rough first draft done

Manuscript/CBD.tex

@@ -232,7 +232,7 @@ Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximat
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 When technically possible, each level of theory is tested with four Gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989} 
 This helps us to assess the convergence of each property with respect to the size of the basis set.
-More importantly, for each studied quantity (i.e., the autoisomerisation barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis. 
+More importantly, for each studied quantity (i.e., the automerization barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis. 
 In most cases, these TBEs are defined using extrapolated CCSDTQ/aug-cc-pVTZ values or, in a single occasion, NEVPT2(12,12). 
 
 The extrapolation of the CCSDTQ/aug-cc-pVTZ values is done via a ``pyramidal'' scheme, where we employ systematically the most accurate level of theory and the largest basis set available.
@@ -261,9 +261,9 @@ For example,
 and so on.
 If neither CC4, nor CCSDT are feasible, then we rely on NEVPT2(12,12). 
 The procedure for each extrapolated value is explicitly mentioned as a footnote.
-Note that, due to error bar inherently linked to the CIPSI calculations (see Subsection \ref{sec:SCI}), these are mostly used as an additional safety net to further check the convergence of the CCSDTQ estimates.
+Note that, due to error bar inherently linked to the CIPSI calculations (see Sec.~\ref{sec:SCI}), these are mostly used as an additional safety net to further check the convergence of the CCSDTQ estimates.
 
-Tables gathering these TBEs as well as literature data for the automerization barrier and the vertical excitation energies can be found in the {\SupInf}.
+Additional tables gathering these TBEs as well as literature data for the automerization barrier and the vertical excitation energies can be found in the {\SupInf}.
  
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@@ -806,38 +806,38 @@ However, it is clear from the inspection of the wave function that, with respect
 
 As for the previous geometry we start by discussing the SF-TD-DFT results (Table \ref{tab:sf_D4h}), and in particular the singlet-triplet gap, \ie, the energy difference between {\sBoneg} and {\Atwog}.
 For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the TBE value is \SI{0.144}{\eV}) but it is worth mentioning that B3LYP and PBE0 predict a negative singlet-triplet gap (hence a triplet ground state).
-Increasing the exact exchange in hybrids or relying on RSHs (even with a small amount of short-range exact exchange) allows to recover a positive gap and a singlet ground state.
+Increasing the fraction of exact exchange in hybrids or relying on RSHs (even with a small amount of short-range exact exchange) allows to recover a positive gap and a singlet ground state.
 At the SF-TD-DFT level, the energy gap between the two singlet excited states, {\Aoneg} and {\Btwog}, is particularly small and grows moderately with the amount of short-range exact exchange.
 The influence of the exact exchange on the singlet energies is quite significant with an energy difference of the order of \SI{1}{\eV} between the functional with the smallest amount of exact exchange (B3LYP) and the functional with the largest amount (M06-2X).
-As for the excitation energies computed at the {\Dtwo} equilibrium structure and the automerization barrier, functionals with a large fraction of short-range exact exchange deliver much more accurate results.
+As for the excitation energies computed at the {\Dtwo} ground-state equilibrium structure and the automerization barrier, functionals with a large fraction of short-range exact exchange deliver much more accurate results.
 Yet, the transition energy to {\Btwog} is off by more than half an \si{\eV} compared to the TBE, while the doubly-excited state is much closer to the reference value (errors of \SI{-0.251}{}, \SI{-0.097}{}, and \SI{-0.312}{\eV} for BH\&HLYP, M06-2X, and M11, respectively)
 Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level.
+Note again that the $\expval*{S^2}$ values reported in {\SupInf} show that there is no significant spin contamination in these excited states.
 
 Next, we discuss the various ADC schemes (Table \ref{tab:sf_D4h}) where we were not able to compute the vertical energies with the aug-cc-pVQZ basis due to our limited computational resources.
 Overall, we observe similar trends than the ones mentioned in Sec.~\ref{sec:D2h}.
 Concerning the singlet-triplet gap, each scheme predicts it to be positive.
 Although it provides a decent singlet-triplet gap value, SF-ADC(2)-x seems to particularly struggle with the singlet excited states ({\Aoneg} and {\Btwog}), especially for the doubly-excited state {\Aoneg} where it underestimates the vertical excitation energy by \SI{0.4}{\eV}.
 Again, averaging the SF-ADC(2)-s and SF-ADC(3) transition energies is beneficial in most cases at the exception of {\Aoneg}.
-Although the basis set effect are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and for any wave function method in general.
+Although the basis set effects are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and for any wave function method in general.
 
