reached butadiene
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@ -142,22 +142,22 @@ were computed on MP2/6-31G(d) structures with several levels of theories, notabl
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first series of TBE was {\CASPT}, \cite{Sch08} the majority of the most recent TBE (so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level, often using a basis set extrapolation technique.
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In more details, CC3/TZVP values were typically corrected for basis set effects by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works used Thiel's TBE for
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assessing low-order methods, \cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18,Hel19} highlighting
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their value for the community. In contrast, the number of extensions of this original set remains quite limited, i.e., {\CCSDT}/TZVP reference energies computed for 17 singlet states of six molecules appeared in 2014, \cite{Kan14}
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their value for the community. In contrast, the number of extensions of this original set remains quite limited, \ie, {\CCSDT}/TZVP reference energies computed for 17 singlet states of six molecules appeared in 2014, \cite{Kan14}
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and 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide) have been described in 2017. \cite{Kan17}
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This has motivated us to propose two years ago a set of 106 transition energies for which it was technically possible to reach the Full Configuration Interaction (FCI) limit by performing high-order CC calculations (up to {\CCSDTQP}) and selected
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CI (sCI) transition energy calculations on {\CCT} GS structures. \cite{Loo18a} We used these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} was on the {\FCI} spot, whereas we could not
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detect significant differences of accuracies between {\CCT} and {\CCSDT}, both being very accurate with mean absolute errors (MAE) as small as 0.03 eV compared to {\FCI} for ES with a single excitation character. These conclusions
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agree well with earlier studies. \cite{Wat13,Kan14,Kan17} We also recently proposed a set of 20 TBE for transitions presenting a large double excitation character. \cite{Loo19c} For such transitions, one can distinguish the ES
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as a function of $\%T_1$, the percentage of single excitation calculated at the {\CCT} level. For ES with a significant yet not dominant double excitation character, such as the famous $A_g$ ES of butadiene ($\%T_1$=75\%),
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CC methods including triples in the GS treatment deliver rather accurate estimates (MAE of 0.11 eV with {\CCT} and 0.06 eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or
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the generally robust $N$-Electron Valence State Perturbation Theory ({\NEV}). In contrast, for ES with a dominant double excitation character, e.g., the low-lying $n,n \ra \pis,\pis$ excitation in nitrosomethane ($\%T_1$=2\%),
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single-reference methods are not suited (MAE of 0.86 eV with {\CCT} and 0.42 eV with {\CCSDT}) and multi-reference methods are, in practice, required to obtain accurate results. \cite{Loo19c} Obviously, a clear limit of our 2018 work\cite{Loo18a}
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as a function of $\%T_1$, the percentage of single excitation calculated at the {\CCT} level. For ES with a significant yet not dominant double excitation character, such as the famous $A_g$ ES of butadiene ($\%T_1 = 75\%$),
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CC methods including triples in the GS treatment deliver rather accurate estimates (MAE of $0.11$ eV with {\CCT} and $0.06$ eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or
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the generally robust $N$-Electron Valence State Perturbation Theory ({\NEV}). In contrast, for ES with a dominant double excitation character, \eg, the low-lying $n,n \ra \pis,\pis$ excitation in nitrosomethane ($\%T_1$=2\%),
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single-reference methods are not suited (MAE of $0.86$ eV with {\CCT} and $0.42$ eV with {\CCSDT}) and multi-reference methods are, in practice, required to obtain accurate results. \cite{Loo19c} Obviously, a clear limit of our 2018 work\cite{Loo18a}
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is that we treated only compounds containing 1--3 non-hydrogen atoms, hence introducing a significant chemical bias. Therefore we have decided to go for larger molecules and we consider in the present contribution organic
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compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large bases remains a dream, and, the convergence of {\sCI} with the number of determinants is slower
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as well, so that extrapolating to the {\FCI} limit with a ca. 0.01 eV error bar is rarely doable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work,\cite{Loo18a} is beyond reach.
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as well, so that extrapolating to the {\FCI} limit with a ca.~$0.01$ eV error bar is rarely doable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work,\cite{Loo18a} is beyond reach.
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Anticipating this problem, we have previously demonstrated that one can very accurately estimate such limit by correcting values obtained with a high-level of theory and a double-$\zeta$ basis set by {\CCT} results obtained with a
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larger basis, \cite{Loo18a} and we glibally follow such strategy here. In addition, we also performed {\NEV} calculations in an effort to check the consistency of our estimates, especially for ES with intermediate $\%T_1$ values.
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Using this protocol, we define a set of more than 200 {\AVTZ} reference transition energies, most being within $\pm$ 0.03 eV for the {\FCI} limit. These reference energies are obtained on {\CCT} geometries and further basis set
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Using this protocol, we define a set of more than 200 {\AVTZ} reference transition energies, most being within $\pm 0.03$ eV for the {\FCI} limit. These reference energies are obtained on {\CCT} geometries and further basis set
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corrections up to at least quadruple-$\zeta$, are also provided with {\CCT}. Together with the results obtained in our two earlier works, \cite{Loo18a,Loo19c} the present TBE will hopefully contribute to climb a further step
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on the ES accuracy staircase.
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@ -231,7 +231,7 @@ Given that the {\sCI} results converges rather slowly for these larger systems,
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consistency of the approaches rather than as definitive TBE (see next Section). We also show the results of {\NEV}/{\AVTZ} calculations for all relevant states to have a further consistency check. We underline that, except
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when specifically discussed, all ES present a dominant single-excitation character (see also next Section), so that we do not expect serious CC breakdowns. This is especially true for the triplet ES that are known to show
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very large \%$T_1$ for the vast majority of states, \cite{Sch08} and we consequently put our maximal computational effort on determining accurate transition energies for singlet states. To assign the different ES, we use literature data, as well as
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usual criteria, i.e., relative energies, symmetries and compositions of the underlying MOs, as well as oscillator strengths. This allows clear-cut assignment for the vast majority of the cases. There are however some
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usual criteria, \ie, relative energies, symmetries and compositions of the underlying MOs, as well as oscillator strengths. This allows clear-cut assignment for the vast majority of the cases. There are however some
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state/method combination for which strong mixing between ES of the same symmetry makes unambiguous assignments beyond reach, which is a typical problem in such works. Such cases are however not statistically
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relevant and are therefore unlikely to change any of our main conclusions.
