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@ -144,7 +144,7 @@ generally come with a dreadful scaling with system size and, in addition, typica
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).
\cite{Ful92,Ser93,Ser93b,Ser93c,Lor95b,Mer96,Mer96b,Roo96,Ser96b} To this end, they relied on experimental GS geometries and the complete-active-space second-order perturbation theory ({\CASPT}) approach with the largest active spaces and basis sets
one could dream of at the time.
These {\CASPT} values were latter used to assess the performances of TD-DFT combined with various exchange-correlation functionals, \cite{Toz99b,Bur02} and remained for a long time the best theoretical references available on the market.
These {\CASPT} values were later used to assess the performances of TD-DFT combined with various exchange-correlation functionals, \cite{Toz99b,Bur02} and remained for a long time the best theoretical references available on the market.
However, beyond comparisons with experiments, which are always challenging when computing vertical transition energies, \cite{San16b} there was no approach available at that time to ascertain the accuracy of these transition energies.
Nowadays, it is common knowledge that CASPT2 has the tendency of underestimating vertical excitation energies in organic molecules.
@ -308,17 +308,16 @@ state is the onset, whereas an estimate of the vertical energy ($4.2 \pm 0.2$ eV
\end{flushleft}
\end{sidewaystable}
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
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 ($\Td > 90\%$, \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
valence $\pi \ra \pis$ transitions, which are all characterized by a strongly dominant single excitation nature ($\Td > 90\%$, \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
energies by approximately $0.10$ eV, when going from {\Pop} to {\AVDZ} and from {\AVDZ} to {\AVTZ}. The lowest triplet state appears less basis set sensitive, though. As expected,
extending further the basis set size (to quadruple and quintuple-$\zeta$) leaves the results pretty much unchanged. The same observation holds when adding a second set of diffuse functions, or when correlating the core electrons (see the SI). Obviously,
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 very neat agreement between the CC/{\AVTZ} and {\NEV}/{\AVTZ}. 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 ES of cyanoacetylene with successive drops of the transition
energies by approximately $0.10$ eV, when going from {\Pop} to {\AVDZ}, and from {\AVDZ} to {\AVTZ}. The lowest triplet state appears less basis set sensitive, though. As expected,
extending further the basis set size (to quadruple- and quintuple-$\zeta$) leaves the results pretty much unchanged. The same observation holds when adding a second set of diffuse functions, or when correlating the core electrons (see the SI). Obviously,
both cyanogen and diacetylene yield very similar trends, with limited methodological effects and quite large basis set effects, except for the $^1\Sigma_g^+ \ra {} ^3\Sigma_u^+$ transitions. We note that all CC3 and CCSDT values are, at worst,
within $\pm 0.02$ eV of the {\FCI} window, \ie, all methods presented in Table \ref{Table-1} provide very consistent estimates. For all the states reported in this Table, the average absolute deviation between {\NEV}/{\AVTZ} and {\CCT}/{\AVTZ}
({\CCSDT}/{\AVTZ}) is as small as $0.02$ ($0.03$) eV, the lowest absorption and emission energies of cyanogen being the only two cases showing significant deviations. As a final note, all our vertical absorption (emission) energies are significantly bigger
({\CCSDT}/{\AVTZ}) is as small as $0.02$ ($0.03$) eV, the lowest absorption and emission energies of cyanogen being the only two cases showing significant deviations. As a final note, all our vertical absorption (emission) energies are significantly larger
(smaller) than the experimentally measured 0-0 energies, as it should. We refer the interested reader to previous works, \cite{Fis03,Loo19a} for comparisons between theoretical ({\CASPT} and {\CCT}) and experimental 0-0 energies for these three compounds.
@ -401,8 +400,8 @@ Our results are listed in Tables \ref{Table-2} and S2. As above, considering the
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 ($\Td = 85\%$),
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
it is clear from the {\FCI} value that only {\CCSDTQ} is energetically 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 this 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,
@ -488,30 +487,30 @@ for the lowest $^1B_g$ ($^1B_u$) ES, a range of $4.2$--$4.5$ ($7.4$--$7.9$) eV i
\end{table}
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 $n \ra \pis$
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
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 all our efforts. For the lowest $n \ra \pis$
transitions of both spin symmetry, 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}).
