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\author{Pierre-Fran{\c c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Denis Jacquemin}
\email{Denis.Jacquemin@univ-nantes.fr}
\affiliation[CEISAM, Nantes]{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
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\title{Is ADC(3) as Accurate as CC3 for Valence and Rydberg Transition Energies?}
\date{\today}
\begin{tocentry}
\vspace{1cm}
\includegraphics[width=\textwidth]{TOC}
\end{tocentry}
\begin{document}
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%%%%%%%%%%%%%%%%
%%% ABSTRACT %%%
%%%%%%%%%%%%%%%%
\begin{abstract}
The search for new models rapidly delivering accurate excited state energies and properties is one of the most active research lines of theoretical chemistry.
Along with these developments, the performances of known methods are constantly reassessed thanks to new benchmark values. In this Letter,
we show that the third-order algebraic diagrammatic construction, ADC(3), does not yield transition energies of the same quality as the third-order coupled cluster method, CC3.
This is demonstrated by extensive comparisons with several hundreds high-quality vertical transition energies obtained with FCI, CCSDTQ, and CCSDT. Direct
comparisons with experimental 0-0 energies of small- and medium-size molecules support the same conclusion, which holds for both valence and Rydberg transitions.
In regards of these results, we introduce a composite approach, ADC(2.5), which consists in averaging the ADC(2) and ADC(3) excitation energies.
Although ADC(2.5) does not match the CC3 accuracy, it significantly improves the ADC(3) results, especially for vertical energies.
\end{abstract}
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\noindent
%%%%%%%%%%%%%%%%%%%%
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
Electronic excited states (ES) play an important role in many technological applications (photovoltaics, photocatalysis, light-emitting diodes, \ldots), but their characterization from
purely experimental data remains often tedious. This has stimulated the developments of various density- and wavefunction-based methods allowing to model accurately ES. Amongst
all these wavefunction approaches, the algebraic diagrammatic construction (ADC), which relies on perturbation theory to access excitation energies and properties, has now become one of
the most popular. \cite{Dre15} The ADC scheme, originally developed by Schirmer and Trofimov, \cite{Sch82,Sch91,Bar95b,Tro97,Tro97b,Sch04d,Sch18} has several advantages over the well-known coupled cluster (CC)
family of methods, e.g., hermiticity and higher compactness for odd expansion orders. These assets have greatly contributed to the ever growing applications of ADC. In particular, its second-order
variant, ADC(2), generally provides valence transition energies as accurate as the one obtained with the second-order CC model, CC2, \cite{Chr95,Hat00} for a smaller computational cost [yet similar $\order*{N^5}$ scaling].
\cite{Win13,Har14,Jac15b}
One of the originality of ADC($n$) lies in its alternative representation, known as intermediate-state representation, of the polarization propagator which poles {provides} the vertical excitation energies. \cite{Sch82}
These intermediate states are generated by applying a set of creation and annihilation operators to the $n$th-order M{\o}ller-Plesset (MP$n$) ground-state wave function, and are then orthogonalized block-wise according
to their excitation class. \cite{Sch91} This explains why ADC($n$) is usually presented as ``MP$n$ for excited states'' in the literature.
One can show that the intermediate states and genuine ES are related by a unitary transformation $\mathbf{X}$, which satisfies the Hermitian eigenvalue problem $\mathbf{M} \mathbf{X} = \mathbf{\Omega} \mathbf{X}$
(with $\mathbf{X}^\dag \mathbf{X} = \mathbf{1}$), where $\mathbf{M}$ is the so-called ADC matrix and $\mathbf{\Omega}$ is a diagonal matrix gathering the corresponding excitation energies.
We refer the interested reader to Ref.~\citenum{Dre15} for a non-technical discussion of the general form of the ADC($n$) matrices.
H\"attig pointed out several interesting theoretical connections between ADC(2), CC2, and an iterative variant of the double correction to configuration interaction singles [CIS(D$_\infty$)]. \cite{Hat05c}
In particular, he showed that ADC(2) is a symmetrized version of CIS(D$_\infty$), and that the only modification required to obtain CIS(D$_\infty$) excitation energies from CC2 is to replace the ground-state CC2 amplitudes by those from MP2.
This idea has been exploited by Dreuw's group to develop the so-called CCD-ADC(2) method where the ADC(2) amplitudes are replaced by those obtained from a coupled cluster doubles (CCD) calculation. \cite{Hod19a,Hod19b}
In addition to improve excitation energies, because CCD-ADC(2) does not rely on perturbation theory anymore, it has been shown to be more robust for molecular dissociation energy curves. \cite{Hod19a}
One of the disadvantages of CC2 compared to ADC(2) is that, due to its non-Hermitian nature, CC2 does not provide a physically correct description of conical intersections between states of the same symmetry, a difficulty absent in ADC(2).
Similarities between the third-order variants, ADC(3) and CC3, \cite{Chr95b} are likely to exist but, to the best of our knowledge, these potential formal connections have never investigated in the literature.
Nonetheless, it is worth mentioning that CC3, which scales as $\order*{N^7}$, treats the ground state at fourth order of perturbation theory and the 2h-2p block at second order, whereas ADC(3) describes the ground state and 2h-2p block at third and first order of perturbation theory, respectively. \cite{Dre15}
This difference becomes particularly apparent in the calculation of double excitations, for which ADC(3) typically yields inaccurate values. \cite{Loo18a}
However, ADC(3), with its $\order*{N^6}$ computational scaling, has the indisputable advantage of being computationally lighter than CC3, and has a more compact configuration space.
In 2014, Harbach \emph{et al.} \cite{Har14} reported an efficient implementation of ADC(3) and benchmarked its accuracy for transition
energies using the theoretical best estimates (TBE) of the famous Thiel set \cite{Sch08} as reference. They concluded that, {using the benchmark data available at that time, it was impossible to
determine whether ADC(3) or CC3 was the most more accurate.} As ADC(3) enjoys a lower formal computational scaling [$\order*{N^6}$] than CC3 [$\order*{N^7}$], and is generally regarded as the logical path for improvement over ADC(2), this
finding contributed to enhance the popularity of ADC(3) in the electronic structure community. ADC(3) was subsequently employed to perform theory \textit{vs} experiment comparisons, \cite{Hol15,Boh16,Kni16,Hol17b,Tik19,Sue19,Avi19}
and to define benchmark values for assessing lower-order methods. \cite{Pla15,Prl16b,Mew16,Aza17b}
%%%%%%%%%%%%%%%%%%%%%%%%%
%%% VERTICAL ENERGIES %%%
%%%%%%%%%%%%%%%%%%%%%%%%%
Given, on the one hand, that ADC(3) was advocated as a benchmark method and, on the other hand, the recent availability of high-accuracy reference energies for a large panel of ES, \cite{Loo18a,Loo19c,Loo20a}
we believe that the time has come to perform a new performance assessment of this method. To this end, we have first considered our most recent set of TBE/\emph{aug}-cc-pVTZ obtained for vertical
transition energies in organic compounds encompassing from one to six non-hydrogen atoms. \cite{Loo18a,Loo20a} These TBE have been computed at very high levels of theory, \ie, mostly FCI
(full configuration interaction) for molecules with up to three non-hydrogen atoms, \cite{Loo18a} CCSDTQ (coupled-cluster with single double triple and quadruple excitations)
for four non-hydrogen atom derivatives, \cite{Loo20a} and CCSDT for compounds containing $5$ or $6$ non-hydrogen atoms. \cite{Loo20a}
Note that, for the smallest compounds where the following comparison is actually possible, the mean absolute errors (MAE) obtained with CCSDTQ and CCSDT compared to FCI are trifling ($0.01$ eV and $0.03$ eV, respectively). \cite{Loo18a}
Table \ref{Table-1} provides a statistical analysis of the performances of the second- and third-order ADC and CC methods, using these TBE as reference. Figure \ref{Fig-1} gives histograms
of the errors for both singlet and triplet states. The full list of data can be found in the Supporting Information. We consider here a set of $328$ ES, that has been divided into three relatively equivalent
subsets of $1$--$3$ non-hydrogen atoms ($106$ ES), $4$ non-hydrogen atoms ($89$ ES) and $5$--$6$ non-hydrogen atoms ($134$ ES). From these data, it is quite clear that CC3 delivers astonishingly accurate transition energies with MAE below or equal to $0.03$ eV for each subset, and no deviation exceeding $\pm 0.20$ eV. This is {in line} with several previous benchmarks. \cite{Tro02,Hat05c,Loo18a,Loo18b,Loo19a,Sue19,Loo20a} Again, consistently with
previous analyses and theoretical considerations (see above), the ADC(2) and CC2 performances are very similar and these second-order methods deliver a global MAE smaller than $0.2$ eV, together with negligible MSE for all subsets.
This confirms that ADC(2) is indeed a very interesting computational tool thanks to its attractive accuracy/cost ratio. Nevertheless, in par with the above-described conclusions, we found that the performance of ADC(3) is rather
average with a significant underestimation (mean signed error, MSE of $-0.11$ eV for the full set) and a MAE around $0.20$ eV for each subset. Overall, ADC(3) underestimates transition energies and provides an
average deviation of the same order of magnitude as ADC(2) and CC2. Strikingly, the MAE of ADC(3) is basically one order of magnitude larger than the MAE of CC3.
%%% TABLE I %%%
\begin{table}[htp]
\footnotesize
\caption{
Mean signed error (MSE), mean absolute error (MAE), maximal positive error [Max($+$)], and maximal negative error [Max($-$)] with respect to the highly-accurate TBE/\emph{aug}-cc-pVTZ of Refs.~\citenum{Loo18a} and \citenum{Loo20a}
(see text for details) for various sets of vertical transition energies. All values are in eV. The raw data, which can be found in Table S1 of the Supporting Information, have been obtained with the \emph{aug}-cc-pVTZ basis set and within the frozen-core approximation.}
\label{Table-1}
\begin{tabular}{cldddd}
\hline
Set & Method & \ctab{MSE} & \ctab{MAE} & \ctab{Max($-$)} & \ctab{Max($+$)} \\
\hline
All & ADC(2) &0.00 &0.16 &-0.76 &0.64\\
& ADC(2.5) &-0.05 &0.08 &-0.33 &0.24\\
& ADC(3) &-0.11 &0.21 &-0.79 &0.55\\
& CC2 &0.02 &0.17 &-0.71 &0.63\\
& CC3 &0.00 &0.02 &-0.09 &0.19\\
\hline
$1$--$3$ non-H & ADC(2) &-0.01 &0.21 &-0.76 &0.57 \\
atoms\cite{Loo18a} & ADC(2.5) &-0.08 &0.10 &-0.33 &0.24 \\
& ADC(3) &-0.15 &0.23 &-0.79 &0.39 \\
& CC2 & 0.03 &0.21 &-0.71 &0.63 \\
& CC3 &-0.01 &0.03 &-0.09 &0.19 \\
$4$ non-H & ADC(2) &-0.03 &0.18 &-0.73 &0.64\\
atoms\cite{Loo20a} & ADC(2.5) &-0.07 &0.08 &-0.29 &0.15\\
& ADC(3) &-0.10 &0.24 &-0.76 &0.49\\
& CC2 & 0.03 &0.20 &-0.68 &0.59\\
& CC3 & 0.00 &0.02 &-0.05 &0.17\\
$5$--$6$ non-H & ADC(2) &0.03 &0.11 &-0.48 &0.45\\
atoms\cite{Loo20a} & ADC(2.5) &-0.02 &0.06 &-0.26 &0.24\\
& ADC(3) &-0.08 &0.18 &-0.46 &0.55\\
& CC2 &0.01 &0.12 &-0.58 &0.31\\
& CC3 &0.00 &0.01 &-0.03 &0.04\\
\hline
Valence & ADC(2) &0.07 &0.13& -0.76& 0.54\\
& ADC(2.5) &-0.05 &0.07& -0.24 & 0.24\\
& ADC(3) &-0.16 &0.23& -0.46& 0.50\\
& CC2 &0.12 &0.16& -0.71& 0.50\\
& CC3 &0.00 &0.02& -0.09& 0.19\\
Rydberg & ADC(2) &-0.14 &0.22& -0.38& 0.64\\
& ADC(2.5) &-0.07 &0.09& -0.33& 0.24\\
& ADC(3) &-0.01 &0.18& -0.79& 0.55\\
& CC2 &-0.17 &0.21& -0.41& 0.63\\
& CC3 &-0.01 &0.02& -0.09& 0.17\\
Singlet & ADC(2) &-0.03 &0.17& -0.76& 0.64\\
& ADC(2.5) &-0.05 &0.09& -0.33& 0.24\\
& ADC(3) &-0.07 &0.21& -0.79& 0.55\\
& CC2 &-0.02 &0.18& -0.71& 0.59\\
& CC3 &0.00 &0.02& -0.09& 0.19\\
Triplet & ADC(2) &0.05 &0.15& -0.70& 0.57\\
& ADC(2.5) &-0.06 &0.07& -0.23& 0.19\\
& ADC(3) &-0.17 &0.22& -0.56& 0.38\\
& CC2 &0.09 &0.16& -0.66& 0.63\\
& CC3 &0.00 &0.01& -0.09& 0.04\\
$n \ra \pis$ & ADC(2) &-0.04 &0.09& -0.38& 0.23\\
& ADC(2.5) &-0.02 &0.06& -0.23& 0.24\\
& ADC(3) &0.00 &0.14& -0.32& 0.40\\
& CC2 &0.02 &0.08& -0.25& 0.21\\
& CC3 &0.00 &0.01& -0.05& 0.04\\
$\pi \ra \pis$ & ADC(2) &0.14 &0.17& -0.31& 0.64\\
& ADC(2.5) &-0.07 &0.08& -0.33& 0.19\\
& ADC(3) &-0.27 &0.29& -0.79& 0.55\\
& CC2 &0.19 &0.21& -0.41& 0.63\\
& CC3 &0.01 &0.02& -0.09& 0.17\\
\hline
\end{tabular}
\end{table}
%%% %%% %%% %%%
As can be seen in Table \ref{Table-1}, the ADC(3) MAE obtained for the singlet ($0.21$ eV) and triplet ($0.23$ eV) ES, as well as for valence ($0.23$ eV) and Rydberg ($0.18$ eV) ES are all rather similar. Interestingly,
ADC(2) exhibits the reverse valence/Rydberg trend with a smaller error for valence transitions ($0.13$ eV) and a larger one for Rydberg ES ($0.22$ eV). It is only for the $n \ra \pis$ transitions ($0.14$ eV) that the ADC(3) MAE
becomes significantly lower than the usual $0.2$ eV error bar. This success is mitigated by the fact that it is also for the $n \ra \pis$ transitions that ADC(2) and CC2 are the most accurate,
as both yield MAE smaller than $0.10$ eV for this ES family. On a more optimistic note, one notices that the ADC(3) errors are smallest for the largest compounds gathered in Table \ref{Table-1}.
This hints that the error might well decrease with system size and become more acceptable for ``real-life'' derivatives. However, a similar trend is observed for both ADC(2) and CC2.
It is therefore difficult to perform a trustworthy extrapolation to larger systems.
Finally, as we have found previously, \cite{Loo18a} ADC(3) seems to overcorrect ADC(2). Therefore, in the spirit of Grimme's and
Hobza's MP2.5 approach tailored to provide accurate interaction energies, \cite{Pit09} we propose here its excited-state equivalent, ADC(2.5), that simply corresponds to the average between the
ADC(2) and ADC(3) transition energies. Indeed, test numerical experiments have shown that such 50/50 ratio is close to optimal for the present set of transitions.
This ADC(2.5) {protocol} delivers a MSE of $-0.05$ eV and a MAE of $0.08$ eV considering the entire set of transitions. It is therefore significantly more accurate than ADC(2) or ADC(3) (taken
separately) for practically the same cost as ADC(3). This is well illustrated in Figure S1 of the Supporting Information. This observation might indicate that a renormalized version of ADC(3) could be an interesting alternative to improve its
overall accuracy, as commonly done for one-electron Green's function methods. \cite{Ced75,Sch83}
%%% FIGURE 1 %%%
\begin{figure}
\includegraphics[width=\linewidth,viewport=2cm 14cm 19cm 27.5cm,clip]{Fig-1.pdf}%DJ to T2: NE PAS CHANGER
\caption{Histograms of the errors (in eV) obtained with ADC(2), ADC(3), CC2, and CC3 taking the TBE/\emph{aug}-cc-pVTZ values of Refs.~\citenum{Loo18a} and \citenum{Loo20a} as reference.
``Count'' refers to the number of transitions in each group.
The full list of data can be found in the Supporting Information.
Note the difference of scaling in the vertical axes.}
\label{Fig-1}
\end{figure}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%
%%% 0-0 ENERGIES %%%
%%%%%%%%%%%%%%%%%%%%
Notwithstanding the high accuracy of the vertical excitation energies presented above, CCSDT and CCSDTQ are not error-free. In addition, the previous analysis is limited to compact compounds with a maximum of $6$
non-hydrogen atoms. Therefore, it is worth investigating the correlation between experiment and theoretical observables. Meaningful theory-experiment comparisons for ES are always challenging but
the simplest and safest strategy is very likely to be comparing 0-0 energies, an approach that has been used many times before, \eg, see our recent review on the topic. \cite{Loo19b} Following this strategy, we then consider
here the (slightly extended) set of compounds defined in Ref.~\citenum{Loo19a}: it encompasses gas-phase measurements for 71 singlet and 30 triplet low-lying transitions. Note that the typical uncertainty of such
experimental gas-phase measurements is of the order of $10^{-4}$ eV (or 1 cm$^{-1}$) only. We select here (EOM-)CCSD/\emph{def2}-TZVPP geometries and (TD-)B3LYP/6-31+G(d) vibrational corrections, as it is
known that the errors in the 0-0 energies are mostly driven by the inaccuracy in the adiabatic energies, rather than the approximate nature of the structures and/or vibrations, \cite{Fur02,Sen11b,Win13,Loo19a}
{e.g., for a given method applied for adiabatic energies, similar statistical errors are obtained when selecting CC2, CCSD, or CC3 geometries.} \cite{Loo19a}
Our calculations are again performed with the \emph{aug}-cc-pVTZ basis set, and within the frozen-core approximation. The full list of raw data are given in the Supporting Information. Statistical data can be found in Table \ref{Table-2}
and Figure \ref{Fig-2}.
First, considering all 101 cases, we notice that the CC3 adiabatic energies produce chemically accurate 0-0 energies in {59}\%\ of the cases, with errors almost systematically smaller than $0.15$ eV.
None of the other approaches can match such a feat. In particular, both ADC(2) and ADC(3) deliver MAE above $0.15$ eV and a chemical accuracy rate smaller than $20\%$. As in the set of vertical transitions
discussed above, ADC(2.5) outperforms ADC(2) and ADC(3), and yields rather small deviations of the same order of magnitude than CC2 (MAE of 0.10 eV).
The fact that CC2 provides more consistent 0-0 energies than ADC(2) while their performances were found similar for vertical
energies might be related to the relatively poorer description of potential energy surfaces with the latter approach. \cite{Bud17}%other ref ??
Turning our attention to the impact of spin symmetry, we note that, although CC3 remains very accurate, we observe a slight decline of its accuracy for triplet ES, a conclusion fitting with our recent study. \cite{Loo19a}
It is also quite clear that ADC(3) has the edge over ADC(2) for triplet ES, whereas the opposite trend is observed for the singlets. Surprisingly, opposite conclusions were drawn for vertical transitions (see above). Despite its tendency to overerestimate (underestimate) singlet (triplet) transition energies (see Figure \ref{Fig-2}),
CC2 is found to be globally more robust than ADC(2) and ADC(3) for both ES families. Probably
more enlightening is the comparison between the results obtained on small (71 molecules with 1--5 non-hydrogen atoms) and medium (30 molecules with $6$--$10$ non-hydrogen atoms) compounds (see Table \ref{Table-2}), the latter set being mostly composed
of (substituted) six-membered rings. One sees a clear improvement of the ADC(3) performance going from the smaller to the larger molecules, with a MAE of $0.12$ eV and a chemical accuracy rate of $43\%$ for
the latter group. These values are definitively promising. Indeed, although such a MAE value remains {three} times larger than its CC3 analogue, this hints that ADC(3) might become significantly more accurate for larger compounds.