 Then, we discuss the multi-reference results (Table \ref{tab:D4h}).
 For both active spaces, expectedly, CASSCF does not provide a quantitive energetic description of the excited states, although it is worth mentioning that the right state ordering is preserved.
-This is, of course, magnified with the (4e,4o) active space for which the second-order perturbative treatment is unable to provide a faithful description due to the restricted active space.
-In particular SC-NEVPT2(4,4)/aug-cc-pVTZ and PC-NEVPT2(4,4)/aug-cc-pVTZ underestimate the singlet-triplet gap by \SI{0.072}{} and \SI{0.097}{\eV} and, more importantly, swap {\Aoneg} and {\Btwog}.
+This is, of course, magnified with the (4e,4o) active space for which the second-order perturbative treatment is unable to provide a satisfying description due to the restricted active space.
+In particular SC-NEVPT2(4,4)/aug-cc-pVTZ and PC-NEVPT2(4,4)/aug-cc-pVTZ underestimate the singlet-triplet gap by \SI{0.072}{} and \SI{0.097}{\eV} and, more importantly, flip the ordering of {\Aoneg} and {\Btwog}.
 Although {\Aoneg} is not badly described, the excitation energy of the ionic state {\Btwog} is off by \SI{1}{\eV}.
-Thanks to the IPEA shift in CASPT2(4,4), the singlet-triplet gap is accurate and the state ordering remains correct but the ionic state is still far from being accurate.
-The (12e,12o) active space significantly damp these effects, and, as usually, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of multi-configurational approaches remains questionable for the ionic state with an error up to \SI{-0.278}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
+Thanks to the IPEA shift in CASPT2(4,4), the singlet-triplet gap is accurate and the state ordering remains correct but the ionic state is still far from being well described.
+The (12e,12o) active space significantly alleviate these effects, and, as usually, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of multi-configurational approaches remains questionable for the ionic state with an error up to \SI{-0.278}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
 
 Finally, let us consider the excitation energies computed with various CC models and gathered in Table \ref{tab:D4h}.
-Now we look at Table \ref{tab:D4h} for vertical energies obtained with standard methods, we have first the various CC methods, for all the CC methods we were only able to reach the aug-cc-pVTZ basis. 
-%For the CCSD method we cannot obtain the vertical energy for the $1\,{}^1B_{2g}$ state, we can notice a small energy variation through the bases for the triplet state with around 0.06 eV. The energy variation is larger for the $2\,{}^1A_{1g}$ state with 0.20 eV.
-Again, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. 
-Note that for CC we started from a restricted Hartree-Fock (RHF) reference and in that case the ground state is the {\Aoneg} state, then the {\sBoneg} state is the single deexcitation and the {\Btwog} state is the double excitation from our ground state. 
-For the CC3 method we do not have the vertical energies for the triplet state {\Atwog}. 
-Considering all bases for the  {\Aoneg}and {\Btwog} states we have an energy difference of about \SI{0.15}{\eV} and \SI{0.12}{\eV}, respectively. 
-The CCSDT energies are close to the CC3 ones for the  {\Aoneg} state with an energy difference of around \SIrange{0.03}{0.06}{\eV} considering all bases. 
-For the{\Btwog} state the energy difference between the CC3 and the CCSDT values is larger with \SIrange{0.18}{0.27}{\eV}. 
-We can make a similar observation between the CC4 and the CCSDTQ values, for the  {\Aoneg} state we have an energy difference of about \SI{0.01}{\eV} and this time we have smaller energy difference for the {\Btwog} with \SI{0.01}{\eV}. 
+As mentioned in Sec.~\ref{sec:CC}, we remind the reader that these calculations are performed by considering the {\Aoneg} state as reference, and that, therefore, 
+{\sBoneg} and {\Btwog} are obtained as a deexcitation and an excitation, respectively. 
+Consequently, with respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character. 
+This explains why one observes a slower convergence of the transition energies in the case of {\sBoneg} as shown in Fig.~\ref{fig:D4h}.
+It is clear from the results of Table \ref{tab:D4h} that, if one wants to reach high accuracy, it is mandatory to include quadruple excitations.
+Indeed, at the CCSDT/aug-cc-pVTZ level, the singlet-triplet gap is already very accurate (off by \SI{0.005}{\eV} only) while the excitation energies of the singlet states are still \SI{0.131}{} and \SI{0.503}{\eV} away from their respective TBE.
+These deviations drop to \SI{0.011}{} and \SI{-0.013}{\eV} at the CC4/aug-cc-pVTZ level.
+As a final comment, we can note that the agreement between our CCSDTQ-based TBEs and the CIPSI calculations are consistent if one takes into account the extrapolation error (see Sec.~\ref{sec:SCI}).
 
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 \section{Conclusion}
@@ -845,34 +845,29 @@ We can make a similar observation between the CC4 and the CCSDTQ values, for the
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 In the present study, we have benchmarked a larger number of computational methods on the automerization barrier and the vertical energies of the cyclobutadiene (CBD) molecule in its square ({\Dfour}) and rectangular ({\Dtwo}) arrangements, for which we have defined theoretical best estimates (TBEs) based on extrapolated CCSDTQ/aug-cc-pVTZ data.
-%Otherwise we got the CCSDTQ/aug-cc-pVTZ values by correcting the CCSDTQ/aug-cc-pVDZ values by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ (Eq.~\eqref{eq:aug-cc-pVTZ}) and we obtain the CCSDTQ/aug-cc-pVDZ values by correcting the CCSDTQ/6-31G+(d) values by the difference between CC4/aug-cc-pVDZ and CC4/6-31G+(d) (Eq.~\eqref{eq:aug-cc-pVDZ}). 
-%When the CC4/aug-cc-pVTZ values were not obtained we corrected the CC4/aug-cc-pVDZ values by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ to obtain them (Eq.~\eqref{eq:CC4_aug-cc-pVTZ}). 
-%If the CC4 values have not been obtained then we used the same scheme that we just described but by using the CCSDT values. 
-%If neither the CC4 and CCSDTQ values were not available then we used the NEVPT2(12,12)/aug-cc-pVTZ values. 
 
