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\begin{document}

% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\CEISAM}{Universit\'e de Nantes, CNRS,  CEISAM UMR 6230, F-44000 Nantes, France}

\title{Benchmarking CASPT3 Vertical Excitation Energies}

\author{Martial \surname{Boggio-Pasqua}}
	\email{martial.boggio@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}
\author{Denis \surname{Jacquemin}}
	\email{denistriou.jacqueminous-aka-gros.lapin@univ-nantes.fr}
	\affiliation{\CEISAM}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

% Abstract
\begin{abstract}
Based on 284 vertical transition energies of various natures extracted from the QUEST database, we assess the accuracy of third-order multireference perturbation theory, CASPT3, in the context of molecular excited states.
When one applies the infamous ionization-potential-electron-affinity (IPEA) shift, we show that CASPT3 provides a similar accuracy as its second-order counterpart, CASPT2, with the same mean absolute error of 0.11 eV.
However, as already reported, we also observe that the accuracy of CASPT3 is almost insensitive to the IPEA shift, irrespectively of the type of the transitions, with a small reduction of the mean absolute errors to 0.09 eV when the IPEA shift is switched off
%\bigskip
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\end{abstract}

% Title
\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Perturbation theory is a relatively inexpensive route towards the exact solution of the Schr\"odinger equation. 
However, it rarely works this way in practice as the perturbative series may exhibit quite a large spectrum of behaviors. \cite{Olsen_1996,Christiansen_1996,Cremer_1996,Olsen_2000,Olsen_2019,Stillinger_2000,Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006,Goodson_2011}
For example, in single-reference M{\o}ller-Plesset (MP) perturbation theory, \cite{Moller_1934} erratic, slowly convergent, and divergent behaviors have been observed. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Leininger_2000a,Malrieu_2003,Damour_2021}
Systematic improvement is thus difficult to achieve and it is extremely challenging to predict, \textit{a priori}, the behavior of the series. \cite{Marie_2021a} 
This has led, in certain specific contexts, to the development of empirical strategy like MP2.5 where one simply averages the second-order (MP2) and third-order (MP3) total energies. \cite{Pitonak_2009}

Extension of single-reference perturbation theory to electronic excited states is far from being trivial, and the algebraic diagrammatic
construction (ADC) approximation of the polarization propagator is probably the most natural. \cite{Schirmer_1982,Schirmer_1991,Barth_1995,Schirmer_2004,Schirmer_2018,Trofimov_1997,Trofimov_1997b,Trofimov_2002,Trofimov_2005,Trofimov_2006,Harbach_2014,Dreuw_2015}
However, the ADC series naturally inherits some of the drawbacks of its MP parent and it has been shown to be not particularly rapidly convergent in the context of vertical excitation energies. \cite{Loos_2018a,Loos_2020a,Veril_2021}
This has led some of the authors to recently propose the ADC(2.5) composite approach, where, in the same spirit as MP2.5, one averages the second-order [ADC(2)] and third-order [ADC(3)] vertical transition energies. \cite{Loos_2020d}

Multi-reference perturbation theory is somewhat easier to generalize to excited states as one selects the states of interest to include in the reference (zeroth-order) space via the so-called complete-active-space self-consistent field (CASSCF) formalism, hence catching effectively static correlation in the zeroth-order model space.
The missing dynamical correlation can then be recovered in the (first-order) outer space via low-order perturbation theory, as performed in the complete-active-space second-order perturbation theory (CASPT2) of Roos and coworkers, \cite{Andersson_1990,Andersson_1992,Roos_1995a} Hirao's multireference second-order M{\o}llet-Plesset (MRMP2) approach, \cite{Hirao_1992} or the $N$-electron valence state second-order perturbation theory (NEVPT2) developed by Angeli, Malrieu, and coworkers. \cite{Angeli_2001a,Angeli_2001b,Angeli_2002,Angeli_2006}
However, these multi-reference formalisms and their implementation are much more involved and costly than their single-reference counterparts.

Although it has well-document weaknesses, CASPT2 is indisputably the most popular of the three approaches mentioned above. 
As such, it has been employed in countless computational studies involving electronic excited states. \cite{Serrano-Andres_1993a,Serrano-Andres_1993b,Serrano-Andres_1993c,Serrano-Andres_1995,Roos_1996,Serrano-Andres_1996a,Serrano-Andres_1996b,Serrano-Andres_1998b,Roos_1999,Merchan_1999,Roos_2002,Serrano-Andres_2002,Serrano-Andres_2005,Tozer_1999,Burcl0_2002,Peach_2008,Faber_2013,Schreiber_2008,Silva-Junior_2008,Sauer_2009,Silva-Junior_2010a,Silva-Junior_2010b,Silva-Junior_2010c}

In the context of excited states, its most severe drawback is certainly the intruder state problem (which is, by construction, absent in NEVPT2) that describes a situation where one or several determinants of the outer (first-order) space, known as perturbers, have an energy close to the zeroth-order CASSCF wave function, hence producing divergences in the denominators of the second-order perturbative energy.
One can then introduce a shift in the denominators to avoid such situations, and correcting afterwards the second-order energy for the use of this shift.
The use of real-valued \cite{Roos_1995b,Roos_1996} or imaginary \cite{Forsberg_1997} level shifts has been successfully tested and is now routine in excited-state calculations. \cite{Schapiro_2013,Zobel_2017,Sarka_2022}