Finally, we wish to recall that these conclusions are made using (EOM-)CCSD geometries and (TD-)DFT harmonic vibrational corrections for all methods. Thus, the overall error is not exclusively
(though probably predominantly) related to the method selected to compute adiabatic energies. It would be definitely interesting to have access to ground- and excited-state ADC(3) geometries in order to investigate if whether or nor it yields an improvement of the ADC(3) performance. \cite{Reh19}
%%% TABLE 2 %%%
\begin{table}[htp]
\footnotesize
\caption{Mean signed error (MSE) and mean absolute error (MAE), as well as percentage of chemical accuracy (\CheA, absolute error below 0.043 eV) and acceptable error (\AccE, absolute error below 0.150 eV) with respect
to experimental 0-0 energies for the (71) singlet and (30) triplet sets of 0-0 energies from Ref.~\citenum{Loo19a}. All values are in eV and have been obtained with the \emph{aug}-cc-pVTZ basis set and within the frozen-core approximation
using (EOM-)CCSD/\emph{def2}-TZVPP geometries and (TD-)B3LYP/6-31+G* vibrational corrections. The full list of data can be found in the Supporting Information.
}
\label{Table-2}
\begin{tabular}{clddcc}
\hline
Set & Method & \ctab{MSE} & \ctab{MAE} & \ctab{\CheA} & \ctab{\AccE} \\
\hline
All & ADC(2) & -0.09 &0.16 &18 &52\\%OK
& ADC(2.5) & -0.08 &0.10 &24 &78\\%OK
& ADC(3) & -0.07 &0.18 &19 &50\\%OK
& CC2 & 0.00 &0.10 &31 &75\\%OK
& CC3 &-0.03 &0.04 &59 &98 \\%PAS OK
\hline
$1$--$5$ non-H & ADC(2) & -0.10 &0.16 &15 &55\\%OK
atoms & ADC(2.5) & -0.11 &0.11 &24 &72\\%OK
& ADC(3) & -0.13 &0.21 &8 &41\\%OK
& CC2 & 0.01 &0.09 &31 &82\\%OK
& CC3 &-0.03 &0.05 &62 &97 \\%OK
$6$--$10$ non-H & ADC(2) & -0.07 &0.17 &23 &47\\%OK
atoms & ADC(2.5) & -0.01 &0.06 &23 &97\\%OK
& ADC(3) & 0.05 &0.12 &43 &70\\%OK
& CC2 & -0.11 &0.11 &30 &60\\%OK
& CC3 & -0.03 &0.04 &53 &100\\%PAS OK
\hline
Singlet & ADC(2) & -0.05 & 0.13 & 23 & 62 \\%OK
& ADC(2.5) & -0.07 &0.09 & 31 &76 \\%OK
& ADC(3) & -0.09 & 0.19 & 18 & 48 \\%OK
& CC2 & +0.05 & 0.09 & 34 & 80 \\%OK
& CC3 & -0.03 & 0.04 & 63 & 99 \\% PAS OK
Triplet & ADC(2) & -0.20 & 0.23 & 7 & 30 \\%OK
& ADC(2.5)& -0.12 &0.12 & 7 & 87 \\%OK
& ADC(3) & -0.04 & 0.17 & 20 & 53 \\%OK
& CC2 & -0.11 & 0.12 & 23 & 63 \\%OK
& CC3 & -0.05 & 0.05 & 50 & 97 \\%OK
\hline
\end{tabular}
\end{table}
%%% %%% %%% %%%
%%% FIGURE 2 %%%
\begin{figure}[htp]
\includegraphics[width=\linewidth,viewport=2cm 14cm 19cm 27.5cm,clip]{Fig-2.pdf}%DJ to T2: NE PAS CHANGER
\caption{Histograms of the errors (in eV) obtained with ADC(2), ADC(3), CC2, and CC3 taking experimental 0-0 energies as reference.
``Count'' refers to the number of transitions in each group.
The full list of data can be found in the Supporting Information.
Note the difference of scaling in the vertical axes.}
\label{Fig-2}
\end{figure}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%
%%% DISCUSSION %%%
%%%%%%%%%%%%%%%%%%
At this stage, it seems natural to wonder why the conclusions of the 2014 ADC(3) assessment \cite{Har14} based on Thiel's set differ significantly from ours although the nature of the molecules belonging to the
two sets are relatively similar. To understand this discrepancy, let us reexamine the data of Ref.~\citenum{Har14}. In this work, Thiel's original TBE (denoted as TBE-1), \cite{Sch08} mostly based on CASPT2/TZVP
but also incorporating some CC3/TZVP (as well as other values), were used as reference rather than Thiel's most recent set of TBE (denoted as TBE-2), \cite{Sil10c} which are mostly basis set corrected CC3/\emph{aug}-cc-pVTZ
values. In addition, given the knowledge at that time, the authors of Ref.~\citenum{Har14} logically decided that {to consider only the non-CC3 TBE values in their comparison of the relative accuracy of ADC(3) and CC3},
which is a very reasonable point. Considering the subset of TBE-1 based on CASTP2 (\ie, excluding the CC3 values from TBE-1),
Ref.~\citenum{Har14} reports, for the singlet states, a MSE (MAE) of $+0.23$ ($0.24$) eV for CC3. This value has to be compared with a MSE (MAE) of $+0.12$ ($0.24$) eV for ADC(3) where the reference was
taken as the entire TBE-1 set. \cite{Har14} Similarly, for the 19 triplet excitation energies of the TBE-1 set not based on CC3, the MSE is $+0.12$ eV with CC3 and $-0.10$ eV with ADC(3). \cite{Har14}
The direct comparison of ADC(3) and CC3 is also instructive. By considering now CC3 as reference, the MSE (MAE) of ADC(3) reported in Ref.~\citenum{Har14} are $-0.20$ ($0.29$) eV for the singlets and
$-0.22$ ($0.25$) eV for the triplets. \cite{Har14} These numbers are consistent with the findings of the present Letter, and show that ADC(3) significantly underestimates both families of transitions.
We can then conclude that the bias in this earlier ADC(3) assessment \cite{Har14} was likely due to the CASPT2 reference values. Indeed, as clearly demonstrated in a recent series of papers
\cite{Loo18a,Loo18b,Loo19a,Sue19,Loo20a}, CC3 is a very robust method which generally delivers chemically accurate excitation energies, while CASPT2 has a clear tendency of underestimating transition energies. \cite{Loo20a}
In this context, we also wish to point out that an early ADC(3) \emph{vs}
FCI benchmark performed for a series of small molecules (\ce{H2O}, \ce{HF}, \ce{N2}, \ce{Ne}, \ce{CH2}, and \ce{BH}), \cite{Tro02} concluded that \emph{``the mean absolute error,
as calibrated versus the FCI results for 41 singlet and triplet transitions, has been found to be smaller than $0.2$ eV''} (more precisely the MAE is equal to $0.18$ eV for the first four compounds) and
that \emph{``the quality of the results [...] does not match the impressive accuracy of the CC3 computations''}. The present results confirm these two earlier assertions.
%%%%%%%%%%%%%%%%%%%%%%%%
%%% LARGER MOLECULES %%%
%%%%%%%%%%%%%%%%%%%%%%%%
An additional aspect to take into account is that previous comparisons between ADC(3) transition energies and experimental $\lambda_{\mathrm{max}}$ values were often performed in the vertical approximation, \cite{Kni16,Mew18}
which means that the geometry relaxation and vibronic effects were neglected, {which is often done, as such vibronic corrections are computationally expensive. However, as} shown in several works, \cite{Die04b,Goe10a,Sen11b,Jac12d,Fan14b,San16b}
this approximation implies a significant bias, because the blueshift between the experimental 0-0 energy and the $\lambda_{\mathrm{max}}$ value is typically smaller than the blueshift between the computed 0-0 and vertical energies.
As a consequence, applying the vertical approximation favors methods delivering smaller transition energies.
As an example, the $Q$-band of \ce{Mg}-porphyrin was studied at various levels of theory
including ADC(3) in 2018. \cite{Mew18} The first experimental maxima appears at $2.07$ eV, \cite{Edw71} a value smaller than the ADC(2), CCSD, and TD-DFT vertical transitions (which are found in the $2.27$--$2.43$ eV range)
as it should. \cite{Mew18} In contrast, the ADC(3) vertical value of $2.00$ eV, {is the closest from experiment but} presents the incorrect error sign and would likely be significantly too redshifted if
vibronic corrections were accounted for. Indeed, according to Durbeej, \cite{Fan14b} the CC2 difference between vertical and 0-0 energies is $-0.05$ eV in the (free-base) porphyrin. This brings the
ADC(3) estimate to $-0.12$ eV compared to experiment and improves the agreement for the other approaches. Again, both the error sign and its magnitude are quite coherent with the present estimates.
Using the same procedure, ADC(2.5) would give a 0-0 energy of $2.11$ eV, in superb agreement with experiment.
In the same work, \cite{Mew18} an ADC(3) value of $4.65$ eV is reported for the lowest $B_u$ state of \emph{trans}-octatetraene, a bright ES with a dominant single-excitation character.
\cite{Mew18} This value is significantly lower than Thiel's CC3 value of $4.84$ eV, \cite{Sil10c} although the latter was obtained on a MP2 geometry that slightly underestimates the bond length alternation,
whereas the ADC(3) estimate relies on a more accurate CCSD(T) structure. The measured gas-phase 0-0 energy for this transition is $4.41$ eV, \cite{Leo84} and the estimated difference between
vertical and 0-0 energies is $-0.45$ eV at the TD-BHHLYP level, \cite{Die04b} and $-0.36$ eV at the CC2 level, \cite{Fan14b} again hinting that the ADC(3) value is in fact slightly too low by a magnitude of $-0.12$ eV
if one naively applies the CC2 correction (determined on a CC2 geometry). In this case, ADC(2.5) would only slightly reduced the error to $-0.10$ eV.
Of course, these two comparisons remain very qualitative and one would greatly benefit from ADC(3) 0-0 energies which, to the best of our knowledge, are not available to date for these compounds.
%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%%
In this Letter, we have provided what we believe are compelling evidences that the transition energies computed with ADC(3) in organic compounds are significantly less accurate than their CC3 counterparts.
This statement is based on i) extensive comparisons with both vertical energies determined with higher levels of theory (CCSDT, CCSDTQ, and FCI), and ii) accurate 0-0 energies measured in gas phase for small-
and medium-size compounds. This conclusion apparently holds almost irrespectively of the nature of the transition, provided that the ES does not exhibit a dominant double excitation character. Of course, given
that the ADC(3) error for 0-0 energies has a clear tendency to significantly drop for the largest compounds considered here (\ie, substituted six-membered rings), one could rightfully speculate that ADC(3)
would become more accurate for even larger compounds, a claim that we cannot honestly verify at this stage. Besides, ADC(3) might also deliver accurate ES properties (such as geometries, transition and total
dipoles, oscillator strengths, two-photon cross-sections, etc). Indeed, these properties are treated at third order of perturbation theory by both ADC(3) and CC3. We believe that comparisons between CC3 and ADC(3) properties is a particular
point that needs to be further investigated in the future.
%%%%%%%%%%%%%%%%%%%%
%%% COMP DETAILS %%%
%%%%%%%%%%%%%%%%%%%%
\section*{Computational details}
For the set of vertical transition energies, the CC3/\emph{aug}-cc-pVTZ geometries of Refs.~\citenum{Loo18a} and \citenum{Loo20a} have been selected because the TBE have been obtained on the
very same structures. The GS and ES structures used in the 0-0 calculations have been obtained at the (EOM-)CCSD/\emph{def2}-TZVPP level and are provided in the Supporting Information of Ref.~\citenum{Loo19a}.
The zero-point vibrational energies used to compute the 0-0 energies have been (mostly) obtained at the (TD-)B3LYP/6-31+G(d) level and are all listed in the Supporting Information of Ref.~\citenum{Loo19a}. The CC and ADC calculations
have been performed with DALTON \cite{dalton} and Q-CHEM, \cite{Sha15} respectively, with the \emph{aug}-cc-pVTZ basis set. The ADC calculations have been performed within the RI approximation. {Test calculations
have shown that this approximation implies only trifling changes in the transition energies ($\leq 0.01$ eV). } We refer the readers to our previous works \cite{Loo18a,Loo19a} for additional details.
%%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%%
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
PFL thanks the \textit{Centre National de la Recherche Scientifique} for funding.
This research used resources of i) the GENCI-CINES/IDRIS; ii) CCIPL (\emph{Centre de Calcul Intensif des Pays de Loire}); iii) a local Troy cluster and iv) HPC resources from ArronaxPlus
(grant ANR-11-EQPX-0004 funded by the French National Agency for Research).
%%%%%%%%%%%%%%%%%
%%% SUPP INFO %%%
%%%%%%%%%%%%%%%%%
%\begin{suppinfo}
\section*{Supporting Information Available}
Full list of transition energies for vertical and 0-0 energies.
%\end{suppinfo}
%%%%%%%%%%%%%%%%%%%%
%%% BIBLIOGRAPHY %%%
%%%%%%%%%%%%%%%%%%%%
\bibliography{biblio-new}
\end{document}

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\documentclass[journal=jpclcd,manuscript=letter,layout=traditional]{achemso}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amsmath,amssymb,amsfonts,physics,float,lscape,soul,rotating,longtable}
\usepackage[version=4]{mhchem}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\newcommand{\mc}{\multicolumn}
\newcommand{\mr}{\multirow}
\newcommand{\EFCI}{E_\text{FCI}}
\newcommand{\EexCI}{E_\text{exCI}}
\newcommand{\EsCI}{E_\text{sCI}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\PsisCI}{\Psi_\text{sCI}}
\newcommand{\Ndet}{N_\text{det}}
\newcommand{\ex}[4]{{#1}\,$^{#2}$#3$_{#4}$}
% methods
\newcommand{\TDDFT}{TD-DFT}
\newcommand{\CASSCF}{CASSCF}
\newcommand{\CASPT}{CASPT2}
\newcommand{\ADC}[1]{ADC(#1)}
\newcommand{\CC}[1]{CC#1}
\newcommand{\CCSD}{CCSD}
\newcommand{\EOMCCSD}{EOM-CCSD}
\newcommand{\CCSDT}{CCSDT}
\newcommand{\CCSDTQ}{CCSDTQ}
\newcommand{\CI}{CI}
\newcommand{\sCI}{sCI}
\newcommand{\exCI}{exCI}
\newcommand{\FCI}{FCI}
% basis
\newcommand{\AVDZ}{\emph{aug}-cc-pVDZ}
\newcommand{\AVTZ}{\emph{aug}-cc-pVTZ}
\newcommand{\DAVTZ}{d-\emph{aug}-cc-pVTZ}
\newcommand{\AVQZ}{\emph{aug}-cc-pVQZ}
\newcommand{\DAVQZ}{d-\emph{aug}-cc-pVQZ}
\newcommand{\TAVQZ}{t-\emph{aug}-cc-pVQZ}
\newcommand{\AVPZ}{\emph{aug}-cc-pV5Z}
\newcommand{\DAVPZ}{d-\emph{aug}-cc-pV5Z}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\Ryd}{\mathrm{R}}
\newcommand{\Val}{\mathrm{V}}
\newcommand{\Fl}{\mathrm{F}}
\newcommand{\ra}{\rightarrow}
\newcommand{\pis}{\pi^\star}
\newcommand{\si}{\sigma}
\newcommand{\sis}{\sigma^\star}
\newcommand{\SI}{Supporting Information}
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\renewcommand{\thesection}{S\arabic{section}}
\renewcommand\floatpagefraction{.99}
\renewcommand\topfraction{.99}
\renewcommand\bottomfraction{.99}
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\title{Is ADC(3) as Accurate as CC3 for Valence and Rydberg Transition Energies?\\Supporting Information}
\author{Pierre-Fran{\c c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Denis Jacquemin}
\email{Denis.Jacquemin@univ-nantes.fr}
\affiliation[UN, Nantes]{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
\begin{document}
\clearpage
\section{Vertical energies}
Below is a list of vertical transition energies obtained for a set of compounds described elsewhere. \cite{Loo18a,Loo20a}. These transition energies, obtained
on CC3/{\AVTZ} geometries, have been computed with the {\AVTZ} basis set and within the frozen-core approximation. Note that the CC2, ADC(2) and CC3 data are already available in these
previous works, and are reproduced below for the sake of completeness. To identify the ES with the different approaches considered here, we used the usual strategies, \textit{i.e.}, relative energies,
spatial and spin symmetries, symmetries and weights of the underlying molecular orbitals, and oscillator strengths. This allows unambiguous assignments for the vast majority of the
states. There are however some state/method combinations for which strong mixing between ES of the same symmetry makes such assignments difficult. These challenging
cases are nonetheless statistically irrelevant.
\renewcommand*{\arraystretch}{.65}
\LTcapwidth=\textwidth
\begin{footnotesize}
\begin{longtable}{p{3.5cm}p{3.3cm}c|cccc}
\caption{Comparisons between the TBE (see Refs.~\citenum{Loo18a} and \citenum{Loo20a} for details and geometries) and the vertical excitation energies obtained with CC2, CC3, ADC(2), and ADC(3).
All the values have been obtained with the \emph{aug}-cc-pVTZ basis set and within the frozen-core approximation.
We have removed from this training set all excited states with a dominant double excitation character.
F, R and V stand for fluorescence, Rydberg and valence states, respectively.
