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@ -113,10 +113,10 @@
Following our previous work focussing on compounds containing up to 3 non-hydrogen atoms [\emph{J. Chem. Theory Comput.} {\bfseries 14} (2018) 4360--4379], we present here highly-accurate vertical transition energies
obtained for 27 molecules encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione,
diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine.
To obtain these energies, we use equation-of-motion coupled cluster theory up to the highest technically possible excitation order for these systems (CCT3, EOM-CCSDT, and EOM-CCSDTQ), selected configuration interaction (SCI) calculations (with tens of millions of determinants in the reference space),
To obtain these energies, we use equation-of-motion coupled cluster theory up to the highest technically possible excitation order for these systems (CC3, EOM-CCSDT, and EOM-CCSDTQ), selected configuration interaction (SCI) calculations (with tens of millions of determinants in the reference space),
as well as the multiconfigurational $n$-electron valence state perturbation theory (NEVPT2) method.
All these approaches are applied in combination with diffuse-containing atomic basis sets. For all transitions, we report at least CC3/aug-cc-pVQZ vertical excitation
energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical methods with a typically deviation of $\pm 0.04$ eV, except for transitions with a dominant double excitation character where the error is much larger.
All these approaches are applied in combination with diffuse-containing atomic basis sets. For all transitions, we report at least CC3/\emph{aug}-cc-pVQZ vertical excitation
energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of $\pm 0.04$ eV, except for transitions with a dominant double excitation character where the error is much larger.
The present contribution gathers a large, diverse and accurate set of more than 200 highly-accurate transition energies for states of various natures
(valence, Rydberg, singlet, triplet, $n \ra \pis$, $\pi \ra \pis$, \ldots).
We use this series of theoretical best estimates to benchmark a series of popular methods for excited state calculations: CIS(D), ADC(2),
@ -137,8 +137,8 @@ However, to obtain a quantitative assessment of the accuracy that can be expecte
available. \cite{Lau13}
While several of these benchmarks rely on experimental data as reference (typically band shapes \cite{Die04,Die04b,Avi13,Cha13,Lat15b,Mun15,Vaz15,San16b} or 0-0 energies
\cite{Die04b,Goe10a,Jac12d,Chi13b,Win13,Fan14b,Jac14a,Jac15b,Loo19b}), reference from theoretical best estimates (TBE) based on state-of-the-art computational methods \cite{Sch08,Sau09,Sil10b,Sil10c,Sch17,Loo18a}
is advantageous as it allows comparisons on a perfectly equal footing (same geometry, vertical transitions, no environmental effects, etc). In such a case, the challenge is in fact to obtain accurate TBE, as these top-notch theoretical models
\cite{Die04b,Goe10a,Jac12d,Chi13b,Win13,Fan14b,Jac14a,Jac15b,Loo19b}), references from theoretical best estimates (TBE) based on state-of-the-art computational methods \cite{Sch08,Sau09,Sil10b,Sil10c,Sch17,Loo18a}
are advantageous as they allow comparisons on a perfectly equal footing (same geometry, vertical transitions, no environmental effects, etc). In such a case, the challenge is in fact to obtain accurate TBE, as these top-notch theoretical models
generally come with a dreadful scaling with system size and, in addition, typically require large atomic basis sets to deliver transition energies close to the complete basis set (CBS) limit.
More than 20 years ago, Serrano-Andr\`es, Roos, and collaborators compiled an impressive series of reference transition energies for several typical conjugated organic molecules (butadiene, furan, pyrrole, tetrazine, \ldots).
@ -150,7 +150,7 @@ Nowadays, it is common knowledge that CASPT2 has the tendency of underestimating
A decade ago, Thiel and coworkers defined TBE for 104 singlet and 63 triplet valence ES in 28 small and medium conjugated CNOH organic molecules. \cite{Sch08,Sil10b,Sil10c} These TBE
were computed on MP2/6-31G(d) structures with several levels of theories, notably {\CASPT} and various coupled cluster (CC) variants ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default theoretical protocol used by Thiel and coworkers to define their
first series of TBE used {\CASPT}, \cite{Sch08} the vast majority of their most recent TBE (the so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level of theory, often using a basis set extrapolation technique.
first series of TBE was {\CASPT}, \cite{Sch08} the vast majority of their most recent TBE (the so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level of theory, often using a basis set extrapolation technique.
More specifically, CC3/TZVP values were corrected for basis set incompleteness errors by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works exploited Thiel's TBE for
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 further
their value for the electronic structure community. In contrast, the number of extensions/improvements of this original set remains quite limited.
@ -158,25 +158,25 @@ For example, K\'ann\'ar and Szalay computed, in 2014, {\CCSDT}/TZVP reference en
Three years later, the same authors reported 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide). \cite{Kan17}
Following the same philosophy, two years ago, we reported 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 (up to {\CCSDTQP}) and selected
CI (SCI) calculations on {\CCT}/{\AVTZ} GS structures. \cite{Loo18a} We exploited these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} yields near-{\FCI} quality excitation energies, whereas we could not
CI (SCI) calculations on {\CCT}/{\AVTZ} GS structures. \cite{Loo18a} We exploited these TBE to benchmark many ES methods. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} yields near-{\FCI} quality excitation energies, whereas we could not
detect any significant differences between {\CCT} and {\CCSDT} transition energies, both being very accurate with mean absolute errors (MAE) as small as $0.03$ eV compared to {\FCI}.
