modifying discussion

Manuscript/CBD-SI.tex

@@ -146,7 +146,7 @@ H            -2.092429    0.000000    0.000000
 %%%%%%%%%%%%%%%%%%%%%%%%
 \end{itemize}
 
-\begin{squeezetable}
+%\begin{squeezetable}
 \begin{table*}
 	\caption{Energy differences between the states computed with various methods and the reference TBE values. 
 	Note that AB stands for the automerization barrier and is reported in \si{\kcalmol}. 
@@ -207,7 +207,7 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
 	\fnt[7]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(3)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
 	\fnt[8]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
 \end{table*}
-\end{squeezetable}
+%\end{squeezetable}
 
 
 %%% %%% %%% %%%

Manuscript/CBD.tex

@@ -603,17 +603,18 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
 %%% %%% %%% %%%
 
 Table \ref{tab:sf_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods. 
-Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation energies with respect to the level of theory is illustrated in Fig.~\ref{fig:D2h}. %\hl{Should we add CCSD ? Would it be interesting to have "normal" ADC values for some states ?}
+Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation energies with respect to the level of theory is illustrated in Fig.~\ref{fig:D2h}. 
 
 At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in the {\tBoneg}, {\sBoneg}, and {\twoAg} are 99\%, 95\%, and 1\%, respectively.
 Therefore, the two formers are dominated by single excitations, while the latter state corresponds to a genuine double excitation.
 
 First, let us discuss basis set effects at the SF-TD-DFT level (Table \ref{tab:sf_D2h}).
 As expected, these are found to be small and the results are basically converged to the complete basis set limit with the triple-$\zeta$ basis, which is definitely not the case for the wave function methods. \cite{Giner_2019}
-Regarding now the accuracy of the vertical excitation energies, again, we clearly see that, for each transition, the functionals with the largest amount of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) are the most accurate.
+Regarding now the accuracy of the vertical excitation energies, again, we see that, for {\tBoneg} and {\sBoneg}, the functionals with the largest amount of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) are the most accurate.
 Functionals with a large share of exact exchange are known to perform best in the SF-TD-DFT framework as the Hartree-Fock exchange term is the only non-vanishing term in the spin-flip block. \cite{Shao_2003}
-However, their overall accuracy remains average especially for the singlet states,  {\sBoneg} and {\twoAg}, with error of the order of \SIrange{0.2}{0.5}{\eV} compared to the TBEs.
+However, their overall accuracy remains average especially for the singlet states, {\sBoneg} and {\twoAg}, with error of the order of \SIrange{0.2}{0.5}{\eV} compared to the TBEs.
 The triplet state, {\tBoneg}, is much better described with errors below \SI{0.1}{\eV}. 
+Surprisingly, for the doubly-excited state, {\twoAg}, the hybrid functionals with a low percentage of exact exchange (B3LYP and PBE0) are the best performers with absolute errors below \SI{0.05}{\eV}.
 Note that, as evidenced by the data reported in {\SupInf}, none of these states exhibit a strong spin contamination.
 
 Second, we discuss the various SF-ADC schemes (Table \ref{tab:sf_D2h}), \ie, SF-ADC(2)-s, SF-ADC(2)-x, and SF-ADC(3). 
@@ -622,7 +623,7 @@ At the SF-ADC(2)-s level, going from the smallest 6-31+G(d) basis to the largest
 These basis set effects are fairly transferable to the other wave function methods that we have considered here. 
 This further motivates the ``pyramidal'' extrapolation scheme that we have employed to produce the TBE values (see Sec.~\ref{sec:TBE}).
 Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the present context with much larger errors than the other schemes.
-Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns making SF-ADC(2.5) particularly accurate with a maximum error of \SI{0.029}{\eV} for the doubly-excited state {\twoAg}.
+Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns making SF-ADC(2.5) particularly accurate except for the doubly-excited state {\twoAg} where the error with respect to the TBE (\SI{0.140}{\eV}) is larger than the SF-ADC(2)-s error (\SI{0.093}{\eV}).
 
 Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table \ref{tab:D2h}.
 Regarding the multi-configurational calculations, the most striking result is the poor description of the {\sBoneg} ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state. 
@@ -630,7 +631,7 @@ Of course, the PT2 correction is able to correct the root-flipping problem but c
 Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}. 
 This feature is characteristic of the inadequacy of the active space to model such a state.
 For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller (below \SI{0.1}{\eV}).
-Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet state) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
+Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet and doubly-excited states) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
 
 Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4.
 This trend is in line with the observations made on the QUEST database. \cite{Veril_2021}
@@ -811,8 +812,9 @@ For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the
 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 exact exchange at short range.
 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 on the {\Dtwo} ground-state equilibrium structure and the automerization barrier, the functionals with a large fraction of short-range exact exchange yield 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).
+As for the excitation energies computed on the {\Dtwo} ground-state equilibrium structure and the automerization barrier, the functionals with a large fraction of short-range exact exchange yield more accurate results.
+Yet, the transition energy to {\Btwog} is off by half an \si{\eV} compared to the TBE for BH\&HLYP and M11, while the doubly-excited state is much closer to the reference value (errors of \SI{-0.251}{} and \SI{-0.312}{\eV} for BH\&HLYP and M11, respectively).
+With errors of \SI{-0.066}{}, \SI{-0.097}{}, and \SI{-0.247}{\eV} for {\Atwog}, {\Aoneg}, and {\Btwog}, M06-2X is the best performer here. 
 Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level.
 We emphasize that the $\expval*{S^2}$ values reported in {\SupInf} indicate again that there is no significant spin contamination in these excited states.
 
@@ -827,9 +829,9 @@ Let us turn to the multi-reference results (Table \ref{tab:D4h}).
 For both active spaces, expectedly, CASSCF does not provide a quantitive energetic description, 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 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}.
+Although {\Aoneg} is not badly described, the excitation energy of the ionic state {\Btwog} is off by almost \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 well described.
-The (12e,12o) active space significantly alleviates these effects, and, as usual now, 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, \emph{e.g.,} an error up to \SI{-0.278}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
+The (12e,12o) active space significantly alleviates these effects, and, as usual now, 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, \emph{e.g.,} an error up to \SI{-0.093}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
 
 Finally, let us analyze the excitation energies computed with various CC models that are gathered in Table \ref{tab:D4h}.
 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, 
@@ -837,7 +839,7 @@ As mentioned in Sec.~\ref{sec:CC}, we remind the reader that these calculations
 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 with such a computational strategy, 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.
+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{0.688}{\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 CCSDTQ-based TBEs and the CIPSI results are consistent if one takes into account the extrapolation error (see Sec.~\ref{sec:SCI}).