} \label{Table-SI-1}\\
\hline
Compound & State & TBE & CC2 & CC3 &ADC(2)& ADC(3) \\
\hline
\endfirsthead
\hline
Compound & State & TBE & CC2 & CC3 &ADC(2)& ADC(3) \\
\hline
\endhead
\hline \multicolumn{7}{r}{{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
Acetaldehyde &$^1A'' (\Val;n \ra \pis)$ &4.31 &4.41 &4.31 &4.24 &4.29 \\%
&$^3A'' (\Val;n \ra \pis)$ &3.97 &3.98 &3.95 &3.83 &3.89 \\%
Acetone &$^1A_2 (\Val; n \ra \pis)$ & 4.47 &4.55 &4.48 &4.37 &4.50\\
&$^1B_2 (\Ryd; n \ra 3s)$ & 6.46 &5.91 &6.43 &5.87 &6.91\\
&$^1A_2 (\Ryd; n \ra 3p)$ & 7.47 &6.84 &7.45 &6.81 &7.90\\
&$^1A_1 (\Ryd; n \ra 3p)$ & 7.51 &6.89 &7.48 &6.85 &7.92\\
&$^1B_2 (\Ryd; n \ra 3p)$ & 7.62 &7.02 &7.59 &6.99 &8.01\\
&$^3A_2 (\Val; n \ra \pis)$ & 4.13 &4.16 &4.15 &4.00 &4.12\\
&$^3A_1 (\Val; \pi \ra \pis)$ & 6.25 &6.50 &6.28 &6.37 &6.01\\
Acetylene &$^1\Sigma_u^- (\Val;\pi \ra \pis)$ &7.10 &7.26 &7.09 &7.24 &6.72 \\%
&$^1\Delta_u (\Val;\pi \ra \pis)$ &7.44 &7.59 &7.42 &7.56 &7.06 \\%
&$^3\Sigma_u^+ (\Val;\pi \ra \pis)$ &5.53 &5.76 &5.50 &5.75 &5.24 \\%
&$^3\Delta_u (\Val;\pi \ra \pis)$ &6.40 &6.60 &6.40 &6.57 &6.06 \\%
&$^3\Sigma_u^- (\Val;\pi \ra \pis)$ &7.08 &7.29 &7.07 &7.27 &6.72 \\%
&$^1A_u [\Fl] (\Val;\pi \ra \pis)$ &3.64 &3.94 &3.64 &3.78 &2.85 \\%
&$^1A_2 [\Fl] (\Val;\pi \ra \pis)$ &3.85 &4.11 &3.84 &3.99 &3.08 \\%
Acrolein &$^1A'' (\Val; n \ra \pis)$ & 3.78 &3.85 &3.74 &3.68 &3.76\\
&$^1A' (\Val; \pi \ra \pis)$ & 6.69 &6.80 &6.65 &6.74 &6.51\\
&$^1A' (\Ryd; n \ra 3s)$ & 7.08 &6.40 &7.07 &6.35 &7.57\\
&$^3A'' (\Val; n \ra \pis)$ & 3.51 &3.49 &3.46 &3.33 &3.45\\
&$^3A' (\Val; \pi \ra \pis)$ & 3.94 &4.06 &3.94 &4.05 &3.66\\
&$^3A' (\Val; \pi \ra \pis)$ & 6.18 &6.37 &6.19 &6.31 &5.86\\
Ammonia &$^1A_2 (\Ryd;n \ra 3s)$ &6.59 &6.39 &6.57 &6.40 &6.63 \\%
&$^1E (\Ryd;n \ra 3p)$ &8.16 &7.85 &8.15 &7.87 &8.21 \\%
&$^1A_1 (\Ryd;n \ra 3p)$ &9.33 &9.05 &9.32 &9.05 &9.38 \\%
&$^1A_2 (\Ryd;n \ra 4s)$ &9.96 &9.65 &9.95 &9.67 &10.00 \\%
&$^3A_2 (\Ryd;n \ra 3s)$ &6.31 &6.14 &6.29 &6.16 &6.31 \\%
Carbon monoxyde &$^1\Pi (\Val;n \ra \pis)$ & 8.49 &8.64 &8.49 &8.69 &8.24 \\%
&$^1\Sigma^- (\Val;\pi \ra \pis)$ & 9.92 &10.30 &9.99 &10.03 &9.73 \\%
&$^1\Delta (\Val;\pi \ra \pis)$ &10.06 &10.60 &10.12 &10.30 &9.82 \\%
&$^1\Sigma^+ (\Ryd)$ &10.95 &11.11 &10.94 &11.32 &10.79 \\%
&$^1\Sigma^+ (\Ryd)$ &11.52 &11.63 &11.49 &11.83 &11.33 \\%
&$^1\Pi (\Ryd)$ &11.72 &11.83 &11.69 &12.03 &11.56 \\%
&$^3\Pi (\Val;n \ra \pis)$ & 6.28 &6.42 &6.30 &6.45 &5.97 \\%
&$^3\Sigma^+ (\Val;\pi \ra \pis)$ & 8.45 &8.72 &8.45 &8.54 &8.21 \\%
&$^3\Delta (\Val;\pi \ra \pis)$ & 9.27 &9.56 &9.30 &9.33 &9.03 \\%
&$^3\Sigma^- (\Val;\pi \ra \pis)$ & 9.80 &10.27 &9.82 &10.01 &9.53 \\%
&$^3\Sigma^+ (\Ryd)$ & 10.47 &10.60 &10.45 &10.83 &10.29 \\%
Benzene &$^1B_{2u} (\Val; \pi \ra \pis)$ & 5.06 &5.26 &5.09 &5.27 &5.01\\
&$^1B_{1u} (\Val; \pi \ra \pis)$ & 6.45 &6.48 &6.44 &6.45 &6.24\\
&$^1E_{1g} (\Ryd; \pi \ra 3s)$ & 6.52 &6.47 &6.52 &6.52 &6.38\\
&$^1A_{2u} (\Ryd; \pi \ra 3p)$ & 7.08 &7.00 &7.08 &7.06 &6.92\\
&$^1E_{2u} (\Ryd; \pi \ra 3p)$ & 7.15 &7.06 &7.15 &7.12 &7.00\\
&$^3B_{1u} (\Val; \pi \ra \pis)$ & 4.16 &4.37 &4.18 &4.37 &3.94\\
&$^3E_{1u}(\Val; \pi \ra \pis)$ & 4.85 &5.08 &4.86 &5.07 &4.60\\
&$^3B_{2u} (\Val; \pi \ra \pis)$ & 5.81 &5.89 &5.81 &5.87 &5.51\\
Butadiene &$^1B_u (\Val; \pi \ra \pis)$ & 6.22 &6.16 &6.22 &6.12 &6.02\\
&$^1B_g (\Ryd; \pi \ra 3s)$ & 6.33 &6.26 &6.33 &6.31 &6.12\\
&$^1A_g (\Val; \pi \ra \pis)$ & 6.50 &7.09 &6.67 &7.14 &5.86\\
&$^1A_u (\Ryd; \pi \ra 3p)$ & 6.64 &6.57 &6.64 &6.63 &6.44\\
&$^1A_u (\Ryd; \pi \ra 3p)$ & 6.80 &6.70 &6.80 &6.76 &6.59\\
&$^1B_u (\Ryd; \pi \ra 3p)$ & 7.68 &7.63 &7.68 &7.48 &7.46\\
&$^3B_u (\Val; \pi \ra \pis)$ & 3.36 &3.45 &3.36 &3.46 &3.09\\
&$^3A_g (\Val; \pi \ra \pis)$ & 5.20 &5.30 &5.20 &5.27 &4.94\\
&$^3B_g (\Ryd; \pi \ra 3s)$ & 6.29 &6.21 &6.28 &6.27 &6.06\\
Cyanoacetylene&$^1\Sigma^- (\Val; \pi \ra \pis)$ & 5.80 &6.03 &5.80 &5.99 &5.37\\
&$^1\Delta (\Val; \pi \ra \pis)$ & 6.07 &6.30 &6.08 &6.25 &5.64\\
&$^3\Sigma^+ (\Val; \pi \ra \pis)$ & 4.44 &4.80 &4.45 &4.77 &4.11\\
&$^3\Delta (\Val; \pi \ra \pis)$ & 5.21 &5.50 &5.22 &5.46 &4.80\\
&$^1A'' [\Fl] (\Val; \pi \ra \pis)$ & 3.54 &3.79 &3.54 &3.73 &2.78\\
Cyanoformaldehyde &$^1A'' (\Val; n \ra \pis)$ & 3.81 &3.97 &3.83 &3.83 &3.77\\
&$^1A'' (\Val; \pi \ra \pis)$ & 6.46 &6.74 &6.42 &6.73 &6.07\\
&$^3A'' (\Val; n \ra \pis)$ & 3.44 &3.51 &3.46 &3.37 &3.38\\
&$^3A' (\Val; \pi \ra \pis)$ & 5.01 &5.34 &5.01 &5.27 &4.63\\
Cyanogen & $^1\Sigma_u^- (\Val; \pi \ra \pis)$ & 6.39 &6.72 &6.39 &6.67 &5.88\\
& $^1\Delta_u (\Val; \pi \ra \pis)$ & 6.66 &7.02 &6.66 &6.95 &6.16\\
& $^3\Sigma_u^+ (\Val; \pi \ra \pis)$ & 4.91 &5.35 &4.90 &5.31 &4.49\\
& $^1\Sigma_u^- [\Fl] (\Val; \pi \ra \pis)$ & 5.05 &5.48 &5.06 &5.39 &4.32\\
Cyclopentadiene&$^1B_2 (\Val; \pi \ra \pis)$ & 5.56 &5.52 &5.54 &5.49 &5.35\\
&$^1A_2 (\Ryd; \pi \ra 3s)$ & 5.78 &5.66 &5.77 &5.71 &5.63\\
&$^1B_1 (\Ryd; \pi \ra 3p)$ & 6.41 &6.26 &6.40 &6.31 &6.25\\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.46 &6.30 &6.45 &6.35 &6.30\\
&$^1B_2 (\Ryd; \pi \ra 3p)$ & 6.56 &6.42 &6.56 &6.48 &6.41\\
&$^3B_2 (\Val; \pi \ra \pis)$ & 3.31 &3.42 &3.32 &3.42 &3.05\\
&$^3A_1 (\Val; \pi \ra \pis)$ & 5.11 &5.36 &5.12 &5.23 &4.86\\
&$^3A_2 (\Ryd; \pi \ra 3s)$ & 5.73 &5.62 &5.73 &5.67 &5.57\\
&$^3B_1 (\Ryd; \pi \ra 3p)$ & 6.36 &6.22 &6.36 &6.27 &6.20\\
Cyclopropene &$^1B_1 (\Val;\sigma \ra \pis)$ &6.68 &6.73 &6.68 &6.75 &6.56 \\%
&$^1B_2 (\Val;\pi \ra \pis)$ &6.79 &6.78 &6.73 &6.86 &6.56 \\%
&$^3B_2 (\Val;\pi \ra \pis)$ &4.38 &4.46 &4.34 &4.45 &4.09 \\%
&$^3B_1 (\Val;\sigma \ra \pis)$ &6.45 &6.44 &6.40 &6.45 &6.26 \\%
Cyclopropenone&$^1B_1 (\Val; n \ra \pis)$ & 4.26 &4.01 &4.21 &3.88 &4.66\\
&$^1A_2 (\Val; n \ra \pis)$ & 5.55 &5.65 &5.57 &5.47 &5.61\\
&$^1B_2 (\Ryd; n \ra 3s)$ & 6.34 &5.84 &6.32 &5.79 &6.64\\
&$^1B_2 (\Val; \pi \ra \pis$) & 6.54 &6.46 &6.54 &6.33 &6.83\\
&$^1B_2 (\Ryd; n \ra 3p)$ & 6.98 &6.56 &6.96 &6.43 &7.33\\
&$^1A_1 (\Ryd; n \ra 3p)$ & 7.02 &6.47 &7.00 &6.41 &7.36\\
&$^1A_1 (\Val; \pi \ra \pis)$ & 8.28 &8.28 &8.28 &8.10 &8.17\\
&$^3B_1 (\Val; n \ra \pis)$ & 3.93 &3.73 &3.91 &3.62 &4.28\\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.88 &4.99 &4.89 &4.90 &4.80\\
&$^3A_2 (\Val; n \ra \pis)$ & 5.35 &5.45 &5.37 &5.28 &5.36\\
&$^3A_1 (\Val; \pi \ra \pis)$ & 6.79 &6.42 &6.83 &6.84 &6.63\\
Cyclopropenethione&$^1A_2 (\Val; n \ra \pis)$ & 3.41 &3.53 &3.43 &3.38 &3.46\\
&$^1B_1 (\Val; n \ra \pis)$ & 3.45 &3.50 &3.43 &3.37 &3.82\\
&$^1B_2 (\Val; \pi \ra \pis)$ & 4.60 &4.91 &4.64 &4.72 &4.72\\
&$^1B_2 (\Ryd; n \ra 3s)$ & 5.34 &5.22 &5.34 &5.17 &5.41\\
&$^1A_1 (\Val; \pi \ra \pis)$ & 5.46 &5.59 &5.49 &5.36 &5.36\\
&$^1B_2 (\Ryd; n \ra 3p)$ & 5.92 &5.82 &5.93 &5.77 &6.02\\
&$^3A_2 (\Val; n \ra \pis)$ & 3.28 &3.37 &3.30 &3.23 &3.30\\
&$^3B_1 (\Val; n \ra \pis)$ & 3.32 &3.38 &3.31 &3.26 &3.65\\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.01 &4.24 &4.02 &4.12 &3.96\\
&$^3A_1 (\Val; \pi \ra \pis)$ & 4.01 &4.16 &4.03 &4.04 &3.83\\
Diacetylene &$^1\Sigma_u^- (\Val; \pi \ra \pis)$ & 5.33 &5.51 &5.34 &5.49 &4.95 \\
&$^1\Delta_u (\Val; \pi \ra \pis)$ & 5.61 &5.76 &5.61 &5.72 &5.22 \\
&$^3\Sigma_u^+ (\Val; \pi \ra \pis)$ & 4.10 &4.39 &4.08 &4.37 &3.79 \\
&$^3\Delta_u (\Val; \pi \ra \pis)$ & 4.78 &5.03 &4.80 &5.01 &4.43 \\
Diazomethane &$^1A_2 (\Val;\pi \ra \pis)$ &3.14 &3.37 &3.07 &3.34 &2.74 \\%
&$^1B_1 (\Ryd;\pi \ra 3s)$ &5.54 &5.53 &5.45 &5.63 &5.23 \\%
&$^1A_1 (\Val;\pi \ra \pis)$ &5.90 &6.00 &5.84 &5.97 &5.48 \\%
&$^3A_2 (\Val;\pi \ra \pis)$ &2.79 &3.08 &2.83 &3.01 &2.44 \\%
&$^3A_1 (\Val;\pi \ra \pis)$ &4.05 &4.25 &4.03 &4.20 &3.64 \\%
&$^3B_1 (\Ryd;\pi \ra 3s)$ &5.35 &5.40 &5.31 &5.50 &5.08 \\%
&$^3A_1 (\Ryd;\pi \ra 3p)$ &6.82 &7.04 &6.80 &7.09 &6.36 \\%
&$^1A'' [\Fl] (\Val;\pi \ra \pis)$ &0.71 &0.90 &0.68 &0.81 &0.24 \\%
Dinitrogen &$^1\Pi_g (\Val;n \ra \pis)$ &9.34 &9.44 &9.34 &9.48 &9.16 \\%
&$^1\Sigma_u^- (\Val;\pi \ra \pis)$ &9.88 &10.32 &9.88 &10.26 &9.33 \\%
&$^1\Delta_u (\Val;\pi \ra \pis)$ &10.29 &10.86 &10.29 &10.79 &9.74 \\%
&$^1\Sigma_g^+ (\Ryd)$ &12.98 &12.83 &13.01 &12.99 &13.01 \\%
&$^1\Pi_u (\Ryd)$ &13.03 &13.15 &13.22 &13.32 &12.98 \\%
&$^1\Sigma_u^+ (\Ryd)$ &13.09 &12.89 &13.12 &13.07 &13.09 \\%
&$^1\Pi_u (\Ryd)$ &13.46 &13.96 &13.49 &14.00 &13.40 \\%
&$^3\Sigma_u^+ (\Val;\pi \ra \pis)$ &7.70 &8.19 &7.68 &8.15 &7.25 \\%
&$^3\Pi_g (\Val;n \ra \pis)$ &8.01 &8.19 &8.04 &8.20 &7.77 \\%
&$^3\Delta_u (\Val;\pi \ra \pis)$ &8.87 &9.30 &8.87 &9.25 &8.36 \\%
&$^3\Sigma_u^- (\Val;\pi \ra \pis)$ &9.66 &10.29 &9.68 &10.23 &9.14 \\%
Ethylene &$^1B_{3u} (\Ryd;\pi \ra 3s)$ &7.39 &7.29 &7.35 &7.34 &7.17 \\%
&$^1B_{1u} (\Val;\pi \ra \pis)$ &7.93 &7.92 &7.91 &7.91 &7.69 \\%
&$^1B_{1g} (\Ryd;\pi \ra 3p)$ &8.08 &7.95 &8.03 &7.99 &7.84 \\%
&$^3B_{1u} (\Val;\pi \ra \pis)$ &4.54 &4.59 &4.53 &4.59 &4.28 \\%
&$^3B_{3u} (\Ryd;\pi \ra 3s)$ &7.23 &7.19 &7.24 &7.23 &7.05 \\%
&$^3B_{1g} (\Ryd;\pi \ra 3p)$ &7.98 &7.91 &7.98 &7.95 &7.80 \\%
Formaldehyde &$^1A_2 (\Val; n \ra \pis)$ &3.98 &4.07 &3.97 &3.92 &3.90 \\%
&$^1B_2 (\Ryd;n \ra 3s)$ &7.23 &6.56 &7.18 &6.50 &7.62 \\%
&$^1B_2 (\Ryd;n \ra 3p)$ &8.13 &7.57 &8.07 &7.53 &8.45 \\%
&$^1A_1 (\Ryd;n \ra 3p)$ &8.23 &7.52 &8.18 &7.47 &8.61 \\%
&$^1A_2 (\Ryd;n \ra 3p)$ &8.67 &8.04 &8.64 &7.99 &9.02 \\%
&$^1B_1 (\Val;\sigma \ra \pis)$ &9.22 &9.32 &9.19 &9.17 &9.17 \\%
&$^1A_1 (\Val;\pi \ra \pis)$ &9.43 &9.54 &9.48 &9.46 &9.05 \\%
&$^3A_2 (\Val;n \ra \pis)$ &3.58 &3.59 &3.57 &3.46 &3.48 \\%
&$^3A_1 (\Val;\pi \ra \pis)$ &6.06 &6.30 &6.05 &6.20 &5.71 \\%
&$^3B_2 (\Ryd;n \ra 3s)$ &7.06 &6.44 &7.03 &6.39 &7.44 \\%
&$^3B_2 (\Ryd;n \ra 3p)$ &7.94 &7.45 &7.92 &7.41 &8.23 \\%
&$^3A_1 (\Ryd;n \ra 3p)$ &8.10 &7.44 &8.08 &7.40 &8.46 \\%
&$^3B_1 (\Ryd;n \ra 3d)$ &8.42 &8.52 &8.41 &8.39 &8.32 \\%
&$^1A^" [\Fl] (\Val;n \ra \pis)$ &2.80 &2.97 &2.84 &2.71 &2.77 \\%
Formamide &$^1A'' (\Val;n \ra \pis)$ &5.65 &5.69 &5.66 &5.45 &5.75 \\%
&$^3A'' (\Val;n \ra \pis)$ &5.38 &5.36 &5.38 &5.15 &5.42 \\%
&$^3A' (\Val;\pi \ra \pis)$ &5.81 &5.99 &5.82 &5.88 &5.63 \\%
Furan &$^1A_2 (\Ryd; \pi \ra 3s)$ & 6.09 &6.06 &6.08 &6.12 &5.95\\
&$^1B_2 (\Val; \pi \ra \pis)$ & 6.37 &6.45 &6.34 &6.47 &6.15\\
&$^1A_1 (\Val; \pi \ra \pis)$ & 6.56 &6.77 &6.58 &6.76 &6.48\\
&$^1B_1 (\Ryd; \pi \ra 3p)$ & 6.64 &6.59 &6.63 &6.64 &6.49\\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.81 &6.75 &6.80 &6.82 &6.67\\
&$^1B_2 (\Ryd; \pi \ra 3p)$ & 7.24 &7.25 &7.23 &7.29 &7.09 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.20 &4.43 &4.22 &4.41 &3.91 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 5.46 &5.66 &5.48 &5.59 &5.23 \\
&$^3A_2 (\Ryd; \pi \ra 3s)$ & 6.02 &6.01 &6.02 &6.08 &5.89 \\
&$^3B_1 (\Ryd; \pi \ra 3p)$ & 6.59 &6.55 &6.59 &6.61 &6.45 \\
Glyoxal &$^1A_u (\Val; n \ra \pis)$ & 2.88 &2.91 &2.88 &2.83 &2.83 \\
&$^1B_g (\Val; n \ra \pis)$ & 4.24 &4.44 &4.27 &4.27 &4.23 \\
&$^1B_g (\Val; n \ra \pis)$ & 6.57 &6.51 &6.58 &6.50 &6.80 \\
&$^1B_u (\Ryd; n \ra 3p)$ & 7.71 &7.16 &7.67 &7.18 &7.86 \\
&$^3A_u (\Val; n \ra \pis)$ & 2.49 &2.47 &2.49 &2.39 &2.40 \\
&$^3B_g (\Val; n \ra \pis)$ & 3.89 &3.96 &3.90 &3.82 &3.85 \\
&$^3B_u (\Val; \pi \ra \pis)$ & 5.15 &5.42 &5.17 &5.33 &4.83 \\
&$^3A_g (\Val; \pi \ra \pis)$ & 6.30 &6.56 &6.30 &6.45 &5.93 \\
Hydrogen chloride & $^1\Pi (\mathrm{CT})$ &7.84 &7.96 &7.84 &7.97 &7.79 \\%