Although these conclusions agree well with earlier studies, \cite{Wat13,Kan14,Kan17} they obviously only hold for single excitations, \ie, transitions with $\Td$ in the range $80$--$100\%$. Therefore, we also recently proposed a set of 20 TBE for transitions exhibiting a significant double-excitation character (\ie, with $\Td$ typically below $80\%$). \cite{Loo19c}
Unsurprisingly, our results clearly evidenced that the error in CC methods is intimately related to the $\Td$ value.
For example, ES with a significant yet \titou{not dominant} double excitation character, such as the infamous $A_g$ ES of butadiene ($\Td = 75\%$),
For example, ES with a significant yet not dominant double excitation character [such as the infamous $A_g$ ES of butadiene ($\Td = 75\%$)]
CC methods including triples 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
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 ($\Td = 2\%$),
single-reference methods have been found to be unsuitable with MAEs of $0.86$ and $0.42$ eV for {\CCT} and {\CCSDT}, respectively.
In this case, multireference methods are required to obtain accurate results. \cite{Loo19c}
A clear limit from our 2018 work \cite{Loo18a} was the sizes of the compounds put together in our set.
A clear limit of our 2018 work \cite{Loo18a} was the size of the compounds put together in our set.
These were limited to $1$--$3$ non-hydrogen atoms, hence introducing a potential ``chemical'' bias. Therefore, we have decided, in the present contribution, to consider larger molecules with organic
compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large one-electron basis sets is elusive. Moreover, the convergence of the {\SCI} energy with respect to the number of determinants is obviously slower for these larger compounds, hence
extrapolating to the {\FCI} limit with an error of $\sim 0.01$ eV is rarely achievable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work, \cite{Loo18a} is definitely out of reach here.
Anticipating this problem, we have recently investigated bootstrap CBS extrapolation techniques. \cite{Loo18a,Loo19c}
In particular, we have demonstrated that, following an ONIOM-like scheme, \cite{Chu15} one can very accurately estimate such limit by correcting high-level values obtained in a small basis by the difference between {\CCT} results obtained in a larger basis and in the same small basis.\cite{Loo18a}
We globally follow such strategy here. In addition, we also perform {\NEV} calculations in an effort to check the consistency of our estimates. It is especially critical for ES with intermediate $\Td$ values.
Using this protocol, we define a set of more than 200 {\AVTZ} reference transition energies, most being within $\pm 0.03$ eV of the {\FCI} limit. These reference energies are obtained on {\CCT}/{\AVTZ} geometries and further basis set
We globally follow such strategy here. In addition, we also perform {\NEV} calculations in an effort to check the consistency of our estimates. This is particularly critical for ES with intermediate $\Td$ values.
Using this protocol, we define a set of more than 200 \emph{aug}-cc-pVQZ reference transition energies, most being within $\pm 0.03$ eV of the {\FCI} limit. These reference energies are obtained on {\CCT}/\emph{aug}-cc-pVTZ geometries and additional basis set
corrections (up to quadruple-$\zeta$ at least) are also provided for {\CCT}. Together with the results obtained in our two earlier works, \cite{Loo18a,Loo19c} the present TBE will hopefully contribute to climb a rung higher on the ES accuracy ladder.
%
@ -187,7 +187,7 @@ corrections (up to quadruple-$\zeta$ at least) are also provided for {\CCT}. Tog
Unless otherwise stated, all transition energies are computed in the frozen-core approximation (with a large core for the sulfur atoms).
Pople's {\Pop} and Dunning's \emph{aug}-cc-pVXZ (X $=$ D, T, Q, and 5) atomic basis sets are systematically employed in our excited-state calculations.
In the following, we employ the aVXZ shorthand notations for these diffuse-containing Dunning basis sets.
In the following, we employ the aVXZ shorthand notations for these diffuse-containing basis sets.
Various statistical quantities are reported in the remaining of this paper: the mean signed error (MSE), mean absolute error (MAE), root mean square error (RMSE), standard deviation of the errors (SDE), as well as the positive [\MaxP] and negative [\MaxN] maximum errors.
Here, we globally follow the same procedure as in Ref.~\citenum{Loo18a}, so that we only briefly outline the various theoretical methods that we have employed in the subsections below.
@ -207,12 +207,12 @@ coworkers. \cite{Boo09} We refer the interested reader to Ref.~\citenum{Gar19} w
Excited-state calculations are performed within a state-averaged formalism which means that the CIPSI algorithm select determinants simultaneously for the GS and ES. Therefore, all electronic states share the same set of determinants with different CI coefficients.
