EOM-SF -> SF-EOM
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@ -227,7 +227,7 @@ Additionally, we have also computed SF-TD-DFT excitation energies using range-se
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Finally, the hybrid meta-GGA functional M06-2X (54\% of exact exchange) \cite{Zhao_2008} and the RSH meta-GGA functional M11 (42.8\% of short-range exact exchange and 100\% at long range) \cite{Peverati_2011} are also employed.
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Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
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\alert{There also exist spin-flip extensions of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} and we consider here the spin-flip version of EOM-CCSD, named EOM-SF-CCSD. \cite{Krylov_2001a}
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\alert{There also exist spin-flip extensions of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} and we consider here the spin-flip version of EOM-CCSD, named SF-EOM-CCSD. \cite{Krylov_2001a}
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Additionally, Manohar and Krylov introduced a non-iterative triples correction to EOM-CCSD and extended it to the spin-flip variant. \cite{Manohar_2008}
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Two types of triples corrections were proposed: (i) EOM-CCSD(dT) that uses the diagonal elements of the similarity-transformed CCSD Hamiltonian, and (ii) EOM-CCSD(fT) where the Hartree-Fock orbital energies are considered instead.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -355,7 +355,7 @@ SF-ADC(2)-s & $6.69$ & $6.98$ & $8.63$ & \\
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SF-ADC(2)-x & $8.63$ & $8.96$ &$10.37$ & \\
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SF-ADC(2.5) & $7.36$ & $7.76$ & $9.11$ & \\
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SF-ADC(3) & $8.03$ & $8.54$ & $9.58$ \\
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\alert{EOM-SF-CCSD} & \alert{$5.86$} & \alert{$6.27$} & \alert{$7.40$} \\[0.1cm]
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\alert{SF-EOM-CCSD} & \alert{$5.86$} & \alert{$6.27$} & \alert{$7.40$} \\[0.1cm]
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CASSCF(4,4) & $6.17$ & $6.59$ & $7.38$ & $7.41$ \\
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CASPT2(4,4) & $6.56$ & $6.87$ & $7.77$ & $7.93$ \\
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SC-NEVPT2(4,4) & $7.95$ & $8.31$ & $9.23$ & $9.42$ \\
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@ -409,7 +409,12 @@ We note that SF-ADC(2)-x [which scales as $\order*{N^6}$] is probably not worth
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This behavior was previously reported by Dreuw's group. \cite{Wormit_2014,Harbach_2014,Dreuw_2015}
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Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models.
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\alert{We observe that EOM-SF-CCSD tends to underestimate of about \SI{1.5}{\kcalmol} the energy barrier compared to the TBE. This can be amended by using the triples correction with the EOM-SF-CCSD(fT) and EOM-SF-CCSD(dT) methods (see {\SupInf}). We also note that the EOM-SF-CCSD values for the energy barrier are close to the CC3 ones. Our results are in agreement with previous studies \cite{Manohar_2008,Lefrancois_2015} and justify to avoid the more expensive calculations of the triples correction as we can expect a similar trend. Note that contrary to a previous statement \cite{Manohar_2008} the (fT) correction performs better than the (dT) one for the energy barrier (however, for the excited states we retrieve the same statement).}
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\alert{We observe that SF-EOM-CCSD tends to underestimate by about \SI{1.5}{\kcalmol} the energy barrier compared to the TBE.
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This can be alleviated by including the triples correction with SF-EOM-CCSD(fT) and SF-EOM-CCSD(dT) (see {\SupInf}).
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We also note that the SF-EOM-CCSD values for the energy barrier are close to the CC3 ones.
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Our results are in agreement with previous studies \cite{Manohar_2008,Lefrancois_2015} and justify to avoid the more expensive calculations of the triples correction as we can expect a similar trend.
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Note that contrary to a previous statement \cite{Manohar_2008} the (fT) correction performs better than the (dT) correction for the energy barrier.
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However, for the excited states, we retrieve the same statement (see below).}
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Concerning the multi-reference approaches with the minimal (4e,4o) active space, the TBEs are bracketed by the CASPT2 and NEVPT2 values that differ by approximately \SI{1.5}{\kcalmol} for all bases.
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In this case, the NEVPT2 values are fairly accurate with differences below half a \si{\kcalmol} compared to the TBEs.