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@ -280,7 +280,7 @@ $^3\Delta_u$ &4.93&4.93&4.92&4.94$\pm$0.01 &4.86&4.85& &4.85$\pm$0.02 &4.8
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\begin{footnotesize}
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$^a${{\CASPT} results from Ref.~\citenum{Luo08};}
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$^b${Experimental 0-0 energies from Refs.~\citenum{Job66a} and \citenum{Job66b} (vacuum UV experiments);}
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$^c${Vertical fluorescence energy from the lowest excited state;}
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$^c${Vertical fluorescence energy of the lowest excited state;}
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$^d${Experimental 0-0 energies from Refs.~\citenum{Cal63} ($^3\Sigma_u^+$), \citenum{Bel69} ($^1\Sigma_u^-$), and \citenum{Fis72} ($^1\Delta_u$), all analyzing vacuum electronic spectra;}
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$^e${Experimental 0-0 energies from Ref.~\citenum{Hai79} (singlet ES, vacuum UV experiment) and Ref.~\citenum{All84} (triplet ES, EELS). In the latter contribution, the $2.7$ eV value for the $^3\Sigma_u^+$
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state is the onset, whereas an estimate of the vertical energy ($4.2 \pm 0.2$ eV) is given for the $^3\Delta_u$ state.}
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@ -305,8 +305,8 @@ within $\pm 0.02$ eV of the {\FCI} window, \ie, all methods presented in Table \
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\subsubsection{Cyclopropenone, cyclopropenethione, and methylenecyclopropene}
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These three related compounds present a three-member $sp^2$ carbon cycle conjugated to an external $\pi$ bond. While the ES of methylenecyclopropene have regularly been investigated with theoretical tools in the past,
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\cite{Mer96,Roo96,Car10b,Lea12,Gua13,Dad14,Gua14,Sch17,Bud17} the only investigations of vertical transitions we could find for the two other derivatives are the detailed 2002 {\CASPT} study of Serrano-Andr\'es and
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coworkers on both compounds, \cite{Ser02} and a more recent work reporting the three lowest-lying singlet states of cyclopropenone at the {\CASPT}/6-31G level.\cite{Liu14b}
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\cite{Mer96,Roo96,Car10b,Lea12,Gua13,Dad14,Gua14,Sch17,Bud17} the only investigations of vertical transitions we could find for the two other derivatives are a detailed {\CASPT} study of Serrano-Andr\'es and
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coworkers in 2002, \cite{Ser02} and a more recent work reporting the three lowest-lying singlet states of cyclopropenone at the {\CASPT}/6-31G level.\cite{Liu14b}
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\begin{table}[htp]
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@ -362,12 +362,12 @@ $^3A_1 (\pi \ra \pis)$ &4.74&4.74& & 4.67$\pm$0.10 &4.74&4.74$^g$&4.74&
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\vspace{-0.5 cm}
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\begin{flushleft}
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\begin{footnotesize}
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$^a${{\CASPT} results of Ref.~\citenum{Ser02};}
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$^b${Electron impact experiment of Ref.~\citenum{Har74}. Note that the $5.5$ eV peak was assigned differently in the original paper, and we follow here the analysis of Serrano-Andr\'es, \cite{Ser02}
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$^a${{\CASPT} results from Ref.~\citenum{Ser02};}
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$^b${Electron impact experiment from Ref.~\citenum{Har74}. Note that the $5.5$ eV peak was assigned differently in the original paper, and we follow here the analysis of Serrano-Andr\'es, \cite{Ser02}
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whereas the $6.1$ eV assignment was ``supposed'' in the original paper; experimental $\lambda_{\mathrm{max}}$ have been measured at $3.62$ eV and $6.52$ eV for the $^1B_1$ ($n \ra \pis$) and
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$^1B_2$ ($\pi \ra \pis$) transitions, respectively; \cite{Bre72}}
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$^c${{\CASPT} results of Refs.~\citenum{Mer96} and \citenum{Roo96};}
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$^d${{\CCT} results of Ref.~\citenum{Sch17};}
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$^c${{\CASPT} results from Refs.~\citenum{Mer96} and \citenum{Roo96};}
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$^d${{\CCT} results from Ref.~\citenum{Sch17};}
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$^e${$\lambda_{\mathrm{max}}$ in pentane at $-78^o$C from Ref.~\citenum{Sta84};}
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$^f${Significant state mixing with the $^1A_1$($\pi \ra 3p$) transition, yielding unambiguous attribution difficult;}
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$^g${As can be seen in the SI, our {\FCI}/{\AVDZ} estimates are $3.45 \pm 0.04$ and $4.79 \pm 0.02$ eV for the two lowest triplet states of methylenecyclopropene hinting that the CC3 and CCSDT
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@ -377,30 +377,32 @@ results might be slightly too low for the second transition. }
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\end{table}
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Our results are listed in Tables \ref{Table-2} and S2. As above, considering the Pople basis set {\Pop}, we note very small differences between {\CCT}, {\CCSDT}, and {\CCSDTQ}, the latter method giving transition energies
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Our results are listed in Tables \ref{Table-2} and S2. As above, considering the {\Pop} basis set, we notice very small differences between {\CCT}, {\CCSDT}, and {\CCSDTQ}, the latter method giving transition energies
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systematically falling within the {\FCI} extrapolation incertitude, except in one case (the lowest totally symmetric state of methylenecyclopropene for which the {\CCSDTQ} value is ``off'' by $0.02$ eV only). Depending on the state, it is
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either {\CCT} or {\CCSDT} that is the closest to {\CCSDTQ}. In fact, considering all {\CCSDTQ}/{\Pop} data listed in Table \ref{Table-2} as benchmark, the mean absolute deviation of {\CCT} and {\CCSDT} are 0.019 and 0.016 eV, respectively,
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hinting that the improvements brought by the latter more computationally intensive method are limited for these compounds. For the lowest $B_2$ state of methylenecyclopropene, one of the most challenging cases (\%$T_1 = 85\%$),
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it is clear from the {\exCI} value that only {\CCSDTQ} is really close to the spot, the {\CCT} and {\CCSDT} results being slightly too large by ca. 0.05 eV. It seems likely that the same pattern appears for the corresponding state in
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cyclopropenethione, although in that case the FCI error bar is large, preventing definitive conclusions. Interestingly, the quite small Pople basis set provides data within ca. 0.10 eV of basis set convergence at {\CCT} level for 80\%\ of
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the transitions. There are of course exceptions to this rule, e.g., the strongly dipole-allowed $^1A_1 (\pi \ra \pis)$ ES of cyclopropenone and the $^1B_1(\pi \ra 3s)$ ES of methylenecyclopropene are significantly
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too blueshifted with the Pople's basis set (Table S2). For cyclopropenone, our {\CCSDT}/{\AVTZ} estimates do agree reasonably well with the {\CASPT} data of Serrano-Andr\'es, but for the $^1B_2 (\pi \ra \pis)$ state
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that we locate significantly higher in energy and the three Rydberg states that CC foresees at significantly smaller energies. The current {\NEV} results are globally in better agreement with the CC values than the older {\CASPT} estimates,
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though some non-negligible deviations pertain. Even if comparisons with experiments should be made very cautiously, we note that the CC data are clearly more coherent with the electron impact measurements\cite{Har74} for the Rydberg
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states than the original {\CASPT} values. For cyclopropenethione, we obtain transition energies typically in agreement or larger than those obtained with {\CASPT}, \cite{Ser02} though there is no obvious relationship between the
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valence/Rydberg nature of the considered ES and the relative {\CASPT} error. The average absolute deviation between our {\NEV} and {\CCT} results is 0.08 eV only. Eventually for methylenecyclopropene, our values logically agree
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very well with the recent estimates of Schwabe and Goerigk, \cite{Sch17} obtained at the {\CCT}/{\AVTZ} level on a different geometry, whereas the available {\CASPT} values \cite{Mer96,Roo96} appear too low as compared to the
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current {\NEV} and {\CCSDT} values. For this compound, the available experimental data being wavelength of maximal absorption determined in condensed phase, \cite{Sta84} only a qualitative match is logically reached between theory and experiment.