Our implementation of the CIPSI algorithm for ES is detailed in Ref.~\citenum{Sce19}. For each system, a preliminary SCI calculation is performed using Hartree-Fock orbitals in order to generate SCI wavefunctions with at least 5,000,000 determinants.
State-averaged natural orbitals are then computed based on this wavefunction, and a new, larger SCI calculation is performed with this new set of orbitals. This has the advantage to produce a smoother and faster convergence of the SCI energy to the FCI limit.
State-averaged natural orbitals are then computed based on this wavefunction, and a new, larger SCI calculation is performed with this new set of orbitals. This has the advantage to produce a smoother and faster convergence of the SCI energy towards the FCI limit.
For the largest systems, an additional iteration is sometimes required in order to obtain better quality natural orbitals and hence well-converged calculations.
The total SCI energy is defined as the sum of the (zeroth-order) variational energy (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction which takes into account the external determinants, \ie,
the determinants which do not belong to the variational space but are linked to the reference space via a non-zero matrix element. The magnitude of this second-order correction, $E^{(2)}$, provides a qualitative idea of the ``distance" to the FCI limit.
For maximum efficiency, the total SCI energy is linearly extrapolated to $E^{(2)} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wavefunctions. These extrapolated total energies simply (labeled as FCI in the remaining of the paper)
For maximum efficiency, the total SCI energy is linearly extrapolated to $E^{(2)} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wavefunctions. These extrapolated total energies (simply labeled as FCI in the remaining of the paper)
are then used to computed vertical excitation energies. Although it is not possible to provide a theoretically-sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wavefunction and its corresponding
extrapolated value. We believe that it provides a very safe estimate of the extrapolation error. Additional information about the SCI wavefunctions and excitation energies as well as their extrapolated values can be found in the SI.
@ -238,7 +238,7 @@ The definition of the active space considered for each system as well as the num
For the other levels of theory, we apply a variety of programs, namely, CFOUR,\cite{cfour} DALTON,\cite{dalton} GAUSSIAN,\cite{Gaussian16} ORCA,\cite{Nee12} MRCC,\cite{Rol13,mrcc} and Q-CHEM. \cite{Sha15} CFOUR is used for
{\CCT}, \cite{Chr95b,Koc97} CCSDT-3, \cite{Wat96,Pro10} {\CCSDT} \cite{Nog87} and {\CCSDTQ}\cite{Kuc91}; Dalton for {\CCD}, \cite{Chr95,Hat00} {\CCSD},\cite{Pur82} CCSDR(3), \cite{Chr96b} and {\CCT} \cite{Chr95b,Koc97}; Gaussian
for CIS(D); \cite{Hea94,Hea95} ORCA for the similarity-transformed EOM-CCSD ({\STEOM})\cite{Noo97,Dut18}; MRCC for {\CCSDT} \cite{Nog87} and {\CCSDTQ}; \cite{Kuc91} and Q-Chem for {\AD}. \cite{Dre15}
Default program settings were applied. We note that for {\STEOM} we report only states that are characterized by an active character percentage of 98\%\ or larger.
Default program settings were applied. We note that for {\STEOM} we only report states that are characterized by an active character percentage of $98\%$ or larger.
%
% III. Results & Discussion
@ -250,7 +250,7 @@ In the following, we present the results obtained for molecules containing four,
Given that the {\SCI} results converges rather slowly for these larger systems, we provide an estimated error bar for these extrapolated {\FCI} values (\emph{vide supra}). In most cases, these extrapolated FCI reference data are used as a safety net to demonstrate the
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
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
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
very large $\Td$ 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
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
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
relevant and are therefore unlikely to change any of our main conclusions.
@ -310,7 +310,7 @@ state is the onset, whereas an estimate of the vertical energy ($4.2 \pm 0.2$ eV
The ES of these three closely-related linear molecules containing two triple bonds have been quite rarely theoretically investigated, \cite{Fis03,Pat06,Luo08,Loo18b,Loo19a} though (rather old) experimental measurements of their
0-0 energies are available for several ES. \cite{Cal63,Job66a,Job66b,Bel69,Fis72,Har77,Hai79,All84} Our main results are collected in Tables \ref{Table-1} and S1. We consider only low-lying
valence $\pi \ra \pis$ transitions, which have all a strongly dominant single excitation character (\%$T_1 > 90\%$, \emph{vide infra}). For cyanoacetylene, the {\FCI}/{\Pop} estimates come with small error bars, and one notices an excellent agreement between these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning's double-$\zeta$ basis set results for which the {\FCI} uncertainties are however
valence $\pi \ra \pis$ transitions, which have all a strongly dominant single excitation character ($\Td > 90\%$, \emph{vide infra}). For cyanoacetylene, the {\FCI}/{\Pop} estimates come with small error bars, and one notices an excellent agreement between these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning's double-$\zeta$ basis set results for which the {\FCI} uncertainties are however
larger. Using the {\CCSDTQ} values as references, it appears that the previously obtained {\CASPT} estimates\cite{Luo08} are, as expected, too low and that the {\CCT} transition energies are slightly more accurate than
their CCSDT counterparts, although all CC estimates of Table \ref{Table-1} come, for a given basis set, in a very tight energetic window. There is also a superb agreement between the CC and {\NEV} values with the
{\AVTZ} basis set. All these facts provide strong evidences that the CC estimates can be fully trusted for these three linear systems. The basis set effects are quite significant for the valence excited-states of cyanoacetylene with successive drops of the transition
@ -400,9 +400,9 @@ results might be slightly too low for the second transition. }
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
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
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 MAE of {\CCT} and {\CCSDT} is $0.019$ and $0.016$ eV, respectively,
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 (\%$T_1 = 85\%$),
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 ($\Td = 85\%$),
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
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
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
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
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
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,
@ -638,17 +638,17 @@ This seems to indicate that {\NEV}, as applied here, has a slight tendency to ov
\subsection{Five-atom molecules}
We now consider five-member rings, and, in particular, five common derivatives that have been considered in several previous theoretical studies (\emph{vide infra}): cyclopentadiene, furan, imidazole, pyrrole, and thiophene. As the most
advanced levels of theory used in the previous Section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds, one has to rely on the nature of the ES and the consistency between the results of
different approaches to deduce TBE.
