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%% This BibTeX bibliography file was created using BibDesk.
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%% https://bibdesk.sourceforge.io/
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%% Created for Pierre-Francois Loos at 2022-05-25 14:31:23 +0200
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%% Created for Pierre-Francois Loos at 2022-06-08 17:31:37 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Knowles_1988,
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abstract = {A new method for evaluating one-particle coupling coefficients in a general configuration interaction calculation is presented. Through repeated application and use of resolutions of the identity, two-, three- and four-body coupling coefficients and density matrices may be built in a simple and efficient way. The method is therefore of use in both multiconfiguration SCF (MC SCF) and multireference configuration interaction (MRCI) calculations. Examples show that the approach is efficient for both these applications.},
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author = {Peter J. Knowles and Hans-Joachim Werner},
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date-added = {2022-06-08 17:31:36 +0200},
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date-modified = {2022-06-08 17:31:36 +0200},
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doi = {https://doi.org/10.1016/0009-2614(88)87412-8},
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journal = {Chem. Phys. Lett.},
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number = {6},
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pages = {514-522},
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title = {An efficient method for the evaluation of coupling coefficients in configuration interaction calculations},
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volume = {145},
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year = {1988},
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bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0009261488874128},
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bdsk-url-2 = {https://doi.org/10.1016/0009-2614(88)87412-8}}
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@article{Werner_1988,
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author = {Werner,Hans‐Joachim and Knowles,Peter J.},
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date-added = {2022-06-08 17:31:36 +0200},
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date-modified = {2022-06-08 17:31:36 +0200},
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doi = {10.1063/1.455556},
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journal = {J. Chem. Phys.},
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number = {9},
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pages = {5803-5814},
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title = {An efficient internally contracted multiconfiguration--reference configuration interaction method},
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volume = {89},
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year = {1988},
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bdsk-url-1 = {https://doi.org/10.1063/1.455556}}
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@article{Head-Gordon_2003,
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author = {Martin, Head-Gordon},
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doi = {10.1016/S0009-2614(03)00422-6},
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journal = {Chem. Phys. Lett.},
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number = {3},
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pages = {508-511},
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title = {Characterizing unpaired electrons from the one-particle density matrix},
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volume = {372},
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number = {3},
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pages = {508-511},
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year = {2003},
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year = {2003},
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bdsk-url-1 = {https://doi.org/10.1016/S0009-2614(03)00422-6}}
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@article{Orms_2018,
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author = {Natalie, Orms and Dirk.R, Rehn and Andreas, Dreuw and Anna I. Krylov},
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doi = {10.1021/acs.jctc.7b01012},
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journal = {J. Chem. Theory Comput.},
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number = {2},
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pages = {638-648},
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title = {Characterizing Bonding Patterns in Diradicals and Triradicals by Density-Based Wave Function Analysis: A Uniform Approach},
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volume = {14},
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number = {2},
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pages = {638-648},
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year = {2018},
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year = {2018},
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bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.7b01012}}
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@article{Gulania_2021,
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@ -119,7 +119,7 @@ The energy of this barrier is estimated, experimentally, in the range of \SIrang
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The lowest-energy excited states of CBD in both symmetries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and the {\sBoneg} and {\Atwog} states for the square one.
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Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown.
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Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be inaccessible with \alert{standard} adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and remain torturous for state-of-the-art methods like \alert{standard} equation-of-motion third-order coupled-cluster (EOM-CC3) \cite{Christiansen_1995,Koch_1997} or even coupled-cluster with singles, doubles, and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
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Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be inaccessible with \alert{standard} adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and \alert{remain challenging for standard hierarchy of EOM-CC methods that are using ground-state Hartree-Fock reference}. \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
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In order to tackle the problem of multi-configurational character and double excitations, we have explored several approaches.
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The most evident way is to rely on \alert{multi-reference} methods, which are naturally designed to address such scenarios.
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@ -138,10 +138,7 @@ Nonetheless, SCI methods remain very expensive and can be applied to a limited n
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Finally, another option to deal with these chemical scenarios is to rely on the spin-flip formalism, established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020} where one accesses the ground and doubly-excited states via a single (spin-flip) de-excitation and excitation from the lowest triplet state, respectively.
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\alert{One drawback of spin-flip methods is spin contamination} (\ie, the artificial mixing of electronic states with different spin multiplicities) due not only to the spin incompleteness in the spin-flip expansion but also to the potential spin contamination of the reference configuration. \cite{Casanova_2020}
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One can address part of this issue by increasing the excitation order or by complementing the spin-incomplete configuration set with the missing configurations. \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
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%both solutions being associated with an increased computational cost.
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\alert{Note that we can quantify the polyradical character associated to a given electronic state using Head-Gordon \cite{Head-Gordon_2003} index that provide a measure of the number of unpaired electrons.\cite{Orms_2018}}
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\alert{Note that one can quantify the polyradical character associated to a given electronic state using Head-Gordon's index \cite{Head-Gordon_2003} that provides a measure of the number of unpaired electrons. \cite{Orms_2018}}
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In the present work, we define highly-accurate reference values and investigate the accuracy of each family of computational methods mentioned above on the automerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
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Computational details are reported in Sec.~\ref{sec:compmet}.