Hydrogen sulfide &$^1A_2 (\Ryd;n \ra 4p)$ &6.18 &6.35 &6.19 &6.37 &6.05 \\%
&$^1B_1 (\Ryd;n \ra 4s)$ &6.24 &6.30 &6.24 &6.34 &6.18 \\%
&$^3A_2 (\Ryd;n \ra 4p)$ &5.81 &5.91 &5.82 &5.91 &5.67 \\%
&$^3B_1 (\Ryd;n \ra 4s)$ &5.88 &5.94 &5.88 &5.96 &5.81 \\%
Imidazole &$^1A'' (\Ryd; \pi \ra 3s)$ & 5.71 &5.69 &5.71 &5.75 &5.61 \\
&$^1A' (\Val; \pi \ra \pis)$ & 6.41 &6.51 &6.41 &6.50 &6.31 \\
&$^1A'' (\Val; n \ra \pis)$ & 6.50 &6.47 &6.50 &6.51 &6.39 \\
&$^3A' (\Val; \pi \ra \pis)$ & 4.73 &4.94 &4.75 &4.92 &4.47 \\
&$^3A'' (\Ryd; \pi \ra 3s)$ & 5.66 &5.66 &5.67 &5.72 &5.57 \\
&$^3A' (\Val; \pi \ra \pis)$ & 5.74 &5.94 &5.74 &5.93 &5.49 \\
&$^3A'' (\Val; n \ra \pis)$ & 6.31 &6.36 &6.33 &6.31 &6.26 \\
Isobutene &$^1B_1 (\Ryd; \pi \ra 3s)$ & 6.46 &6.37 &6.45 &6.43 &6.33 \\
&$^1A_1 (\Ryd; \pi \ra 3p)$ & 7.01 &6.95 &7.00 &6.97 &6.82 \\
&$^3A_1 (\Val; (\pi \ra \pis)$ & 4.53 &4.62 &4.53 &4.62 &4.30 \\
Ketene &$^1A_2 (\Val;\pi \ra \pis)$ &3.86 &4.17 &3.88 &4.11 &3.67 \\%
&$^1B_1 (\Ryd;n \ra 3s)$ &6.01 &5.94 &5.96 &6.03 &5.87 \\%
&$^1A_2 (\Ryd;\pi \ra 3p)$ &7.18 &7.09 &7.16 &7.18 &7.07 \\%
&$^3A_2 (\Val;n \ra \pis)$ &3.77 &3.98 &3.78 &3.92 &3.56 \\%
&$^3A_1 (\Val;\pi \ra \pis)$ &5.61 &5.72 &5.61 &5.67 &5.39 \\%
&$^3B_1 (\Ryd;n \ra 3s)$ &5.79 &5.77 &5.76 &5.85 &5.67 \\%
&$^3A_2 (\Ryd;\pi \ra 3p)$ &7.12 &7.06 &7.12 &7.15 &7.03 \\%
&$^1A^" [\Fl] (\Val;\pi \ra \pis)$ &1.00 &1.26 &1.00 &1.19 &0.67 \\%
Methanimine &$^1A^" (\Val; n \ra \pis)$ &5.23 &5.32 &5.20 &5.29 &5.05 \\%
&$^3A^" (\Val; n \ra \pis)$ &4.65 &4.65 &4.61 &4.61 &4.44 \\%
Methylenecyclopropene& $^1B_2 (\Val; \pi \ra \pis)$ & 4.28 &4.51 &4.31 &4.46 &4.18 \\
&$^1B_1 (\Ryd; \pi \ra 3s)$ & 5.44 &5.35 &5.44 &5.38 &5.26 \\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 5.96 &5.85 &5.95 &5.87 &5.78 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 3.49 &3.64 &3.50 &3.61 &3.30\\
&$^3A_1 (\Val; \pi \ra \pis)$ & 4.74 &4.81 &4.74 &4.80 &4.51\\
Nitrosomethane&$^1A'' (\Val;n \ra \pis)$ &1.96 &1.98 &1.96 &1.88 &1.72 \\%
&$^1A' (\Ryd;n \ra 3s/3p)$ &6.40 &5.84 &6.31 &5.86 &6.48 \\%
&$^3A'' (\Val;n \ra \pis)$ &1.16 &1.12 &1.14 &1.03 &0.84 \\%
&$^3A' (\Val;\pi \ra \pis)$ &5.60 &5.74 &5.51 &5.75 &5.04 \\%
&$^1A'' [\Fl] (\Val;n \ra \pis)$ &1.67 &1.68 &1.69 &1.55 &1.40 \\%
Propynal & $^1A'' (\Val; n \ra \pis)$ & 3.80 &3.96 &3.82 &3.78 &3.81\\
&$^1A'' (\Val; \pi \ra \pis)$ & 5.54 &5.71 &5.51 &5.73 &5.20\\
&$^3A'' (\Val; n \ra \pis)$ & 3.47 &3.53 &3.49 &3.38 &3.45\\
&$^3A' (\Val; \pi \ra \pis)$ & 4.47 &4.71 &4.43 &4.67 &4.10\\
Pyrazine &$^1B_{3u} (\Val; n \ra \pis)$ & 4.15 &4.14 &4.14 &4.17 &4.13\\
&$^1A_{u} (\Val; n \ra \pis)$ & 4.98 &4.86 &4.97 &4.88 &5.21\\
&$^1B_{2u} (\Val; \pi \ra \pis)$ & 5.02 &5.14 &5.03 &5.47 &4.88\\
&$^1B_{2g} (\Val; n \ra \pis)$ & 5.71 &5.86 &5.71 &5.87 &5.67\\
&$^1A_{g} (\Ryd; n \ra 3s)$ & 6.65 &6.20 &6.66 &6.30 &6.96\\
&$^1B_{1g} (\Val; n \ra \pis)$ & 6.74 &6.67 &6.73 &6.67 &7.00\\
&$^1B_{1u} (\Val; \pi \ra \pis)$ & 6.88 &6.89 &6.86 &6.88 &6.66\\
&$^1B_{1g} (\Ryd; \pi \ra 3s)$ & 7.21 &7.21 &7.20 &7.27 &7.18\\
&$^1B_{2u} (\Ryd; n \ra 3p)$ & 7.24 &6.74 &7.25 & &\\
&$^1B_{1u} (\Ryd; n \ra 3p)$ & 7.44 &7.03 &7.45 & &\\
&$^3B_{3u} (\Val; n \ra \pis)$ & 3.59 &3.60 &3.59 &3.62 &3.52\\
&$^3B_{1u} (\Val; \pi \ra \pis)$ & 4.35 &4.60 &4.39 &4.57 &4.05\\
&$^3B_{2u} (\Val; (\pi \ra \pis)$ & 4.39 &4.57 &4.40 &4.59 &4.10 \\
&$^3A_{u} (\Val; n \ra \pis)$ & 4.93 &4.82 &4.93 &4.84 &5.15 \\
&$^3B_{2g} (\Val; n \ra \pis)$ & 5.08 &5.19 &5.08 & &\\
&$^3B_{1u} (\Val; \pi \ra \pis)$ & 5.28 &5.59 &5.29 & &\\
Pyridazine &$^1B_1 (\Val; n \ra \pis)$ & 3.83 &3.78 &3.83 &3.79 &3.86 \\
&$^1A_2 (\Val; n \ra \pis)$ & 4.37 &4.26 &4.37 &4.27 &4.60 \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 5.26 &5.43 &5.29 &5.44 &5.18 \\
&$^1A_2 (\Val; n \ra \pis)$ & 5.72 &5.79 &5.74 &5.81 &5.74 \\
&$^1B_2 (\Ryd; n \ra 3s)$ & 6.17 &5.59 &6.17 &5.69 &6.67 \\
&$^1B_1 (\Val; n \ra \pis)$ & 6.37 &6.33 &6.37 &6.35 &6.62 \\
&$^1B_2 (\Val; \pi \ra \pis)$ & 6.75 &6.86 &6.74 &6.85 & \\
&$^3B_1 (\Val; n \ra \pis)$ & 3.19 &3.18 &3.19 &3.19 &3.12 \\
&$^3A_2 (\Val; n \ra \pis)$ & 4.11 &4.01 &4.11 &4.02 &4.22 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 4.82 &5.07 &4.83 &5.06 &4.46 \\
Pyridine &$^1B_1 (\Val; n \ra \pis)$ & 4.95 &4.99 &4.96 &4.98 &4.99 \\
&$^1B_2 (\Val; \pi \ra \pis)$ & 5.14 &5.32 &5.17 &5.33 &5.08 \\
&$^1A_2 (\Val; n \ra \pis)$ & 5.40 &5.28 &5.40 &5.27 &5.70 \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 6.62 &6.21 &6.63 &6.31 &7.17 \\
&$^1A_1 (\Ryd; n \ra 3s)$ & 6.76 &6.68 &6.76 &6.65 &6.39 \\
&$^1A_2 (\Ryd; \pi \ra 3s)$ & 6.82 &6.79 &6.81 &6.83 &6.65 \\
&$^1B_1 (\Ryd; \pi \ra 3p)$ & 7.39 &7.34 &7.38 &7.38 &7.21 \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 7.39 &7.45 &7.39 &7.48 &7.27 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 4.30 &4.53 &4.33 &4.53 &4.06 \\
&$^3B_1 (\Val; n \ra \pis)$ & 4.46 &4.48 &4.46 &4.47 &4.43 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.79 &4.98 &4.79 &4.98 &4.49 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 5.04 &5.29 &5.05 &5.28 &4.75 \\
&$^3A_2 (\Val; n \ra \pis)$ & 5.36 &5.24 &5.35 &5.23 &5.62 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 6.24 &6.39 &6.25 &6.35 &5.98 \\
Pyrimidine &$^1B_1 (\Val; n \ra \pis)$ & 4.44 &4.41 &4.44 &4.37 &4.54 \\
&$^1A_2 (\Val; n \ra \pis)$ & 4.85 &4.77 &4.86 &4.73 &5.06 \\
&$^1B_2 (\Val; \pi \ra \pis)$ & 5.38 &5.54 &5.41 &5.52 &5.33 \\
&$^1A_2 (\Val; n \ra \pis)$ & 5.92 &5.96 &5.93 &5.93 &6.08 \\
&$^1B_1 (\Val; n \ra \pis)$ & 6.26 &6.25 &6.26 &6.22 &6.52 \\
&$^1B_2 (\Ryd; n \ra 3s)$ & 6.70 &6.20 &6.72 &6.25 &7.11 \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 6.88 &6.84 &6.87 &6.83 &6.52 \\
&$^3B_1 (\Val; n \ra \pis)$ & 4.09 &4.07 &4.10 &4.05 &4.12 \\
&$^3A_2 (\Val; n \ra \pis)$ & 4.66 &4.60 &4.66 &4.58 &4.73 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.96 &5.17 &4.96 &5.14 &4.63 \\
Pyrrole &$^1A_2 (\Ryd; \pi \ra 3s)$ & 5.24 &5.23 &5.24 &5.30 &5.14 \\
&$^1B_1 (\Ryd; \pi \ra 3p)$ & 6.00 &5.91 &5.98 &5.94 &5.89 \\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.00 &5.96 &6.01 &6.03 &5.91 \\
&$^1B_2 (\Val; (\pi \ra \pis)$ & 6.26 &6.30 &6.25 &6.35 &6.11 \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 6.30 &6.47 &6.32 &6.47 &6.29 \\
&$^1B_2 (\Ryd; \pi \ra 3p)$ & 6.83 &6.89 &6.83 &6.91 &6.69 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.51 &4.72 &4.53 &4.71 &4.26 \\
&$^3A_2 (\Ryd; \pi \ra 3s)$ & 5.21 &5.20 &5.21 &5.27 &5.11 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 5.45 &5.66 &5.46 &5.62 &5.23 \\
&$^3B_1 (\Ryd; \pi \ra 3p)$ & 5.91 &5.86 &5.92 &5.89 &5.84 \\
Streptocyanine &$^1B_2 (\Val;\pi \ra \pis)$ &7.13 &7.20 &7.13 &7.00 &7.16 \\%
&$^3B_2 (\Val;\pi \ra \pis)$ & 5.47 &5.60 &5.48 &5.55 &5.33 \\%
Tetrazine &$^1B_{3u} (\Val; n \ra \pis)$ & 2.47 &2.38 &2.46 &2.42 &2.42 \\
&$^1A_{u} (\Val; n \ra \pis)$ & 3.69 &3.53 &3.67 &3.58 &3.87 \\
&$^1B_{1g} (\Val; n \ra \pis)$ & 4.93 &5.02 &4.91 &5.04 &4.97 \\
&$^1B_{2u} (\Val; \pi \ra \pis)$ & 5.21 &5.31 &5.23 &5.31 &5.08 \\
&$^1B_{2g} (\Val; n \ra \pis)$ & 5.45 &5.64 &5.46 &5.68 &5.13 \\
&$^1A_{u} (\Val; n \ra \pis)$ & 5.53 &5.56 &5.52 &5.59 &5.49 \\
&$^1B_{2g} (\Val; n \ra \pis)$ & 6.12 &6.18 &6.13 &6.21 &6.50 \\
&$^1B_{1g} (\Val; n \ra \pis)$ & 6.91 &6.95 &6.92 &6.97 &6.59 \\
&$^3B_{3u} (\Val; n \ra \pis)$ & 1.85 &1.81 &1.85 &1.85 &1.74 \\
&$^3A_{u} (\Val; n \ra \pis)$ & 3.45 &3.31 &3.44 &3.35 &3.54 \\
&$^3B_{1g} (\Val; n \ra \pis)$ & 4.20 &4.27 &4.20 &4.27 &4.06 \\
&$^3B_{2u} (\Val; \pi \ra \pis)$ & 4.52 &4.77 &4.52 &4.76 &4.06 \\
&$^3B_{2g} (\Val; n \ra \pis)$ & 5.04 &5.15 &5.05 &5.16 &4.86 \\
&$^3A_{u} (\Val; n \ra \pis)$ & 5.11 &5.13 &5.11 &5.16 &5.07 \\
&$^3B_{1u} (\Val; \pi \ra \pis)$ & 5.42 &5.70 &5.42 &5.67 &5.06 \\
Thioacetone &$^1A_2 (\Val; n \ra \pis)$ & 2.53 &2.63 &2.55 &2.47 &2.50 \\
&$^1B_2 (\Ryd; n \ra 4s)$ & 5.56 &5.50 &5.55 &5.47 &5.65\\
&$^1A_1 (\Val; \pi \ra \pis)$ & 5.88 &6.09 &5.90 &5.87 &5.53 \\
&$^1B_2 (\Ryd; n \ra 4p)$ & 6.51 &6.44 &6.51 &6.43 &6.53 \\
&$^1A_1 (\Ryd; n \ra 4p)$ & 6.61 &6.53 &6.61 &6.48 &6.64 \\
&$^3A_2 (\Val; n \ra \pis)$ & 2.33 &2.33 &2.34 &2.20 &2.26 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 3.45 &3.59 &3.46 &3.52 &3.17 \\
Thioformaldehyde&$^1A_2 (\Val;n \ra \pis)$ &2.22 &2.34 &2.23 &2.24 &2.05 \\%
&$^1B_2 (\Ryd;n \ra 4s)$ &5.96 &5.82 &5.91 &5.80 &5.94 \\%
&$^1A_1 (\Val;\pi \ra \pis)$ &6.38 &6.71 &6.48 &6.57 &5.98 \\%
&$^3A_2 (\Val;n \ra \pis)$ &1.94 &1.94 &1.94 &1.86 &1.77 \\%
&$^3A_1 (\Val;\pi \ra \pis)$ & 3.43 &3.48 &3.38 &3.45 &3.07 \\%
&$^3B_2 (\Ryd;n \ra 4s)$ &5.72 &5.64 &5.72 &5.62 &5.71 \\%
&$^1A_2 [\Fl] (\Val;n \ra \pis)$ &1.95 &2.09 &1.97 &1.92 &1.80 \\%
Thiophene &$^1A_1 (\Val; \pi \ra \pis)$ & 5.64 &5.75 &5.65 &5.72 &5.61 \\
&$^1B_2 (\Val; \pi \ra \pis)$ & 5.98 &6.07 &5.96 &6.07 &5.79 \\
&$^1A_2 (\Ryd; \pi \ra 3s)$ & 6.14 &6.07 &6.14 &6.15 &6.03 \\
&$^1B_1 (\Ryd; \pi \ra 3p)$ & 6.14 &6.15 &6.14 &6.24 &6.02 \\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.21 &6.35 &6.25 &6.35 &6.14 \\
&$^1B_1 (\Ryd; \pi \ra 3s)$ & 6.49 &6.48 &6.50 &6.51 &6.43 \\
&$^1B_2 (\Ryd; \pi \ra 3p)$ & 7.29 &7.26 &7.29 &7.34 &7.18 \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 3.92 &4.12 &3.94 &4.11 &3.65 \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 4.76 &4.91 &4.77 &4.86 &4.56 \\
&$^3B_1 (\Ryd; \pi \ra 3p)$ & 5.93 &6.00 &5.95 &6.09 &5.83 \\
&$^3A_2 (\Ryd; \pi \ra 3s)$ & 6.08 &6.03 &6.09 &6.11 &5.97 \\
Thiopropynal &$^1A'' (\Val; n \ra \pis)$ & 2.03 &2.20 &2.05 &2.08 &1.86 \\
&$^3A'' (\Val; n \ra \pis)$ & 1.80 &1.84 &1.81 &1.74 &1.63 \\
Triazine &$^1A_1'' (\Val; n \ra \pis)$ & 4.72 &4.64 &4.73 &4.58 &4.83 \\
&$^1A_2'' (\Val; n \ra \pis)$ & 4.75 &4.75 &4.74 &4.69 &4.99 \\
&$^1E'' (\Val; n \ra \pis)$ & 4.78 &4.72 &4.78 &4.66 &4.95 \\
&$^1A_2' (\Val; \pi \ra \pis)$ & 5.75 &5.89 &5.78 &5.83 &5.78 \\
&$^1A_1' (\Val; \pi \ra \pis)$ & 7.24 &7.32 &7.24 &7.18 &6.78 \\
&$^1E' (\Ryd; n \ra 3s)$ & 7.32 &6.87 &7.35 &6.89 &7.68 \\
&$^1E'' (\Val; n \ra \pis)$ & 7.78 &7.71 &7.79 & &\\
&$^1E' (\Val; \pi \ra \pis)$ & 7.94 &7.63 &7.92 &7.65 &7.88 \\
&$^3A_2'' (\Val; n \ra \pis)$ & 4.33 &4.32 &4.33 &4.29 &4.35 \\
&$^3E'' (\Val; n \ra \pis)$ & 4.51 &4.46 &4.51 &4.42 &4.59 \\
&$^3A_1'' (\Val; n \ra \pis)$ & 4.73 &4.65 &4.75 &4.59 &4.53 \\
&$^3A_1' (\Val; \pi \ra \pis)$ & 4.85 &5.12 &4.88 &5.10 &4.97 \\
&$^3E' (\Val; \pi \ra \pis)$ & 5.59 &5.88 &5.61 &5.82 &5.32 \\
&$^3A_2' (\Val; (\pi \ra \pis)$ & 6.62 &6.76 &6.63 &6.63 &6.27 \\
Water & $^1B_1 (\Ryd; n \ra 3s)$ &7.62 &7.23 &7.65 &7.18 &7.84 \\%
& $^1A_2 (\Ryd; n \ra 3p)$ &9.41 &8.89 &9.43 &8.84 &9.63 \\%
& $^1A_1 (\Ryd; n \ra 3s)$ &9.99 &9.58 &10.00 &9.52 &10.22 \\%
& $^3B_1 (\Ryd; n \ra 3s)$ &7.25 &6.91 &7.28 &6.86 &7.41 \\%
& $^3A_2 (\Ryd; n \ra 3p)$ &9.24 &8.77 &9.26 &8.72 &9.43 \\%
& $^3A_1 (\Ryd; n \ra 3s)$ &9.54 &9.20 &9.56 &9.15 &9.70 \\%
\end{longtable}
\end{footnotesize}
\begin{figure}
\includegraphics[width=\linewidth,viewport=2cm 22cm 19cm 27.5cm,clip]{Fig-S1.pdf}%DJ to T2: NE PAS CHANGER
\caption{Histograms of the errors (in eV) obtained with ADC(2), ADC(2.5), and ADC(3) taking the TBE/\emph{aug}-cc-pVTZ values of Refs.~\citenum{Loo18a} and \citenum{Loo20a} as reference (as in Fig.~1 in the main text).
