done with 4 atoms
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@ -595,37 +595,36 @@ $^i${0-0 energies from Ref.~\citenum{Jud84c}.}
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\end{table}
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\titou{HERE}
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For acetone, one should clearly distinguish the valence ES, for which both methodological and basis set effects are small, and the Rydberg transitions that, not only are very
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sensitive to the basis set, but are upshifted by ca.~$0.04$ eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} transition energies of Merch\'an and coworkers listed on the
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r.h.s. of Table \ref{Table-4} are quite clearly too small, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the details of the calculations, as the Urban group obtained {\CASPT}
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excitation energies of $4.40$, $4.09$ and $6.22$ eV for the $^1A_2$, $^3A_2$, and $^3A_1$ ES, \cite{Pas12} in much better agreement with ours. Their estimates for the three $n \ra 3p$ transitions of $7.52$, $7.57$, and $7.53$ eV
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for the $^1A_2$, $^1A_1$, and $^1B_2$ ES also systematically fall within $0.10$ eV of our current CC values, whereas for these three ES, the current {\NEV} values are quite clearly too large.
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For acetone, one should clearly distinguish the valence ES, for which both methodological and basis set effects are small, and the Rydberg transitions that both very
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sensitive to the basis set, and upshifted by ca.~$0.04$ eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} transition energies of Merch\'an and coworkers listed on the
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right panel of Table \ref{Table-4} are clearly too low, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the \titou{details of the calculations}, as the Urban group obtained {\CASPT}
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excitation energies of $4.40$, $4.09$ and $6.22$ eV for the $^1A_2$, $^3A_2$, and $^3A_1$ ES, \cite{Pas12} in much better agreement with ours. Their estimates of the three $n \ra 3p$ transitions, $7.52$, $7.57$, and $7.53$ eV
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for the $^1A_2$, $^1A_1$, and $^1B_2$ ES, also systematically fall within $0.10$ eV of our current CC values, whereas for these three ES, the current {\NEV} values are clearly too large.
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In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to the CC expansion, as illustrated by the maximal discrepancies of $\pm$0.02 eV between the {\CCT} and {\CCSDTQ} results
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with the {\Pop} basis set. While the lowest $n \ra \pis$ transition of both spin symmetries are rather insensitive to the selected basis set, all other states need quite large bases to be correctly described (Table S4).
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As expected our theoretical vertical transition energies show the same ranking but are systematically larger than the available experimental 0-0 energies.
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In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to the CC expansion, as illustrated by the maximal discrepancies of $\pm$0.02 eV between the {\CCT}/{\Pop} and {\CCSDTQ}/{\Pop} results.
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While the lowest $n \ra \pis$ transition of both spin symmetries are rather basis set insensitive, all the other states need quite large one-electron bases to be correctly described (Table S4).
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As expected, our theoretical vertical transition energies show the same ranking but are systematically larger than the available experimental 0-0 energies.
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For the isoleectronic isobutene, we considered two singlet Rydberg and one triplet valence ES. For all three cases, we note very nice agreement between {\CCT} and {\CCSDT} results for all considered basis sets, the
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CC results being also within or very close to the {\FCI} estimates with Pople's basis set. The match with the {\CCSD} results of Caricato and coworkers, \cite{Car10} is also very satisfying.
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For the isoleectronic isobutene molecule, we have considered two singlet Rydberg state and one triplet valence ES. For these three cases, we note, for each basis, a very nice agreement between {\CCT} and {\CCSDT}, the
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CC results being also very close to the {\FCI} estimates obtained with the Pople basis set. The similarity with the {\CCSD} results of Caricato and coworkers \cite{Car10} is also very satisfying.