-\titou{Within the SF-TD-DFT framework, we advice the use of exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange. 
-This has been shown to be beneficial for the automerization barrier and the vertical excitation energies in the {\Dtwo} and {\Dfour} arrangements.}
+The main take-home messages of the present work can be summarized as follows:
+\begin{itemize}
 
-\titou{At the SF-ADC level, we have found that the extended scheme SF-ADC(2)-x is not good, while SF-ADC(2)-s and SF-ADC(3) have opposite behavior which means that SF-ADC(2.5) is really good.}
+\item Within the SF-TD-DFT framework, we advice to use exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange. 
+This has been shown to be beneficial for the automerization barrier and the vertical excitation energies in the {\Dtwo} and {\Dfour} arrangements.
 
-\titou{For the multireference methods, we have found that NEVPT2 and CASPT2 can provide different results for the small active space, but they becomes very similar when the larger active space is considered. 
-Fro ma more general perspective, a singificant difference between NEVPT2 and CASPT2 can be then seen as a warning that the active space has been poorly chosen.
-Also, the ionic state is usually significantly worse than the other state.
-CASSCF cannot be advised for such a purpose.}
+\item At the SF-ADC level, we have found that the extended scheme, SF-ADC(2)-x, systematically worsen the results compared to the cheaper standard version, SF-ADC(2)-s. 
+Moreover, as previously reported, SF-ADC(2)-s and SF-ADC(3) have opposite error patterns which means that SF-ADC(2.5) is an excellent compromise.
 
-In order to provide a benchmark of the automerization barrier and vertical energies we used coupled-cluster (CC) methods with doubles (CCSD), with triples (CCSDT and CC3) and with quadruples (CCSTQ and CC4). 
-Due to the presence of multi-configurational states we used multi-reference methods (CASSCF, CASPT2 and NEVPT2) with two active spaces ((4,4) and (12,12)). 
-We also used spin-flip (SF-) within two frameworks, in TD-DFT with various global and range-separated hybrids functionals, and in ADC with the ADC(2)-s, ADC(2)-x and ADC(3) schemes. 
-The CC methods provide good results for the AB and vertical energies, however in the case of multi-configurational states CC with only triples is not sufficient and we have to include the quadruples to correctly describe these states. 
-multi-configurational methods also provide very solid results for the largest active space with second order correction (CASPT2 and NEVPT2).
+\item For the {\Dfour} square planar structure, a faithful energetic description of the excited states is harder to reach at the SF-TD-DFT level because of the strong multi-configurational character.
+In such scenario, the SF-TD-DFT excitation energies can exhibit errors of the order of \SI{1}{\eV} compared to the TBEs. 
+However, it was satisfying to see that the spin-flip version of ADC can lower these errors to \SIrange{0.1}{0.2}{\eV}.
 
-With SF-TD-DFT the quality of the results are, of course, dependent on the functional but for the doubly excited states we have solid results. 
-In SF-ADC we have very good results compared to the TBEs even for the doubly excited states, nevertheless the ADC(2)-x scheme give almost systematically worse results than the ADC(2)-s ones and using the ADC(3) scheme does not always provide better values. 
+\item Concerning the multi-configurational methods, we have found that NEVPT2 and CASPT2 can provide different excitation energies for the small (4e,4o) active space, but they becomes very similar when the larger (12e,12o) active space is considered. 
+From a more general perspective, a significant difference between NEVPT2 and CASPT2 can be seen as a warning that the active space has been poorly chosen.
 
-The description of the excited states of the {\Dtwo} structure give rise to good agreement between the single reference and multi-configurational methods due to the large $\%T_1$ percentage of the first two excited states. 
-When this percentage is much smaller as in the case of the doubly excited state {\twoAg} the spin-flip methods show very good results within the ADC framework but even at the TD-DFT level spin-flip display good results compared to the TBE value. 
-As said in the discussion, for the {\Dfour} geometry, the description of excited states is harder because of the strong multi-configurational character where SF-TD-DFT can present more than 1 eV of error compared to the TBE. 
-However, SF-ADC can show error of around \SIrange{0.1}{0.2}{\eV} which can be better than the multi-configurational methods results.
+\item The ionic states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space.
+
+\item In the context of CC methods, although the inclusion of triple excitations is very satisfactory in most cases, the inclusion of quadruples excitation (via CC4 or CCSDTQ) is mandatory to reach high accuracy (especially in the case of doubly-excited states).
+We also point out that, within the error bar linked to the CIPSI extrapolation procedure, CCSDTQ and CIPSI yield similar excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states.
+
+\end{itemize}
 
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 \acknowledgements{