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@ -39,14 +39,14 @@
\newcommand{\SupInf}{supporting information}
%
\newcommand{\Pop}{6-31+G(d)}
\newcommand{\AVDZ}{aVDZ}
\newcommand{\AVTZ}{aVTZ}
\newcommand{\DAVTZ}{daVTZ}
\newcommand{\AVQZ}{aVQZ}
\newcommand{\DAVQZ}{daVQZ}
\newcommand{\TAVQZ}{taVQZ}
\newcommand{\AVPZ}{aV5Z}
\newcommand{\DAVPZ}{dapV5Z}
\newcommand{\AVDZ}{aug-cc-pVDZ}
\newcommand{\AVTZ}{aug-cc-pVTZ}
\newcommand{\DAVTZ}{d-aug-cc-pVTZ}
\newcommand{\AVQZ}{aug-cc-pVQZ}
\newcommand{\DAVQZ}{d-aug-cc-pVQZ}
\newcommand{\TAVQZ}{t-aug-cc-pVQZ}
\newcommand{\AVPZ}{aug-cc-pV5Z}
\newcommand{\DAVPZ}{d-aug-cc-pV5Z}
%Vector
@ -116,7 +116,7 @@ Nevertheless, given the large \%$T_1$ values, one can likely consider them accu
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in aza-naphthalene (1,4,5,8-tetraazanaphthalene).
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -173,7 +173,7 @@ Although one cannot consider this 4.57 eV estimate as chemically accurate (the e
The next transition of $^1B_{3g}$ symmetry is the first of $\pi \ra \pis$ character.
Although its associated \%$T_1$ value is rather large, there is substantial difference between CC3 and CCSDT, making our TBE of 4.58 eV falling in the ``unsafe'' category, though it is obviously more accurate than previous CASPT2 values that are too close from the experimental 0-0 energies to be trustworthy.
For the next strongly-allowed transition ($^1B_{1u}$), one also notices small yet non-negligible differences between CC3 and CCSDT.
Our TBE of 5.62 eV is slightly larger than Thiel's one (5.47 eV obtained using CC3), \cite{Silva-Junior_2010}
Our TBE of 5.62 eV is slightly larger than Thiel's one (5.47 eV obtained using CC3) \cite{Silva-Junior_2010},
It is objectively hard to determined which one is the most accurate, and if the difference in the ground-state geometries (CC3 here \emph{vs} MP2 in Thiel's work) also plays a significant role in this discrepancy.
The situation is similar for the $B_{3u}$ transition, although in that case the present TBE of 5.79 eV is close to Thiel's CC3 value of 5.71 eV.
Note that Thiel selected a CASPT2 value of 5.55 eV as TBE due to the rather small \%$T_1$ value for that state.
@ -191,7 +191,7 @@ For the two $\pi \ra \pis$ excitations, exactly the same trends are found, but n
\begin{landscape}
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in $p$-benzoquinone.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -240,29 +240,31 @@ There is a good to excellent agreement between the CC3 and CCSDT values for both
Thus, one can likely trust the obtained TBEs as these transitions are unproblematic.
Experimentally, a $t$Bu substituted cyclopentadienone
shows a weakly-allowed band peaking at 3.22 eV in vapour \cite{Garbisch_1966}, which is likely the $^1B_2$ state.
For the lowest triplet state of cyclopentadienone, the experimental triplet energy was (indirectly) estimated experimentally to be $1.50 \pm 0.02$ eV, \cite{Khuseynov_2014} but this value corresponds to the triplet lowest-energy geometry, so that direct comparisons with our data is unreasonable.
For the lowest triplet state of cyclopentadienone, the experimental triplet energy was (indirectly) estimated experimentally to be $1.50 \pm 0.02$ eV \cite{Khuseynov_2014}, but this value corresponds to the triplet lowest-energy geometry, so that direct comparisons with our data is unreasonable.
This 2014 work also contains CCSD(T) estimates of the adiabatic electron affinities for the two lowest triplet states.
In the singlet manifold of that molecule, one next finds one dark transition of purely double
$n,\pi \ra \pis,\pis$ character, for which CC theory is not well suited, as clearly illustrated by the huge difference between CC3 and CCSDT results. Using a minimal active space ($\pi$ space and lone pairs),
we obtained with NEVPT2 a value for this transition energy of 5.02 eV, ca. 0.7 eV below the CCSDT estimate and ca. 0.5 eV the earlier CASPT2 values. This 5.02 eV estimate although not chemically
accurate is likely the best available today. The fourth singlet state is an interesting yet challenging $pi,\pi \ra \pis,\pis$ showing a \%$T_1$ of 49.9 \%. For that state, the NEVPT2 value is 6.02 eV, which is
likely again the most realistic value available. The fifth transition shows a butadiene-$A_g$-like character, that is a all-symmetric $\pi \ra \pis$ transition with a significant double excitation character of ca. 25\%.
For this transition, we based our TBE on the CCSDT estimate, but it should be too large by ca. 0.10 eV. The experimental spectrum of the related compounds shows a strong peak at 5.93 eV \cite{Garbisch_1966},
likely corresponding to the overlap between these two $^1A_1$ transitions. In cyclopentadienone, the third triplet is an unproblematic $\pi \ra \pis$ transition, for which there is a remarkable consistency
between our CC estimates, again significantly above the previous CASPT2 data \cite{Serrano-Andres_2002}. Finally, the highest triplet considered has originally a highly dominant multi-excitation character
and our best estimate of 4.91 eV was obtained with NEVPT2, the CC values being much too large.