can be seen in Table S3, the {\AVTZ} basis set delivers excitation energies very close to the CBS limit: the largest variation when upgrading from {\AVTZ} 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 ultra 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 bamboozled theoretical chemists for many years. It is beyond the scope of the present study to provide an exhaustive
list of previous calculations and experimental measurements for these two hallmark ES, and we refer the readers to Refs.~\citenum{Wat12} and \citenum{Shu17} for a general and broader overview and the corresponding references. For the $B_u$ transition,
we believe that the most solid TBE 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 their 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 set corrected {\FCI} estimate of $6.50$ eV \cite{Loo19c} remains
the most accurate available to date. These two values are slightly smaller than the semi-stochastic 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} For these two thoroughly studied ES, one can, of course, find many other estimates, \eg, at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} \cite{Sok17} levels.
the most accurate available to date. These two values are slightly lower than the semi-stochastic 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} For these two thoroughly studied ES, one can of course find many other estimates, \eg, at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} \cite{Sok17} levels.
Globally, for 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 to the
already mentioned $^1A_g$ ES that is $0.2$ and $0.1$ eV too high with the two former CC methods, \titou{consistent with the large electronic reorganization taking place in that state.} For all the other butadiene ES listed in
already mentioned $^1A_g$ ES that is $0.2$ and $0.1$ eV too high with the two former CC methods, \titou{a direct consequence of the large electronic reorganization taking place during this transition.} 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, except for the lowest $B_u$ ES for which the associated excitation energy is highly dependent on the selected
active space (see the SI). Finally, as can be seen in Table S3, {\AVTZ} produces near-CBS excitation energies for most ES.
However, a significant basis set effect exists for the Rydberg $^1B_u (\pi \ra 3p)$ ES with an energy lowering 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 as our calculations.
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, \ie, highly consistent CC estimates, reasonable agreement between {\NEV} and {\CCT}, and limited basis set effects beyond {\AVTZ}, except for the $^1B_u (n \ra 3p)$ Rydberg state (see Tables \ref{Table-3} and S3). This Rydberg state also indicates a unexpectedly large deviation of $0.04$ eV between {\CCT} and {\CCSDTQ}. More interestingly,
glyoxal presents a genuine low-lying double ES of $^1A_g$ symmetry. The corresponding $(n,n \ra \pis,\pis)$ transition is totally unseen by approaches that cannot model double excitations, \eg, TD-DFT, {\CCSD},
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, \ie, highly consistent CC estimates, reasonable agreement between {\NEV} and {\CCT}, and limited basis set effects beyond {\AVTZ}, except for the $^1B_u (n \ra 3p)$ Rydberg state (see Tables \ref{Table-3} and S3). This Rydberg state also exhibits an unexpectedly large deviation of $0.04$ eV between {\CCT} and {\CCSDTQ}. More interestingly,
glyoxal presents a genuine low-lying double ES of $^1A_g$ symmetry. The corresponding $(n,n) \ra (\pis,\pis)$ transition is totally unseen by approaches that cannot model double excitations, \eg, TD-DFT, {\CCSD},
or {\AD}. Compared to the {\FCI} values, the {\CCT} and {\CCSDT} estimates associated with this transition are too large by $\sim 1.0$ and $\sim 0.5$ eV, respectively, whereas both the {\CCSDTQ} and {\NEV} approaches are much closer, 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} are unhelpful in view of the targeted accuracy.
@ -605,22 +604,22 @@ $^i${0-0 energies from Ref.~\citenum{Jud84c}.}
\end{flushleft}
\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 both very
sensitive to the basis set, and upshifted by ca.~$0.04$ eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} transition energies of Merch\'an and coworkers listed on the
right panel of Table \ref{Table-4} are clearly too low, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the \titou{details of the calculations}, as the Urban group obtained {\CASPT}
For acetone, one should clearly distinguish the valence ES, for which both methodological and basis set effects are small, and the Rydberg transitions that are both very
basis set sensitiv, and upshifted by ca.~$0.04$ eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} transition energies of Merch\'an and coworkers listed on the
right panel of Table \ref{Table-4} are clearly too low, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the computational set-up, 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 of the three $n \ra 3p$ transitions, $7.52$, $7.57$, 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 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}/{\Pop} and {\CCSDTQ}/{\Pop} results.
In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to the excitation order of the CC expansion, as illustrated by the maximal discrepancies of $\pm 0.02$ eV between the {\CCT}/{\Pop} and {\CCSDTQ}/{\Pop} results.
While the lowest $n \ra \pis$ transition of both spin symmetries are rather basis set insensitive, all the other states need quite large one-electron bases to be correctly described (Table S4).