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either {\CCT} or {\CCSDT} that is closest to {\CCSDTQ}. In fact, considering the {\CCSDTQ}/{\Pop} data listed in Table \ref{Table-2} as reference, the mean absolute deviation of {\CCT} and {\CCSDT} is $0.019$ and $0.016$ eV, respectively,
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hinting that the improvement brought by the latter, more expensive method is limited for this set of compounds. For the lowest $B_2$ state of methylenecyclopropene, one of the most challenging cases (\%$T_1 = 85\%$),
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it is clear from the {\FCI} value that only {\CCSDTQ} is close, the {\CCT} and {\CCSDT} results being slightly too large by $\sim 0.05$ eV. It seems reasonable to believe that the same observation can be made for the corresponding state of
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cyclopropenethione, although in that case the FCI error bar is too large to prevent any definitive conclusion. Interestingly, at the {\CCT} level of theory, the rather small {\Pop} basis set provides data within $0.10$ eV of the CBS limit for 80\%\ of
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the transitions. There are, of course, exceptions to this rule, \eg, the strongly dipole-allowed $^1A_1 (\pi \ra \pis)$ ES of cyclopropenone and the $^1B_1(\pi \ra 3s)$ ES of methylenecyclopropene which are significantly
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over blueshifted with the Pople basis set (Table S2). For cyclopropenone, our {\CCSDT}/{\AVTZ} estimates do agree reasonably well with the {\CASPT} data of Serrano-Andr\'es, except for the $^1B_2 (\pi \ra \pis)$ state
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that we locate significantly higher in energy and the three Rydberg states that our CC calculations predict at significantly lower energies. The present {\NEV} results are globally in better agreement with the CC values,
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though non-negligible deviations pertain. Even if comparisons with experiment should be made very cautiously, we note that, for the Rydberg states, the present CC data are clearly more coherent with the electron impact measurements\cite{Har74} than the original {\CASPT} values. For cyclopropenethione, we typically obtain transition energies in agreement or larger than those obtained with {\CASPT}, \cite{Ser02} though there is no obvious relationship between the
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valence/Rydberg nature of the ES and the relative {\CASPT} error. The average absolute deviation between our {\NEV} and {\CCT} results is $0.08$ eV only. Finally, in the case of methylenecyclopropene, our values logically agree
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very well with the recent estimates of Schwabe and Goerigk, \cite{Sch17} obtained at the {\CCT}/{\AVTZ} level of theory on a different geometry.
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As anticipated, the available {\CASPT} values \cite{Mer96,Roo96} appear too low as compared to the
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present {\NEV} and {\CCSDT} values. For this compound, the available experimental data are based on the wavelength of maximal absorption determined in condensed phase. \cite{Sta84} Hence, only a qualitative match is reached between theory and experiment.
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\subsubsection{Acrolein, butadiene, and glyoxal}
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Let us now turn to three pseudo-linear $\pi$-conjugated systems that have been the subject to several ES investigations before, namely, acrolein, \cite{Aqu03,Sah06,Car10b,Lea12,Gua13,Mai14,Aza17b,Sch17,Bat17}
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butadiene, \cite{Dal04,Sah06,Sch08,Sil10c,Li11,Wat12,Dad12,Lea12,Ise12,Ise13,Sch17,Shu17,Sok17,Chi18,Cop18,Tra19,Loo19c} and glyoxal, \cite{Sta97b,Koh03,Hat05c,Sah06,Lea12,Poo14,Sch17,Aza17b,Loo18b}
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that we all consider in their most stable \emph{trans} conformation in the following. Amongst these works, it is worth highlighting the detailed theoretical investigation by Saha, Ehara, and Nakatsuji, who reported a huge
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number of ES in these three systems using a coherent theoretical Symmetry-Adapted-Cluster Configuration-Interaction (SAC-CI) protocol. \cite{Sah06} Our results are listed in Tables \ref{Table-3} and S3.
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Let us now turn our attention to the excited states of three pseudo-linear $\pi$-conjugated systems that have been the subject to several investigations in the past, namely, acrolein, \cite{Aqu03,Sah06,Car10b,Lea12,Gua13,Mai14,Aza17b,Sch17,Bat17}
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butadiene, \cite{Dal04,Sah06,Sch08,Sil10c,Li11,Wat12,Dad12,Lea12,Ise12,Ise13,Sch17,Shu17,Sok17,Chi18,Cop18,Tra19,Loo19c} and glyoxal. \cite{Sta97b,Koh03,Hat05c,Sah06,Lea12,Poo14,Sch17,Aza17b,Loo18b}
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Amongst these works, it is worth highlighting the detailed theoretical investigation of Saha, Ehara, and Nakatsuji, who reported a huge
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number of ES for these three systems using a coherent theoretical protocol based on the symmetry-adapted-cluster configuration interaction (SAC-CI) method. \cite{Sah06}
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In the following, these three molecules are considered in their most stable \emph{trans} conformation.
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Our results are listed in Tables \ref{Table-3} and S3.