In the singlet manifold of this molecule, one next finds one dark transition of purely double $(n,\pi) \ra (\pis,\pis)$ character, for which CC theory is not well suited, as clearly illustrated by the huge difference between CC3 and CCSDT.
Using a minimal active space ($\pi$ space and lone pairs), we obtained with NEVPT2 value of 5.02 eV, i.e., 0.7 eV below the CCSDT estimate and roughly half an eV higher than the earlier CASPT2 values. This 5.02 eV estimate although not chemically accurate is likely the best available today.
The fourth singlet state is an interesting yet challenging $(pi,\pi) \ra (\pis,\pis)$ showing a \%$T_1$ of 49.9 \%.
For this particular state, the NEVPT2 value is 6.02 eV, which is likely again the most realistic value available.
The fifth transition shows a butadiene-$A_g$-like character, that is a totally-symmetric $\pi \ra \pis$ transition with a significant double excitation character around 25\%.
For this transition, we based our TBE on the CCSDT estimate, but it might be too large by roughly 0.10 eV.
The experimental spectrum of the related compounds shows a strong peak at 5.93 eV \cite{Garbisch_1966}, likely corresponding to the overlap between these two $^1A_1$ transitions.
In cyclopentadienone, the third triplet is an unproblematic $\pi \ra \pis$ transition, for which there is a remarkable consistency between our CC estimates, again significantly above the previous CASPT2 data \cite{Serrano-Andres_2002}.
Finally, the highest triplet considered has originally a highly dominant multi-excitation character and our best estimate of 4.91 eV was obtained with NEVPT2, the CC values being too large.
In the singlet manyfold of cyclopentadienethione on finds again a $^1B_1 (n,\pi \ra \pis,\pis)$ purely double transition and a 50/50 single/double $^1A_1 (\pi,\pi \ra \pis,\pis)$ transition. The methodological trends
are similar to those noted for the oxygen-derivatives, and the TBE listed in the QUEST database are obtained with NEVPT2: 3.16 eV and 5.43 eV. For the latter, it should be noted that there are several transitions
of mixed character close in energy, so that definitive attribution is challenging if not unsettled. In contrast to cyclopentadienone, the $^1A_1 (\pi \ra \pis)$ transition shows a very large \%$T_1$ and small
differences between CC3 and CCSDT, so that we have been able to define a ``safe'' TBE of 4.96 eV. For the records our NEVPT2 estimate for that state is consistent, 4.90 eV. For the triplets of cyclopentadienethione,
the trends are totally similar to those of the oxygen structure. For the fourth triplet of double character, our TBE is 3.13 eV, a value again obtained with NEVPT2.
For the singlet manifold of cyclopentadienethione, on finds again a $^1B_1 (n,\pi \ra \pis,\pis)$ purely double transition and a 50/50 single/double $^1A_1 (\pi,\pi) \ra (\pis,\pis)$ transition.
The methodological trends are similar to those noted for the oxygen-derivatives, and the TBE listed in the QUEST database are obtained with NEVPT2: 3.16 eV and 5.43 eV.
For the latter, it should be noted that there are several transitions of mixed character close in energy, so that definitive attribution is challenging.
In contrast to cyclopentadienone, the $^1A_1 (\pi \ra \pis)$ transition shows a very large \%$T_1$ value and small differences between CC3 and CCSDT, so that we have been able to define a ``safe'' TBE of 4.96 eV.
For the records, our NEVPT2 estimate for that state is consistent, 4.90 eV.
For the triplets of cyclopentadienethione, the trends are totally similar to those of the oxygen structure.
For the fourth triplet of double character, our TBE is 3.13 eV, a value again obtained with NEVPT2.
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in cyclopentadienone and cyclopentadienethione.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -304,20 +306,20 @@ Cyclopentadienethione&$^1A_2 (n \ra \pis)$ & &87.2 &1.76 &1.74 &1.71 &1.72 &1.
\subsection{Diazirine}
This small compact molecule is the diazo equivalent of cycloproprene, and it has been introduced in one of our last works \cite{Sarkar_2021}. It is a rather elusive compound experimentally
so that the most complete studies of transitions energies are theoretical and have been performed at the MCQDPT2 \cite{Han_1999} and EOM-CCSD \cite{Fedorov_2009} levels. For the
eight considered transitions (Table \ref{tab:diazirine}), the \%$T_1$ are larger than 90\%\ and the differences between the CC3, CCSDT, and CCSDTQ values are at most 0.02 eV.
In addition, but possibly for the $^1B_2$ transition, the basis set effects are rather limited. In short, one can very confident that the TBE/{\AVTZ} given in the database are chemically
accurate at least for the selected geometry and basis set. If we compare to the previously published values, one notes that the MCQDPT2 results are likely slightly off target, whereas
there is a quite good agreement with the EOM-CCSD values of Krylov and coworkers \cite{Fedorov_2009}. This latter work used a very diffuse basis set, so that one cannot be definitive
that the remaining differences are purely related to the level of theory. Finally, we are aware of only one experimental data: the 0-0 energy of the lowest singlet state: 3.87 eV \cite{Robertson_1966},
a value slightly smaller than the computed vertical transition energy, as it should.
This compact molecule is the diazo equivalent of cycloproprene, and it has been introduced in our latest work \cite{Sarkar_2021}.
It is a rather elusive compound experimentally so that the most complete study of its transitions energies are theoretical and have been performed at the MCQDPT2 \cite{Han_1999} and EOM-CCSD \cite{Fedorov_2009} levels.
For the eight considered transitions (Table \ref{tab:diazirine}), the \%$T_1$ values are larger than 90\%\ and the differences between the CC3, CCSDT, and CCSDTQ values are at most 0.02 eV.