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For the three remaining compounds, namely, cyanoformaldehyde, propynal, and thiopropynal, we report low-lying valence transitions all showing a largely dominant single excitation character. The basis set
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For the three remaining compounds, namely, cyanoformaldehyde, propynal, and thiopropynal, we report low-lying valence transitions with a definite single excitation character. The basis set
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effects are clearly under control (they are only significant for the second $^1A''$ ES of cyanoformaldehyde) and we could not detect any variation larger than $0.03$ eV between the {\CCT} and {\CCSDT} values for
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a given basis, hinting that the CC values should be close to the spot, as confirmed by the {\FCI} data.
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a given basis, alluding that the CC values are very accurate.
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This is further confirmed by the {\FCI} data.
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\subsubsection{Intermediate conclusions}
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\label{sec-ic}
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As we have seen for the 15 four-atom molecules considered here, we found extremely consistent transition energies between the tested CC approaches and the {\FCI} estimates in the vast majority of cases,
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Importantly, we confirm the previous conclusions obtained on smaller compounds:\cite{Loo18a} i) {\CCSDTQ} values systematically fall within, or are extremely close from, the {\FCI} error bar,
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ii) both {\CCT} and {\CCSDT} are also highly trustable when the considered ES does not show a very strong double excitation character. Indeed, considering all the 54 ``single transitions''
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for which {\CCSDTQ} estimates could be obtained (only excluding the lowest $^1A_g$ ES of butadiene and glyoxal), we determined trifling mean signed errors (MSE of 0.00 eV), tiny
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As we have seen for the 15 four-atom molecules considered here, we found extremely consistent transition energies between CC and {\FCI} estimates in the vast majority of the cases.
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Importantly, we confirm our previous conclusions obtained on smaller compounds: \cite{Loo18a} i) {\CCSDTQ} values systematically fall within (or are extremely close to) the {\FCI} error bar,
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ii) both {\CCT} and {\CCSDT} are also highly trustable when the considered ES does not exhibit a strong double excitation character. Indeed, considering the 54 ``single'' excitations
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for which {\CCSDTQ} estimates could be obtained (only excluding the lowest $^1A_g$ ES of butadiene and glyoxal), we determined negligible mean signed errors (MSE of $0.00$ eV), tiny
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MAE ($0.01$ and $0.02$ eV), and small maximal deviations ($0.05$ and $0.04$ eV) for {\CCT} and {\CCSDT}, respectively. This clearly indicates that these two approaches provide chemically-accurate
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estimates (errors $< 1$ kcal.mol$^{-1}$ or $0.043$ eV) for most electronic transitions. Interestingly, some of us have shown that {\CCT} also provides chemically-accurate 0-0 energies as compared
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to experimental values for most valence transitions. \cite{Loo18b,Loo19a,Sue19} When comparing the {\NEV} and {\CCT} ({\CCSDT}) results obtained with {\AVTZ} for all transitions in four-atom molecules,
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one obtains a mean signed deviation of $+0.09$ ($+0.09$) eV and a mean absolute deviation of $0.11$ ($0.12$) eV, considering all 91 (65) ES for which comparisons are possible, again excluding only
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the lowest $^1A_g$ states of butadiene and glyoxal. Although the error cannot be fully ascribed to the multi-reference method, that is additionally dependent of the selected active space, it seems to
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indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with the {\CASPT} approach that, from the comparisons discussed above,
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generally undershoots the transition energies.
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estimates (errors below $1$ kcal.mol$^{-1}$ or $0.043$ eV) for most electronic transitions. Interestingly, some of us have shown that {\CCT} also provides chemically-accurate 0-0 energies as compared
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to experimental values for most valence transitions. \cite{Loo18b,Loo19a,Sue19} When comparing the {\NEV} and {\CCT} ({\CCSDT}) results obtained with {\AVTZ} for the 91 (65) ES for which comparisons are possible (again excluding only the lowest $^1A_g$ states of butadiene and glyoxal),
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one obtains a mean signed deviation of $+0.09$ ($+0.09$) eV and a mean absolute deviation of $0.11$ ($0.12$) eV.