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\begin{table}[htp]
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\caption{\small Vertical transition energies determined in acrolein, butadiene, and glyoxal. All values are in eV.}
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\caption{\small Vertical transition energies (in eV) of acrolein, butadiene, and glyoxal.}
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\label{Table-3}
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\begin{footnotesize}
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\begin{tabular}{p{2.9cm}|p{.5cm}p{.9cm}p{1.1cm}p{1.45cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
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@ -451,45 +453,46 @@ $^3A_g (\pi \ra \pis)$ &6.35&6.35& & &6.34&6.34& 6.30&6.30 &6.33 & &
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\vspace{-0.5 cm}
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\begin{flushleft}
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\begin{footnotesize}
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$^a${{\CASPT} of Ref.~\citenum{Aqu03};}
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$^b${SAC-CI of Ref.~\citenum{Sah06};}
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$^c${Vacuum UV spectra from Ref.~\citenum{Wal45}; for the lowest state, the same 3.71 eV value is reported in Ref.~\citenum{Bec70}.}
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$^d${MR-AQCC of Ref.~\citenum{Dal04}, theoretical best estimates listed for the lowest $B_u$ and $A_g$ states;}
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$^a${{\CASPT} results from Ref.~\citenum{Aqu03};}
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$^b${SAC-CI results from Ref.~\citenum{Sah06};}
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$^c${Vacuum UV spectra from Ref.~\citenum{Wal45}; for the lowest state, the same $3.71$ eV value is reported in Ref.~\citenum{Bec70}.}
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$^d${MR-AQCC results from Ref.~\citenum{Dal04}, theoretical best estimates listed for the lowest $B_u$ and $A_g$ states;}
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$^e${Electron impact experiment from Refs.~\citenum{Fli78} and \citenum{Doe81} for the singlet states and from Ref.~\citenum{Mos73} for the two lowest triplet transitions;
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note that for the lowest $B_u$ state, there is a vibrational structure with peaks at 5.76, 5.92, and 6.05 eV;}
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note that for the lowest $B_u$ state, there is a vibrational structure with peaks at $5.76$, $5.92$, and $6.05$ eV;}
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$^f${From Ref.~\citenum{Loo19c};}
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$^g${{\CCT} of Ref.~\citenum{Sch17};}
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$^h${Electron impact experiment from Ref.~\citenum{Ver80} but for the second $^1B_g$ ES for which the value is from another work; \cite{Rob85b} note that
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for the lowest $^1B_g$ ($^1B_u$) ES, a range of 4.2--4.5 (7.4--7.9) eV is given Ref.~\citenum{Ver80}. }
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$^g${{\CCT} results from Ref.~\citenum{Sch17};}
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$^h${Electron impact experiment from Ref.~\citenum{Ver80} except for the second $^1B_g$ ES for which the value is from another work; \cite{Rob85b} note that
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for the lowest $^1B_g$ ($^1B_u$) ES, a range of $4.2$--$4.5$ ($7.4$--$7.9$) eV is given in Ref.~\citenum{Ver80}. }
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\end{footnotesize}
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\end{flushleft}
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\end{table}
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Acrolein, due to its lower symmetry and high density of ES with mixed characters, is challenging for theory and {\CCSDTQ} calculations were technically impossible despite our efforts. For the lowest $n \ra \pis$
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transitions of both spin symmetries, the {\FCI} estimates come with a tiny error bar, and it is obvious that the CC estimates are slightly too low, especially with {\CCSDT}. Nevertheless, at the exception the second singlet
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and triplet $A''$ ES, the {\CCT} and {\CCSDT} estimates are within $\pm$0.03 eV of each other. These $A''$ ES are also the only two transitions for which the discrepancies between {\CCT} and {\NEV} exceed 0.20 eV. This hints
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at a good accuracy for all other transitions. This statement is supported by the fact that the current CC values are nearly systematically bracketed by previous {\CASPT} (lower bound)\cite{Aqu03} and SAC-CI (higher bound)\cite{Sah06} results,
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consistently with the typical error signs of these two models. For the two lowest triplet states, the present {\CCT}/{\AVTZ} values are also within $\pm$0.05 eV of recent MR-CI estimates (3.50 and 3.89 eV). \cite{Mai14} As
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can be seen in Table S3, {\AVTZ} allows being very close from basis set convergence, the largest variation when going to {\AVQZ} (+0.04 eV) is obtained for the second $^1A'$ ES of Rydberg nature. As the experimental data
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are limited to measured UV spectra, \cite{Wal45,Bec70} one has therefore to be cautious in establishing TBE for acrolein (\emph{vide infra}).
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Acrolein, due to its lower symmetry and high density of ES with mixed characters, is challenging from a theoretical point of view, and {\CCSDTQ} calculations were technically impossible despite our efforts. For the lowest $n \ra \pis$
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transitions of both spin manifolds, the {\FCI} estimates come with a tiny error bar, and it is obvious that the CC excitation energies are slightly too low, especially with {\CCSDT}. Nevertheless, at the exception of the second singlet
|
||||
and triplet $A''$ ES, the {\CCT} and {\CCSDT} transition energies are within $\pm 0.03$ eV of each other. These $A''$ ES are also the only two transitions for which the discrepancies between {\CCT} and {\NEV} exceed $0.20$ eV.
|
||||
\titou{This hints at a good accuracy for all other transitions.}
|
||||
This statement is supported by the fact that the present CC values are nearly systematically bracketed by previous {\CASPT} (lower bound)\cite{Aqu03} and SAC-CI (upper bound)\cite{Sah06} results,
|
||||
consistently with the typical error sign of these two models. For the two lowest triplet states, the present {\CCT}/{\AVTZ} values are also within $\pm 0.05$ eV of recent MRCI estimates ($3.50$ and $3.89$ eV). \cite{Mai14} As
|
||||
can be seen in Table S3, the {\AVTZ} basis set delivers excitation energies very close to the CBS limit: the largest variation \titou{when going to {\AVQZ}} ($+0.04$ eV) is obtained for the second $^1A'$ Rydberg ES. As experimental data
|
||||
are limited to measured UV spectra, \cite{Wal45,Bec70} one has to be cautious in establishing TBE for acrolein (\emph{vide infra}).
|
||||
|
||||
The nature and relative energies of the lowest bright $B_u$ and dark $A_g$ ES of butadiene have been puzzling theoretical chemists for many years. It is beyond our scope to provide an exhaustive
|
||||
The nature and relative energies of the lowest bright $B_u$ and dark $A_g$ ES of butadiene have bamboozled theoretical chemists for many years. It is beyond our scope to provide an exhaustive
|
||||
list of previous calculations and experimental estimates for these two hallmark ES, and we refer the readers to Refs.~\citenum{Wat12} and \citenum{Shu17} for overviews and references. For the $B_u$ transition
|
||||
the best previous TBE we are aware of is the 6.21 eV value obtained by Watson and Chan using a computational strategy similar to ours. \cite{Wat12} Our {\CCSDT}/{\AVTZ} value of 6.24 eV is obviously compatible
|
||||
with this reference value, and our TBE value is actually 6.21 eV as well (\emph{vide infra}). For the $A_g$ state, we believe that our previous basis-corrected exCI estimate of 6.50 eV \cite{Loo19c} remains
|
||||
the most accurate available to date. These two values are slightly smaller than the heath-bath CI data obtained by Chien \emph{et al.} with a double-$\zeta$ basis and a slightly different geometry: 6.45 and 6.58 eV for $B_u$ and $A_g$,
|
||||
respectively. \cite{Chi18} One can of course find many other estimates, e.g., at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} levels, \cite{Sok17} for these two ES.