In addition, except possibly for the $^1B_2$ transition, the basis set effects are rather limited.
In short, one can be very confident that the TBEs/{\AVTZ} given in the database are chemically accurate at least for the selected geometry and basis set.
If we compare to the previously published values, one notes that the MCQDPT2 results are likely slightly off target, whereas there is quite a good agreement with the EOM-CCSD values of Krylov and coworkers \cite{Fedorov_2009}.
This latter work used a very diffuse basis set, so that one cannot be definitive that the remaining differences are purely related to the level of theory.
Finally, we are aware of only one experimental data: the 0-0 energy of the lowest singlet state at 3.87 eV \cite{Robertson_1966}, a value slightly smaller than the computed vertical transition energy, as it should.
%%% TABLE %%%
\begin{landscape}
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in diazirine with CC3, CCSDT, and CCSDTQ.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -348,34 +350,42 @@ Diazirine &$^1B_1 (n \ra \pis)$ & 0.002 &92.5 &4.20 &4.16 &4.11 &4.11 &4.18 &4
\subsection{Hexatriene and octatetraene}
We have treated ethylene and butadiene, the two shortest members of the polyenes in QUEST\#1 and QUEST\#3, respectively \cite{Loos_2018a,Loos_2020b}. The evolution of excited state energies in longer polyenic chains
is obviously of interest and this is why we considered hexatriene and octatetraene here. It should be noted that the transition energies are rather sensitive to the bond length alternation in polyenes, so that we trust that our choice
of CC3 geometries is an asset as compared to previous estimates. Our results are collected in Table \ref{tab:hexaocta}. Let us start by the famous singlet $B_u$ and $A_g$ valence states. The transition to $B_u$ is bright and
has a strong single-excitation character, so that one expects CC to be an adequate methodology, and one indeed finds small differences between CC3 and CCSDT values (ca. 0.03 eV). Our TBE/{\AVTZ} are 5.37 and 4.78 eV for
the hexatriene and octatetraene, respectively. These values are compared to selected previous theoretical estimates in Table \ref{tab:hexaocta} and one clearly notice quite a large spread. Experimentally, the 0-0 transition has
been measured to be 4.95 eV by electron impact \cite{Flicker_1977} and 4.93 eV by optical spectroscopy \cite{Leopold_1984} for hexatriene, and 4.41 eV with the latter technique for the longer oligomer \cite{Leopold_1984b}, values
logically smaller than our vertical estimates. However, one clearly notices a decrease of ca. 0.53 eV when increasing the chain length, as compared to 0.59 eV with our TBE/{\AVTZ} highlighting the consistency of quantum and
measured trends. The $A_g$ transitions are well-known to be much more challenging: the states are dark in one-photon absorption, and it have a very significant multi-excitation character (\%$T_1$ of ca. 65\%\ for both compounds).
On a positive note, the basis set effects are very limited for the $A_g$ state, {\Pop} being apparently sufficient. In contrast, as expected for such transition, there is a significant drop of the theoretical estimate in going from CC3 to CCSDT.
From the analysis performed for double excitations in Ref.~\cite{Loos_2019}, it is unclear if NEVPT2 or CASPT2 would in fact outperform CCSDT for such ``mixed-character'' state, so that we cannot define a trustworthy TBE on this basis.
However, based on our experience for butadiene \cite{Loos_2020b}, one can widely estimate the transition energies to be in the ca. 5.55--5.60 eV (hexatriene) and 4.80--4.85 eV (octratetraene) domains. Interestingly the FCI value of
Chien and coworkers with a small basis set for hexatriene, 5.59 eV, is compatible with such an estimate. Experimentally, for hexatriene, multiphoton experiments allowed to estimate the $A_g$ state to be slightly above the $B_u$ transition
\cite{Fujii_1985}, an outcome that theory reproduces.
Ethylene and butadiene, the two shortest members of the polyenes, have been treated in QUEST\#1 and QUEST\#3, respectively \cite{Loos_2018a,Loos_2020b}.
The evolution of excited state energies in longer polyenic chains is obviously of interest and this is why we considered hexatriene and octatetraene here.
It should be noted that the transition energies are rather sensitive to the bond length alternation in polyenes, so that we trust that our choice of CC3 geometries is an asset as compared to previous estimates.
Our results are collected in Table \ref{tab:hexaocta}.
Let us start by the famous singlet $B_u$ and $A_g$ valence states.
The transition to $B_u$ is bright and has a strong single-excitation character, so that one expects CC to be an adequate methodology, and one indeed finds small differences between CC3 and CCSDT values (ca.~0.03 eV).
Our TBE/{\AVTZ} are 5.37 and 4.78 eV for hexatriene and octatetraene, respectively.
These values are compared to selected previous theoretical estimates in Table \ref{tab:hexaocta} and one clearly notices quite a large spread.
Experimentally, the 0-0 transition has been measured to be 4.95 eV by electron impact \cite{Flicker_1977} and 4.93 eV by optical spectroscopy \cite{Leopold_1984} for hexatriene, and 4.41 eV with the latter technique for the longer oligomer \cite{Leopold_1984b}, values logically smaller than our vertical estimates.
However, one clearly notices a decrease of 0.53 eV when increasing the chain length, as compared to 0.59 eV with our TBE/{\AVTZ} highlighting the consistency of quantum and
measured trends.
The $A_g$ transitions are known to be much more challenging: the states are dark in one-photon absorption, and it has a very significant multi-excitation character (\%$T_1$ of roughly 65\%\ for both compounds).
On a positive note, the basis set effects are very limited for the $A_g$ state, {\Pop} being apparently sufficient.