``Count'' refers to the number of transitions in each group. Note the difference of scaling in the vertical axes.}
\label{Fig-1}
\end{figure}
\clearpage
\section{0-0 energies}
For the 0-0 energies, we used the (EOM-)CCSD/\emph{def2}-TZVPP geometries available in the Supporting Information of Ref.~\citenum{Loo19a}. We reproduce below the experimental values together with the literature references and the ZPVE
corrections (mostly determined at the B3LYP/6-31+G(d) level, see Ref. \citenum{Loo19a} for details). Two ``large'' (10 non-hydrogen atoms) molecules have been added to the original
set: \emph{p}-diisocyano-benzene and tetrafluorobenzene. The geometries are given below in the same format as in Ref.~\citenum{Loo19a}.
%
% Table-S-2
%
\renewcommand*{\arraystretch}{.65}
\LTcapwidth=\textwidth
\begin{footnotesize}
\begin{longtable}{ll|cccc}
\caption{Experimental 0-0 energies and symmetries for the set of 0-0 energies. The experimental 0-0 energies are reported in both cm$^{-1}$ (the unit used in most experimental works) and
eV. The ZPVE are also indicated in eV in the rightmost column.
} \label{Table-SI-2}\\
\hline
& & \multicolumn{3}{c}{Experimental reference} &\\
Molecule & State & cm$^{-1}$ & eV & Ref. & $\Delta^{\mathrm{ZPVE}}$\\
\hline
\endfirsthead
\hline
& & \multicolumn{3}{c}{Experimental reference}& \\
Molecule & State & cm$^{-1}$ & eV & Ref. & $\Delta^{\mathrm{ZPVE}}$\\
\hline
\endhead
\hline \multicolumn{6}{r}{{Continued on next page}} \\
\endfoot
\hline
\begin{scriptsize}
$^a$ in nm; $^b$ "best estimate"
\end{scriptsize}
\endlastfoot
Acetaldehyde &$^1A^{''}$ ($n \rightarrow \pi^\star$) &29769 &3.691 &\citenum{Liu95} &-0.070 \\
Acetone &$^1A_2$ ($n \rightarrow \pi^\star$) &30435 &3.773 &\citenum{Bab83} &-0.058 \\
Acetyl cyanide &$^1A^{''}$ ($n \rightarrow \pi^\star$) &27511 &3.411 &\citenum{Yoo99} &-0.053 \\
Acetylene &$^1\Sigma_u^-$ ($\pi \rightarrow \pi^\star$) &42197.7 &5.232 &\citenum{Her66} &-0.077 \\
&$^1\Delta_u$ ($\pi \rightarrow \pi^\star$) &54116 &6.710 &\citenum{Foo73} &-0.136 \\
Acrolein &$^1A^{''}$ ($n \rightarrow \pi^\star$) &25858.1 &3.206 &\citenum{Hol63} &-0.089 \\
Aniline &$^1A^{''}$ ($\pi \rightarrow \pi^\star$) &34029 &4.219 &\citenum{Sin96} &-0.191 \\
Benzene &$^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &38086.7 &4.722 &\citenum{Chr98} &-0.162 \\
Benzonitrile &$^1B_1$ ($\pi \rightarrow \pi^\star$) &36513 &4.527 &\citenum{Bor07d} &-0.145 \\
Benzoquinone &$^1B_{1g}$ ($n \rightarrow \pi^\star$) &20045 &2.485 &\citenum{Ter79} &-0.080 \\
CCl$_2$ &$^1B_1$ &17256.9 &2.140 &\citenum{Ric08} &0.010 \\
CClF &$^1A^{''}$ &25277.8 &3.134 &\citenum{Kar93} &-0.001 \\
CF$_2$ &$^1B_1$ &37226 &4.615 &\citenum{Mat67} &-0.008 \\
Cyanoacetylene &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &38484.9 &4.772 &\citenum{Job66a} &-0.118 \\
&$^1\Delta$ ($\pi \rightarrow \pi^\star$) &44221 &5.483 &\citenum{Job66b} &-0.119 \\
Cyanoformaldehyde &$^1A^{''}$ ($n \rightarrow \pi^\star$) &26283.37 &3.259 &\citenum{Kar91b} &-0.063 \\
Cyanogen &$^1\Sigma_u^-$ ($\pi \rightarrow \pi^\star$) &45399.85 &5.629 &\citenum{Fis72} &-0.086 \\
&$^1\Delta_u$ ($\pi \rightarrow \pi^\star$) & &5.96 &\citenum{Hal97} &-0.078 \\
2-Cyclopenten-1-one &$^1A^{''}$ ($n \rightarrow \pi^\star$) &27210 &3.374 &\citenum{Che98c} &-0.096 \\
Diacetylene &$^1\Sigma_u^-$ ($\pi \rightarrow \pi^\star$) &34912.37 &4.329 &\citenum{Har77} &-0.125 \\
&$^1\Delta_u$ ($\pi \rightarrow \pi^\star$) &40845 &5.064 &\citenum{Ban92} &-0.192 \\
\emph{p}-Dicyano-benzene& $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &35120 &4.354 &\citenum{Fuj92} &-0.140 \\%N
\emph{p}-Diethynylbenzene& $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &34255 &4.247 &\citenum{Ste03} &-0.134 \\%N
\emph{p}-Difluoro-benzene&$^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &36838 &4.567 &\citenum{Kni88} &-0.160 \\
Difluorodiazirine &$^1B_1$ ($n \rightarrow \pi^\star$) &28374.21 &3.518 &\citenum{Sie90} &-0.072 \\
2,6-Difluoro-pyridine & $^1B_2$ ($\pi \rightarrow \pi^\star$) &37820 &4.689 &\citenum{Nib03} &-0.149 \\
\emph{p}-Diisocyano-benzene& $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &35566 &4.410 &\citenum{Meh15} &-0.137 \\
Fluoro-benzene &$^1B_2$ ($\pi \rightarrow \pi^\star$) &37813 &4.688 &\citenum{But07} &-0.161 \\
Formaldehyde &$^1A_2$ ($n \rightarrow \pi^\star$) &28188.02 &3.495 &\citenum{Clo83} &-0.085 \\
Formic acid &$^1A^{''}$ ($n \rightarrow \pi^\star$) &37413.39 &4.639 &\citenum{Bea01} &-0.096 \\
Formylchloride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &32760 &4.062 &\citenum{Din99} &-0.069 \\
Formylfluoride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &37491.7 &4.648 &\citenum{Cra97} &-0.063 \\
Glyoxal &$^1A_u$ ($n \rightarrow \pi^\star$) &21973.43 &2.724 &\citenum{Pad67} &-0.060 \\
H$_2$C$_3$ &$^1A_2$ ($\pi \rightarrow \pi^\star$) &13677 &1.696 &\citenum{Sta12} &-0.059 \\
HCN &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &52256.4 &6.479 &\citenum{Her66} &-0.120 \\
HCP &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &34769.9 &4.311 &\citenum{Her66} &-0.087 \\
HNO &$^1A^{''}$ ($n \rightarrow \pi^\star$) &13154.4 &1.631 &\citenum{Her66} &-0.029 \\
HPO &$^1A^{''}$ ($n \rightarrow \pi^\star$) &19032.78 &2.360 &\citenum{Tac02} &-0.043 \\
HPS &$^1A^{''}$ ($\sigma \rightarrow \pi^\star$) &12291.4 &1.524 &\citenum{Gri13b} &-0.019 \\
HSiF &$^1A^{''}$ &23260.021 &2.884 &\citenum{Har95} &-0.037 \\
\emph{cis}-Hydroquinone & $^1A_1$ ($\pi \rightarrow \pi^\star$) &33534.782 &4.158 &\citenum{Hum93} &-0.156 \\%N
\emph{trans}-Hydroquinone& $^1B_u$ ($\pi \rightarrow \pi^\star$) &33500.054 &4.153 &\citenum{Hum93} &-0.159 \\%N
Isocyanogen &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &42523 &5.272 &\citenum{Lyn07} &-0.077 \\
Nitrosomethane &$^1A^{''}$ ($n \rightarrow \pi^\star$) &14408 &1.786 &\citenum{Ern78} &-0.026 \\
Nitrosylcyanide &$^1A^{''}$ ($n \rightarrow \pi^\star$) &11339.9 &1.406 &\citenum{Dix85} &0.004 \\
Oxalyl fluoride &$^1A_u$ ($n \rightarrow \pi^\star$) &32445 &4.023 &\citenum{Liv79} &-0.104 \\
Phenylacetylene &$^1B_2$ ($\pi \rightarrow \pi^\star$) &35877.18 &4.448 &\citenum{Rib99} &-0.158 \\
Phosgene &$^1A_2$ ($n \rightarrow \pi^\star$) &32730 &4.058 &\citenum{Gid62} &-0.089 \\
Propynal &$^1A^{''}$ ($n \rightarrow \pi^\star$) &26162.94 &3.244 &\citenum{Bra74} &-0.092 \\
4H-pyran-4-one & $^1A^2$ ($n \rightarrow \pi^\star$) &28360 &3.516 &\citenum{Gor93} &-0.122 \\
Pyrazine &$^1B_{3u}$ ($n \rightarrow \pi^\star$) &30875.78 &3.828 &\citenum{Sie89} &-0.209 \\
Pyrimidine &$^1B_1$ ($n \rightarrow p$) &31188 &3.867 &\citenum{Fis03b} &-0.174 \\
Selenoformaldehyde &$^1A_2$ ($n \rightarrow \pi^\star$) &13635 &1.691 &\citenum{Clo87} &-0.062 \\
SiCl$_2$ &$^1B_1$ &30013.5 &3.721 &\citenum{Kar93b} &-0.015 \\
Silylidene &$^1A_2$ (Ryd) &15132.97 &1.876 &\citenum{Smi03} &0.009 \\
&$^1B_2$ (Ryd) &29312.88 &3.634 &\citenum{Har97} &0.018 \\
Tetrafluorobenzene & $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &36555 &4.532 &\citenum{Oku86} &-0.136 \\
Tetrazine &$^1B_{3u}$ ($n \rightarrow \pi^\star$) &18128.07 &2.248 &\citenum{Ker97} &-0.093 \\
Thioacetaldehyde &$^1A^{''}$ ($n \rightarrow \pi^\star$) & &2.22 &\citenum{Jud83} &-0.064 \\
Thioacetone &$^1A_2$ ($n \rightarrow \pi^\star$) & &2.33 &\citenum{Jud83} &-0.047 \\
Thioacrolein &$^1A^{''}$ ($n \rightarrow \pi^\star$) &15124.6 &1.875 &\citenum{Jud84b} &-0.050 \\
Thiocarbonyllbromide &$^1A_2$ ($n \rightarrow \pi^\star$) &17992 &2.231 &\citenum{Sim87} &-0.033 \\
Thiocarbonylchlorofluoride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &21657.4 &2.685 &\citenum{Sub74} &-0.032 \\
&$^2A^{'}$ ($\pi \rightarrow \pi^\star$) &35277 &4.374 &\citenum{Clo80} &-0.066 \\
Thiocarbonylfluoride &$^1A_2$ ($n \rightarrow \pi^\star$) &23477.1 &2.911 &\citenum{Mou70} &-0.034 \\
Thioformaldehyde &$^1A_2$ ($n \rightarrow \pi^\star$) &16395.6 &2.033 &\citenum{Clo83} &-0.066 \\
Thioformylchloride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &18792 &2.330 &\citenum{Jud85b} &-0.054 \\
Thiophosgene &$^1A_2$ ($n \rightarrow \pi^\star$) &18712.2 &2.320 &\citenum{Fuj07} &-0.030 \\
& $^2A_1$ ($\pi \rightarrow \pi^\star$) &34277.32 &4.250 &\citenum{Fuj07} &-0.069 \\
Thiopropynal &$^1A^{''}$ ($n \rightarrow \pi^\star$) &14656.4 &1.817 &\citenum{Jud84c} &-0.048 \\
Trifluoronitrosomethane &$^1A^{''}$ ($n \rightarrow \pi^\star$) &13929.7 &1.727 &\citenum{Dye87} &-0.017 \\
Acetaldehyde &$^3A^{''}$ ($n \rightarrow \pi^\star$) &27240 &3.377& \citenum{Mou85} &-0.074\\
Acrolein &$^3 A^{''}$ ($n \rightarrow \pi^\star$) &24247.3 &3.006& \citenum{Hla13} &-0.109\\
Benzaldehyde &$^3 A^{''}$ ($n \rightarrow \pi^\star$) &25183 &3.122& \citenum{Ohm88} &-0.111\\
Benzoquinone &$^3B_{1g}$ ($n \rightarrow \pi^\star$) &18370 &2.278& \citenum{Koy71} &-0.075\\
CHCl &$^3 A^{''}$ &2163.28 &0.268& \citenum{Tao08b} &0.007\\
Cyanogen &$^3\Sigma_u^+$ ($\pi \rightarrow \pi^\star$) &33289.9 &4.127& \citenum{Cal63} &-0.110\\
4-Cyclopentene-1,3-dione &$^3B_1$ ($n \rightarrow \pi^\star$) &20540 &2.547& \citenum{Spr09} &-0.074\\
2-Cyclopenten-1-one &$^3A^{''}$ ($n \rightarrow \pi^\star$) &25956.29 &3.218& \citenum{Pil07} &-0.111\\
Formaldehyde &$^3A_2$ ($n \rightarrow \pi^\star$) &25194.34 &3.124& \citenum{Clo83} &-0.092\\
Glyoxal & $^3A_u$ ($n \rightarrow \pi^\star$) &19199 &2.380& \citenum{Ott99c} &-0.056\\
H$_2$C$_3$ &$^3B_1$ ($\pi \rightarrow \pi^\star$) &10354 &1.284& \citenum{Sta12} &-0.066\\
Oxalyl Chloride &$^3A_u$ ($n \rightarrow \pi^\star$) &410.02$^a$ &3.024& \citenum{Yos96} &-0.085\\
Ozone &$^3A_2$ &9553.021 &1.184& \citenum{Bou98} &-0.061\\
Propynal &$^3A^{''}$ ($n \rightarrow \pi^\star$) &24127.1 &2.991& \citenum{Bir73} &-0.106\\
4H-pyran-4-one &$^3A^2$ ($n \rightarrow \pi^\star$) &27291.5 &3.384& \citenum{Hof08} &-0.134\\
4H-pyran-4-thione &$^3A^2$ ($n \rightarrow \pi^\star$) &16846.4 &2.089& \citenum{Rut02} &-0.097\\
Pyrazine &$^3B_{3u}$ ($n \rightarrow \pi^\star$) &26820.3 &3.325& \citenum{Ott95} &-0.177\\
Pyrimidine &$^3 B_1$ ($n \rightarrow \pi^\star$) &28534.0 &3.538& \citenum{Ott93} &-0.176\\
Selenoformaldehyde &$^3A_2$ ($n \rightarrow \pi^\star$) &12171.0 &1.509& \citenum{Jud88} &-0.071\\
SiF$_2$ &$^3 B_1$ &26319.5 &3.263& \citenum{Kar95} &-0.002\\
SO$_2$ &$^3 B_1$ &25765.737 &3.195& \citenum{Hua00} &-0.055\\
Tetrazine &$^3B_{3u}$ ($n \rightarrow \pi^\star$) &13608.0 &1.687& \citenum{Liv71} &-0.061\\
Thioacetaldehyde &$^3A^{''}$ ($n \rightarrow \pi^\star$) &16293.8 &2.020& \citenum{Jud87} &-0.060\\
Thioacetone &$^3A_2$ ($n \rightarrow \pi^\star$) &17327.8 &2.148& \citenum{Mou91} &-0.045\\
Thioacrolein &$^3A^{''}$ ($n \rightarrow \pi^\star$) &14036.2 &1.740& \citenum{Jud84b} &-0.070\\
Thioformaldehyde &$^3A_2$ ($n \rightarrow \pi^\star$) &14507.39 &1.799& \citenum{Clo83} &-0.080\\
Thioformylchloride &$^3A^{''}$ ($n \rightarrow \pi^\star$) &17233.9 &2.137& \citenum{Jud85b} &-0.050\\
Thiophosgene &$^3A_2$ ($n \rightarrow \pi^\star$) &17493.788 &2.169& \citenum{Fuj06} &-0.035\\
Thiopropynal &$^1A^{''}$ ($n \rightarrow \pi^\star$) &13257.4 &1.644& \citenum{Jud84c} &-0.064\\
Triazine &$^3E^{"}$ ($n \rightarrow \pi^\star$) &335$^{a,b}$ &3.701& \citenum{Oht83} &-0.148\\
\end{longtable}
\end{footnotesize}
Cartesian coordinates (in \AA) obtained at the (EOM-)CCSD/\emph{def2}-TZVPP level of theory for the two additional compounds.
The notations of the excited states refer to the ground-state point group symmetry.