We now consider five-membered rings, and, in particular, five common derivatives that have been considered in several previous theoretical studies (\emph{vide infra}): cyclopentadiene, furan, imidazole, pyrrole, and thiophene.
As the most advanced levels of theory employed in the previous Section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds, one has to rely on the nature of the ES and the consistency between results 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 $\Td$ 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}
Eventually, we note that the experimental data, \cite{Vee76b,Fli76,Rob85b} provide the same state ordering as our calculations.
For furan, \textit{ab initio} calculations have been performed with almost any possible wavefunction approach.\cite{Ser93b,Nak96,Tro97b,Chr98b,Chr98c,Wan00,Gro03,Pas06b,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Hol15,Sch17}
However, 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 ES states are listed in Tables \ref{Table-5} and S5.
All singlet (triplet) transitions show $\Td$ in the $92$--$94\%$ ($97$--$99\%$) range. Consistently, the maximal discrepancy between {\CCT} and {\CCSDT} is small ($0.04$ eV). In addition, there is a
decent consistency between the present data and both the {\NEV} results of Ref.~\citenum{Pas06b} and the MR-CC values of Ref.~\citenum{Li10c}.
This holds 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 believe that our estimate is the most accurate to date for this particularly tricky ES.
Interestingly, the recent {\AT} values of Ref.~\citenum{Hol15} are consistently smaller by ca.~$-0.2$ eV as compared to {\CCSDT} (see Table \ref{Table-6}), consistently with the error sign we observed in smaller compounds for {\AT}. \cite{Loo18a}
Again, 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 (in eV) of furan and pyrrole.}
@ -707,13 +707,13 @@ $^l${Vapour UV spectra from Refs.~\citenum{Pal03b}, \citenum{Hor67}, and \citenu
\end{flushleft}
\end{table}
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 $\Td$ but for the totally symmetric $\pi \ra \pis$ excitation ($\Td = 86\%$). 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 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.
Like furan, pyrrole has been extensively investigated in the literature using a large panel of theoretical methods. \cite{Ser93b,Nak96,Tro97,Pal98,Chr99,Wan00,Roo02,Pal03b,Pas06c,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Nev14,Sch17,Hei19}
Here, we report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All these transitions have very large $\Td$ except for the totally symmetric $\pi \ra \pis$ excitation ($\Td = 86\%$). For all the 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 twenty years old extrapolated CC values of Christiansen and coworkers \cite{Chr99} is quite remarkable. The only exceptions are the two
$B_2$ transitions that were reported as significantly mixed in this 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 this transition.
As can be seen in Table S5, {\AVTZ} yields basis-set converged transition energies, except, 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 \titou{broad}, and the rare available experiments \cite{Hor67,Bav76,Fli76b,Vee76b,Pal98,Pal03b} can only be considered as general guidelines.
