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EOM-SF-CCSD discussion

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Manuscript/CBD.tex
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@ -230,8 +230,10 @@ Additionally, we have also computed SF-TD-DFT excitation energies using range-se
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. 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.
Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999} Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
Although there also exist spin-flip extension of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} they are not considered here. %Although there also exist spin-flip extension of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} they are not considered here.
%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1. %EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
\alert{There also exist spin-flip extension of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} and we considered the spin-flip variant of EOM-CCSD called EOM-SF-CCSD.\cite{Krylov_2001a}
Manohar and Krylov \cite{Manohar_2008} presented a noniterative triples correction to EOM-CCSD and extended it to the spin-flip variant. Two types of triples correction were given, first the EOM-CCSD(dT) obtained using the full similarity-transformed CCSD Hamiltonian diagonal and the second one EOM-CCSD(fT) using Hartree-Fock orbital energy differences.}
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@ -409,6 +411,7 @@ In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which respectively sca
Nonetheless, at a $\order*{N^5}$ computational scaling, SF-ADC(2)-s is particularly accurate, even compared to high-order CC methods (see below). Nonetheless, at a $\order*{N^5}$ computational scaling, SF-ADC(2)-s is particularly accurate, even compared to high-order CC methods (see below).
We note that SF-ADC(2)-x [which scales as $\order*{N^6}$] is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3). We note that SF-ADC(2)-x [which scales as $\order*{N^6}$] is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3).
This behavior was previously reported by Dreuw's group. \cite{Wormit_2014,Harbach_2014,Dreuw_2015} This behavior was previously reported by Dreuw's group. \cite{Wormit_2014,Harbach_2014,Dreuw_2015}
\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).}
Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models. Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models.
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. 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.
@ -629,7 +632,10 @@ 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. 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}). 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. 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 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}). 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}).
\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}
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}
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}}
Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table \ref{tab:D2h}. 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 \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. 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.
@ -754,6 +760,10 @@ PC-NEVPT2(4,4) & 6-31+G(d) & $0.085$ & $1.496$ & $1.329$ \\
% & aug-cc-pVDZ & $0.273$ & $1.823$ & $2.419$ \\ % & aug-cc-pVDZ & $0.273$ & $1.823$ & $2.419$ \\
% & aug-cc-pVTZ & $0.271$ & $1.824$ & $2.415$ \\ % & aug-cc-pVTZ & $0.271$ & $1.824$ & $2.415$ \\
% & aug-cc-pVQZ & $0.273$ & $1.825$ & $2.413$ \\[0.1cm] % & aug-cc-pVQZ & $0.273$ & $1.825$ & $2.413$ \\[0.1cm]
%MRCI(4,4)+Q & 6-31+G(d) & $0.260$ & $1.728$ & $2.272$ \\
% & aug-cc-pVDZ & $0.225$ & $1.669$ & $2.073$ \\
% & aug-cc-pVTZ & $0.219$ & $1.667$ & $2.054$ \\
% & aug-cc-pVQZ & $0.220$ & $1.667$ & $2.048$ \\[0.1cm]
CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\ CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
& aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\ & aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\
& aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\ & aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\
@ -831,6 +841,7 @@ Globally, we observe similar trends as those noted in Sec.~\ref{sec:D2h}.
Concerning the singlet-triplet gap, each scheme predicts it to be positive. 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}. 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}. Again, averaging the SF-ADC(2)-s and SF-ADC(3) transition energies is beneficial in most cases at the exception of {\Aoneg}.
\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.}
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. 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.
Let us turn to the multi-reference results (Table \ref{tab:D4h}). Let us turn to the multi-reference results (Table \ref{tab:D4h}).
@ -867,6 +878,8 @@ This has been shown to be clearly beneficial for the automerization barrier and
\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. \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.
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. 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.
\alert{\item The EOM-SF-CCSD method has shown, as previously stated, similar results to the SF-ADC(2)-s scheme, especially for the excitation energies. As previously reported, EOM-SF-CCSD(dT) (with (dT) triples correction to the EOM-SF-CCSD) and EOM-SF-CCSD(fT) (with (fT) triples correction to the EOM-SF-CCSD) improve the results and the (dT) correction performs better for the vertical excitation energies on both the {\Dtwo} and {\Dfour} equilibrium geometries.}
\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. \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. 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}. However, it was satisfying to see that the spin-flip version of ADC can lower these errors to \SIrange{0.1}{0.2}{\eV}.