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@ -497,7 +502,7 @@ SF-ADC(2.5) & 6-31+G(d) & $1.496$ & $3.328$ & $4.219$ \\
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SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
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& aug-cc-pVDZ & $1.422$ & $3.180$ & $4.208$ \\
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& aug-cc-pVTZ & $1.419$ & $3.162$ & $4.224$ \\[0.1cm]
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\alert{EOM-SF-CCSD} & \alert{6-31+G(d)} & \alert{$1.663$} & \alert{$3.515$} & \alert{$4.275$} \\
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\alert{SF-EOM-CCSD} & \alert{6-31+G(d)} & \alert{$1.663$} & \alert{$3.515$} & \alert{$4.275$} \\
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& \alert{aug-cc-pVDZ} & \alert{$1.611$} & \alert{$3.315$} & \alert{$4.216$} \\
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& \alert{aug-cc-pVTZ} & \alert{$1.609$} & \alert{$3.293$} & \alert{$4.245$} \\[0.1cm]
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%SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
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@ -635,9 +640,9 @@ This further motivates the ``pyramidal'' extrapolation scheme that we have emplo
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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.
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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}).
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\alert{We observe that EOM-SF-CCSD excitation energies are closer to the SF-ADC(2)-s, with an energy difference of about \SI{0.1}{\eV}, than the other schemes as it was already noticed by LeFrançois and co-workers. \cite{Lefrancois_2015}
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We see that the EOM-SF-CCSD excitations energies for the triplet state are larger of about (\SI{0.3}{\eV} compared to the CCSD ones, this was also present in the study of Manohar and Krylov. \cite{Manohar_2008}
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Again, we have similar results, with EOM-SF-CCSD, than previous studies \cite{Manohar_2008,Lefrancois_2015} for the excited states. We can logically expect similar trend for EOM-SF-CCSD(fT) and EOM-SF-CCSD(dT) that lower the excitation energies and tend to be in a better agreement with respect to the TBE (see {\SupInf}). Note that the (dT) correction demonstrates better performance than the (fT) one as previously observed. \cite{Manohar_2008}}
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\alert{We observe that SF-EOM-CCSD excitation energies are closer to the SF-ADC(2)-s, with an energy difference of about \SI{0.1}{\eV}, than the other schemes as it was already noticed by LeFrançois and co-workers. \cite{Lefrancois_2015}
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We see that the SF-EOM-CCSD excitations energies for the triplet state are larger of about (\SI{0.3}{\eV} compared to the CCSD ones, this was also present in the study of Manohar and Krylov. \cite{Manohar_2008}
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Again, we have similar results, with SF-EOM-CCSD, than previous studies \cite{Manohar_2008,Lefrancois_2015} for the excited states. We can logically expect similar trend for SF-EOM-CCSD(fT) and SF-EOM-CCSD(dT) that lower the excitation energies and tend to be in a better agreement with respect to the TBE (see {\SupInf}). Note that the (dT) correction demonstrates better performance than the (fT) one as previously observed. \cite{Manohar_2008}}
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Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table \ref{tab:D2h}.
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Regarding the \alert{multi-reference} 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.
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@ -717,7 +722,7 @@ SF-ADC(2.5) & 6-31+G(d) & $0.234$ & $1.705$ & $2.087$ \\
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SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
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& aug-cc-pVDZ & $0.088$ & $1.571$ & $1.878$ \\
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& aug-cc-pVTZ & $0.079$ & $1.575$ & $1.853$ \\[0.1cm]
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\alert{EOM-SF-CCSD} & \alert{6-31+G(d)} & \alert{$0.446$} & \alert{$1.875$} & \alert{$2.326$} \\
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\alert{SF-EOM-CCSD} & \alert{6-31+G(d)} & \alert{$0.446$} & \alert{$1.875$} & \alert{$2.326$} \\
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& \alert{aug-cc-pVDZ} & \alert{$0.375$} & \alert{$1.776$} & \alert{$2.102$} \\
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& \alert{aug-cc-pVTZ}& \alert{$0.354$} & \alert{$1.768$} & \alert{$2.060$} \\
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\end{tabular}
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@ -845,7 +850,7 @@ Although it provides a decent singlet-triplet gap value, SF-ADC(2)-x seems to pa
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Again, averaging the SF-ADC(2)-s and SF-ADC(3) transition energies is beneficial in most cases at the exception of {\Aoneg}.
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Although the basis set effects are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and this holds for wave function methods in general.