In contrast, as expected for such transition, there is a significant drop of the theoretical estimate in going from CC3 to CCSDT.
From the analysis performed for double excitations in Ref.~\cite{Loos_2019}, it is unclear if NEVPT2 or CASPT2 would in fact outperform CCSDT for such ``mixed-character'' state, so that we cannot define a trustworthy TBE on this basis.
However, based on our experience for butadiene \cite{Loos_2020b}, one can widely estimate the transition energies to be in the range 5.55--5.60 eV for hexatriene and in the range 4.80--4.85 eV for octratetraene.
Interestingly the FCI value of Chien \textit{et al.}~with a small basis set for hexatriene (5.59 eV) is compatible with such an estimate.
Experimentally, for hexatriene, multiphoton experiments estimate the $A_g$ state to be slightly above the $B_u$ transition \cite{Fujii_1985}, an outcome that theory reproduces.
For the two Rydberg transitions of hexatriene, the differences between CC3 and CCSDT estimates are very small, the \%$T_1$ large, so that CC values can likely be trusted. However, the basis set effects are rather large, and {\AVTZ} might
be insufficient to reach convergence. Our values are reasonably similar to those obtained with CASPT2 almost thirty years ago \cite{Serrano-Andres_1993}. The experimental 0-0 energies are 5.68 eV and 6.06 eV \cite{Sabljic_1985}, but
the assignments of the Rydberg transitions is a matter of discussion \cite{Serrano-Andres_1993}, so that we prefer again not to use measured data as reference.
For the two Rydberg transitions of hexatriene, the differences between CC3 and CCSDT estimates are very small, \%$T_1$ values are large, so that CC estimates can likely be trusted.
However, the basis set effects are rather large, and {\AVTZ} might be insufficient to reach basis set convergence.
Our values are reasonably similar to those obtained with CASPT2 almost thirty years ago \cite{Serrano-Andres_1993}.
The experimental 0-0 energies are 5.68 eV and 6.06 eV \cite{Sabljic_1985}, but the assignments of Rydberg transitions is a matter of discussion \cite{Serrano-Andres_1993}, so that we prefer again not to use measured data as reference.
For the triplet ESs, given their very large \%$T_1$, it seems logically to trust the CC estimates. We note that, to our knowledge, this work is the first to report true CC3/{\AVTZ} values for these two systems. Indeed, the previous CC3 estimates
provided by Thiel \cite{Silva-Junior_2010,Silva-Junior_2010b} were obtained by correcting CC3/TZVP values thanks to CC2 calculations using a larger basis set. These authors nevertheless provided very close estimates to ours: 2.71, 4.33, 2.32,
and 3.69 eV (going down the list of triplet ESs in Table \ref{tab:hexaocta}, i.e., data within 0.04 eV of the current values. Comparatively, the previous MRMP and CASPT2 results \cite{Serrano-Andres_1993,Nakayama_1998,Silva-Junior_2010},
are therefore slightly too low for the triplet transition energies. For hexatriene (octatetraene), electron impact studies return maxima at 2.61 (2.10) and 4.11 (3.55) eV for the two lowest transition \cite{Flicker_1977} (\cite{Allan_1984}), in reasonable
agreement with the values listed in the QUEST database.
For the triplet excited states, given their very large \%$T_1$ values, we logically trust the CC estimates.
We note that, to the best of our knowledge, this work is the first to report true CC3/{\AVTZ} values for these two systems.
Indeed, the previous CC3 estimates provided by Thiel \cite{Silva-Junior_2010,Silva-Junior_2010b} were obtained by correcting CC3/TZVP values thanks to CC2 calculations using a larger basis set.
These authors nevertheless provided very close estimates to ours: 2.71, 4.33, 2.32, and 3.69 eV (going down the list of triplet excited states in Table \ref{tab:hexaocta}).
These data are within 0.04 eV of the current values.
Comparatively, the previous MRMP and CASPT2 results \cite{Serrano-Andres_1993,Nakayama_1998,Silva-Junior_2010} are therefore slightly too low for the triplet transition energies.
For hexatriene (octatetraene), electron impact studies return maxima at 2.61 (2.10) and 4.11 (3.55) eV for the two lowest transitions \cite{Flicker_1977} (\cite{Allan_1984}), in reasonable agreement with the values listed in the QUEST database.
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in hexatriene and octatetraene.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -409,21 +419,24 @@ Octatetraene &$^1 B_u (\pi \ra \pis)$ &n.d. &91.5 &4.95 &4.78 &4.75 &4.98 &
\subsection{Maleimide}
Maleimide was quite surprisingly much less studied theoretically than other similar compounds. We are only aware of the 2003 CASPT2 nalaysis of Climent and coworkers \cite{Climent_2003},
a refined 2020 joint theory/experiment study using CASPT2, ADC(3), and EOM-CCSD \cite{Lehr_2020}, as well as two quite recent investifgations focussed on the geometries of specific
states \cite{Tuna_2016,Budzak_2017} rather than on the transition energies. Our results are listed in Table \ref{tab:malei}. For all considered singlet (triplet) transitions, we obtained a
\%$T_1$ larger than 85\%\ (95\%), and one indeed notices very consistent estimates with CC3 and CCSDT, the largest difference being 0.03 eV. All transitions can therefore be considered
as rather ``safe''. Comparing the 2003 and 2020 CASPT2 values (see Table \ref{tab:malei}), one notices large differences between the two, and our present estimates are (much) closer from the
most recent values. Nevertheless, even the 2020 CASPT2 results seem rather too low as compared to the values provided here. The experimental data are limited. Interestingly, Climent and coworkers
attributed the experimental 0-0 absorption at 3.33 eV (see footnotes in Table \ref{tab:malei}) to the second transition, but given our data, we trust this is more likely the $B_1$ transition, an assignment
consistent with the fact that this band shows some non-zero intensity experimentally. The $A_2$ transition seems indeed dsignificantly too high with CCSDT to be attributed to the 3.33 eV measurement.
