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@ -272,7 +272,7 @@ State-averaged complete-active-space self-consistent field (SA-CASSCF) calculati
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\subsection{Spin-Flip}
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\label{sec:sf}
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In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore algebraic-diagrammatic construction \cite{schirmer_1982} (ADC) using standard ADC(2)-s \cite{trofimov_1997,dreuw_2015} and extended ADC(2)-x \cite{dreuw_2015} schemes as well as the ADC(3) \cite{dreuw_2015,trofimov_2002,harbach_2014} scheme. We also use spin-flip within the TD-DFT \cite{casida_1995} framework. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1 \cite{shao_2015}. Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The B3LYP \cite{becke_1988b,lee_1988a,becke_1993b}, PBE0 \cite{adamo_1999a,ernzerhof_1999} and BH\&HLYP hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using range-separated hybrid (RSH) functionals as: CAM-B3LYP \cite{yanai_2004a}, LC-$\omega$PBE08 \cite{weintraub_2009a} and $\omega$B97X-V \cite{mardirossian_2014}. The main difference here between these RSH functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. To complete the use of spin-flip TD-DFT we also considered the hybrid meta-GGA functional M06-2X \cite{zhao_2008} and the RSH meta-GGA functional M11 \cite{peverati_2011}.
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In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore algebraic-diagrammatic construction \cite{schirmer_1982} (ADC) using standard ADC(2)-s \cite{trofimov_1997,dreuw_2015} and extended ADC(2)-x \cite{dreuw_2015} schemes as well as the ADC(3) \cite{dreuw_2015,trofimov_2002,harbach_2014} scheme. We also use spin-flip within the TD-DFT \cite{casida_1995} framework. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1 \cite{shao_2015}. Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The B3LYP \cite{becke_1988b,lee_1988a,becke_1993b}, PBE0 \cite{adamo_1999a,ernzerhof_1999} and BH\&HLYP hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using range-separated hybrid (RSH) functionals as: CAM-B3LYP \cite{yanai_2004a}, LC-$\omega$PBE08 \cite{weintraub_2009a} and $\omega$B97X-V \cite{mardirossian_2014}. The main difference here between these RSH functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. To complete the use of spin-flip TD-DFT we also considered the hybrid meta-GGA functional M06-2X \cite{zhao_2008} and the RSH meta-GGA functional M11 \cite{peverati_2011}. Note that all SF-TD-DFT calculations are done within the TDA approximation.
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -770,7 +770,7 @@ For the $1\,{}^1B_{1g} $ state of the $(D_{2h})$ structure we see that all the x
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Then, for the $2\,{}^1A_{g} $ state we obtain closer results of the SF-TD-DFT methods to the TBE than in the case of the $1\,{}^1B_{1g} $ state. Indeed, we have an energy difference of about 0.01-0.34 eV for the $2\,{}^1A_{g} $ state whereas we have 0.35-0.93 eV for the $1\,{}^1B_{1g} $ state. The ADC schemes give the same error to the TBE value than for the other singlet state with 0.02 eV for the ADC(2) scheme and 0.07 eV for the ADC(3) one. The ADC(2)-x scheme provides a larger error with 0.45 eV of energy difference. Here, the CC methods manifest more variations with 0.63 eV for the CC3 value and 0.28 eV for the CCSDT compared to the TBE values. The CC4 method provides a small error with less than 0.01 eV of energy difference. The multiconfigurational methods globally give smaller error than for the other singlet state with, for the two active spaces, 0.03-0.12 eV compared to the TBE value. We can notice that CC3 and CCSDT provide larger energy errors for the $2\,{}^1A_{g} $ state than for the $1\,{}^1B_{1g} $ state, this is due to the strong multiconfigurational character of the $2\,{}^1A_{g} $ state whereas the $1\,{}^1B_{1g} $ state has a very weak multiconfigurational character. It is interesting to see that SF-TD-DFT and SF-ADC methods give better results compared to the TBE than CC3 and even CCSDT meaning that spin-flip methods are able to describe double excited states. Note that multireference methods obviously give better results too for the $2\,{}^1A_{g} $ state.