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@ -206,7 +203,7 @@ For ionic excited states, like the {\sBoneg} state of CBD, it is particularly im
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On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
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Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
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In order to avoid the intruder state problem in CASPT2, a real-valued level shift of \SI{0.3}{\hartree} is set, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.25}{\hartree} to avoid systematic underestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022}
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%For the sake of comparison, in some cases, we have also performed multi-reference CI (MRCI) calculations.
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\alert{For the sake of comparison, in some cases, we have also performed multi-reference CI calculations including Davidson correction (MRCI+Q). \cite{Knowles_1988,Werner_1988}}
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All these calculations are carried out with MOLPRO. \cite{Werner_2020}
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%and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -230,10 +227,9 @@ 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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%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.
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%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
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\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}
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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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\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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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -411,9 +407,10 @@ In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which respectively sca
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Nonetheless, at a $\order*{N^5}$ computational scaling, SF-ADC(2)-s is particularly accurate, even compared to high-order CC methods (see below).
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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).
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This behavior was previously reported by Dreuw's group. \cite{Wormit_2014,Harbach_2014,Dreuw_2015}
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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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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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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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The CASSCF results predict an even lower barrier than CASPT2 due to the well known lack of dynamical correlation at the CASSCF level.
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@ -549,10 +546,10 @@ PC-NEVPT2(4,4) &6-31+G(d)& $1.409$ & $2.652$ & $4.120$ \\
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%& aug-cc-pVDZ & $1.558$ & $3.670$ & $4.254$ \\
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%& aug-cc-pVTZ & $1.568$ & $3.678$ & $4.270$ \\
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%& aug-cc-pVQZ & $1.574$ & $3.681$ & $4.280$ \\[0.1cm]
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%MRCI(4,4)+Q & 6-31+G(d) & $1.525$ & $3.515$ & $4.165$ \\
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% & aug-cc-pVDZ & $1.510$ & $3.347$ & $4.142$ \\
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% & aug-cc-pVTZ & $1.519$ & $3.342$ & $4.159$ \\
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% & aug-cc-pVQZ & $1.525$ & $3.342$ & $4.169$ \\[0.1cm]
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\alert{MRCI(4,4)+Q} & 6-31+G(d) & $1.525$ & $3.515$ & $4.165$ \\
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& aug-cc-pVDZ & $1.510$ & $3.347$ & $4.142$ \\
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& aug-cc-pVTZ & $1.519$ & $3.342$ & $4.159$ \\
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& aug-cc-pVQZ & $1.525$ & $3.342$ & $4.169$ \\[0.1cm]
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CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.220$ \\
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& aug-cc-pVDZ & $1.685$ & $3.856$ & $4.221$ \\
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& aug-cc-pVTZ & $1.686$ & $3.844$ & $4.217$ \\
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@ -637,6 +634,7 @@ These basis set effects are fairly transferable to the other wave function metho
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This further motivates the ``pyramidal'' extrapolation scheme that we have employed to produce the TBE values (see Sec.~\ref{sec:TBE}).
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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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@ -764,10 +762,10 @@ PC-NEVPT2(4,4) & 6-31+G(d) & $0.085$ & $1.496$ & $1.329$ \\
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% & aug-cc-pVDZ & $0.273$ & $1.823$ & $2.419$ \\
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% & aug-cc-pVTZ & $0.271$ & $1.824$ & $2.415$ \\
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% & aug-cc-pVQZ & $0.273$ & $1.825$ & $2.413$ \\[0.1cm]
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%MRCI(4,4)+Q & 6-31+G(d) & $0.260$ & $1.728$ & $2.272$ \\
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% & aug-cc-pVDZ & $0.225$ & $1.669$ & $2.073$ \\
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% & aug-cc-pVTZ & $0.219$ & $1.667$ & $2.054$ \\
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% & aug-cc-pVQZ & $0.220$ & $1.667$ & $2.048$ \\[0.1cm]
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\alert{MRCI(4,4)+Q} & 6-31+G(d) & $0.260$ & $1.728$ & $2.272$ \\
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& aug-cc-pVDZ & $0.225$ & $1.669$ & $2.073$ \\
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& aug-cc-pVTZ & $0.219$ & $1.667$ & $2.054$ \\
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& aug-cc-pVQZ & $0.220$ & $1.667$ & $2.048$ \\[0.1cm]
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CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
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& aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\
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& aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\
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@ -845,9 +843,10 @@ Globally, we observe similar trends as those noted in Sec.~\ref{sec:D2h}.
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Concerning the singlet-triplet gap, each scheme predicts it to be positive.
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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}.
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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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\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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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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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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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 limited active space.
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@ -882,7 +881,8 @@ 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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\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.}
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\item \alert{EOM-SF-CCSD shows similar performance than 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 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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