|
||||
More globally, in butadiene, we find an excellent coherence between the {\CCT}, {\CCSDT}, and {\CCSDTQ} estimates, that all fall in a $\pm$0.02 eV window. Unsurprisingly, this does not apply for the
|
||||
already mentioned $^1A_g$ ES that is 0.2 and 0.1 eV too high with the two former CC methods, consistent with the large electronic reorganization taking place in that state. For all the other butadiene ES listed in
|
||||
Table \ref{Table-3}, both {\CCT} and {\CCSDT} can be trusted. We also note that the {\NEV} estimates are within 0.1--0.2 eV of the CC values, but for the lowest $B_u$ ES, which is very dependent on the selected
|
||||
active space (see the SI). Finally, as can be seen in Table S3, {\AVTZ} is sufficient for most ES, but a significant basis set effect exists for the Rydberg $^1B_u (\pi \ra 3p)$ ES with an energy decrease as large as -0.12 eV
|
||||
the best previous TBE we are aware of is the $6.21$ eV value obtained by Watson and Chan using a computational strategy similar to ours. \cite{Wat12} Our {\CCSDT}/{\AVTZ} value of $6.24$ eV is obviously compatible
|
||||
with this reference value, and our TBE value is actually $6.21$ eV as well (\emph{vide infra}). For the $A_g$ state, we believe that our previous basis-corrected {\FCI} estimate of $6.50$ eV \cite{Loo19c} remains
|
||||
the most accurate available to date. These two values are slightly smaller than the heath-bath CI data obtained by Chien \emph{et al.} with a double-$\zeta$ basis and a slightly different geometry: $6.45$ and $6.58$ eV for $B_u$ and $A_g$,
|
||||
respectively. \cite{Chi18} One can of course find many other estimates, \eg, at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} levels, \cite{Sok17} for these two ES.
|
||||
More globally, in butadiene, we find an excellent coherence between the {\CCT}, {\CCSDT}, and {\CCSDTQ} estimates, that all fall in a $\pm 0.02$ eV window. Unsurprisingly, this does not apply for the
|
||||
already mentioned $^1A_g$ ES that is $0.2$ and $0.1$ eV too high with the two former CC methods, consistent with the large electronic reorganization taking place in that state. For all the other butadiene ES listed in
|
||||
Table \ref{Table-3}, both {\CCT} and {\CCSDT} can be trusted. We also note that the {\NEV} estimates are within $0.1$--$0.2$ eV of the CC values, but for the lowest $B_u$ ES, which is very dependent on the selected
|
||||
active space (see the SI). Finally, as can be seen in Table S3, {\AVTZ} is sufficient for most ES, but a significant basis set effect exists for the Rydberg $^1B_u (\pi \ra 3p)$ ES with an energy decrease as large as $-0.12$ eV
|
||||
when going from {\AVTZ} to {\AVQZ}. For the records, we note that the available electron impact data \cite{Mos73,Fli78,Doe81} provide the very same ES ordering value as our calculations.
|
||||
|
||||
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, i.e., highly consistent CC estimates, reasonable agreement between {\NEV} and {\CCT} estimates, and limited basis set effects beyond {\AVTZ}
|
||||
but for the considered Rydberg state (see Tables \ref{Table-3} and S3). This Rydberg $^1B_u (n \ra 3p)$ state also hows a comparatively large deviation between {\CCT} and {\CCSDTQ}, that is 0.04 eV. More interestingly,
|
||||
glyoxal presents a low-lying ``true'' double ES, $^1A_g (n,n \ra \pis,\pis)$, a transition that is totally unseen by approaches that do not explicitly include double excitations during the calculation of transition energies, e.g., TD-DFT, {\CCSD},
|
||||
or {\AD}. Compared to the {\exCI} values, the {\CCT} and {\CCSDT} estimates for this transition are too large by ca. 1.0 and 0.5 eV, respectively, whereas both the {\CCSDTQ} and {\NEV} approaches are much closer to the spot, as already
|
||||
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, \ie, highly consistent CC estimates, reasonable agreement between {\NEV} and {\CCT} estimates, and limited basis set effects beyond {\AVTZ}
|
||||
but for the considered Rydberg state (see Tables \ref{Table-3} and S3). This Rydberg $^1B_u (n \ra 3p)$ state also hows a comparatively large deviation between {\CCT} and {\CCSDTQ}, that is $0.04$ eV. More interestingly,
|
||||
glyoxal presents a low-lying ``true'' double ES, $^1A_g (n,n \ra \pis,\pis)$, a transition that is totally unseen by approaches that do not explicitly include double excitations during the calculation of transition energies, \eg, TD-DFT, {\CCSD},
|
||||
or {\AD}. Compared to the {\FCI} values, the {\CCT} and {\CCSDT} estimates for this transition are too large by ca.~$1.0$ and $0.5$ eV, respectively, whereas both the {\CCSDTQ} and {\NEV} approaches are much closer to the spot, as already
|
||||
mentioned in our previous work. \cite{Loo19c} For the other transitions, the present {\CCT} estimates are logically coherent with the values of Ref.~\citenum{Sch17} obtained with the same approach on a different
|
||||
geometry, and remain slightly lower than the SAC-CI estimates of Ref.~\citenum{Sah06}. Once more, the experimental data \cite{Ver80,Rob85b} make an unhelpful guide in view of the targeted accuracy.
|
||||
|
||||
@ -501,7 +504,7 @@ There are also a few experimental values available for all six derivatives. \cit
|
||||
and S4.
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in acetone, cyanonformaldehyde, isobutene, propynal, thioacetone, and thiopropynal. All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of acetone, cyanonformaldehyde, isobutene, propynal, thioacetone, and thiopropynal.}
|
||||
\label{Table-4}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{l|p{.5cm}p{.9cm}p{1.1cm}p{1.4cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||||
@ -583,10 +586,10 @@ $^i${0-0 energies from Ref.~\citenum{Jud84c}.}
|
||||
\end{table}
|
||||
|
||||
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
|
||||
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
|
||||
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
|
||||
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}
|
||||
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
|
||||
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.
|
||||
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
|
||||
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.
|
||||
|
||||
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
|
||||
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).
|
||||
@ -596,7 +599,7 @@ For the isoleectronic isobutene, we considered two singlet Rydberg and one tripl
|
||||
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.
|
||||
|
||||
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
|
||||
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
|
||||
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
|
||||
a given basis, hinting that the CC values should be close to the spot, as confirmed by the {\FCI} data.
|
||||
|
||||
\subsubsection{Intermediate conclusions}
|
||||
@ -606,10 +609,10 @@ As we have seen for the 15 four-atom molecules considered here, we found extreme
|
||||
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,
|
||||
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''
|
||||
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
|
||||
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
|
||||
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
|
||||
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
|
||||
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
|
||||
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,
|
||||
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
|
||||
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
|
||||
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
|
||||
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,
|
||||
generally undershoots the transition energies.
|
||||
@ -622,14 +625,14 @@ different approaches to deduce TBE.