\subsubsection{\emph{p}-Diisocyano-benzene}
\begin{singlespace}
Ground state
\begin{verbatim}
C 0.000000 1.210459 0.692565
C 0.000000 1.210459 -0.692565
C 0.000000 0.000000 1.377022
C 0.000000 0.000000 -1.377022
C 0.000000 -1.210459 0.692565
C 0.000000 -1.210459 -0.692565
C 0.000000 0.000000 3.932314
C 0.000000 0.000000 -3.932314
N 0.000000 0.000000 2.763434
N 0.000000 0.000000 -2.763434
H 0.000000 2.135125 1.244480
H 0.000000 2.135125 -1.244480
H 0.000000 -2.135125 1.244480
H 0.000000 -2.135125 -1.244480
\end{verbatim}
Excited state [$^1B_{2u}$ ($\pi \rightarrow \pi^\star$)]
\begin{verbatim}
C 0.000000 1.238754 0.709424
C 0.000000 1.238754 -0.709424
C 0.000000 -0.000000 1.404087
C 0.000000 -0.000000 -1.404087
C 0.000000 -1.238754 0.709424
C 0.000000 -1.238754 -0.709424
C 0.000000 -0.000000 3.944346
C 0.000000 -0.000000 -3.944346
N 0.000000 -0.000000 2.771103
N 0.000000 -0.000000 -2.771103
H 0.000000 2.159635 1.265112
H 0.000000 2.159635 -1.265112
H 0.000000 -2.159635 1.265112
H 0.000000 -2.159635 -1.265112
\end{verbatim}
\subsubsection{Tetrafluorobenzene}
Ground state
\begin{verbatim}
C 0.000000 1.190995 0.692100
C 0.000000 1.190995 -0.692100
C 0.000000 0.000000 -1.393333
C 0.000000 -1.190995 -0.692100
C 0.000000 -1.190995 0.692100
C 0.000000 0.000000 1.393333
F 0.000000 2.348323 -1.348963
F 0.000000 2.348323 1.348963
F 0.000000 -2.348323 1.348963
F 0.000000 -2.348323 -1.348963
H 0.000000 0.000000 -2.469482
H 0.000000 0.000000 2.469482
\end{verbatim}
Excited state [$^1B_{2u}$ ($\pi \rightarrow \pi^\star$)]
\begin{verbatim}
C 1.178195 0.707583 0.063424
C 1.178195 -0.707583 0.063424
C 0.000000 -1.473594 0.163267
C -1.178195 -0.707583 0.063424
C -1.178195 0.707583 0.063424
C 0.000000 1.473594 0.163267
F 2.339183 -1.319369 -0.118329
F 2.339183 1.319369 -0.118329
F -2.339183 1.319369 -0.118329
F -2.339183 -1.319369 -0.118329
H 0.000000 -2.523792 0.389226
H 0.000000 2.523792 0.389226
\end{verbatim}
\end{singlespace}
\clearpage
%
% Table-S-2
%
\renewcommand*{\arraystretch}{.65}
\LTcapwidth=\textwidth
\begin{footnotesize}
\begin{longtable}{ll|cccc}
\caption{0-0 energies computed with ADC(2), ADC(3), CC2, and CC3. All values are in eV, and they have been obtained on the basis of the corresponding \emph{aug}-cc-pVTZ frozen-core adiabatic energies
determined on the (EOM-)CCSD/\emph{def2}-TZVPP geometries. The ZPVE corrections are listed above.} \label{Table-SI-3}\\
\hline
Molecule & State & ADC(2) & ADC(3) & CC2 & CC3 \\
\hline
\endfirsthead
\hline
Molecule & State & ADC(2) & ADC(3) & CC2 & CC3 \\
\hline
\endhead
\hline \multicolumn{6}{r}{{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
Acetaldehyde &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.455 &3.830 &3.671 &3.663 \\
Acetone &$^1A_2$ ($n \rightarrow \pi^\star$) &3.430 &4.007 &3.671 &3.733 \\
Acetyl cyanide &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.136 &3.607 &3.348 &3.381 \\
Acetylene &$^1\Sigma_u^-$ ($\pi \rightarrow \pi^\star$) &5.323 &4.753 &5.322 &5.163 \\
&$^1\Delta_u$ ($\pi \rightarrow \pi^\star$) &6.797 &6.247 &6.808 &6.627 \\
Acrolein &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.040 &3.445 &3.254 &3.229 \\
Aniline &$^1A^{''}$ ($\pi \rightarrow \pi^\star$) &4.264 &4.165 &4.261 &4.169 \\
Benzene &$^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &4.930 &4.684 &4.904 &4.709 \\
Benzonitrile &$^1B_1$ ($\pi \rightarrow \pi^\star$) &4.735 &4.494 &4.711 &4.526 \\
Benzoquinone &$^1B_{1g}$ ($n \rightarrow \pi^\star$) &2.310 &2.788 &2.443 &2.502 \\
CCl$_2$ &$^1B_1$ &2.066 &1.986 &2.180 &2.232 \\
CClF &$^1A^{''}$ &3.048 &2.978 &3.170 &3.198 \\
CF$_2$ &$^1B_1$ &4.504 &4.474 &4.632 &4.642 \\
Cyanoacetylene &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &4.815 &4.323 &4.780 &4.693 \\
&$^1\Delta$ ($\pi \rightarrow \pi^\star$) &5.483 &5.060 &5.480 &5.413 \\
Cyanoformaldehyde &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.133 &3.371 &3.309 &3.270 \\
Cyanogen &$^1\Sigma_u^-$ ($\pi \rightarrow \pi^\star$) &5.721 &5.212 &5.696 &5.595 \\
&$^1\Delta_u$ ($\pi \rightarrow \pi^\star$) &6.051 &5.529 &6.045 &5.907 \\
2-Cyclopenten-1-one &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.065 &3.734 &3.275 &3.355 \\
Diacetylene &$^1\Sigma_u^-$ ($\pi \rightarrow \pi^\star$) &4.268 &3.927 &4.237 &4.243 \\
&$^1\Delta_u$ ($\pi \rightarrow \pi^\star$) &4.966 &4.613 &4.958 &4.935 \\
\emph{p}-Dicyano-benzene& $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &4.546 &4.308 &4.522 &4.351 \\
\emph{p}-Diethynylbenzene& $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &4.455 &4.220 &4.434 &4.265 \\
\emph{p}-Difluoro-benzene&$^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &4.631 &4.519 &4.741 &4.525 \\
Difluorodiazirine &$^1B_1$ ($n \rightarrow \pi^\star$) &3.488 &3.304 &3.477 &3.501 \\
2,6-Difluoro-pyridine & $^1B_2$ ($\pi \rightarrow \pi^\star$) &4.744 &4.647 &4.758 &4.644 \\
\emph{p}-Diisocyano-benzene& $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &4.606 &4.372 &4.590 &4.407 \\
Fluoro-benzene &$^1B_2$ ($\pi \rightarrow \pi^\star$) &4.841 &4.648 &4.831 &4.664 \\
Formaldehyde &$^1A_2$ ($n \rightarrow \pi^\star$) &3.365 &3.548 &3.541 &3.482 \\
Formic acid &$^1A^{''}$ ($n \rightarrow \pi^\star$) &4.308 &4.886 &4.563 &4.588 \\
Formylchloride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.839 &4.219 &4.075 &4.047 \\
Formylfluoride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &4.410 &4.820 &4.661 &4.611 \\
Glyoxal &$^1A_u$ ($n \rightarrow \pi^\star$) &2.662 &2.733 &2.727 &2.726 \\
H$_2$C$_3$ &$^1A_2$ ($\pi \rightarrow \pi^\star$) &1.952 &1.418 &1.930 &1.713 \\
HCN &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &6.760 &5.963 &6.681 &6.409 \\
HCP &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &4.458 &3.794 &4.432 &4.250 \\
HNO &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.555 &1.403 &1.600 &1.637 \\
HPO &$^1A^{''}$ ($n \rightarrow \pi^\star$) &2.087 &2.291 &2.242 &2.277 \\
HPS &$^1A^{''}$ ($\sigma \rightarrow \pi^\star$) &1.460 &1.250 &1.522 &1.426 \\
HSiF &$^1A^{''}$ &2.923 &2.684 &2.955 &2.886 \\
\emph{cis}-Hydroquinone & $^1A_1$ ($\pi \rightarrow \pi^\star$) &4.119 &4.103 &4.125 &4.092 \\
\emph{trans}-Hydroquinone& $^1B_u$ ($\pi \rightarrow \pi^\star$) &4.111 &4.097 &4.127 &4.085 \\
Isocyanogen &$^1\Sigma^-$ ($\pi \rightarrow \pi^\star$) &5.209 &5.168 &5.405 &5.108 \\
Nitrosomethane &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.666 &1.569 &1.749 &1.781 \\
Nitrosylcyanide &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.441 &1.165 &1.417 &1.446 \\
Oxalyl fluoride &$^1A_u$ ($n \rightarrow \pi^\star$) &3.758 &4.209 &3.928 &3.988 \\
Phenylacetylene &$^1B_2$ ($\pi \rightarrow \pi^\star$) &4.652 &4.409 &4.628 &4.445 \\
Phosgene &$^1A_2$ ($n \rightarrow \pi^\star$) &3.727 &4.394 &3.956 &4.039 \\
Propynal &$^1A^{''}$ ($n \rightarrow \pi^\star$) &3.039 &3.430 &3.251 &3.241 \\
4H-pyran-4-one & $^1A^2$ ($n \rightarrow \pi^\star$) &3.168 &3.971 &3.422 &3.487 \\
Pyrazine &$^1B_{3u}$ ($n \rightarrow \pi^\star$) &3.827 &3.797 &3.795 &3.799 \\
Pyrimidine &$^1B_1$ ($n \rightarrow p$) &3.717 &3.909 &3.770 &3.818 \\
Selenoformaldehyde &$^1A_2$ ($n \rightarrow \pi^\star$) &1.659 &1.625 &1.812 &1.708 \\
SiCl$_2$ &$^1B_1$ &3.749 &3.541 &3.794 &3.697 \\
Silylidene &$^1A_2$ (Ryd) &2.127 &1.557 &2.089 &1.850 \\
&$^1B_2$ (Ryd) &3.692 &3.224 &3.633 &3.579 \\
Tetrafluorobenzene & $^1B_{2u}$ ($\pi \rightarrow \pi^\star$) &4.532 &4.501 &4.553 &4.487 \\
Tetrazine &$^1B_{3u}$ ($n \rightarrow \pi^\star$) &2.207 &2.185 &2.177 &2.246 \\
Thioacetaldehyde &$^1A^{''}$ ($n \rightarrow \pi^\star$) &2.145 &2.169 &2.300 &2.204 \\
Thioacetone &$^1A_2$ ($n \rightarrow \pi^\star$) &2.171 &2.361 &2.358 &2.298 \\
Thioacrolein &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.913 &1.828 &2.062 &1.923 \\
Thiocarbonyllbromide &$^1A_2$ ($n \rightarrow \pi^\star$) &2.114 &2.237 &2.284 &2.233 \\
Thiocarbonylchlorofluoride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &2.582 &2.640 &2.756 &2.618 \\
&$^2A^{'}$ ($\pi \rightarrow \pi^\star$) &4.112 &4.007 &4.542 &4.324 \\
Thiocarbonylfluoride &$^1A_2$ ($n \rightarrow \pi^\star$) &2.889 &2.825 &3.029 &2.815 \\
Thioformaldehyde &$^1A_2$ ($n \rightarrow \pi^\star$) &2.042 &1.911 &2.154 &2.038 \\
Thioformylchloride &$^1A^{''}$ ($n \rightarrow \pi^\star$) &2.282 &2.277 &2.448 &2.337 \\
Thiophosgene &$^1A_2$ ($n \rightarrow \pi^\star$) &2.240 &2.368 &2.425 &2.349 \\
& $^2A_1$ ($\pi \rightarrow \pi^\star$) &3.948 &3.878 &4.411 &4.229 \\
Thiopropynal &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.845 &1.715 &1.971 &1.849 \\
Trifluoronitrosomethane &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.666 &1.470 &1.717 &1.739 \\
Acetaldehyde &$^3A^{''}$ ($n \rightarrow \pi^\star$) &3.037 &3.429 &3.214 &3.308 \\
Acrolein &$^3 A^{''}$ ($n \rightarrow \pi^\star$) &2.664 &3.172 &2.849 &2.946 \\
Benzaldehyde &$^3 A^{''}$ ($n \rightarrow \pi^\star$) &2.711 &3.333 &2.900 &3.050 \\
Benzoquinone &$^3B_{1g}$ ($n \rightarrow \pi^\star$) &2.161 &2.522 &2.229 &2.351 \\
CHCl &$^3 A^{''}$ &0.091 &0.010 &0.190 &0.270 \\
Cyanogen &$^3\Sigma_u^+$ ($\pi \rightarrow \pi^\star$) &4.329 &3.790 &4.263 &4.006 \\
4-Cyclopentene-1,3-dione &$^3B_1$ ($n \rightarrow \pi^\star$) &2.219 &2.765 &2.297 &2.499 \\
2-Cyclopenten-1-one &$^3A^{''}$ ($n \rightarrow \pi^\star$) &2.767 &3.549 &2.960 &3.137 \\
Formaldehyde &$^3A_2$ ($n \rightarrow \pi^\star$) &2.878 &3.092 &3.012 &3.067 \\
Glyoxal & $^3A_u$ ($n \rightarrow \pi^\star$) &2.247 &2.320 &2.308 &2.360 \\
H$_2$C$_3$ &$^3B_1$ ($\pi \rightarrow \pi^\star$) &1.395 &1.115 &1.378 &1.308 \\
Oxalyl Chloride &$^3A_u$ ($n \rightarrow \pi^\star$) &2.663 &3.238 &2.808 &3.004 \\
Ozone &$^3A_2$ &1.377 &0.459 &1.051 &1.164 \\
Propynal &$^3A^{''}$ ($n \rightarrow \pi^\star$) &2.628 &3.104 &2.804 &2.922 \\
4H-pyran-4-one &$^3A^2$ ($n \rightarrow \pi^\star$) &2.892 &3.770 &3.122 &3.293 \\
4H-pyran-4-thione &$^3A^2$ ($n \rightarrow \pi^\star$) &1.934 &2.117 &2.095 &2.044 \\
Pyrazine &$^3B_{3u}$ ($n \rightarrow \pi^\star$) &3.333 &3.242 &3.312 &3.301 \\
Pyrimidine &$^3 B_1$ ($n \rightarrow \pi^\star$) &3.389 &3.527 &3.432 &3.486 \\
Selenoformaldehyde &$^3A_2$ ($n \rightarrow \pi^\star$) &1.340 &1.403 &1.461 &1.475 \\
SiF$_2$ &$^3 B_1$ &2.932 &3.019 &2.992 &3.176 \\
SO$_2$ &$^3 B_1$ &2.774 &2.829 &2.804 &2.925 \\
Tetrazine &$^3B_{3u}$ ($n \rightarrow \pi^\star$) &1.666 &1.543 &1.642 &1.673 \\
Thioacetaldehyde &$^3A^{''}$ ($n \rightarrow \pi^\star$) &1.855 &1.941 &1.977 &1.986 \\
Thioacetone &$^3A_2$ ($n \rightarrow \pi^\star$) &1.923 &2.149 &2.075 &2.108 \\
Thioacrolein &$^3A^{''}$ ($n \rightarrow \pi^\star$) &1.624 &1.628 &1.742 &1.712 \\
Thioformaldehyde &$^3A_2$ ($n \rightarrow \pi^\star$) &1.675 &1.646 &1.761 &1.763 \\
Thioformylchloride &$^3A^{''}$ ($n \rightarrow \pi^\star$) &1.980 &2.050 &2.117 &2.115 \\
Thiophosgene &$^3A_2$ ($n \rightarrow \pi^\star$) &1.960 &2.158 &2.117 &2.141 \\
Thiopropynal &$^1A^{''}$ ($n \rightarrow \pi^\star$) &1.515 &1.501 &1.613 &1.613 \\
Triazine &$^3E^{"}$ ($n \rightarrow \pi^\star$) &3.451 &3.730 &3.530 &3.643 \\
\end{longtable}
\end{footnotesize}
\clearpage
\bibliography{biblio-new}
\end{document}

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\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Anthony Scemama}
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Denis Jacquemin}
\email{Denis.Jacquemin@univ-nantes.fr}
\affiliation[CEISAM, Nantes]{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
\let\oldmaketitle\maketitle
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\title{The Quest For Highly Accurate Excitation Energies: A Computational Perspective}
\date{\today}
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%%%%%%%%%%%%%%%%
%%% ABSTRACT %%% 150 WORd MAX !!!!!!! 149 lˆ.
%%%%%%%%%%%%%%%%
\begin{abstract}
We provide an overview of the successive steps that made possible to obtain increasingly accurate excitation energies with computational chemistry tools, eventually leading to chemically accurate vertical transition energies for small- and medium-size molecules.
First, we describe the evolution of \textit{ab initio} methods employed to define benchmark values, with originally Roos' CASPT2 method, then the CC3 method as in the renowned Thiel set, and more recently the resurgence of selected configuration interaction methods.
The latter method has been able to deliver consistently, for both single and double excitations, highly accurate excitation energies for small molecules, as well as medium-size molecules with compact basis sets.
Second, we describe how these high-level methods and the creation of representative benchmark sets of excitation energies have allowed to assess fairly and accurately the performance of computationally lighter methods.
We conclude by discussing the future theoretical and technological developments in the field.
\end{abstract}
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\noindent
%%%%%%%%%%%%%%%%%%%%
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
The accurate modeling of excited-state properties with \textit{ab initio} quantum chemistry methods is a clear ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come
(see, for example, Refs.~\citenum{Dre05,Gon12,Gho18} and references therein). Of particular interest is the access to precise excitation energies, \ie, the energy difference between ground and excited electronic states, and their intimate link with photophysical and
photochemical processes. The factors that makes this quest for high accuracy particularly delicate are very diverse.
First of all (and maybe surprisingly), it is, in most cases, tricky to obtain reliable and accurate experimental data that one can straightforwardly compare to theoretical values. In the case of vertical excitation energies, \ie, excitation energies at a fixed geometry, band maxima
do not usually correspond to theoretical values as one needs to take into account both geometric relaxation and zero-point vibrational energy motion. Even more problematic, experimental spectra might not be available in gas phase, and, in the worst-case scenario, no clear
assignment could be made. For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Die04b,Win13,Fan14b,Loo19b}
Second, developing theories suited for excited states is usually more complex and costly than their ground-state equivalent, as one might lack a proper variational principle for excited-state energies.
As a consequence, for a given level of theory, excited-state methods are usually less accurate than their ground-state counterpart, potentially creating a ground-state bias that leads to inaccurate excitation energies.
Another feature that makes excited states particularly fascinating and challenging is that they can be both very close in energy from each other and have very different natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet,
triplet, etc). Therefore, it would be highly desirable to possess a computational method (or protocol) that provides a balanced treatment of the entire ``spectrum'' of excited states.
We think that, at this stage, none of the existing methods does provide such a feat at an affordable cost for chemically-meaningful compounds.
What are the requirement of the ``perfect'' theoretical model? As mentioned above, a balanced treatment of excited states with different character is highly desirable. Moreover, chemically accurate excitation energies (\ie, with error smaller than $1$~kcal/mol or $0.043$~eV)
would be also beneficial in order to provide a quantitative chemical picture. The access to other properties, such as oscillator strengths, dipole moments, and analytical energy gradients, is also an asset if one wants to compare with experimental data.
Let us not forget about the requirements of minimal user input and minimal chemical intuition (\ie, black box models are preferable) in order to minimize the potential bias brought by the user appreciation of the problem complexity. Finally, low computational scaling with
respect to system size and small memory footprint cannot be disregarded. Although the simultaneous fulfillment of all these requirements seems elusive, it is useful to keep these criteria in mind. Table \ref{tab:method} is here for fulfill such a purpose.
In this Table, we also provide the typical error bar associated with each of these methods.
Table S1 of the {\SI} reports additional details about (some of the) existing BSE and wave function theory benchmarks, whereas a review of TD-DFT benchmark studies
can be found elsewhere. \cite{Lau13}
As can be seen in Table S1, the actual error bar obtained for a given method strongly depends on the actual type of excited states and compounds.
Hence, the values listed in Table \ref{tab:method} should be viewed as ``typical'' errors for organic molecules, nothing more.
%%% TABLE I %%%
\begin{table}
\footnotesize
\caption{Formal computational scaling of various excited-state methods with respect to the number of one-electron basis functions $N$ and the accessibility of various key properties in popular computational software packages.