\begin{table}[htp]
\caption{\small Vertical transition energies (in eV) of cyclopentadiene, imidazole, and thiophene.}
@ -792,48 +792,45 @@ $^q${0-0 energies from Ref.~\citenum{Hol14}.}
\end{flushleft}
\end{table}
Although a quite significant array of previous wavefunction studies has been performed for cyclopentadiene not only at the {\CASPT}, \cite{Ser93b,Sch08,Sil10c} and CC \cite{Sch08,Sil10b,Sch17} levels but also with
SAC-CI \cite{Wan00b} and various multi-reference approaches, \cite{Nak96,She09b} this compound has been less intensively studied than furan and pyrrole (\emph{vide infra}), probably due to the presence of the
methylene group that renders the computations significantly more expensive. All transitions listed in Tables \ref{Table-6} and S6 are characterized by $\Td$ exceeding 93\%\ but for the $^1A_1 (\pi \ra \pis)$
excitation that has a nature similar to the lowest $A_g$ state of butadiene ($\Td = 79\%$). Consistently, the {\CCT} and {\CCSDT} results are nearly identical for all ES but for that transition. By comparing the results
obtained for this $A_1 (\pi \ra \pis)$ transition to its butadiene counterpart, one can infer that the {\CCSDT} estimate is probably too large by ca.~$0.04$--$0.08$ eV, and that the {\NEV} value is likely not more accurate
than the {\CCSDT} one. This statement is also in line with the results of Ref.~\citenum{Loo19c}. For the two $B_2 (\pi \ra \pis)$ transitions, we could obtain a {\FCI}/{\Pop} estimates of $5.78 \pm 0.02$ eV and
(singlet) $3.33 \pm 0.05$ eV (triplet); the {\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As can be seen in Tables \ref{Table-6} and S6, the basis set effects are rather moderate
for all transitions, with no variations larger than $0.10$ eV ($0.02$ eV) between {\AVDZ} and {\AVTZ} ({\AVTZ} and {\AVQZ}). When comparing to literature data, our values are unsurprisingly consistent with the {\CCT} values of
Although a diverse array of wavefunction studies has been performed on cyclopentadiene, including {\CASPT}, \cite{Ser93b,Sch08,Sil10c} CC, \cite{Sch08,Sil10b,Sch17} SAC-CI \cite{Wan00b} and various multi-reference approaches \cite{Nak96,She09b}), this compound has received less attention that other members of the five-membered ring family, namely furan and pyrrole (\emph{vide infra}). This is probably due to the presence of the
methylene group that renders the computations significantly more expensive. Most transitions listed in Tables \ref{Table-6} and S6 are characterized by $\Td$ exceeding $93\%$, the only exception being the $^1A_1 (\pi \ra \pis)$
excitation that has a similar nature to the lowest $A_g$ state of butadiene ($\Td = 79\%$). Consistently, the {\CCT} and {\CCSDT} results are nearly identical for all ES except for the $^1A_1$ ES. By comparing the results
obtained for this $A_1 (\pi \ra \pis)$ transition to its butadiene counterpart, one can infer that the {\CCSDT} estimate is probably too large by roughly $0.04$--$0.08$ eV, and that the {\NEV} value is unlikely to be accurate enough. This statement is also in line with the results of Ref.~\citenum{Loo19c}. For the two $B_2 (\pi \ra \pis)$ transitions, we could obtain a {\FCI}/{\Pop} estimates of $5.78 \pm 0.02$ eV (singlet)
and $3.33 \pm 0.05$ eV (triplet), the {\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As one can see in Tables \ref{Table-6} and S6, the basis set effects are rather moderate
for the transitions of cyclopentadiene, with no variation larger than $0.10$ eV ($0.02$ eV) between {\AVDZ} and {\AVTZ} ({\AVTZ} and {\AVQZ}). When comparing to literature data, our values are unsurprisingly consistent with the {\CCT} values of
Schwabe and Goerigk, \cite{Sch17} and tend to be significantly larger than earlier {\CASPT} \cite{Ser93b,Sil10c} and MR-MP \cite{Nak96} estimates. As expected, a few gas-phase experiments are available as well for this
derivative, \cite{Fru79,McD85,McD91b,Sab92} but hardly allow to make the final call.
derivative, \cite{Fru79,McD85,McD91b,Sab92} but they hardly represent grounds for comparison.
Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remains the {\CASPT} work of Serrano-Andr\`es and coworkers from 1996, \cite{Ser96b} and the
basis-set extrapolated {\CCT} results of Silva-Junior \emph{et al.} for the valence transitions. \cite{Sil10c} The experimental data in gas-phase are also limited.\cite{Dev06}
Our results are displayed in Tables \ref{Table-6} and S6. The {\CCT} and {\CCSDT} values are quite consistent despite the fact that the $\Td$ values of the two singlet $A'$ states
are slightly smaller than $90\%$. For the eight transitions considered here, the basis set effects are moderate and {\AVTZ} yield results within $0.03$ eV of {\AVQZ} (Table S6 in the SI). \hl{NEV ?}
Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remaining the 1996 {\CASPT} work of Serrano-Andres \emph{et al}, \cite{Ser96b} and the
basis-set extrapolated {\CCT} investigation of Silva-Junior \emph{et al} for the valence transitions. \cite{Sil10c} The experimental data in gas-phase are also limited.\cite{Dev06}
Our results are displayed in Tables \ref{Table-6} and S6. The {\CCT} and {\CCSDT} values are quite consistent despite the fact that the $\Td$ of the two singlet $A'$ states
are slightly smaller 90\%. For all eight considered transitions, the basis set effects are moderate and {\AVTZ} yield results within $0.03$ eV of {\AVQZ} (Table S6 in the SI). \hl{NEV ?}
Finally, the ES thiophene, which is one of the most important building block in organic electronic devices, were the subject of a few previous theoretical investigations, \cite{Ser93c,Pal99,Wan01,Kle02,Pas07,Hol14} that unveiled a series of
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)
Finally, the ES of thiophene, which is one of the most important building block in organic electronic devices, were the subject of a previous theoretical investigations, \cite{Ser93c,Pal99,Wan01,Kle02,Pas07,Hol14} that unveiled a series of
transitions that were not yet characterized in the available measurements. \cite{Dil72,Fli76,Fli76b,Var82,Hab03,Pal99,Hol14} To the best of 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 $\Td$ above
90\%\ but for the two $^1A_1$ transitions that show $\Td$ 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 \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
$90\%$ except for the two $^1A_1$ transitions for which $\Td = 88\%$ and $87\%$, respectively. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for the lowest totally symmetric transition. Thiophene is
also one of these compounds for which the unambiguous characterization of the nature of the ES is difficult, with, \eg, a strong mixing between the second and third singlet ES of $B_2$ symmetry. This makes the assignment of the valence
($\pi \ra \pis$) or Rydberg ($\pi \ra 3p$) character of this transition particularly tricky 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.