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Finally we look at the vertical energy errors for the $D_{4h}$ structure. First, we consider the $1\,{}^3A_{2g} $ state, the SF-TD-DFT methods give errors of about 0.07-1.6 eV where the largest energy differences are provided by the hybrid functionals. The ADC schemes give similar error with around 0.06-1.1 eV of energy difference. For the CC methods we have an energy error of 0.06 eV for CCSD and less than 0.01 eV for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) 0.29 eV of error and 0.02 eV for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about 0.12-0.13 eV. A larger active space shows again an improvement with 0.23 eV of error for CASSCF(12,12) and around 0.01-0.04 eV for the other multireference methods. CIPSI provides similar error with 0.02 eV. Then, we look at the $2\,{}^1A_{1g}$ state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about 0.10-1.03 eV. The ADC schemes give better errors with around 0.07-0.41 eV and where again the ADC(2)-x does not improve the ADC(2)-s energy but also gives worse results. For the CC methods we have energy errors of about 0.10-0.16 eV and CC4 provides really close energy to the TBE one with 0.01 eV of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of 0.01-0.73 eV and 0.02-0.44 eV respectively with the largest errors coming from the CASSCF method. Lastly, we look at the $1\,{}^1B_{2g}$ state where we have globally larger errors. The SF-TD-DFT exhibits errors of 0.43-1.50 eV whereas ADC schemes give errors of 0.18-0.30 eV. CC3 and CCSDT provide energy differences of 0.50-0.69 eV and the CC4 shows again close energy to the CCSDTQ TBE energy with 0.01 eV of error. The multireference methods give energy differences of 0.38-1.39 eV and 0.11-0.60 eV for the four by four and twelve by twelve active spaces respectively. We can notice that we have larger variations for the vertical energies of the square CBD than for the rectangular one. This can be explained by the fact that because of the degeneracy in the $D_{4h}$ structure it leads to strong multiconfigurational character states where single reference methods are unreliable. We can also see that for the CC methods we have a better description of the $2\,{}^1A_{1g}$ state than the $1\,{}^1B_{2g}$ state, this can be related, as previously said in Subsubsection \ref{sec:D4h}, to the fact that $1\,{}^1B_{2g}$ corresponds to a double excitation from the reference state. To obtain an improved description of the $1\,{}^1B_{2g}$ state we have to include quadruples.
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Finally we look at the vertical energy errors for the $D_{4h}$ structure. First, we consider the $1\,{}^3A_{2g} $ state, the SF-TD-DFT methods give errors of about 0.07-1.6 eV where the largest energy differences are provided by the hybrid functionals. The ADC schemes give similar error with around 0.06-1.1 eV of energy difference. For the CC methods we have an energy error of 0.06 eV for CCSD and less than 0.01 eV for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) 0.29 eV of error and 0.02 eV for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about 0.12-0.13 eV. A larger active space shows again an improvement with 0.23 eV of error for CASSCF(12,12) and around 0.01-0.04 eV for the other multireference methods. CIPSI provides similar error with 0.02 eV. Then, we look at the $2\,{}^1A_{1g}$ state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about 0.10-1.03 eV. The ADC schemes give better errors with around 0.07-0.41 eV and where again the ADC(2)-x does not improve the ADC(2)-s energy but also gives worse results. For the CC methods we have energy errors of about 0.10-0.16 eV and CC4 provides really close energy to the TBE one with 0.01 eV of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of 0.01-0.73 eV and 0.02-0.44 eV respectively with the largest errors coming from the CASSCF method. Lastly, we look at the $1\,{}^1B_{2g}$ state where we have globally larger errors. The SF-TD-DFT exhibits errors of 0.43-1.50 eV whereas ADC schemes give errors of 0.18-0.30 eV. CC3 and CCSDT provide energy differences of 0.50-0.69 eV and the CC4 shows again close energy to the CCSDTQ TBE energy with 0.01 eV of error. The multireference methods give energy differences of 0.38-1.39 eV and 0.11-0.60 eV for the four by four and twelve by twelve active spaces respectively. We can notice that we have larger variations for the vertical energies of the square CBD than for the rectangular one. This can be explained by the fact that because of the degeneracy in the $D_{4h}$ structure it leads to strong multiconfigurational character states where single reference methods are unreliable. We can also see that for the CC methods we have a better description of the $2\,{}^1A_{1g}$ state than the $1\,{}^1B_{2g}$ state, this can be related, as previously said in Subsubsection \ref{sec:D4h}, to the fact that $1\,{}^1B_{2g}$ corresponds to a double excitation from the reference state. To obtain an improved description of the $1\,{}^1B_{2g}$ state we have to include quadruples. At the end of Table \ref{tab:TBE} we show some literature results obtain from Ref ~\cite{lefrancois_2015,manohar_2008} where the cc-pVTZ basis is used. The SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3)results are presented and are consistent with our results with the exact same schemes but with the AVTZ basis.
