EOM-SF-CCSD discussion

Manuscript/CBD.tex

@@ -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.
 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. 
+\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).
 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}
+\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.
 
 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. 
 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 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}.
 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-pVTZ & $0.271$ & $1.824$ & $2.415$ \\
 % & 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) & $0.386$ & $1.974$ & $2.736$ \\
  & aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\
  & 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.
 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}.
+\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.
 
 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. 
 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.
 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}.