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This seems to indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with {\CASPT} that is known to generally underestimate transition energies, as further illustrated and discussed above.
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\subsection{Five-atom molecules}
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@ -688,7 +687,7 @@ $^c${MR-CC results from Ref.~\citenum{Li10c};}
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$^d${{\AT} results from Ref.~\citenum{Hol15};}
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$^e${{\CCT} results from Ref.~\citenum{Sch17};}
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$^f${Various experiments summarized in Ref.~\citenum{Wan00};}
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$^g${Electron impact from Ref.~\citenum{Vee76b}: for the $^1A_1$ state two values (6.44 and 6.61 eV) are reported, whereas for the two lowest triplet states, 3.99 eV and 5.22 eV values can be found in Ref.~\citenum{Fli76};}
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$^g${Electron impact from Ref.~\citenum{Vee76b}: for the $^1A_1$ state two values (6.44 and 6.61 eV) are reported, whereas for the two lowest triplet states, Two values (3.99 eV and 5.22 eV) can be found in Ref.~\citenum{Fli76};}
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$^h${{\NEV} results from Ref.~\citenum{Pas06c};}
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$^i${Best estimate from Ref.~\citenum{Chr99}, based on CC calculations;}
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$^j${XMS-{\CASPT} results from Ref.~\citenum{Hei19};}
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@ -768,7 +767,7 @@ $^c${MR-MP results from Ref.~\citenum{Nak96};}
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$^d${{\CCT} results from Ref.~\citenum{Sch17};}
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$^e${Electron impact from Ref.~\citenum{Fru79};}
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$^f${Gas phase absorption from Ref.~\citenum{McD91b};}
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$^g${Energy loss from Ref.~\citenum{McD85} for the two valence state; two-photon resonant experiment from Ref.~\citenum{Sab92} for the $^1A_2$ Rydberg ES;}
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$^g${Energy loss from Ref.~\citenum{McD85} for the two valence states; two-photon resonant experiment from Ref.~\citenum{Sab92} for the $^1A_2$ Rydberg ES;}
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$^h${{\CASPT} results from Ref.~\citenum{Ser96b};}
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$^i${{\CCT} results from Ref.~\citenum{Sil10c};}
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$^j${Gas-phase experimental estimates from Ref.~\citenum{Dev06};}
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@ -777,7 +776,7 @@ $^l${SAC-CI results from Ref.~\citenum{Wan01};}
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$^m${CCSDR(3) results from Ref.~\citenum{Pas07};}%, this work also contains {\NEV} estimates;}
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$^n${TBE from Ref.~\citenum{Hol14}, based on EOM-CCSD for singlet and ADC(2) for triplets;}
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$^o${0-0 energies from Ref.~\citenum{Dil72};}
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$^p${0-0 energies from Ref.~\citenum{Var82} for the singlets and energy loss experiment from Ref.~\citenum{Hab03} for the triplet ES;}
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$^p${0-0 energies from Ref.~\citenum{Var82} for the singlets and energy loss experiment from Ref.~\citenum{Hab03} for the triplets;}
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$^q${0-0 energies from Ref.~\citenum{Hol14}.}
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\end{footnotesize}
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\end{flushleft}
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@ -854,7 +853,7 @@ $^b${{\CCT} results from Ref.~\citenum{Chr96c};}
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$^c${SAC-CI results from Ref.~\citenum{Li07b};}
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$^d${RASPT2(18,18) results from Ref.~\citenum{Sha19};}
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$^e${Electron impact from Ref.~\citenum{Doe69};}
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$^f${Jet-cooled experiment from Ref.~\citenum{Hir91} for the two lowest states, multi-photon experiments from Refs.~ \citenum{Joh76} and \citenum{Joh83} for the Rydberg states.}
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$^f${Jet-cooled experiment from Ref.~\citenum{Hir91} for the two lowest states, multi-photon experiments from Refs.~\citenum{Joh76} and \citenum{Joh83} for the Rydberg states.}
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\end{footnotesize}
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\end{flushleft}
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\end{table}
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@ -922,14 +921,14 @@ $^a${{\CASPT} results from Ref.~\citenum{Web99};}
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$^b${{\STEOM} results from Ref.~\citenum{Noo99};}