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%Here again we can make the same comment for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states of the square CBD than the $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states of the rectangular CBD. The first state ($2\,{}^1A_{1g}$) has a strong multiconfigurational character
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@ -779,6 +779,59 @@ Finally we look at the vertical energy errors for the $D_{4h}$ structure. First,
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%%% TABLE I %%%
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%\begin{squeezetable}
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%\begin{table*}
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% \caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage T1 calculated at the CC3/AVTZ level.}
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%
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% \label{tab:TBE}
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% \begin{ruledtabular}
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% \begin{tabular}{lrrrrrrr}
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%%\begin{tabular}{*{1}{*{8}{l}}}
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%&\mc{3}{r}{$D_{2h}$ excitation energies (eV)} & \mc{3}{r}{$D_{4h}$ excitation energies (eV)} \\
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% \cline{3-5} \cline{6-8}
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%Method & AB & $1\,{}^3B_{1g} $~(98.7 \%) & $1\,{}^1B_{1g} $ (95.0 \%)& $2\,{}^1A_{g} $(0.84 \%) & $1\,{}^3A_{2g} $ & $2\,{}^1A_{1g} $ & $1\,{}^1B_{2g} $ \\
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% \hline
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%SF-TD-B3LYP & $10.41$ & $0.241$ & $-0.926$ & $-0.161$ & $-0.164$ & $-1.028$ & $-1.501$ \\
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%SF-TD-PBE0 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\
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%SF-TD-BHHLYP & $8.95$ & $0.078$ & $-0.393$ & $0.343$ & $-0.099$ & $-0.251$ & $-0.603$ \\
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%SF-TD-M06-2X & $1.42$ & $0.000$ & $-0.354$ & $0.208$ & $-0.066$ & $-0.097$ & $-0.432$ \\
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%SF-TD-CAM-B3LYP & $9.90$ & $0.280$ & $-0.807$ & $-0.011$ & $-0.134$ & $-0.920$ & $-1.370$ \\
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%SF-TD-$\omega $B97X-V & $10.01$ & $0.335$ & $-0.774$ & $0.064$ & $-0.118$ & $-0.928$ & $-1.372$ \\
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%SF-TD-M11 & $2.29$ & $0.097$ & $-0.474$ & $0.151$ & $-0.063$ & $-0.312$ & $-0.675$ \\
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%SF-TD-LC-$\omega $PBE08 & $10.81$ & $0.435$ & $-0.710$ & $0.199$ & $-0.086$ & $-0.939$ & $-1.376$ \\[0.1cm]
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%SF-ADC(2)-s & $-0.30$ & $0.069$ & $-0.026$ & $-0.018$ & $0.112$ & $0.112$ & $-0.190$ \\
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%SF-ADC(2)-x & $1.44$ & $0.077$ & $-0.094$ & $-0.446$ & $0.068$ & $-0.409$ & $-0.303$ \\
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%SF-ADC(3) & $0.65$ & $-0.043$ & $0.037$ & $0.075$ & $-0.065$ & $0.075$ & $-0.181$ \\[0.1cm]
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%CCSD & $0.95$ & & & & $-0.059$ & $0.100$ & \\
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%CC3 & $-1.05$ & $-0.060$ & $-0.006$ & $0.628$ & & $0.162$ & $0.686$ \\
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%CCSDT & $-0.25$ & $-0.051$ & $0.014$ & $0.280$ & $0.005$ & $0.131$ & $0.503$ \\
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%CC4 & $-0.11$ & & $0.003$ & $-0.006$ & & $0.011$ & $-0.013$ \\
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%CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
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%SA2-CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
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%CASPT2(4,4) & & $-0.050$ & $-0.202$ & $-0.077$ & $-0.016$ & $0.006$ & $-0.399$ \\
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%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
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%SC-NEVPT2(4,4) & & $-0.083$ & $-0.703$ & $-0.041$ & $-0.120$ & $-0.072$ & $-0.979$ \\
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%PC-NEVPT2(4,4) & & $-0.080$ & $-0.757$ & $-0.066$ & $-0.118$ & $-0.097$ & $-1.031$ \\
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%MRCI(4,4) & & $0.106$ & $0.553$ & $0.121$ & $0.127$ & $0.324$ & $0.381$ \\[0.1cm]
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%SA2-CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
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%CASPT2(12,12) & & $0.018$ & $0.058$ & $-0.106$ & $0.039$ & $0.038$ & $-0.108$ \\
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%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
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%SC-NEVPT2(12,12) & $-0.65$ & $0.039$ & $0.063$ & $-0.063$ & $0.021$ & $0.046$ & $-0.142$ \\