For this assignment, we therefore agree with the analysis of Ref.~\citenum{Lehr_2020}. Globally, our CCSDT values are typically bracketed by the EOM-CCSD and CASPT2 values of that recent study,
which we consider a good hint of accuracy.
Maleimide was quite surprisingly much less studied theoretically than other similar compounds.
We are only aware of the 2003 CASPT2 analysis of Climent and coworkers \cite{Climent_2003}, a refined 2020 joint theory/experiment study using CASPT2, ADC(3), and EOM-CCSD \cite{Lehr_2020}, as well as two quite recent investigations focussed on the geometries of specific states \cite{Tuna_2016,Budzak_2017} rather than on the transition energies.
Our results are listed in Table \ref{tab:malei}.
For all considered singlet (triplet) transitions, we obtained \%$T_1$ values larger than 85\%\ (95\%), and one indeed notices very consistent estimates with CC3 and CCSDT, the largest difference being 0.03 eV.
All transitions can therefore be considered as rather ``safe''.
Comparing the 2003 and 2020 CASPT2 values (see Table \ref{tab:malei}), one notices large differences between the two, and our present estimates are (much) closer from the
most recent values.
Nevertheless, even the 2020 CASPT2 results seem rather too low as compared to the values provided here.
The experimental data are limited.
Interestingly, Climent and coworkers attributed the experimental 0-0 absorption at 3.33 eV (see footnotes in Table \ref{tab:malei}) to the second transition, but given our data, we believe that it is more likely the $B_1$ transition, an assignment consistent with the fact that this band shows non-zero experimental intensities.
The $A_2$ transition seems indeed significantly too high with CCSDT to be attributed to the 3.33 eV measurement.
For this assignment, we therefore agree with the analysis of Ref.~\citenum{Lehr_2020}.
Globally, our CCSDT values are typically bracketed by the EOM-CCSD and CASPT2 values of this recent study, which we consider a good hint of accuracy.
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in maleimide.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -459,47 +472,52 @@ observed in vapour for $N$-Me-maleimide, and the 5.53 eV value is a 0-0 energy i
\subsection{Naphthalene}
Naphthalene due to its high-symmetry and significance for organic electronics is also a popular benchmark molecule \cite{Rubio_1994,Packer_1996,Schreiber_2008,Silva-Junior_2010,Silva-Junior_2010b,Sauri_2011,Harbach_2014,Fliegl_2014},
although some studies are focussed on the lowest states ``only'' \cite{Prlj_2016,Bettanin_2017}. Our results are listed in Table \ref{tab:naphth}. We believe that the convincing work of Fliegl and Sundholm remains the most
complete analysis to date \cite{Fliegl_2014}.
Naphthalene, due to its high-symmetry and significance for organic electronics, is also a popular benchmark molecule \cite{Rubio_1994,Packer_1996,Schreiber_2008,Silva-Junior_2010,Silva-Junior_2010b,Sauri_2011,Harbach_2014,Fliegl_2014}, although some studies are focussed on the lowest-energy states ``only'' \cite{Prlj_2016,Bettanin_2017}.
Our results are listed in Table \ref{tab:naphth}.
We believe that the convincing work of Fliegl and Sundholm remains the most complete analysis to date \cite{Fliegl_2014}.
For the singlet transitions, we could obtain CCSDT, albeit only with the Pople's basis set. However, these likely remain quite significant, as none of the considered excited state, even those with Rydberg
character seems to be very strongly affected by the basis set. Indeed, the MAD between CC3/{\Pop} and CC3/{\AVTZ} is 0.16 eV, the maximal discrepancy being 0.23 eV. When one compares
the CCSDT and CC3 values, we note that there are only two transitions (the lowest $^1A_g$ and the second $^1B_{3u}$) for which changes exceeding 0.03 eV can be found between the two basis.
It is also striking that for the higher-lying $A_g$, both CC3 and CCSDT transition energies are the same despite the \%$T_1$ being 72\%\ only. It therefore appears that naphthalene provides a
series of well-behaved transitions for which CC theory is well suited. For the valence transition, the TBEs we obtained are very close from the previous CC3 values of Thiel. Let us now
discuss the valence transitions in more details. For the lowest $^1B_{3u}$ and $^1B_{2u}$ transition, our TBEs{\AVTZ} are 4.27 eV and 4.90 eV. On the theoretical side, this can be compared to Thiel's
4.25 eV and 4.82 eV values \cite{Silva-Junior_2010}, or Fliegl and Sundholm 4.16 and 4.80 eV estimates (obtained with larger basis sets) \cite{Fliegl_2014}. On the experimental side, the
we are aware of vapour phase energy loss values of 4.0 and 4.45 eV \cite{Huebner_1972} and (optical) 0-0 energies of 3.97 and 4.45 eV \cite{George_1968}, as well as a 3.93 and 4.35 eV 0-0 energy
measurement in cyclohexane \cite{Mikami_1975}. The next transition, is the lowest Rydberg state ($^1A_u (\pi \ra 3s)$), and our TBE/{\AVTZ} is 5.65 eV, which fits very well the old CASPT2 estimates
of Rubio and coworkers (5.54 eV \cite{Rubio_1994}), the more recent CC-derived value (5.56 eV \cite{Fliegl_2014}) and the only experimental value we are aware off (5.60 eV by energy loss
\cite{Huebner_1972}). It is also likely that the use of even large basis set than {\AVTZ} would decrease a bit our estimate. Next come the valence $^1B_{1g}$ and $A_g$ with our TBE/{\AVTZ} of 5.84
and 5.89 eV (we recall we rated the second one as ``unsafe''). On the theoretical side previous calculations led 5.75 and 5.90 eV with exCC3 \cite{Silva-Junior_2010}, 5.87 and 6.00 eV with RASPT2
\cite{Sauri_2011}, 5.64 and 5.77 eV with exCC3 \cite{Fliegl_2014}. On the experimental side, the measured 0-0 energies are 5.22 and 5.52 eV \cite{Dick_1981} and 5.28 and 5.50 eV both
in solution \cite{Mikami_1975}. All these estimates are rather consistent with one another. Next come two Rydberg $\pi \ra 3p$ transition, for which our TBE of 6.07 and 6.09 eV are slightly
larger than the CASPT2 values of Ref.~\cite{Rubio_1994} and the CC2/extrapolated estimates of 5.94 and 5.96 eV. To our knowledge, no experimental measurement exists for these two transitions.