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\alert{Again, the EOM-SF-CCSD excitation energies are closer to the SF-ADC(2)-s ones than the other schemes and (dT) and (fT) corrections tend to give a better agreement with respect to the TBE (see {\SupInf}). As for the {\Dtwo} excitation energies, the (dT) correction performs better than the (fT) one.}
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\alert{Again, the SF-EOM-CCSD excitation energies are closer to the SF-ADC(2)-s ones than the other schemes and (dT) and (fT) corrections tend to give a better agreement with respect to the TBE (see {\SupInf}). As for the {\Dtwo} excitation energies, the (dT) correction performs better than the (fT) one.}
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Let us turn to the multi-reference results (Table \ref{tab:D4h}).
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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.
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@ -881,8 +886,8 @@ This has been shown to be clearly beneficial for the automerization barrier and
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\item At the SF-ADC level, we have found that, as expected, the extended scheme, SF-ADC(2)-x, systematically worsen the results compared to the cheaper standard version, SF-ADC(2)-s.
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Moreover, as previously reported, SF-ADC(2)-s and SF-ADC(3) have opposite error patterns which means that SF-ADC(2.5) emerges as an excellent compromise.
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\item \alert{EOM-SF-CCSD shows similar performance than the cheaper SF-ADC(2)-s formalism, especially for the excitation energies.
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As previously reported, the two variants including non-iterative triples corrections, EOM-SF-CCSD(dT) and EOM-SF-CCSD(fT), improve the results, the (dT) correction performing slightly better for the vertical excitation energies computed at the {\Dtwo} and {\Dfour} equilibrium geometries.}
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\item \alert{SF-EOM-CCSD shows similar performance than the cheaper SF-ADC(2)-s formalism, especially for the excitation energies.
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As previously reported, the two variants including non-iterative triples corrections, SF-EOM-CCSD(dT) and SF-EOM-CCSD(fT), improve the results, the (dT) correction performing slightly better for the vertical excitation energies computed at the {\Dtwo} and {\Dfour} equilibrium geometries.}
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\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.
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In such scenario, the SF-TD-DFT excitation energies can exhibit errors of the order of \SI{1}{\eV} compared to the TBEs.
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@ -214,7 +214,7 @@ SF-ADC(2)-s & $-0.30$ & \alert{$0.098$} & $-0.026$ & $0.093$ & $0.112$ & $0.112$
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SF-ADC(2)-x & $1.44$ & \alert{$0.106$} & $-0.094$ & $-0.335$ & $0.068$ & $-0.409$ & $-0.118$ \\
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SF-ADC(2.5) & $0.18$ & \alert{$0.042$} & $0.006$ &$0.140$ & $0.024$ & $0.094$ & $0.000$ \\
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SF-ADC(3) & $0.65$ & \alert{$-0.014$} & $0.037$ & $0.186$ & $-0.065$ & $0.075$ & $0.004$ \\
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\alert{EOM-SF-CCSD} & \alert{$-1.53$} & \alert{$0.176$} & \alert{$0.168$} & \alert{$0.207$} & \alert{$0.210$} & \alert{$0.268$} & \alert{$0.211$} \\[0.1cm]
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\alert{SF-EOM-CCSD} & \alert{$-1.53$} & \alert{$0.176$} & \alert{$0.168$} & \alert{$0.207$} & \alert{$0.210$} & \alert{$0.268$} & \alert{$0.211$} \\[0.1cm]
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CASSCF(4,4) & $-1.55$ & \alert{$0.237$} & $1.421$ & $0.403$& $0.290$ & $0.734$ & $1.575$ \\
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CASPT2(4,4) & $-1.16$ & \alert{$-0.021$} & $-0.202$ & $0.034$ & $-0.016$ & $0.006$ & $-0.214$ \\
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%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
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@ -252,9 +252,9 @@ Literature & $8.53$\fnm[3] & $1.573$\fnm[3] & $3.208$\fnm[3] & $4.247$\fnm[3] &
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\fnt[3]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-s/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
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\fnt[4]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-x/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
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\fnt[5]{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.}
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\fnt[6]{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.}
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\fnt[7]{\alert{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD(fT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
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\fnt[8]{\alert{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD(dT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
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\fnt[6]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
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\fnt[7]{\alert{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD(fT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
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\fnt[8]{\alert{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD(dT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
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\fnt[9]{\alert{Value obtained from Ref.~\onlinecite{Gulania_2021} at the EOM-DEA-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
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\fnt[10]{\alert{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(3p-1h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}}
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\fnt[11]{\alert{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(4p-2h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}}
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@ -29,7 +29,9 @@ We look forward to hearing from you.
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%%% REVIEWER 1 %%%
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\noindent \textbf{\large Authors' answer to Reviewer \#1}
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{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.
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{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).
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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.
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The authors reach some interesting and useful conclusions concerning the tested methods.
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}
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\\
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\alert{
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