As expected, our theoretical vertical transition energies show the same ranking but are systematically larger than the available experimental 0-0 energies.
For the isoleectronic isobutene molecule, we have considered two singlet Rydberg state and one triplet valence ES. For these three cases, we note, for each basis, a very nice agreement between {\CCT} and {\CCSDT}, the
For the isoelectronic isobutene molecule, we have considered two singlet Rydberg and one triplet valence ES. For these three cases, we note, for each basis, a very nice agreement between {\CCT} and {\CCSDT}, the
CC results being also very close to the {\FCI} estimates obtained with the Pople basis set. The similarity with the {\CCSD} results of Caricato and coworkers \cite{Car10} is also very satisfying.
For the three remaining compounds, namely, cyanoformaldehyde, propynal, and thiopropynal, we report low-lying valence transitions with a definite single excitation character. The basis set
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, alluding that the CC values are very accurate.
a given basis, eluding that the CC values are very accurate.
This is further confirmed by the {\FCI} data.
\subsubsection{Intermediate conclusions}
@ -634,19 +633,19 @@ MAE ($0.01$ and $0.02$ eV), and small maximal deviations ($0.05$ and $0.04$ eV)
estimates (errors below $1$ kcal.mol$^{-1}$ or $0.043$ eV) for most electronic transitions. Interestingly, some of us have shown that {\CCT} also provides chemically-accurate 0-0 energies as compared
to experimental values for most valence transitions. \cite{Loo18b,Loo19a,Sue19} When comparing the {\NEV} and {\CCT} ({\CCSDT}) results obtained with {\AVTZ} for the 91 (65) ES for which comparisons are possible (again excluding only the lowest $^1A_g$ states of butadiene and glyoxal),
one obtains a MSE of $+0.09$ ($+0.09$) eV and a MAE of $0.11$ ($0.12$) eV.
This seems to indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with {\CASPT} that is known to generally underestimate transition energies, as further illustrated and discussed above.
This seems to indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with {\CASPT} that is known to generally underestimate transition energies, as further illustrated and discussed above and below.
\subsection{Five-atom molecules}
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.
As the most advanced levels of theory employed in the previous Section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds (except in very rare occasions), one has to rely on the nature of the ES and the consistency between results to deduce TBE.
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}
For furan, \textit{ab initio} calculations have been performed with almost every single wavefunction methods.\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
All singlet (triplet) transitions are characterized by $\Td$ values 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.
be significantly higher than in most previous works, even after accounting for the quite large basis set effects ($-0.10$ eV between the {\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.
@ -708,12 +707,12 @@ $^l${Vapour UV spectra from Refs.~\citenum{Pal03b}, \citenum{Hor67}, and \citenu
\end{table}
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
Here, we report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All these transitions have very large $\Td$ values 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.
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 pushing to the
quadruple-$\zeta$ basis set. As mentioned in Thiel's work, \cite{Sch08} the experimental spectra of pyrrole are quite ``disparate'', 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.}
@ -792,19 +791,19 @@ $^q${0-0 energies from Ref.~\citenum{Hol14}.}
\end{flushleft}
\end{table}
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
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 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 as 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
for the electronic 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 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
Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remain 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 ?}
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 their {\AVQZ} counterparts (Table S6 in the SI). \hl{NEV ?}
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)
@ -818,8 +817,8 @@ The data of Table \ref{Table-6} are globally in good agreement with the previous
\subsection{Six-atom molecules}
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.
is the first to propose {\CCSDT} reference energies as well as {\CCT} values obtained with {\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 pointing 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}
@ -830,7 +829,7 @@ These three highly-symmetric systems allow to directly perform {\CCSDT}/{\AVTZ}
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 $90\%$ 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}
For both spin symmetry, 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.}
@ -945,15 +944,15 @@ $^m${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 $\Td > 85\%$ (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.
Again, the previous {\CASPT} estimates \cite{Ful92,Web99,Sch08} appear to be globally too low, while the (unconventional) CASPT3 results \cite{Woy10} seem too high.
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.
The available experimental data \cite{Bol84,Oku90,Wal91,Ste11c} do not report all transitions, but provide a similar ranking for the triplets.
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.
Consequently, the {\CCT} and {\CCSDT} results are very consistent, the sole 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.