We estimate the next valence $^1B_{3u}$ excited state is to be close to 6.19 eV with the {\AVTZ} basis set. This value is consistent with recent estimates of 6.11 \cite{Silva-Junior_2010},
6.20 \cite{Sauri_2011} and 6.06 eV \cite{Fliegl_2014}. For this bright state, there are several available experimental values for the energies: 5.55 eV (crystal) \cite{Bree_1962}, 5.62/5.63 eV (solution)
\cite{Klevens_1949,Bree_1962} and 5.89 (vapour) eV \cite{George_1968}, the latter value being also found by energy loss \cite{Huebner_1972}. The last of the four Rydberg states considered
herein, $^1B_{1u} (\pi \ra 3s)$, is located by us at 6.33 eV, a value 0.3 eV above the CASPT2 value \cite{Rubio_1994} but consistent with a recent CC2 estimate (6.26 eV \cite{Fliegl_2014}).
For the second $^1B_{2u} (\pi \ra \pis)$ transition, our vertical best estimates is 6.42 eV, fitting Thiel's (6.36 eV) \cite{Silva-Junior_2010} and Sundholm's (6.30 eV) results \cite{Fliegl_2014}. The
experimental values are and ca. 6.0 eV (energy loss \cite{Huebner_1972}) and 6.14 eV (optical spectroscopy) \cite{George_1968}. Finally, for the higher-lying $^1B_{1g} (\pi \ra \pis)$ and $^1A_g (\pi \ra \pis)$
states our TBE/{\AVTZ} values of 6.48 and 6.87 eV appear too high as compared to the estimates of Ref.~\cite{Fliegl_2014} (6.19 and 6.40 eV), which is likely due to the strong basis set effects
for these two excited states. The $^1A_g$ transition was estimated at 6.05 eV \cite{Dick_1981} by two-photon spectroscopy.
For the singlet transitions, we could obtain CCSDT, albeit only with Pople's basis set.
However, these remain quite significant as none of the considered excited state (even those with Rydberg character) seems to be strongly affected by the basis set.
Indeed, the mean absolute deviation between CC3/{\Pop} and CC3/{\AVTZ} is 0.16 eV, the maximal discrepancy being 0.23 eV.
When one compares the CCSDT and CC3 values, we note that there are only two transitions (the lowest $^1A_g$ and the second $^1B_{3u}$) for which changes exceeding 0.03 eV can be found between the two bases.
It is also striking that for the higher-lying $A_g$ state, both CC3 and CCSDT transition energies are the same despite \%$T_1$ being 72\%\ only.
It therefore appears that naphthalene provides a series of well-behaved transitions for which CC theory is well suited.
For the valence transition, the TBEs that we obtained are very close from the previous Thiel's CC3 values.
Let us now discuss the valence transitions in more details.
For the lowest $^1B_{3u}$ and $^1B_{2u}$ transitions, our TBEs{\AVTZ} are 4.27 eV and 4.90 eV.
On the theoretical side, these can be compared to Thiel's 4.25 eV and 4.82 eV values \cite{Silva-Junior_2010}, or Fliegl and Sundholm 4.16 and 4.80 eV estimates (obtained with larger basis sets) \cite{Fliegl_2014}.
On the experimental side, we are aware of vapour phase energy loss values of 4.0 and 4.45 eV \cite{Huebner_1972} and (optical) 0-0 energies of 3.97 and 4.45 eV \cite{George_1968}, as well as a 3.93 and 4.35 eV 0-0 energy measurement in cyclohexane \cite{Mikami_1975}.
The next transition is the lowest Rydberg state [$^1A_u (\pi \ra 3s)$], and our TBE/{\AVTZ} is 5.65 eV, which fits very well the old CASPT2 estimates of Rubio and coworkers (5.54 eV \cite{Rubio_1994}), the more recent CC-derived value (5.56 eV \cite{Fliegl_2014}) and the only experimental value we are aware off (5.60 eV by energy loss \cite{Huebner_1972}).
It is also likely that the use of even large basis set than {\AVTZ} would decrease a bit our estimate.
Next come the valence $^1B_{1g}$ and $A_g$ states for which our TBE/{\AVTZ} are 5.84 and 5.89 eV (we recall that we labeled the second one as ``unsafe'').
On the theoretical side, previous calculations led 5.75 and 5.90 eV with exCC3 \cite{Silva-Junior_2010}, 5.87 and 6.00 eV with RASPT2 \cite{Sauri_2011}, 5.64 and 5.77 eV with exCC3 \cite{Fliegl_2014}.