|
||||
|
||||
For furan, previous theoretical works have been performed with almost all possible wavefunction approaches, \cite{Ser93b,Nak96,Tro97b,Chr98b,Chr98c,Wan00,Gro03,Pas06b,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Hol15,Sch17} but the
|
||||
present work is, to the best of our knowledge, the first to disclose {\CCSDT} values as well as {\CCT} energies obtained with a quadruple-$\zeta$ basis set. Our results for ten low-lying states are listed in Tables \ref{Table-5} and S5.
|
||||
All computed singlet (triplet) transitions show $\%T_1$ in the 92--94 \%\ (97--99) \%\ range, and consistently the maximal discrepancies between the {\CCT} and {\CCSDT} transition energies are small (0.04 eV). In addition there is a
|
||||
All computed singlet (triplet) transitions show $\%T_1$ in the 92--94 \%\ (97--99) \%\ range, and consistently the maximal discrepancies between the {\CCT} and {\CCSDT} transition energies are small ($0.04$ eV). In addition there is a
|
||||
good consistency between the present data and both the {\NEV} results of Ref.~\citenum{Pas06b} and the MR-CC values of Ref.~\citenum{Li10c} for almost all transitions, but the $^1B_2 (\pi \ra 3p)$ excitation that we predict to
|
||||
be significantly higher than in most previous works, even after accounting for the quite large basis set effects (-0.10 eV between {\AVTZ} and {\AVQZ} estimates, see Table S5). We trust that our estimate is the most accurate to date for that ES.
|
||||
Interestingly, the recent {\AT} values of Ref.~\citenum{Hol15} are smaller by ca. -0.2 eV as compared to {\CCSDT} values for all transitions (see Table \ref{Table-6}), consistent with the error sign we found in smaller compounds with ADC(3). \cite{Loo18a}
|
||||
be significantly higher than in most previous works, even after accounting for the quite large basis set effects ($-0.10$ eV between {\AVTZ} and {\AVQZ} estimates, see Table S5). We trust that our estimate is the most accurate to date for that ES.
|
||||
Interestingly, the recent {\AT} values of Ref.~\citenum{Hol15} are smaller by ca.~$-0.2$ eV as compared to {\CCSDT} values for all transitions (see Table \ref{Table-6}), consistent with the error sign we found in smaller compounds with ADC(3). \cite{Loo18a}
|
||||
Eventually, we note that the experimental data, \cite{Vee76b,Fli76,Rob85b} provide the same state ordering as our calculations.
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in furan and pyrrole. All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of furan and pyrrole.}
|
||||
\label{Table-5}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||||
@ -669,16 +672,16 @@ $^3B_1 (\pi \ra 3p)$ &5.91&5.90 &5.82&5.81 &5.92& && 5.82& &5.74&
|
||||
\vspace{-0.5 cm}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${{\CASPT} from Ref.~\citenum{Ser93b};}
|
||||
$^b${{\NEV} from Ref.~\citenum{Pas06b};}
|
||||
$^c${MR-CC from Ref.~\citenum{Li10c};}
|
||||
$^d${{\AT} from Ref.~\citenum{Hol15};}
|
||||
$^e${{\CCT} from Ref.~\citenum{Sch17};}
|
||||
$^a${{\CASPT} results from Ref.~\citenum{Ser93b};}
|
||||
$^b${{\NEV} results from Ref.~\citenum{Pas06b};}
|
||||
$^c${MR-CC results from Ref.~\citenum{Li10c};}
|
||||
$^d${{\AT} results from Ref.~\citenum{Hol15};}
|
||||
$^e${{\CCT} results from Ref.~\citenum{Sch17};}
|
||||
$^f${Various experiments summarized in Ref.~\citenum{Wan00};}
|
||||
$^g${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};}
|
||||
$^h${{\NEV} from Ref.~\citenum{Pas06c};}
|
||||
$^h${{\NEV} results from Ref.~\citenum{Pas06c};}
|
||||
$^i${Best estimate from Ref.~\citenum{Chr99}, based on CC calculations;}
|
||||
$^j${XMS-{\CASPT} from Ref.~\citenum{Hei19};}
|
||||
$^j${XMS-{\CASPT} results from Ref.~\citenum{Hei19};}
|
||||
$^k${Electron impact from Refs.~\citenum{Vee76b} and \citenum{Fli76b};}
|
||||
$^l${Vapour UV spectra from Refs.~\citenum{Pal03b}, \citenum{Hor67}, and \citenum{Bav76}.}
|
||||
\end{footnotesize}
|
||||
@ -688,13 +691,13 @@ $^l${Vapour UV spectra from Refs.~\citenum{Pal03b}, \citenum{Hor67}, and \citenu
|
||||
Like furan, pyrrole has been extensively investigated previously using a large palette of approaches. \cite{Ser93b,Nak96,Tro97,Pal98,Chr99,Wan00,Roo02,Pal03b,Pas06c,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Nev14,Sch17,Hei19}
|
||||
We report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All considered transitions have very large $\%T_1$ but for the totally symmetric $\pi \ra \pis$ excitation ($\%T_1 = 86\%$). For all states, we found
|
||||
highly consistent {\CCT} and {\CCSDT} results, often significantly larger than older multi-reference estimates, \cite{Ser93b,Roo02,Li10c} but in nice agreement with the very recent XMS-{\CASPT} results of the Gonzalez'
|
||||
group, \cite{Hei19} at the exception of the $^1A_2 (\pi \ra 3p)$ transition. The match obtained with the 20-years old extrapolated CC values of Christiansen and coworkers \cite{Chr99} is also remarkable but for the two
|
||||
$B_2$ transitions that were reported as significantly mixed in that venerable work. For the lowest singlet ES, the {\FCI}/{\Pop} value is 5.24$\pm$0.02 eV confirming the performances of both {\CCT} and {\CCSDT} for that transition.
|
||||
As can be seen in Table S5, {\AVTZ} yields basis-set converged transition energies, but, like in furan, for the Rydberg $^1B_2 (\pi \ra 3p)$ transition that is significantly redshifted (-0.09 eV) when going to the
|
||||
group, \cite{Hei19} at the exception of the $^1A_2 (\pi \ra 3p)$ transition. The match obtained with the twenty years old extrapolated CC values of Christiansen and coworkers \cite{Chr99} is also remarkable but for the two
|
||||
$B_2$ transitions that were reported as significantly mixed in that venerable work. For the lowest singlet ES, the {\FCI}/{\Pop} value is $5.24 \pm 0.02$ eV confirming the performances of both {\CCT} and {\CCSDT} for that transition.
|
||||
As can be seen in Table S5, {\AVTZ} yields basis-set converged transition energies, but, like in furan, for the Rydberg $^1B_2 (\pi \ra 3p)$ transition that is significantly redshifted ($-0.09$ eV) when going to the
|
||||
quadruple-$\zeta$ basis set. As mentioned in Thiel's work, \cite{Sch08} the experimental spectra of pyrrole are quite broad and the few available experiments \cite{Hor67,Bav76,Fli76b,Vee76b,Pal98,Pal03b} can only be used as general guidelines.
