the orientation.
This discussion about standard and non-standard orientations has been added to the supporting information alongside the corresponding figure.}
\alert{We agree with the reviewer that at the $D_{4h}$ geometry the (2e,2o) active space would be enough to describe the pure static correlation.
However, to calculate the automerization barrier, we need to make the energy difference between the energy obtained for the ground state at the $D_{4h}$ geometry and that at the $D_{2h}$ geometry.
At this last geometry, the correct description of the static correlation requires including (4e,4o) in the active space (i.e., all valence $\pi$ orbitals).
In addition, there are states with ionic character which required including the dynamic electron correlation (in particular the $\sigma$-$\pi$ polarization).
Thus, the improvement of our results by including all $\sigma_{CC}$ is rather expected.
Note also that we have minimized the intruder state problem by using an appropriate level shift and that this potential problem is not present at the NEVPT2 level.
\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.}
\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.}
{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.}
{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{The authors thanks the referee for this valuable comment.
Unfortunately, in order to obtain Head-Gordon's indices for the different spin-flip methods we would have to redo most of our calculations which would take too much time and resources.
Nonetheless, we have mentioned these indices in the text and we will definitely use them in future works.}
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.
{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'.}