On the experimental side, the measured 0-0 energies are 5.22 and 5.52 eV \cite{Dick_1981} and 5.28 and 5.50 eV, both in solution \cite{Mikami_1975}.
All these estimates are rather consistent with one another.
Next come two $\pi \ra 3p$ Rydberg transitions, for which our TBE of 6.07 and 6.09 eV are slightly larger than the CASPT2 values of Ref.~\cite{Rubio_1994} and the extrapolated CC2 estimates of 5.94 and 5.96 eV.
To our knowledge, no experimental measurement exists for these two transitions.
We estimate the next valence $^1B_{3u}$ excited state to be close to 6.19 eV with the {\AVTZ} basis set.
This value is consistent with recent estimates of 6.11 \cite{Silva-Junior_2010},
6.20 \cite{Sauri_2011} and 6.06 eV \cite{Fliegl_2014}.
For this bright state, there are several available experimental values: 5.55 eV (crystal) \cite{Bree_1962}, 5.62/5.63 eV (solution)
\cite{Klevens_1949,Bree_1962} and 5.89 (vapour) eV \cite{George_1968}, the latter value being also found by energy loss \cite{Huebner_1972}.
The last of the four Rydberg states considered herein, $^1B_{1u} (\pi \ra 3s)$, is located by us at 6.33 eV, a value 0.3 eV above the CASPT2 value \cite{Rubio_1994} but consistent with a recent CC2 estimate (6.26 eV \cite{Fliegl_2014}).
For the second $^1B_{2u} (\pi \ra \pis)$ transition, our vertical best estimate is 6.42 eV, fitting Thiel's (6.36 eV) \cite{Silva-Junior_2010} and Sundholm's (6.30 eV) results \cite{Fliegl_2014}.
The experimental values are around 6.0 eV (energy loss \cite{Huebner_1972}) and 6.14 eV (optical spectroscopy) \cite{George_1968}.
Finally, for the higher-lying $^1B_{1g} (\pi \ra \pis)$ and $^1A_g (\pi \ra \pis)$ states, our TBE/{\AVTZ} values of 6.48 and 6.87 eV appear too high as compared to the estimates of Ref.~\cite{Fliegl_2014} (6.19 and 6.40 eV), which is likely due to the strong basis set effects for these two excited states.
The $^1A_g$ transition was estimated at 6.05 eV \cite{Dick_1981} by two-photon spectroscopy.
For the triplet transitions, we have investigated almost the same valence transitions as in Thiel's set \cite{Schreiber_2008,Silva-Junior_2010,Silva-Junior_2010b}. We have to note that for all
states, CC3 returns very large \%$T_1$, and that the differences between {\AVDZ} and {\AVTZ} estimates is at most 0.04 eV. This clearly hints that the present estimates are trustworthy,
but as we have been unable to perform CCSDT calculations, we nevertheless rate all of them as ``unsafe'' in the QUEST database, which is a conservative choice. The present values are
also very typically similar to those obtained by basis set extrapolation by Thiel \cite{Silva-Junior_2010}, but for the highest triplet transition considered here in which a significant 0.2 eV difference
is found. For most transitions, one also find a good consistency with earlier RASPT2 calculations \cite{Sauri_2011}. Experimentally, the available data are typically T-T absorption, and
this includes values of +2.25 eV, +2.93 eV for the two $^3A_g$ states \cite{Hunziker_1972} and 3.12 eV for the intermediate $^3B_{1g}$ transition \cite{Hunziker_1969}, that we can compare to
our +2.32, +3.22, and +3.00 eV values, respectively.
For the triplet transitions, we have investigated almost the same valence transitions as in Thiel's set \cite{Schreiber_2008,Silva-Junior_2010,Silva-Junior_2010b}.
Let us note that for all states, CC3 returns very large \%$T_1$, and that the differences between {\AVDZ} and {\AVTZ} estimates is at most 0.04 eV.
This clearly hints that the present estimates are trustworthy, but as we have been unable to perform CCSDT calculations, we nevertheless rate all of them as ``unsafe'' in the QUEST database, which is a conservative choice.
The present values are also very similar to those obtained by basis set extrapolation thanks to the work of Thiel \cite{Silva-Junior_2010}, except for the highest triplet transition considered here in which a significant difference of 0.2 eV is found.
For most transitions, one also find a good consistency with earlier RASPT2 calculations \cite{Sauri_2011}.
Experimentally, the available data are typically T-T absorption, and this includes values of +2.25 eV, +2.93 eV for the two $^3A_g$ states \cite{Hunziker_1972} and 3.12 eV for the intermediate $^3B_{1g}$ transition \cite{Hunziker_1969}, that we can compare to our +2.32, +3.22, and +3.00 eV values, respectively.
%
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in naphthalene.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -542,15 +560,17 @@ Naphthalene &$^1B_{3u} (\pi \ra \pis)$ &0.000 &85.8 &4.36 &4.33 &4.30 &4.33 &
\subsection{Nitroxyl (HNO)}
In QUEST\#2\ \cite{Loos_2019}, we treated only one ES, of pure double-excitation character, of that compact molecule. We have used a large panel of high-level methods here, considering five ESs, see
Table \ref{tab:hno}. For all transitions, but the Rydberg ones, we could nicely converge FCI/{\AVTZ} values that can be used a solid references. For the Rydberg transition, that is more sensitive to the
basis set, the CCSDTQ/{\AVDZ} and FCI{\AVDZ} also perfectly match. A previous CASPT2 work \cite{Luna_1995}, reported transition energies to the lowest singlet and triplet of 0.67 eV and 1.53 eV, two
In QUEST\#2\ \cite{Loos_2019}, we treated only one excited state, of pure double-excitation character, of this compact molecule.