@ -966,8 +965,8 @@ Those four azabenzenes with $C_{2v}$ and $D_{3h}$ spatial symmetry are also popu
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 is the $^3B_2 (\pi \ra \pis)$ transition. \hl{BASIS SETS}
For the singlet valence ES, we find again a rather good agreement with previous {\STEOM} \cite{Noo99} and CC \cite{Del97b,Sil10c} estimates, which are again significantly higher than the {\CASPT} estimates reported in Refs.~\citenum{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 locates
For the singlet valence ES, we find again a rather good agreement with previous {\STEOM} \cite{Noo99} and CC \cite{Del97b,Sil10c} estimates, which are again significantly higher than the {\CASPT} estimates reported in Refs.~\citenum{Ful92} and \citenum{Sil10c}. For the triplets, the
present data represents the best results 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 locates
almost all ES. The transition energies reported in this very recent work are systematically smaller than our CC estimates by approximately $-0.20$ eV.
Nonetheless, this study provides exactly the same ES ranking as our theoretical protocol.
@ -1039,9 +1038,9 @@ $^j${Significant state mixing with a close-lying Rydberg transition, rendering u
The results obtained for both pyrimidine and triazine are listed in Tables \ref{Table-10} and S10. For the former derivative, previous theoretical \cite{Ful92,Del97b,Ser97b,Noo99,Ohr01,Fis03b,Li07b,Sch08,Sil10b,Sil10c,Car10,Kan14,Sch17,Dut18}
and experimental \cite{Bol84,Pal90,Lin15} studies are rather extensive, as it can be viewed as the smallest model of DNA bases. For triazine, which does not have an abelian point group, theoretical studies are scarcer,
\cite{Ful92,Wal92,Pal95,Del97b,Noo99,Oli05,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18}, especially for the triplet transitions. \cite{Wal92,Pal95,Oli05} Experimental data are also limited. \cite{Bol84,Wal92}
\cite{Ful92,Wal92,Pal95,Del97b,Noo99,Oli05,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18} especially for the triplets. \cite{Wal92,Pal95,Oli05} Experimental data are also limited. \cite{Bol84,Wal92}
As in pyridazine and pyridine, all the ES listed in Table \ref{Table-10} show $\Td$ values larger than $85\%$ for singlets and $95\%$ for triplets, so that {\CCT} and {\CCSDT} are highly coherent, except maybe for the $^3A_1 (\pi \ra \pis)$ transitions in pyrimidine.
The basis set effects are also small, with no variation larger than $0.10$ ($0.03$) eV between double-$\zeta$ and triple-$\zeta$ (triple-$\zeta$ and quadruple-$\zeta$) and only
The basis set effects are also small, with no variation larger than $0.10$ ($0.03$) eV between double- and triple-$\zeta$ (triple- and quadruple-$\zeta$) and only
slightly larger variations for the two Rydberg transitions. For both compounds, the current values are almost systematically larger than previously published data. For the triplets of triazine, the three lowest ES previously estimated by {\CASPT} are too low by roughly half an eV.
\begin{table}[htp]
@ -1441,7 +1440,7 @@ CCSDT-3 &126 &0.05 &0.05 &0.07 &0.04 &0.26 &0.00\\
\label{Fig-1}
\end{figure}
Let us analyse the global performances of all methods, starting with the most accurate and computationally demanding models. The relative accuracies of {\CCT} and {\CCSDT}-3 as compared to {\CCSDT} remains an open question in the literature. \cite{Wat13,Dem13}
Let us analyze the global performances of all methods, starting with the most accurate and computationally demanding models. The relative accuracies of {\CCT} and {\CCSDT}-3 as compared to {\CCSDT} remains an open question in the literature. \cite{Wat13,Dem13}
Indeed, to the best of our knowledge, the only two previous studies discussing this specific aspect are limited to very small compounds. \cite{Kan17,Loo18a} According to the results gathered in Table \ref{Table-bench}, it appears that {\CCT} has a slight edge over {\CCSDT}-3, although
{\CCSDT}-3 is closer to {\CCSDT} in \titou{formal} terms. Indeed, {\CCSDT}-3 seems to provide slightly too large transition energies (MSE of $+0.05$ eV). These conclusions are qualitatively consistent with the analyses performed on smaller derivatives,
\cite{Kan17,Loo18a} but the amplitude of the {\CCSDT}-3 errors is larger in the present set. Although the performances of {\CCT} might be unduly inflated by the use of {\CCSDT} and {\CCSDTQ} reference values, it is also clear that this