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in cyclopentadiene, imidazole, and thiophene. All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of cyclopentadiene, imidazole, and thiophene.}
|
||||
\label{Table-6}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}p{.6cm}}
|
||||
@ -749,20 +752,20 @@ $^3A_2 (\pi \ra 3s)$ &6.20&6.20 &6.01&6.00 &6.09& & &5.88&5.75& &5.83&\\
|
||||
\vspace{-0.5 cm}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${{\CASPT} from Ref.~\citenum{Ser93b};}
|
||||
$^b${SAC-CI from Ref.~\citenum{Wan00b};}
|
||||
$^c${MR-MP from Ref.~\citenum{Nak96};}
|
||||
$^d${{\CCT} from Ref.~\citenum{Sch17};}
|
||||
$^a${{\CASPT} results from Ref.~\citenum{Ser93b};}
|
||||
$^b${SAC-CI results from Ref.~\citenum{Wan00b};}
|
||||
$^c${MR-MP results from Ref.~\citenum{Nak96};}
|
||||
$^d${{\CCT} results from Ref.~\citenum{Sch17};}
|
||||
$^e${Electron impact from Ref.~\citenum{Fru79};}
|
||||
$^f${Gas phase absorption from Ref.~\citenum{McD91b};}
|
||||
$^g${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;}
|
||||
$^h${{\CASPT} from Ref.~\citenum{Ser96b};}
|
||||
$^i${{\CCT} from Ref.~\citenum{Sil10c};}
|
||||
$^h${{\CASPT} results from Ref.~\citenum{Ser96b};}
|
||||
$^i${{\CCT} results from Ref.~\citenum{Sil10c};}
|
||||
$^j${Gas-phase experimental estimates from Ref.~\citenum{Dev06};}
|
||||
$^k${{\CASPT} from Ref.~\citenum{Ser93c};}
|
||||
$^l${SAC-CI from Ref.~\citenum{Wan01};}
|
||||
$^m${CCSDR(3) from Ref.~\citenum{Pas07};}%, this work also contains {\NEV} estimates;}
|
||||
$^n${TBE from Ref.~\citenum{Hol14}, based on EOM-CCSD for singlet and ADC(2) for triplets;}
|
||||
$^k${{\CASPT} results from Ref.~\citenum{Ser93c};}
|
||||
$^l${SAC-CI results from Ref.~\citenum{Wan01};}
|
||||
$^m${CCSDR(3) results from Ref.~\citenum{Pas07};}%, this work also contains {\NEV} estimates;}
|
||||
$^n${TBE from Ref.~\citenum{Hol14}, based on EOM-CCSD for singlet and ADC(2) for triplets;}
|
||||
$^o${0-0 energies from Ref.~\citenum{Dil72};}
|
||||
$^p${0-0 energies from Ref.~\citenum{Var82} for the singlets and energy loss experiment from Ref.~\citenum{Hab03} for the triplet ES;}
|
||||
$^q${0-0 energies from Ref.~\citenum{Hol14}.}
|
||||
@ -791,7 +794,7 @@ Finally, the ES thiophene, which is one of the most important building block in
|
||||
transitions that were not yet characterized in the available measurements. \cite{Dil72,Fli76,Fli76b,Var82,Hab03,Pal99,Hol14} To our knowledge, the present work is the first to report CC calculations obtained with (iterative)
|
||||
triples and therefore constitutes the most accurate estimates to date. Indeed, all the transitions listed in Tables \ref{Table-6} and S6 are characterized by a largely dominant single excitation character, with $\%T_1$ above
|
||||
90\%\ but for the two $^1A_1$ transitions that show $\%T_1$ of 88\%\ and 87\%. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for the lowest totally symmetric transition. Thiophene is
|
||||
also a typical compound in which unambiguous characterization of the nature of the ES is difficult, with e.g., a strong mixing between the second and third singlet $B_2$ ES rendering the assignment of the valence
|
||||
also a typical compound in which unambiguous characterization of the nature of the ES is difficult, with \eg, a strong mixing between the second and third singlet $B_2$ ES rendering the assignment of the valence
|
||||
($\pi \ra \pis$) or Rydberg ($\pi \ra 3p$) character of that transittion uneasy at the {\CCT} level. We note that contradictory assignments can be found in the literature. \cite{Ser93c,Wan01,Pas07} As for the
|
||||
previously discussed isostructural systems, the only ES that undergoes significant basis set effects beyond {\AVTZ} is the Rydberg $^1B_2 (\pi \ra 3p)$ (-0.09 eV when upgrading to {\AVQZ}, see Table S6).
|
||||
The data of Table \ref{Table-6} are globally in good agreement with the previously reported values with discrepancies that are however significant for the three highest-lying singlet states.