We have used a large panel of high-level methods here, considering five excited states (see Table \ref{tab:hno}).
For all transitions (except the Rydberg one), we could nicely converge FCI/{\AVTZ} values that can be used a solid references.
For the Rydberg transition, which is naturally more sensitive to the basis set effect, the CCSDTQ/{\AVDZ} and FCI{\AVDZ} also perfectly match.
A previous CASPT2 work \cite{Luna_1995}, reported transition energies to the lowest singlet and triplet of 0.67 eV and 1.53 eV, two
values that now appear rather too low, a usual trend for CASPT2 when no IPEA shift is applied.
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in nitroxyl with CC3, CCSDT, CCSDTQ and FCI.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -577,19 +597,19 @@ HNO & $^1 A'' (n \ra \pis)$ &0.000 &93.2 &1.78 & 1.75 &1.77 &1.74 &1.77 &1
\subsection{Streptocyanines}
In addition to the smallest streptocyanine treated earlier \cite{Loos_2018a}, we have investigated here the properties of the two next members of the cationic series in which 3 and 5 carbon atoms
are bracketed by NH$_2$ groups (Table \ref{tab:cyanine}). These systems are of specific interest because it is well-known that there are challenging for TD-DFT \cite{LeGuennic_2015}. We report in Table \ref{tab:cyanine}
the transition energies to the lowest ESs of both spin symmetries, of (valence) $\pi \ra \pis$ character. For the shortest of the two compounds treated here, we have been able to converge a
CIPSI calculation with the Pople basis set, and it clearly gives confidence that both CC3 and CCSDT values are accurate, the former method being actually even closer to the FCI extrapolation for that
specific molecules. The most refined previous investigation of these compounds likely remains the one of Send, Valsson and Filippi \cite{Send_2011b}. These authors reported
for the singlet states of these two cyanines: i) exCC3 values of 4.84 and 3.65 eV; ii) DMC values of 5.03(2) and 3.83(2) eV; and iii) CASPT2 estimates of 4.69 and 3.53 eV. The present
estimates match more accurately the previous CC estimates, the DMC and CASPT2 transition energies appearing sightly too large (too low). Once more, given the results in Table \ref{tab:cyanine},
we trust out TBEs are the most accurate to date, at least for the used geometries.
In addition to the smallest streptocyanine treated in our earliest work \cite{Loos_2018a}, we have investigated here the properties of the two next members of the cationic series which contain 3 and 5 carbon atoms respectively bracketed by NH$_2$ groups (Table \ref{tab:cyanine}).
These systems are of specific interest because it is well-known that there are challenging for TD-DFT \cite{LeGuennic_2015}.
We report in Table \ref{tab:cyanine} the transition energies to the lowest excited states of both spin symmetries with (valence) $\pi \ra \pis$ character.
For the smallest of the two compounds treated here, we have been able to converge a CIPSI calculation with the 6-31+G(d) basis set, and it clearly gives us confidence that both CC3 and CCSDT values are accurate, the former method being actually even closer to the FCI extrapolation for that specific molecules.
The detailed investigation of these compounds likely remains the one of Send, Valsson, and Filippi \cite{Send_2011b}.
These authors reported for the singlet states of these two cyanines: i) exCC3 values of 4.84 and 3.65 eV; ii) DMC values of 5.03(2) and 3.83(2) eV; and iii) CASPT2 estimates of 4.69 and 3.53 eV.
The present estimates better match the previous CC estimates, the DMC (CASPT2) transition energies appearing sightly too large (too low).
Once more, given the results in Table \ref{tab:cyanine}, we believe that our TBEs are the most accurate to date, at least for the considered geometries.
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in two cyanines with CC3, CCSDT and FCI.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}
@ -615,15 +635,15 @@ Streptocyanine-5 & $^1 B_2 (\pi \ra \pis)$&1.182 &85.8 &3.63 &3.66 &3.66 & &3.
\subsection{Thioacrolein}
This heavier analogue to acrolein was not much studied theoretically before, but for calculations of its 0-0 energies \cite{Loos_2018} and rather old TD-DFT calculations \cite{Fabian_2001}.
We report in Table \ref{tab:thioacrolein} the transition energies to the lowest ESs of both spin symmetries, of clear valence $n \ra \pis$ character. As can be see, there is for both ESs, a
remarkable insensibility to the basis set size, and also very similar CC3 and CCSDT estimates. The experimental 0-0 energies are 1.88 eV (singlet) and 1.74 eV (triplet) \cite{Judge_1984},
both slightly below our vertical estimates as it should.
This heavier analog of acrolein was not much studied theoretically before, except for calculations of its 0-0 energies \cite{Loos_2018} and rather old TD-DFT calculations \cite{Fabian_2001}.
We report in Table \ref{tab:thioacrolein} the transition energies to the lowest excited states of both spin symmetries, of clear valence $n \ra \pis$ character.
As one can see, there is for both excited states, a remarkable insensibility to the basis set size, and also very similar CC3 and CCSDT estimates.
The experimental 0-0 energies are 1.88 eV (singlet) and 1.74 eV (triplet) \cite{Judge_1984}, both slightly below our vertical estimates as it should.
%%% TABLE %%%
\begin{table}[htp]
\centering
\footnotesize
\scriptsize
\caption{Transition energies (in eV) determined in thioacrolein with CC3 and CCSDT.
For all transitions, we provide the single-excitation character \%$T_1$ obtained at LR-CC3/{\AVTZ} level.
For the dipole-allowed transitions, we provide the corresponding values of the oscillator strength at the same level of theory.}