The data of Table \ref{Table-6} are globally in good agreement with the previously reported values with discrepancies that are significant only for the three highest-lying singlet states.
\subsection{Six-atom molecules}
Let us now turn towards six-member cycles playing a key role in chemistry: benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine. To the best of our knowledge, the present work
Let us now turn towards seven six-membered rings which plays a key role in chemistry: benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine. To the best of our knowledge, the present work
is the first to propose {\CCSDT} reference energies as well as {\CCT} values obtained with a {\AVQZ} for all these compounds. Of course, these systems have been investigated before, and beyond Thiel's
benchmarks, \cite{Sch08,Sil10b,Sil10c} it is worth to point out the investigations of Del Bene and coworkers, \cite{Del97b} performed with a CC approach including perturbative corrections for the triples, and of Nooijen, \cite{Noo99}
who used {\STEOM}, to study the ES of all these derivatives with a theoretically consistent protocol. However, these two works considered the singlet ES only.
benchmarks, \cite{Sch08,Sil10b,Sil10c} it is worth to point out the investigations of Del Bene and coworkers, \cite{Del97b} performed with a CC approach including perturbative corrections for the triples.
Following a theoretically consistent protocol, Nooijen \cite{Noo99} also performed {\STEOM} calculations to study the ES of all these derivatives. However, these two works only considered singlet ES.
\subsubsection{Benzene, pyrazine, and tetrazine}
\hl{Martial: please do check Rydberg assignments throughout}
These three highly-symmetric systems allow to directly perform {\CCSDT}/{\AVTZ} calculations for singlet states without the need for basis set extrapolation. Benzene was studied many times
These three highly-symmetric systems allow to directly perform {\CCSDT}/{\AVTZ} calculations for singlet states without the need of basis set extrapolations. Benzene was studied many times
before, \cite{Sob93,Lor95b,Chr96c,Pac96,Del97b,Noo99,Hal02,Li07b,Sch08,Dev08,Sil10b,Sil10c,Li11,Lea12,Kan14,Sch17,Dut18,Sha19,Loo19c} and we report in Tables \ref{Table-7} and S7 estimates obtained for
five singlet and three triplet ES, all characterized by $\Td$ exceeding 90\%\ but the lowest singlet (86\%). As can be seen, the two CC approaches are again yielding very consistent transitions energies
and {\AVTZ} is essentially providing basis set converged transition energies. The present estimates are also very consistent with early {\CCT}\cite{Chr96c} and very recent RASPT2 values. \cite{Sha19}
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}
five singlet and three triplet ES, all characterized by $\Td$ exceeding $90\%$ except for the lowest singlet ($86\%$). As one can see, the two CC approaches are again yielding very consistent transitions energies
and {\AVTZ} is essentially providing basis set converged transition energies. The present estimates are also very consistent with earlier {\CCT} \cite{Chr96c} and very recent RASPT2 \cite{Sha19} values.
For both spin manifolds, our values are slightly larger than the available electron impact/multi-photon measurements. \cite{Doe69,Nak80,Joh76,Joh83,Hir91}
\begin{table}[htp]
\caption{\small Vertical transition energies (in eV) of benzene.}
@ -946,27 +943,31 @@ $^m${all these doubly ES have a $(n,n \ra \pis, \pis)$ character.}
Numerous previous theoretical estimates are available for both pyrazine, \cite{Ful92,Del97b,Web99,Noo99,Li07b,Sch08,Sau09,Sil10c,Woy10,Car10,Lea12,Kan14,Sch17,Dut18} and tetrazine, \cite{Sta96,Del97b,Rub99,Noo99,Ada00,Noo00,Ang09,Sch08,Sau09,Sil10b,Sil10c,Car10,Lea12,Kan14,Sch17,Dut18,Pas18b}
for which the $D_{2h}$ symmetry helps distinguishing the different ES. Our results are collected in Tables \ref{Table-8} and S8. In pyrazine, all transitions are characterized by $\Td > 85\%$, but for $^1B_{1g} (n \ra \pis)$, and the
changes in going from {\CCT} to {\CCSDT} are always trifling but for the highest-lying singlet state considered here. When going from triple-$\zeta$ to quadruple-$\zeta$, the variations do not exceed $0.04$ eV, even for the four considered Rydberg
ES. This indicates that one can probably be highly confident in the present estimates. Again, the previous {\CASPT} estimates, \cite{Ful92,Web99,Sch08} appear to be globally too low, whereas the unconventional CASPT3 results that are
available, \cite{Woy10} are too large. The same holds for the SAC-CI results. \cite{Li07b} In fact we obtain globally our best match with the {\STEOM} values of Nooijen (but for the highest ES), \cite{Noo99} and recent {\CCT} estimates.