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$^c${SAC-CI results from Ref.~\citenum{Li07b};}
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$^d${{\CCT} results from Ref.~\citenum{Sch17};}
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$^e${Dip spectroscopy from Ref.~\citenum{Oku90} ($B_{3u}$ and $B_{2g}$ states) and EEL from Ref.~\citenum{Wal91} (other states);}
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$^e${\titou{Dip} spectroscopy from Ref.~\citenum{Oku90} ($B_{3u}$ and $B_{2g}$ states) and EEL from Ref.~\citenum{Wal91} (other states);}
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$^f${UV max from Ref.~\citenum{Bol84};}
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$^g${{\CASPT} results from Ref.~\citenum{Rub99};}
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$^h${Ext-{\STEOM} results from Ref.~\citenum{Noo00};}
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$^i${GVVPT2 results from Ref.~\citenum{Dev08};}
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$^j${{\NEV} results from Ref.~\citenum{Ang09};}
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$^k${{\CCT} results from Ref.~\citenum{Sil10c};}
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$^l${From Ref.~\citenum{Pal97}, the singlets are from EEL, but for the 4.97 and 5.92 eV values that are from VUV; the triplets are from EEL, and other triplet peaks are mentioned at 4.21, 4.6, and 5.2 eV but not identified;}
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$^l${From Ref.~\citenum{Pal97}, the singlets are from EEL, except for the $4.97$ and $5.92$ eV values that are from VUV; the triplets are from EEL, and other triplet peaks are mentioned at $4.21$, $4.6$, and $5.2$ eV but not identified;}
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$^m${all these doubly ES have a $(n,n \ra \pis, \pis)$ character.}
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\end{footnotesize}
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\end{flushleft}
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@ -1089,7 +1088,7 @@ $^d${EOM-CCSD({$\tilde{{T}}$}) from Ref.~\citenum{Del97b};}
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$^e${UV max from Ref.~\citenum{Bol84};}
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$^f${EEL from Ref.~\citenum{Lin15};}
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$^g${{\CASPT} from Ref.~\citenum{Oli05};}
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$^h${CC3-ext. from Ref.~\citenum{Sil10c}.}
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$^h${CC3-ext.~from Ref.~\citenum{Sil10c}.}
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\end{footnotesize}
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\end{flushleft}
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\end{table}
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@ -1381,12 +1380,12 @@ Triazine &$^1A_1'' (\Val; n \ra \pis)$ & & 88.3 & 4.72 & {\CCSDT}/AVTZ
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\begin{flushleft}\begin{footnotesize}\begin{singlespace}
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\vspace{-0.6 cm}
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$^a${
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Method A: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\AVDZ} and {\CCSDT}/{\AVDZ} results;
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Method B: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCSDT}/{\Pop} results;
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Method C: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCT}/{\Pop} results;
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Method D: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\AVDZ} and {\CCT}/{\AVDZ} results;
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Method E: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\Pop} and {\CCT}/{\Pop} results;
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Method F: {exCI}/{\AVDZ} value (from Ref.~\citenum{Loo19c}) corrected by the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ} results.
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Method A: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\AVDZ} and {\CCSDT}/{\AVDZ};
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Method B: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCSDT}/{\Pop};
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Method C: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCT}/{\Pop};
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Method D: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\AVDZ} and {\CCT}/{\AVDZ};
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Method E: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\Pop} and {\CCT}/{\Pop};
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Method F: {\FCI}/{\AVDZ} value (from Ref.~\citenum{Loo19c}) corrected by the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ}.
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}
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\end{singlespace}\end{footnotesize}\end{flushleft}
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