|
||||
@ -814,7 +817,7 @@ and {\AVTZ} is essentially providing basis set converged transition energies. Th
|
||||
For both the singlet and triplet transitions, our values are slightly larger than available electron impact/multi-photon measurements. \cite{Doe69,Nak80,Joh76,Joh83,Hir91}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in benzene All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of benzene.}
|
||||
\label{Table-7}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||||
@ -836,10 +839,10 @@ $^3B_{2u} (\pi \ra \pis)$ &6.06&6.06 &5.86&5.86 &5.81& & &5.49&5.88&5.54&5.67&
|
||||
\vspace{-0.5 cm}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${{\CASPT} from Ref.~\citenum{Lor95b};}
|
||||
$^b${{\CCT} from Ref.~\citenum{Chr96c};}
|
||||
$^c${SAC-CI from Ref.~\citenum{Li07b};}
|
||||
$^d${RASPT2(18,18) from Ref.~\citenum{Sha19};}
|
||||
$^a${{\CASPT} results from Ref.~\citenum{Lor95b};}
|
||||
$^b${{\CCT} results from Ref.~\citenum{Chr96c};}
|
||||
$^c${SAC-CI results from Ref.~\citenum{Li07b};}
|
||||
$^d${RASPT2(18,18) results from Ref.~\citenum{Sha19};}
|
||||
$^e${Electron impact from Ref.~\citenum{Doe69};}
|
||||
$^f${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.}
|
||||
\end{footnotesize}
|
||||
@ -847,7 +850,7 @@ $^f${Jet-cooled experiment from Ref.~\citenum{Hir91} for the two lowest states,
|
||||
\end{table}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in pyrazine and tetrazine. All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of pyrazine and tetrazine.}
|
||||
\label{Table-8}
|
||||
\begin{footnotesize}
|
||||
\vspace{-0.3 cm}
|
||||
@ -905,17 +908,17 @@ $^3B_{1u} (\pi \ra \pis)$ &5.51&5.50 &5.46&5.46 &5.42& & &5.09&5.31&
|
||||
\vspace{-0.5 cm}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${{\CASPT} from Ref.~\citenum{Web99};}
|
||||
$^b${{\STEOM} from Ref.~\citenum{Noo99};}
|
||||
$^c${SAC-CI from Ref.~\citenum{Li07b};}
|
||||
$^d${{\CCT} from Ref.~\citenum{Sch17};}
|
||||
$^a${{\CASPT} results from Ref.~\citenum{Web99};}
|
||||
$^b${{\STEOM} results from Ref.~\citenum{Noo99};}
|
||||
$^c${SAC-CI results from Ref.~\citenum{Li07b};}
|
||||
$^d${{\CCT} results from Ref.~\citenum{Sch17};}
|
||||
$^e${Dip spectroscopy from Ref.~\citenum{Oku90} ($B_{3u}$ and $B_{2g}$ states) and EEL from Ref.~\citenum{Wal91} (other states);}
|
||||
$^f${UV max from Ref.~\citenum{Bol84};}
|
||||
$^g${{\CASPT} from Ref.~\citenum{Rub99};}
|
||||
$^h${Ext-{\STEOM} from Ref.~\citenum{Noo00};}
|
||||
$^i${GVVPT2 from Ref.~\citenum{Dev08};}
|
||||
$^j${{\NEV} from Ref.~\citenum{Ang09};}
|
||||
$^k${{\CCT} from Ref.~\citenum{Sil10c};}
|
||||
$^g${{\CASPT} results from Ref.~\citenum{Rub99};}
|
||||
$^h${Ext-{\STEOM} results from Ref.~\citenum{Noo00};}
|
||||
$^i${GVVPT2 results from Ref.~\citenum{Dev08};}
|
||||
$^j${{\NEV} results from Ref.~\citenum{Ang09};}
|
||||
$^k${{\CCT} results from Ref.~\citenum{Sil10c};}
|
||||
$^l${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;}
|
||||
$^m${all these doubly ES have a $(n,n \ra \pis, \pis)$ character.}
|
||||
\end{footnotesize}
|
||||
@ -954,7 +957,7 @@ CC estimates; and iv) same ranking of the ES as in the most recent measurements.
|
||||
Rydberg transition that is separated by only 0.03 eV at the {\CCT}/{AVTZ} level, making the analysis particularly challenging for that specific transition. \hl{Keep or not A1 transitiion}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in pyridazine and pyridine. All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of pyridazine and pyridine.}
|
||||
\label{Table-9}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||||
@ -1000,10 +1003,10 @@ $^3B_2 (\pi \ra \pis)$ &6.46&6.45 &6.30&6.29 &6.25& &6.02& &
|
||||
\vspace{-0.4 cm}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${{\CASPT} from Ref.~\citenum{Ful92};}
|
||||
$^b${{\STEOM} from Ref.~\citenum{Noo99};}
|
||||
$^a${{\CASPT} results from Ref.~\citenum{Ful92};}
|
||||
$^b${{\STEOM} results from Ref.~\citenum{Noo99};}
|
||||
$^c${EOM-CCSD({$\tilde{{T}}$}) from Ref.~\citenum{Del97b};}
|
||||
$^d${CC3-ext. from Ref.~\citenum{Sil10c};}
|
||||
$^d${CC3-ext.~from Ref.~\citenum{Sil10c};}
|
||||
$^e${EEL from Ref.~\citenum{Pal91};}
|
||||
$^f${EEL from Ref.~\citenum{Lin19};}
|
||||
$^g${{\CASPT} from Ref.~\citenum{Lor95};}
|
||||
@ -1024,7 +1027,7 @@ $^j${Significant state mixing with a close-lying Rydberg transition, rendering u
|
||||
|
||||
s
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical transition energies determined in pyrimidine and triazine. All values are in eV.}
|
||||
\caption{\small Vertical transition energies (in eV) of pyrimidine and triazine.}
|
||||
\label{Table-10}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||||
@ -1434,12 +1437,12 @@ in many of the proposed examples, {\CASPT} transitions energies tend to be signi
|
||||
aware of one detailed benchmark \cite{Dut18} only -- provides a smaller MSE than {\CCSD} and comparable MAE and RMS. The spread of the error is however slightly larger as can be seen in Figure \ref{Fig-1} and from the SD values
|
||||
in Table \ref{Table-bench}. These trends are the same as for smaller compounds. \cite{Loo18a} For Thiel's set using {\CCT}/TZVP results as references, Dutta and coworkers also reported a rather good performance
|
||||
of {\STEOM}, though in that case a slightly negative MSE is obtained, \cite{Dut18} which could possibly be due to the different basis sets used. It should be nevertheless stressed that we consider here only ``clean'' {\STEOM} results
|
||||
(see Computational details), therefore removing several difficult cases that are included in the {\CCSD} statistics, e.g., the $A_g$ excitation in butadiene, which can slightly bias the relative accuracies when comparing the two models. Finally, for the three
|
||||
(see Computational details), therefore removing several difficult cases that are included in the {\CCSD} statistics, \eg, the $A_g$ excitation in butadiene, which can slightly bias the relative accuracies when comparing the two models. Finally, for the three
|
||||
second-order methods, namely CIS(D), {\AD}, and {\CCD}, that are often used as reference to benchmark TD-DFT for ``real-life'' applications, we obtain clearly worse performances for the former approach than for the two latter, that show very
|
||||
similar statistical behaviors. These trends were also reported in several previous works. \cite{Hat05c,Jac18a,Sch08,Sil10c,Win13,Har14,Jac15b,Kan17,Loo18a} Interestingly, the {\CCD} MAE obtained here, 0.15 eV, is significantly
|
||||
smaller than the one we found for the smaller compounds (0.22 eV): \cite{Loo18a} in contrast to {\CCSD}, {\CCD} seems to improve with molecular size. As above, Thiel's original MAE for {\CCD} (0.29 eV) was likely too large due
|
||||
to the selection of {\CASPT} reference values. \cite{Sch08} As already noticed by Szalay's group, \cite{Kan14,Kan17} although the MSE of {\CCD} is smaller than the one of {\CCSD}, the standard deviation is significantly larger
|
||||
with the former model, i.e., {\CCD} is less consistent in terms of trends than {\CCSD}.
|
||||
with the former model, \ie, {\CCD} is less consistent in terms of trends than {\CCSD}.
|
||||
|
||||
In Table \ref{Table-bench2}, we report a decomposition of the MAE for different subsets of ES. Only singlet ES could be determined with both CCSDR(3) and CCSDT-3, which is why no value appears in the triplet
|
||||
column for these two methods. A few interesting conclusions emerge from the displayed data. First, the errors for the singlet and triplet transitions are rather similar with all models, but with {\CCSD} that
|
||||
|
||||
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Reference in New Issue
Block a user