\cite{Sch17}. The experimental data we are aware of, \cite{Bol84,Oku90,Wal91,Ste11c} do not report all transitions, but provide globally a similar ranking for the triplet transitions.
for which the $D_{2h}$ symmetry helps distinguishing the different ES. Our results are collected in Tables \ref{Table-8} and S8. In pyrazine, all transitions are characterized by $\Td > 85\%$ (except for the $^1B_{1g} (n \ra \pis)$ transition).
The excitation energies are basically unchanged going from {\CCT} to {\CCSDT} with one exception (the highest-lying singlet state). Going from triple- to quadruple-$\zeta$ basis, the variations do not exceed $0.04$ eV, even for the four Rydberg states, indicating that one can probably be highly confident in the present estimates.
Again, the previous {\CASPT} estimates \cite{Ful92,Web99,Sch08} appear to be globally too low, while the (unconventional) CASPT3 results \cite{Woy10} available are too large.
The same holds for the SAC-CI results. \cite{Li07b} In fact, our best match is obtained with Nooijen's {\STEOM} values (except for the highest ES), \cite{Noo99} and recent {\CCT} estimates. \cite{Sch17}
The available experimental data \cite{Bol84,Oku90,Wal91,Ste11c} do not report all transitions, but provide a similar ranking for the triplet transitions.
For tetrazine, we consider valence ES only, but three transitions present a true double excitation nature ($\Td < 10\%$), for which {\CCT} nor {\CCSDT} can not be viewed as reliable, and the best approach is likely {\NEV}. \cite{Loo19c}
For all other transitions, the $\Td$ are in the 80-90\%\ range for singlets and larger than 95\%\ for triplets, and the results of the two CC approaches are very consistent, but for the lowest $^3B_{1u} (\pi \ra \pis)$ excitation.
In all other cases, there is a good consistency between the values we obtained with the two CC models, and the basis set effects are very small beyond {\AVTZ} with maximal variations of $0.02$ eV only (Table S8). The present values are
almost systematically larger than previous {\CASPT},\cite{Rub99} {\STEOM}, \cite{Noo00} and GVVPT2 \cite{Dev08} estimates, and are globally in agreement with Thiel's {\CCT}/{\AVTZ} values, \cite{Sil10c} although we note variations
of ca.~$0.20$ eV for some specific transitions like the $B_{2g}$ transitions, likely due to the use of different geometries in that work. The experimental EEL values from Palmer's work, \cite{Pal97} show a reasonable agreement with our estimates.
For tetrazine, we consider valence ES only, as well as three transitions exhibiting a true double excitation nature ($\Td < 10\%$).
Of course, for these double excitations, {\CCT} nor {\CCSDT} can be viewed as reliable, and the best approach is likely {\NEV}. \cite{Loo19c}
For all the other transitions, the $\Td$ values are in the $80$--$90\%$ range for singlets and larger than $95\%$ for triplets.
Consequently, the {\CCT} and {\CCSDT} results are very consistent, a single exception being the lowest $^3B_{1u} (\pi \ra \pis)$ transition.
In all other cases, there is a global consistency between our CC values. Moreover, the basis set effects are very small beyond {\AVTZ} with maximal variations of $0.02$ eV only (Table S8). The present values are
almost systematically larger than previous {\CASPT}, \cite{Rub99} {\STEOM}, \cite{Noo00} and GVVPT2 \cite{Dev08} estimates. They are also globally in agreement with Thiel's {\CCT}/{\AVTZ} values, \cite{Sil10c} although we note variations
of approximately $0.20$ eV for specific excitations like the $B_{2g}$ transitions.
This feature is likely due to the use of distinct geometries in the two studies. The experimental EEL values from Palmer's work, \cite{Pal97} show a reasonable agreement with our estimates.