For organic derivatives, the typical error range for single excitations is also provided as a qualitative indicator of the method accuracy.}
\label{tab:method}
\begin{tabular}{p{2.1cm}cccc}
\hline
\mr{2}{*}{Method} & Formal & Oscillator & Analytical & Typical \\
& scaling & strength & gradients & error (eV) \\
\hline
TD-DFT & $N^4$ & \cmark & \cmark & $0.2$--$0.4$$^a$ \\
BSE@\textit{GW} & $N^4$ & \cmark & \xmark & $0.1$--$0.3$$^b$ \\
\\
CIS & $N^5$ & \cmark & \cmark & $\sim 1.0$ \\
CIS(D) & $N^5$ & \xmark & \cmark & $0.2$--$0.3$ \\
ADC(2) & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
CC2 & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
\\
ADC(3) & $N^6$ & \cmark & \xmark & $0.2$ \\
EOM-CCSD & $N^6$ & \cmark & \cmark & $0.1$--$0.3$ \\
\\
CC3 & $N^7$ & \cmark & \xmark & $\sim 0.04$ \\
\\
EOM-CCSDT & $N^8$ & \xmark & \xmark & $\sim 0.03$ \\
EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark & $\sim 0.01$ \\
\\
CASPT2/NEVPT2 & $N!$ & \cmark & \cmark & $0.1$--$0.2$ \\
SCI & $N!$ & \xmark & \xmark & $\sim 0.03$ \\
FCI & $N!$ & \cmark & \cmark & $0.0$ \\
\hline
\end{tabular}
\begin{flushleft}
$^a${The error range is strongly functional and state dependent. The values reported here are for well-behaved cases;}
$^b${Typical error bar for singlet transitions. Larger errors are often observed for triplet excitations.}
\end{flushleft}
\end{table}
%**************
%** HISTORY **%
%**************
Before detailing some key past and present contributions aiming at obtaining highly accurate excitation energies, we start by giving a historical overview of the various excited-state \textit{ab initio} methods that have emerged in the last fifty years.
Interestingly, for pretty much every single method, the theory was derived much earlier than their actual implementation in electronic structure software packages and the same applies to the analytical gradients when available.
%%%%%%%%%%%%%%%%%%%%%
%%% POPLE'S GROUP %%%
%%%%%%%%%%%%%%%%%%%%%
The first mainstream \textit{ab initio} method for excited states was probably CIS (configuration interaction with singles) which has been around since the 1970's. \cite{Ben71}
CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states.
It is not unusual to have errors of the order of $1$ eV which precludes the usage of CIS as a quantitative quantum chemistry method.
Twenty years later, CIS(D) which adds a second-order perturbative correction to CIS was developed and implemented thanks to the efforts of Head-Gordon and coworkers. \cite{Hea94,Ish95}
This second-order correction greatly reduces the magnitude of the error compared to CIS, with a typical error range of $0.2$--$0.3$ eV.
%%%%%%%%%%%%%%%%%%%
%%% ROOS' GROUP %%%
%%%%%%%%%%%%%%%%%%%
In the early 1990's, the complete-active-space self-consistent field (CASSCF) method \cite{Roo80,And90} and its second-order perturbation-corrected variant CASPT2 \cite{And92} (originally developed in Roos' group) became very popular.
This was a real breakthrough.
Although it took more than ten years to obtain analytical gradients, \cite{Cel03} CASPT2 was probably the first method that could provide quantitative results for molecular excited states of genuine photochemical interest. \cite{Roo96}
Nonetheless, it is of common knowledge that CASPT2 has the clear tendency of underestimating vertical excitation energies in organic molecules.
Driven by Angeli and Malrieu, \cite{Ang01} the creation of the second-order $n$-electron valence state perturbation theory (NEVPT2) method several years later was able to cure some of the main theoretical deficiencies of CASPT2.
For example, NEVPT2 is known to be intruder state free and size consistent.
The limited applicability of these multiconfigurational methods is mainly due to the need of carefully defining an active space based on the desired transition(s) in order to obtain meaningful results, as well as their factorial computational growth with the number of active electrons and orbitals.
With a typical minimal valence active space tailored for the desired transitions, the usual error with CASPT2 or NEVPT2 calculations is $0.1$--$0.2$ eV, with the additional complication of the possible IPEA correction for the former method. \cite{Zob17}
We also point out that some emergent approaches, like DMRG (density matrix renormalization group), \cite{Bai19} offer a new path for the development of these multiconfigurational methods.
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%%% TDDFT %%%
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The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Cas95} was a significant step for the community as TD-DFT was able to provide accurate excitation energies at a much lower cost than its predecessors in a black-box way.
For low-lying valence excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$--$0.4$ eV.
However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Mai04,Dre05,Lev06,Eli11}
In the present context, one of the most annoying feature of TD-DFT --- in its most standard (adiabatic) approximation --- is its inability to describe, even qualitatively, charge-transfer states, \cite{Toz99,Dre04} Rydberg states, \cite{Toz98} and double excitations. \cite{Mai04,Lev06,Eli11}
These issues, as well as other well-documented shortcomings of DFT and TD-DFT, are related to the so-called delocalization error. \cite{Aut14a}
One closely related issue is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals. \cite{Goe19,Sue19}
More specifically, despite the development of new, more robust approaches (including the so-called range-separated \cite{Sav96,IIk01,Yan04,Vyd06} and double \cite{Goe10a,Bre16,Sch17} hybrids), it is still difficult (not to say impossible) to select a functional adequate for all families of transitions. \cite{Lau13}
Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
Despite all of this, TD-DFT remains nowadays the most employed excited-state method in the electronic structure community (and beyond).
%%%%%%%%%%%%%%%%%%
%%% CC METHODS %%%
%%%%%%%%%%%%%%%%%%
Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the growth of computational resources, equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) \cite{Sta93} became mainstream in the 2000's.
EOM-CCSD gradients were also quickly available. \cite{Sta95}
With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV for small compounds and generally $0.2$ eV for larger ones, with a typical overestimation of the vertical transition energies.
Its third-order version, EOM-CCSDT, was also implemented and provides, at a significantly higher cost, high accuracy for single excitations. \cite{Kuc01}
Although extremely expensive and tedious to implement, higher orders are also technically possible for small systems thanks to automatically generated code. \cite{Kuc91,Hir04}
For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Perspective} keeping in mind that these CC methods are applied to excited states in the present context.
The original CC family of methods was quickly completed by an approximated and computationally lighter family with, in front line, the second-order CC2 model \cite{Chr95} and its third-order extension, CC3. \cite{Chr95b}
As a $N^7$ method (where $N$ is the number of basis functions), CC3 has a particularly interesting accuracy/cost ratio with errors usually below the chemical accuracy threshold. \cite{Hat05c,Loo18a,Loo18b,Loo19a}
The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively.
It is also noteworthy that CCSDT and CC3 are also able to detect the presence of double excitations, a feature that is absent from both CCSD and CC2. \cite{Loo19c}
%%%%%%%%%%%%%%%%%%%
%%% ADC METHODS %%%
%%%%%%%%%%%%%%%%%%%
It is also important to mention the recent rejuvenation of the second- and third-order algebraic diagrammatic construction [ADC(2) \cite{Sch82} and ADC(3) \cite{Tro99,Har14}] methods that scale as $N^5$ and $N^6$, respectively.
These methods are related to the older second- and third-order polarization propagator approaches (SOPPA and TOPPA). \cite{Odd78,Pac96}
This renaissance was certainly initiated by the enormous amount of work invested by Dreuw's group in order to provide a fast and efficient implementation of these methods, \cite{Dre15} including the analytical gradients, \cite{Reh19} as well as other interesting variants. \cite{Dre15,Hod19a}
These Green's function one-electron propagator techniques indeed represent valuable alternatives thanks to their reduced cost compared to their CC equivalents.
In that regard, ADC(2) is particularly attractive with an error around $0.1$--$0.2$ eV.
However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies and is significantly less accurate than CC3. \cite{Tro02,Loo18a,Loo20a,Loo20b}
%%%%%%%%%%%%%%
%%% BSE@GW %%%
%%%%%%%%%%%%%%
Finally, let us mention the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Str88} (which is usually performed on top of a \textit{GW} calculation). \cite{Hed65}
BSE has gained momentum in the past few years and is a serious candidate as a computationally inexpensive electronic structure theory method that can effectively model excited states with a typical error of $0.1$--$0.3$ eV, as well as some related properties. \cite{Jac17b,Bla18}
One of the main advantage of BSE compared to TD-DFT (with a similar computational cost) is that it allows a faithful description of charge-transfer states and, when performed on top of a (partially) self-consistently \textit{GW} calculation, BSE@\textit{GW} has been shown to be weakly dependent on its starting point (\ie, on the functional selected for the underlying DFT calculation). \cite{Jac16,Gui18}
However, due to the adiabatic (\ie, static) approximation, doubly excited states are completely absent from the BSE spectrum.
%%%%%%%%%%%%%%%%%%%
%%% SCI METHODS %%%
%%%%%%%%%%%%%%%%%%%
In the past five years, \cite{Gin13,Gin15} we have witnessed a resurgence of the so-called selected CI (SCI) methods \cite{Ben69,Whi69,Hur73} thanks to the development and implementation of new, fast, and efficient algorithms to select cleverly
determinants in the full CI (FCI) space (see Refs.~\citenum{Gar18,Gar19} and references therein).
SCI methods rely on the same principle as the usual CI approach, except that determinants are not chosen \textit{a priori} based on occupation or excitation criteria but selected among the entire set of determinants based on their estimated contribution to the FCI wave function or energy.
Indeed, it has been noticed long ago that, even inside a predefined subspace of determinants, only a small number of them significantly contributes.
The main advantage of SCI methods is that no \textit{a priori} assumption is made on the type of electron correlation.
Therefore, at the price of a brute force calculation, a SCI calculation is not, or at least less, biased by the user appreciation of the problem's complexity.
One of the strength of one of the implementation, based on the CIPSI (configuration interaction using a perturbative selection made iteratively) algorithm developed by Huron, Rancurel, and Malrieu \cite{Hur73} is its parallel efficiency which makes possible to run on thousands of CPU cores. \cite{Gar19}
Thanks to these tremendous features, SCI methods deliver near FCI quality excitation energies for both singly and doubly excited states, \cite{Hol17,Chi18,Loo18a,Loo19c} with an error of roughly $0.03$ eV, mostly originating from the extrapolation procedure. \cite{Gar19}
However, although the \textit{``exponential wall''} is pushed back, this type of method is only applicable to molecules with a small number of heavy atoms and/or relatively compact basis sets.
%%%%%%%%%%%%%%%%%
%%% COMPUTERS %%%
%%%%%%%%%%%%%%%%%
For someone who has never worked with SCI methods, it might be surprising to see that one is able to compute near-FCI excitation energies for molecules as big as benzene. \cite{Chi18,Loo19c,Loo20a}
This is mainly due to some specific choices in terms of implementation as explained below.
Indeed, to keep up with Moore's ``Law'' in the early 2000's, the processor designers had no other choice than to propose multi-core chips to avoid an explosion of the energy requirements.
Increasing the number of floating-point operations per second by doubling the number of CPU cores only requires to double the required energy, while doubling the frequency multiplies the required energy by a factor of $\sim$ 8.
This bifurcation in hardware design implied a \textit{change of paradigm} \cite{Sut05} in the implementation and design of computational algorithms. A large degree of parallelism is now required to benefit from a significant acceleration.
Fifteen years later, the community has made a significant effort to redesign the methods with parallel-friendly algorithms. \cite{Val10,Cle10,Gar17b,Pen16,Kri13,Sce13}
In particular, the change of paradigm to reach FCI accuracy with SCI methods came
from the use of determinant-driven algorithms which were considered for long as inefficient
with respect to integral-driven algorithms.
The first important element making these algorithms efficient is the introduction of new bit manipulation instructions (BMI) in the hardware that enable an extremely fast evaluation of Slater-Condon rules \cite{Sce13b} for the direct calculation of
the Hamiltonian matrix elements over arbitrary determinants.
Then massive parallelism can be harnessed to compute the second-order perturbative correction with semi-stochatic algorithms, \cite{Gar17b,Sha17} and perform the sparse matrix multiplications required in Davidson's algorithm to find the eigenvectors associated with the lowest eigenvalues.
Block-Davidson methods can require a large amount of memory, and the recent introduction of byte-addressable non-volatile memory as a new tier in the memory hierarchy \cite{Pen19} will enable SCI calculations on larger molecules.
The next generation of supercomputers is going to generalize the presence of accelerators (graphical processing units, GPUs), leading to a new software crisis.
Fortunately, some authors have already prepared this transition. \cite{Dep11,Kim18,Sny15,Ufi08,Kal17}
%%%%%%%%%%%%%%%
%%% SUMMARY %%%
%%%%%%%%%%%%%%%
In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate, balanced, and reliable excitation energies for all classes of electronic excited states at an affordable cost.
%%% FIG 1 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{QUEST1}
\caption{Mean absolute error (MAE, top) and mean signed error (MSE, bottom) with respect to the TBE/\textit{aug}-cc-pVTZ values from the {\SetA} set (as described in Ref.~\citenum{Loo18a}) for various methods and types of excited states.
The corresponding graph for the maximum positive and negative errors can be found in the {\SI}.
}
\label{fig:Set1}
\end{figure*}
%%% %%% %%%
%%% FIG 2 %%%
\begin{figure}
\includegraphics[width=\linewidth]{QUEST2}
\caption{Mean absolute error (MAE, top) and maximum absolute error (MAX, bottom) with respect to FCI excitation energies for the doubly excited states reported in Ref.~\citenum{Loo19c} for various methods taking into account at least triple excitations.
$\%T_1$ corresponds to single excitation percentage in the transition calculated at the CC3 level.
For this particular set and methods, the mean signed error is equal to the MAE.}
\label{fig:Set2}
\end{figure}
%%% %%% %%%
%%% FIG 3 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{QUEST3}
\caption{Mean absolute error (MAE, top) and mean signed error (MSE, bottom) with respect to the TBE/\textit{aug}-cc-pVTZ values from the {\SetC} set (as described in Ref.~\citenum{Loo20a}) for various methods and types of excited states.
The corresponding graph for the maximum positive and negative errors can be found in the {\SI}.
}
\label{fig:Set3}
\end{figure*}
%%% %%% %%%
%*****************
%** BENCHMARKS ***
%*****************
Although sometimes decried, benchmark sets of molecules and their corresponding reference data are essential for the validation of existing theoretical models and to bring to light and subsequently understand their strengths and, more importantly, their limitations.
These sets have started to emerge at the end of the 1990's for ground-state properties with the acclaimed G2 test set designed by the Pople group. \cite{Cur97}
For excited states, things started moving a little later but some major contributions were able to put things back on track.
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%%% THIEL'S SET %%%
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One of these major contributions was provided by the group of Walter Thiel \cite{Sch08,Sil08,Sau09,Sil10b,Sil10c} with the introduction of the so-called Thiel (or M\"ulheim) set of excitation energies. \cite{Sch08}
For the first time, this set was large, diverse, consistent, and accurate enough to be used as a proper benchmarking set for excited-state methods.
More specifically, it gathers a large number of excitation energies consisting of 28 medium-size organic molecules with a total of 223 valence excited states (152 singlet and 71 triplet states) for which theoretical best estimates (TBEs) were defined.
In their first study Thiel and collaborators performed CC2, CCSD, CC3 and CASPT2 calculations (with the TZVP basis) in order to provide (based on additional high-quality literature data) TBEs for these transitions.
Their main conclusion was that \textit{``CC3 and CASPT2 excitation energies are in excellent agreement for states which are dominated by single excitations''}.
These TBEs were quickly refined with the larger \textit{aug}-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions.
As a direct evidence of the actual value of reference data, these TBEs were quickly picked up to benchmark various computationally effective methods from semi-empirical to state-of-the-art \textit{ab initio} methods (see the \textit{Introduction} of Ref.~\citenum{Loo18a} and references therein).
Theoretical improvements of Thiel's set were slow but steady, highlighting further its quality. \cite{Wat13,Kan14,Har14,Kan17}
In 2013, Watson \textit{et al.} \cite{Wat13} computed CCSDT-3/TZVP (an iterative approximation of the triples of CCSDT \cite{Wat96}) excitation energies for the Thiel set.
Their quality were very similar to the CC3 values reported in Ref.~\citenum{Sau09} and the authors could not appreciate which model was the most accurate.
Similarly, Dreuw and coworkers performed ADC(3) calculations on Thiel's set and arrived at the same kind of conclusion: \cite{Har14}
\textit{``based on the quality of the existing benchmark set it is practically not possible to judge whether ADC(3) or CC3 is more accurate''}.
These two studies clearly demonstrate and motivate the need for higher accuracy benchmark excited-states energies.
%%%%%%%%%%%%%%%%%%%%%%%
%%% JACQUEMIN'S SET %%%
%%%%%%%%%%%%%%%%%%%%%%%
Recently, we made, what we think, is a significant contribution to this quest for highly accurate vertical excitation energies. \cite{Loo18a}
More specifically, we studied 18 small molecules with sizes ranging from one to three non-hydrogen atoms.
For such systems, using a combination of high-order CC methods, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various natures (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly excited) based on CC3/\textit{aug}-cc-pVTZ geometries.
In the following, we label this set of TBEs as {\SetA}.
Importantly, it allowed us to benchmark a series of popular excited-state wave function methods partially or fully accounting for double and triple excitations (see Fig.~\ref{fig:Set1}): CIS(D), CC2, CCSD, STEOM-CCSD, \cite{Noo97} CCSDR(3), \cite{Chr77} CCSDT-3, \cite{Wat96} CC3, ADC(2), and ADC(3).
Our main conclusion was that CC3 is extremely accurate (with a mean absolute error of only $\sim 0.03$ eV), and that, although slightly less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies.
Quite surprisingly, ADC(3) was found to have a clear tendency to overcorrect its second-order version ADC(2).
The mean absolute errors (MAEs) obtained for this set can be found in Fig.~\ref{fig:Set1}.
In a second study, \cite{Loo19c} using a similar combination of theoretical models (but mostly extrapolated SCI energies), we provided accurate reference excitation energies for transitions involving a substantial amount of double excitations using a series of increasingly large diffuse-containing atomic basis sets (up to \textit{aug}-cc-pVQZ when technically feasible).
This set gathers 20 vertical transitions from 14 small- and medium-sized molecules, a set we label as {\SetB} in the remaining of this \textit{Perspective}.
An important addition to this second study was the inclusion of various flavors of multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including, at least, perturbative triples (see Fig.~\ref{fig:Set2}).
Our results clearly evidence that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition.
For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and CCSDT can easily reach $1$ and $0.5$ eV, respectively, while it goes down to a few tenths of an electronvolt for more common transitions (such as in \textit{trans}-butadiene and benzene) involving a significant amount of singles.\cite{Shu17,Bar18b,Bar18a}
The quality of the excitation energies obtained with multiconfigurational methods was harder to predict as the overall accuracy of these methods is highly dependent on both the system and the selected active space.
Nevertheless, CASPT2 and NEVPT2 were found to be more accurate for transition with a small percentage of single excitations (error usually below $0.1$ eV) than for excitations dominated by single excitations where the error is closer from $0.1$--$0.2$ eV (see Fig.~\ref{fig:Set2}).
In our latest study, \cite{Loo20a} in order to provide more general conclusions, we generated highly accurate vertical transition energies for larger compounds with a set composed by 27 organic molecules encompassing from four to six non-hydrogen atoms for a total of 223 vertical transition energies of various natures.
This set, labeled as {\SetC} and still based on CC3/\textit{aug}-cc-pVTZ geometries, is constituted by a reasonably good balance of singlet, triplet, valence, and Rydberg states.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with up to hundred million determinants), as well as the most robust multiconfigurational method, NEVPT2.
Each approach was applied in combination with diffuse-containing atomic basis sets.
For all the transitions of the {\SetC} set, we reported at least CCSDT/\textit{aug}-cc-pVTZ (sometimes with basis set extrapolation) and CC3/\textit{aug}-cc-pVQZ transition energies as well as CC3/\textit{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition.
Pursuing our previous benchmarking efforts, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a dominant double excitation character (see Fig.~\ref{fig:Set3}).
This settles down, at least for now, the debate by demonstrating the superiority of CC3 (in terms of accuracy) compared to methods like CCSDT-3 or ADC(3), see Fig.~\ref{fig:Set3}.