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%PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.278$ \\[0.1cm]
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%%CIPSI & & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ & \\
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%\bf{TBE} & $\left[\bf{8.93}\right]$\fnm[1] & $\left[\bf{1.462}\right]$\fnm[2] & $\left[\bf{3.125}\right]$\fnm[3] & $\left[\bf{4.149}\right]$\fnm[3] & $\left[\bf{0.144}\right]$\fnm[4] & $\left[\bf{1.500}\right]$\fnm[3] & $\left[\bf{2.034}\right]$\fnm[3] \\
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%\end{tabular}
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%
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% \end{ruledtabular}
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% \fnt[1]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CC4/AVTZ and CC4/AVDZ.}
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% \fnt[2]{Value obtained using the NEVPT2(12,12) one.}
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% \fnt[3]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CC4/AVTZ and CC4/AVDZ.}
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% \fnt[4]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CCSDT/AVTZ and CCSDT/AVDZ.}
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%
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%\end{table*}
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%\end{squeezetable}
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%%% %%% %%% %%%
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\begin{squeezetable}
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\begin{table*}
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\caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage T1 calculated at the CC3/AVTZ level.}
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@ -819,7 +872,11 @@ XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
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SC-NEVPT2(12,12) & $-0.65$ & $0.039$ & $0.063$ & $-0.063$ & $0.021$ & $0.046$ & $-0.142$ \\
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PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.278$ \\[0.1cm]
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%CIPSI & & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ & \\
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\bf{TBE} & $\left[\bf{8.93}\right]$\fnm[1] & $\left[\bf{1.462}\right]$\fnm[2] & $\left[\bf{3.125}\right]$\fnm[3] & $\left[\bf{4.149}\right]$\fnm[3] & $\left[\bf{0.144}\right]$\fnm[4] & $\left[\bf{1.500}\right]$\fnm[3] & $\left[\bf{2.034}\right]$\fnm[3] \\
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\bf{TBE} & $\left[\bf{8.93}\right]$\fnm[1] & $\left[\bf{1.462}\right]$\fnm[2] & $\left[\bf{3.125}\right]$\fnm[3] & $\left[\bf{4.149}\right]$\fnm[3] & $\left[\bf{0.144}\right]$\fnm[4] & $\left[\bf{1.500}\right]$\fnm[3] & $\left[\bf{2.034}\right]$\fnm[3] \\[0.1cm]
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lit. & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] & $0.266$\fnm[5] & $1.664$\fnm[5] & $1.910$\fnm[5] \\
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& $10.35$\fnm[6] & $1.576$\fnm[6] & $3.141$\fnm[6] & $3.796$\fnm[6] & $0.217$\fnm[6] & $1.123$\fnm[6] & $1.799$\fnm[6]\\
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& $9.58$ \fnm[7]& $1.456$\fnm[7] & $3.285$\fnm[7] & $4.334$\fnm[7] & $0.083$\fnm[7] & $1.621$\fnm[7] & $1.930$\fnm[7] \\
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& $7.48$\fnm[8]& $1.654$\fnm[8] & $3.416$\fnm[8] & $4.360$\fnm[8] & $0.369$\fnm[8] & $1.824$\fnm[8] & $2.143$\fnm[8] \\
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\end{tabular}
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\end{ruledtabular}
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@ -827,10 +884,13 @@ PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.
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\fnt[2]{Value obtained using the NEVPT2(12,12) one.}
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\fnt[3]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CC4/AVTZ and CC4/AVDZ.}
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\fnt[4]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CCSDT/AVTZ and CCSDT/AVDZ.}
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\fnt[5]{Value obtained from Ref ~\cite{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[6]{Value obtained from Ref ~\cite{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[7]{Value obtained from Ref ~\cite{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[8]{Value obtained from Ref ~\cite{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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\end{table*}
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\end{squeezetable}
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%%% %%% %%% %%%
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%================================================
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