\subsubsection{Pyridazine, pyridine, pyrimidine, and triazine}
\titou{HERE}
Those four azabenzenes, of $C_{2v}$ and $D_{3h}$ symmetry, are also popular molecules for ES calculations. \cite{Pal91,Ful92,Wal92,Lor95,Del97b,Noo97,Noo99,Fis00,Cai00b,Wan01b,Sch08,Sil10b,Sil10c,Car10,Lea12,Kan14,Sch17,Dut18}
Our results for pyridazine and pyridine are collected in Tables \ref{Table-9} and S9. For the former compounds, the available wavefunction results \cite{Pal91,Ful92,Del97b,Noo99,Fis00,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18}
have all considered the singlet transitions only, at the exception of rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} The $\Td$ are larger than 85\%\ (95\%) for the singlet (triplet) transitions,
and the only state for which there is a variation larger than $0.03$ eV between the {\AVDZ} {\CCT} and {\CCSDT} energies, but for the $^3B_2 (\pi \ra \pis)$ transition. \hl{BASIS SETS} For the valence singlet
ES, we find again a quite good agreement with previous {\STEOM} \cite{Noo99} and CC \cite{Del97b,Sil10c} estimates, but are again significantly higher than {\CASPT} estimates. \cite{Ful92,Sil10c} For the triplets, the
Our results for pyridazine and pyridine are gathered in Tables \ref{Table-9} and S9. For the former compound, the available wavefunction results \cite{Pal91,Ful92,Del97b,Noo99,Fis00,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18}
only considered singlet transitions, at the exception of a rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} Again, the $\Td$ values are larger than $85\%$ ($95\%$) for the singlet (triplet) transitions,
and the only state for which there is a variation larger than $0.03$ eV between the {\CCT}/{\AVDZ} and {\CCSDT}/{\AVDZ} energies, but for the $^3B_2 (\pi \ra \pis)$ transition. \hl{BASIS SETS} For the valence singlet
ES, we find again a quite good agreement with previous {\STEOM} \cite{Noo99} and CC \cite{Del97b,Sil10c} estimates, but are again significantly higher than {\CASPT} estimates. \cite{Ful92,Sil10c} For the triplets, the
present data represents the best published to date. Interestingly, beyond the usually cited experiments, \cite{Inn88,Pal91} there is a very recent experimental EEL analysis for pyridazine, \cite{Lin19} that localized
almost all ES. The transition energies reported in this very recent effort are systematically smaller than our CC estimates, by ca.~$-0.20$ eV, but remarkably show exactly the exact same ranking.
@ -1044,8 +1045,7 @@ $^j${Significant state mixing with a close-lying Rydberg transition, rendering u
slightly larger variations for the two Rydberg transitions. For both compounds, the current values are almost systematically larger than most previously published data. For the triplets of triazine, the three lowest ES estimated
by {\CASPT} previously are too low by ca.~$-0.5$ eV.
s
\begin{table}[htp]
\begin{table}[htp]
\caption{\small Vertical transition energies (in eV) of pyrimidine and triazine.}
\label{Table-10}
\begin{footnotesize}
@ -1509,7 +1509,7 @@ CCSDT-3 &0.05 & &0.06 &0.03 &0.08 &0.04\\
\section{Conclusions and outlook}
We have computed highly-accurate vertical transition energies for a set of 27 medium-sized organic molecules containing from 4 to 6 (non-hydrogen) atoms. To this end, we employed several state-of-the-art theoretical models with increasingly large diffuse basis sets.
However, most of our theoretical best estimates are based on {\CCSDTQ} (4 atoms) or {\CCSDT} (5 and 6 atoms) excitation energies. For the vast majority of the
Most of our theoretical best estimates are based on {\CCSDTQ} (4 atoms) or {\CCSDT} (5 and 6 atoms) excitation energies. For the vast majority of the
listed excited states, the present contribution is the very first to disclose (sometimes basis-set extrapolated) {\CCSDT}/{\AVTZ} and (true) {\CCT}/{\AVQZ} transition energies as well as {\CCT}/{\AVTZ} oscillator strengths
for each dipole-allowed transition. Our set contains a total of 238 transition energies and 90 oscillator strengths, with a reasonably good balance between singlet, triplet, valence,
and Rydberg states. Amongst these 238 transitions, we believe that 224 are ``solid'' TBE, \ie, they are chemically accurate (MAE below $0.043$ eV or $1$ kcal.mol$^{-1}$) for the considered geometry.
@ -1520,12 +1520,12 @@ in the $0.12$--$0.23$ eV range.
Paraphrasing Thiel and coworkers, \cite{Sch08} we hope that this new set of vertical transition energies, combined or not with the ones described in our previous works, \cite{Loo18a,Loo19c} will be useful for the community,
will stimulate further developments and analyses in the field, and will provide new grounds for appraising the \emph{pros} and \emph{cons} of ES models already available or currently under development. We can
crystal-ball that the emergence of new {\SCI} algorithms optimized for modern computer architectures will likely lead to the revision of some the present TBE, allowing to climb even higher on the accuracy ladder.
crystal-ball that the emergence of new {\SCI} algorithms optimized for modern supercomputer architectures will likely lead to the revision of some the present TBE, allowing to climb even higher on the accuracy ladder.
\begin{suppinfo}
Geometries.
Basis set and frozen core effects.
Definition of the active spaces for the multi-reference calculations.
Definition of the active spaces for the multi-configurational calculations.
Additional details about the {\SCI} calculations and their extrapolation.
Benchmark data and further statistical analysis.
\end{suppinfo}