Moreover, thanks to the exhaustive and detailed comparisons made in Ref.~\citenum{Loo20a}, we could safely conclude that CC3 also regularly outperforms CASPT2 (which often underestimates excitation energies) and NEVPT2 (which typically overestimates excitation energies) as long as the corresponding transition does not show any strong multiple excitation character.
Our current efforts are now focussing on expanding and merging these sets to create an complete test set of highly accurate excitations energies.
In particular, we are currently generating reference excitations energies for radicals as well as more ``exotic'' molecules containing heavier atoms (such as \ce{Cl}, \ce{P}, and \ce{Si}).
The combination of these various sets would potentially create an ensemble of more than 400 vertical transition energies for small- and medium-size molecules based on accurate ground-state geometries.
Such a set would likely be a valuable asset for the electronic structure community.
It would likely stimulate further theoretical developments in excited-state methods and provide a fair ground for the assessments of the currently available and under development excited-state models.
%%%%%%%%%%%%%%%%%%
%%% Properties
%%%%%%%%%%%%%%%%%%
Besides all the studies described above aiming at reaching chemically accurate vertical transition energies, it should be pointed out that an increasing amount of effort is currently devoted to the obtention of highly-trustable excited-state properties.
This includes, first, 0-0 energies, \cite{Die04b,Hat05c,Goe10a,Sen11b,Win13,Fan14b,Loo18b,Loo19a,Loo19b} which, as mentioned above, offer well-grounded comparisons with experiment.
However, because 0-0 energies are fairly insensitive to the underlying molecular geometries, \cite{Sen11b,Win13,Loo19a} they are not a good indicator of their overall quality.
Consequently, one can find in the literature several sets of excited-state geometries obtained at various levels of theory, \cite{Pag03,Gua13,Bou13,Tun16,Bud17} some of them being determined using state-of-the-art models. \cite{Gua13,Bud17}
There are also investigations of the accuracy of the nuclear gradients at the Franck-Condon point. \cite{Taj18,Taj19}
The interested reader may find useful several investigations reporting sets of reference oscillator strengths. \cite{Sil10c,Har14,Kan14,Loo18a,Loo20b}
Up to now, these investigations focusing on geometries and oscillator strengths have been mostly based on theory-vs-theory comparisons. Indeed, while for small compounds (\ie, typically from di- to tetra-atomic molecules),
one can find very accurate experimental measurements (excited state dipole moments, oscillator strengths, vibrational frequencies, etc), these data are usually not accessible for larger compounds.
Nevertheless, the emergence of X-ray free electron lasers might soon allow to obtain accurate experimental excited state densities and geometrical structures through diffraction experiments.
Such new experimental developments will likely offer new opportunities for experiment-vs-theory comparisons going beyond standard energetics.
Finally, more complex properties, such as two-photon cross-sections and vibrations, have been mostly determined at lower levels of theory, hinting at future studies on this particular subject.
%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%%
As concluding remarks, we would like to highlight once again the major contribution brought by Roos' and Thiel's groups in an effort to define benchmark values for excited states.
Following their footsteps, we have recently proposed a larger, even more accurate set of vertical transitions energies for various types of excited states (including double excitations). \cite{Loo18a,Loo19c,Loo20a}
This was made possible thanks to a technological renaissance of SCI methods which can now routinely produce near-FCI excitation energies for small- and medium-size organic molecules. \cite{Chi18,Gar18,Gar19}
We hope that new technological advances will enable us to push further, in years to come, our quest to highly accurate excitation energies, and, importantly, of other excited-state properties.
%%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%%
%%%%%%%%%%%%%%%%%%%%%%%%
PFL would like to thank Peter Gill for useful discussions.
He also acknowledges funding from the \textit{``Centre National de la Recherche Scientifique''}.
DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support.
%%%%%%%%%%%%%%%%%%%%
%%% BIBLIOGRAPHY %%%
%%%%%%%%%%%%%%%%%%%%
\bibliography{ExPerspective}
\clearpage
\section*{Biographies}
\noindent{\bfseries P.F. Loos} was born in Nancy, France in 1982. He received his M.S.~in Computational and Theoretical Chemistry from the Universit\'e Henri Poincar\'e (Nancy, France) in 2005 and his Ph.D.~from the same university in 2008. From 2009 to 2013, He was undertaking postdoctoral research with Peter M.W.~Gill at the Australian National University (ANU). From 2013 to 2017, he was a \textit{``Discovery Early Career Researcher Award''} recipient at the ANU. Since 2017, he holds a researcher position from the \textit{``Centre National de la Recherche Scientifique (CNRS)} at the \textit{Laboratoire de Chimie et Physique Quantiques} in Toulouse (France), and was awarded, in 2019, an ERC consolidator grant for the development of new excited-state methodologies.
\begin{center}
\includegraphics[width=3cm]{PFLoos.png}
\end{center}
\noindent{\bfseries A. Scemama} received his Ph.D.~in Computational and Theoretical Chemistry from the Universit\'e Pierre et Marie Curie (Paris, France) in 2004.
He then moved to the Netherlands for a one-year postdoctoral stay in the group of Claudia Filippi, and came back in France for another year in the group of Eric Canc\`es.
In 2006, he obtained a Research Engineer position from the \textit{``Centre National de la Recherche Scientifique (CNRS)} at the \textit{Laboratoire de Chimie et Physique Quantiques} in Toulouse (France) to work on computational methods and high-performance computing for quantum chemistry. He was awarded the Crystal medal of the CNRS in 2019.
\begin{center}
\includegraphics[width=3cm]{AScemama.jpg}
\end{center}
\noindent{\bfseries D. Jacquemin} received his PhD in Chemistry from the University of Namur in 1998, before moving to the University of Florida for his postdoctoral stay. He is currently full Professor at the University of Nantes (France).
His research is focused on modeling electronically excited-state processes in organic and inorganic dyes as well as photochromes using a large panel of \emph{ab initio} approaches. His group collaborates with many experimental
and theoretical groups. He is the author of more than 500 scientific papers. He has been ERC grantee (2011--2016), member of Institut Universitaire de France (2012--2017) and received the WATOC's Dirac Medal (2014).
\begin{center}
\includegraphics[width=4cm]{DJacquemin.jpg}
\end{center}
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\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Anthony Scemama}
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Denis Jacquemin}
\email{Denis.Jacquemin@univ-nantes.fr}
\affiliation[CEISAM, Nantes]{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
\let\oldmaketitle\maketitle
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\title{The Quest For Highly Accurate Excitation Energies: A Computational Perspective: SI}
\date{\today}
\begin{document}
\clearpage
%
% Table-1
%
\begin{scriptsize}
\renewcommand{\arraystretch}{0.7}
\begin{longtable}{lccllcc}
\caption{
\label{Table-1}
Non-exhaustive list of previous benchmark studies of wavefunction (and BSE) methods.
We report the method, the year of publication, the characteristics of the set (both number of excited states and nature of the molecules), the benchmark set, the MAE (in eV), and the reference values.
S ans T stands for singlet and triplet respectively, and are specified only when the separated statistics are provided in the original work.
Further details can be found in the corresponding studies (references provided).}
\\
\hline
\hline
Method & Year & No.~of ES & No.~of molecules & Benchmark set & MAE (eV) & Ref. \\
\hline
\endfirsthead
\hline
\hline
Method & Year & No.~of ES & No.~of molecules & Benchmark set & MAE (eV) & Ref. \\
\hline
\endhead
\hline \multicolumn{7}{l}{Continued on next page} \\
\endfoot
\hline
\hline
\multicolumn{7}{l}{$^a$Depending on the selected solvent model: LR/cLR.}\\
\multicolumn{7}{l}{$^b$With the largest considered basis set, 6-311G, using (or not) various IPEA, see the original work.}\\
\endlastfoot
BSE@ev$GW$ & 2015 & 91 &28 (medium-size organic)& Th. $\Ea$ (Thiel's TBE-2) & 0.25 &\citenum{Jac15a} \\
& 2015 & 80 & 80 (organic dyes) & Exp. $\EOO$ (solvated) & 0.19/0.15$^a$ &\citenum{Jac15b} \\
& 2017 & 23 &18 (various) & Exp. excitation energies & 0.18 &\citenum{Jac17b} \\
& 2017 & 62 (T) &20 (medium-size organic)& Th. $\Ea$ (CC3) & 0.58 & \citenum{Jac17a} \\
& 2018 & 104 (S) &28 (medium-size organic )& Th. $\Ea$ (CC3) & 0.26 &\citenum{Gui18} \\
& 2018 & 63 (T) &20 (medium-size organic)& Th. $\Ea$ (CC3) & 0.84 &\citenum{Gui18} \\
CIS & 1995 & 6 &6 (diatomics) & Exp. $\Ead$ & 0.73 & \citenum{Sta95} \\
& 2002 & 34 &28 (mostly di/triatomics) & Exp. $\Ead$ and $\EOO$ & 0.63 & \citenum{Fur02} \\
& 2005 & 19 & 4 (diatomics) & Exp. $\Ead$ & 0.57 & \citenum{Hat05c} \\
& 2007 & 32 & 22 (diverse) &Exp. $\EOO$ & 0.71 &\citenum{Rhe07} \\
& 2009 & 20 & 29 (mostly di/triatomics) &Exp. $\Ead$ and $\EOO$ & 0.58 & \citenum{Rhe09} \\
& 2010 & 69 &11 (medium-size organic) &Exp. excitation energies & 1.07 & \citenum{Car10} \\
& 2010 & 12 &12 (organic dyes) & Exp. $\EOO$ (solvated) & 0.77 &\citenum{Goe10a} \\
& 2011 & 91 & 109 (diverse) &Exp. $\EOO$ & 0.98 & \citenum{Sen11b} \\
& 2013 & 7 & 7 (organic dyes) & Exp. $\EOO$ (solvated) & 0.75 &\citenum{Cha13c} \\
& 2014 & 79 & 96 (various) &Exp. $\EOO$ & 0.88 & \citenum{Fan14b} \\
& 2014 & 29 & 15 (small radicals) &Exp. $\EOO$ & 1.75 &\citenum{Bar14b} \\
& 2017 & 66 & 46 (aromatics) &Exp. $\EOO$ & 1.08 & \citenum{Oru16} \\
CIS(D) & 1995 & 6 &6 (diatomics) & Exp. $\Ead$ & 0.27 & \citenum{Sta95} \\
& 2004 & 32 & 22 (diverse) & Exp. $\EOO$ & 0.19 & \citenum{Gri04b} \\
& 2005 & 19 & 4 (diatomics) & Exp. $\Ead$ & 0.26 & \citenum{Hat05c} \\
& 2007 & 32 & 22 (diverse) &Exp. $\EOO$ & 0.22 &\citenum{Rhe07} \\
& 2010 & 69 &11 (medium-size organic) &Exp. excitation energies & 0.49 & \citenum{Car10} \\
& 2010 & 12 &12 (organic dyes) & Exp. $\EOO$ (solvated) & 0.25 &\citenum{Goe10a} \\
& 2015 & 80 & 80 (organic dyes) & Exp. $\EOO$ (solvated) & 0.18/0.26$^a$ &\citenum{Jac15b} \\
& 2018 & 106 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.25 & \citenum{Loo18a}\\
& 2020 & 221 & 27 (medium-size organic)& Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.23 & \citenum{Loo20a} \\
ADC(2) & 2002 & 43 & 4 (small) & Th. $\Ea$ (FCI) & 0.64 & \citenum{Tro02} \\
& 2005 & 19 & 4 (diatomics) & Exp. $\Ead$ & 0.21 & \citenum{Hat05c} \\
& 2013 & 66 & 46 (aromatics) & Exp. $\EOO$ & 0.08 & \citenum{Win13} \\
& 2014 & 104 & 28 (medium-size organic)&Th. $\Ea$ (Thiel's TBE) & 0.29 & \citenum{Har14} \\
& 2015 & 80 & 80 (organic dyes) & Exp. $\EOO$ (solvated) & 0.22/0.14$^a$ &\citenum{Jac15b} \\
& 2018 & 106 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.21 & \citenum{Loo18a}\\
& 2020 & 101 & 94 (medium-size organic)& Exp. $\EOO$ & 0.16 &\citenum{Loo20b} \\
& 2020 & 328 & 45 (1--6 non-H atoms) & Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.16 &\citenum{Loo20b} \\
& 2020 & 218 & 27 (medium-size organic)& Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.14 & \citenum{Loo20a} \\
ADC(3) & 2002 & 43 & 4 (small) & Th. $\Ea$ (FCI) & 0.17 & \citenum{Tro02} \\
& 2014 & 104 & 28 (medium-size organic)&Th. $\Ea$ (Thiel's TBE) & 0.24 & \citenum{Har14} \\
& 2018 & 106 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.23 & \citenum{Loo18a}\\
& 2020 & 101 & 94 (medium-size organic)& Exp. $\EOO$ & 0.18 &\citenum{Loo20b} \\
& 2020 & 328 & 45 (1--6 non-H atoms) & Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.21 &\citenum{Loo20b} \\
CC2 & 2002 & 43 & 4 (small) & Th. $\Ea$ (FCI) & 0.53 & \citenum{Tro02} \\
& 2003 & 20 & 29 (mostly di/triatomics)& Exp. $\Ead$ and $\EOO$ & 0.17 & \citenum{Koh03} \\
& 2005 & 19 & 4 (diatomics) & Exp. $\Ead$ & 0.16 & \citenum{Hat05c} \\
& 2008 & 26 & 19 (di/triatomics) & Exp. $\Ead$ and $\EOO$ & 0.17 & \citenum{Hel08} \\
& 2008 & 32 & 22 (diverse) & Exp. $\EOO$ & 0.14 & \citenum{Hel08} \\
& 2008 & 152 (S) & 28 (medium-size organic)&Th. $\Ea$ (CASPT2) & 0.32 & \citenum{Sch08} \\
& 2008 & 71 (T) & 20 (medium-size organic)&Th. $\Ea$ (CASPT2) & 0.19 & \citenum{Sch08} \\
& 2009 & 20 & 29 (mostly di/triatomics) &Exp. $\Ead$ and $\EOO$ & 0.18 & \citenum{Rhe09} \\
& 2011 & 15 & 15 (diverse) &Exp. $\EOO$ & 0.17 & \citenum{Sen11b} \\
& 2013 & 66 & 46 (aromatics) &Exp. $\EOO$ & 0.07 & \citenum{Win13} \\
& 2014 & 79 & 96 (various) &Exp. $\EOO$ & 0.19 & \citenum{Fan14b} \\
& 2015 & 80 & 80 (organic dyes) & Exp. $\EOO$ (solvated) & 0.16/0.13$^a$ &\citenum{Jac15b} \\
& 2016 & 132 & 25 (medium-size organic)& Th. $\Ea$ (CC3) & 0.22 & \citenum{Taj16} \\
& 2017 & 66 & 46 (aromatics) &Exp. $\EOO$ & 0.11 & \citenum{Oru16} \\
& 2018 & 106 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.22 & \citenum{Loo18a}\\
& 2018 & 35 & 31 (medium-size organic)&Exp. $\EOO$ & 0.08 & \citenum{Loo18b} \\
& 2020 & 101 & 94 (medium-size organic)& Exp. $\EOO$ & 0.10 &\citenum{Loo20b} \\
& 2020 & 328 & 45 (1--6 non-H atoms) & Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.17 &\citenum{Loo20b} \\
& 2020 & 223 & 27 (medium-size organic)& Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.15 & \citenum{Loo20a} \\
CCSD & 1995 & 6 &6 (diatomics) & Exp. $\Ead$ & 0.19 & \citenum{Sta95} \\
& 2002 & 43 & 4 (small) & Th. $\Ea$ (FCI) & 0.15 & \citenum{Tro02} \\
& 2005 & 19 & 4 (diatomics) & Exp. $\Ead$ & 0.20 & \citenum{Hat05c} \\
& 2008 & 152 (S) & 28 (medium-size organic)&Th. $\Ea$ (CASPT2) & 0.50 & \citenum{Sch08} \\
& 2008 & 71 (T) & 20 (medium-size organic)&Th. $\Ea$ (CASPT2) & 0.16 & \citenum{Sch08} \\
& 2010 & 69 &11 (medium-size organic) &Exp. excitation energies & 0.27 & \citenum{Car10} \\
& 2016 & 132 & 25 (medium-size organic)& Th. $\Ea$ (CC3) & 0.15 & \citenum{Taj16} \\
& 2017 & 23 &18 (various) & Exp. excitation energies & 0.31 &\citenum{Jac17b} \\
& 2018 & 106 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.08 & \citenum{Loo18a}\\
& 2018 & 35 & 31 (medium-size organic)&Exp. $\EOO$ & 0.21 & \citenum{Loo18b} \\
& 2020 & 223 & 27 (medium-size organic)& Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.13 & \citenum{Loo20a} \\
CC3 & 2002 & 43 & 4 (small) & Th. $\Ea$ (FCI) & 0.02 & \citenum{Tro02} \\
& 2005 & 19 & 4 (diatomics) & Exp. $\Ead$ & 0.04 & \citenum{Hat05c} \\
& 2008 & 121 (S) & 28 (medium-size organic)&Th. $\Ea$ (CASPT2) & 0.20 & \citenum{Sch08} \\
& 2008 & 71 (T) & 20 (medium-size organic)&Th. $\Ea$ (CASPT2) & 0.08 & \citenum{Sch08} \\
& 2018 & 35 & 31 (medium-size organic)&Exp. $\EOO$ & 0.02 & \citenum{Loo18b} \\
& 2018 & 106 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.04 & \citenum{Loo18a}\\
& 2019 & 119 & 109 (diverse) &Exp. $\EOO$ & 0.03 & \citenum{Loo19a} \\
& 2020 & 101 & 94 (medium-size organic)& Exp. $\EOO$ & 0.04 &\citenum{Loo20b} \\
& 2020 & 328 & 45 (1--6 non-H atoms) & Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.02 &\citenum{Loo20b} \\
& 2020 & 223 & 27 (medium-size organic)& Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.01 & \citenum{Loo20a} \\
CCSDT & 2018 & 104 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.03 & \citenum{Loo18a}\\
CCSDTQ & 2018 & 73 & 18 (small compounds) & Th. $\Ea$ (FCI) & 0.01 & \citenum{Loo18a}\\
CASPT2 & 2013 & 121 &28 (medium-size organic) & Th. $\Ea$ (CC3) & 0.21 & \citenum{Sch13b} \\
& 2014 & 29 & 15 (small radicals) &Exp. $\EOO$ & 0.12 &\citenum{Bar14b} \\
& 2016 & 23 &18 (various) & Exp. excitation energies & 0.21 &\citenum{Hoy16} \\
& 2017 & 124 & 13 (di and triatomics) & Th. $\Ea$ (FCI) & 0.10/0.11$^b$ &\citenum{Zob17} \\
& 2017 & 130 & 28 (medium-size organic) & Exp. excitation energies & 0.25--0.33$^b$ &\citenum{Zob17} \\
(PC-)NEVPT2 & 2013 & 121 &28 (medium-size organic) & Th. $\Ea$ (CC3) & 0.28 & \citenum{Sch13b} \\
& 2020 & 223 & 27 (medium-size organic)& Th. $\Ea$ (FCI, CCSDTQ, CCSDT) & 0.13 & \citenum{Loo20a} \\
\end{longtable}
\end{scriptsize}
%%% FIG 1 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{MAX}
\caption{Maximum positive and maximum negative errors (in eV) for the {\SetA} and {\SetC} sets of excitation energies with respect to the TBE/\textit{aug}-cc-pVTZ reference values (as described in Refs.~\citenum{Loo18a,Loo20a}) for various methods.
}
\label{fig:Set1}
\end{figure*}
%%% %%% %%%
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\end{document}

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