A second pitfall was revealed by Andersson \textit{et al.} \cite{Andersson_1993,Andersson_1995} and explained by the unbalanced treatment in the zeroth-order Hamiltonian of the open- and closed-shell electronic configurations. 
A cure was quickly proposed via the introduction of an additional parameter in the zeroth-order Hamiltonian, the infamous ionization-potential-electron-affinity (IPEA) shift. \cite{Ghigo_2004}
Although the introduction of an IPEA shift can provide a better agreement between experiment and theory, \cite{Pierloot_2006,Pierloot_2008,Suaud_2009,Kepenekian_2009,Daku_2012,Rudavskyi_2014,Vela_2016,Wen_2018} it has been shown that its application is not systematically justified and has been found to be fairly basis set dependent. \cite{Zobel_2017}

Very recently, based on the highly-accurate vertical excitation energies of the QUEST database, \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c,Veril_2021,Loos_2021c,Loos_2021b} we have reported an exhaustive benchmark of CASPT2 and NEVPT2 for 284 excited states of diverse natures (singlet, triplet, valence, Rydberg, $n\to\pis$, $\pi\to\pis$, and double excitations) computed in 35 small- and medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarka_2022}
Our main take-home message was that both CASPT2 with IPEA shift and the partially-contracted version of NEVPT2 provide fairly reliable vertical transition energy estimates, with slight overestimations and mean absolute errors of \SI{0.11}{} and \SI{0.13}{\eV}, respectively.
Importantly, the introduction of the IPEA shift in CASPT2 was found to lower the mean absolute errors from \SI{0.27}{} to \SI{0.11}{eV}.

In the electronic structure community, third-order perturbation theory has a fairly bad reputation especially within MP perturbation theory where it is rarely worth its extra computational cost. \cite{Rettig_2020} 
Nonetheless, going against popular beliefs and one step further in the perturbative expansion, we propose here to assess the performance of the complete-active-space third-order perturbation theory (CASPT3) method developed by Werner \cite{Werner_1996} and implemented in MOLPRO \cite{Werner_2020} for a significant set of electronic transitions.
Although few CASPT3 calculations have been reported in the literature, 
\cite{Angeli_2006,Yanai_2007,Grabarek_2016,Li_2017,Li_2018,Li_2021,Bittererova_2001,Bokarev_2009,Frankcombe_2011,Gu_2008,Kerkines_2005,Lampart_2008,Leininger_2000,Maranzana_2020,Papakondylis_1999,Schild_2013,Sun_2018,Takatani_2009,Takatani_2010,Verma_2018,Woywod_2010,Yan_2004,Zhang_2020,Zhu_2005,Zhu_2007,Zhu_2013,Zou_2009}
the present study provides a comprehensive benchmark of CASPT3 as well as definite answers regarding its overall accuracy in the framework of electronically excited states.

Based on the same 284 highly-accurate vertical excitation energies from the QUEST database, we show that CASPT3 provides a significant improvement compared to CASPT2.
Moreover, as already reported in Ref.~\onlinecite{Grabarek_2016} where CASPT3 excitation energies are reported for retinal chromophore minimal models, we also observe that the accuracy of CASPT3 is much less sensitive to the IPEA shift.
Note that, although a third-order version of NEVPT has been developed \cite{Angeli_2006} and has been used in several applications \cite{Pastore_2006a,Pastore_2006b,Pastore_2007,Angeli_2007,Camacho_2010,Angeli_2011,Angeli_2012} by Angeli and coworkers, as far as we are aware of, only standalone implementation of NEVPT3 exists.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% FIGURE 1 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{mol.pdf}
	\caption{Various molecular systems considered in this study.
	\label{fig:mol}}
\end{figure}
%%% %%% %%% %%%

For each compound represented in Fig.~\ref{fig:mol}, we have computed the CASPT2 and CASPT3 vertical excitation energies with Dunning's aug-cc-pVTZ 
basis set. \cite{Kendall_1992} 
Geometries and reference theoretical best estimates (TBEs) for the vertical excitation energies have been extracted from the QUEST database \cite{Veril_2021} and can be downloaded at \url{https://lcpq.github.io/QUESTDB_website}.

All the CASPT2 and CASPT3 calculations have been carried out with MOLPRO within the RS2 and RS3 contraction schemes as described in Refs.~\onlinecite{Werner_1996} and \onlinecite{Werner_2020}. 
Both methods have been tested with and without IPEA (labeled as NOIPEA).
The MOLPRO implementation of CASPT3 is based on a modification of the multi-reference configuration interaction (MRCI) module. \cite{Werner_1988,Knowles_1988}
For the sake of computational efficiency, the doubly-excited external configurations are internally contracted while the singly-excited internal and semi-internal configurations are left uncontracted. \cite{Werner_1996}
When an IPEA shift is applied, its value is set to the default value of \SI{0.25}{\hartree} as discussed in Ref.~\onlinecite{Ghigo_2004}.
These perturbative calculations have been performed by considering a state-averaged (SA) CASSCF wave function where we have included the ground state and (at least) the excited states of interest.
In several occasions, we have included additional excited states to avoid convergence and/or root-flipping issues.

For each system and transition, we report in the {\SupInf} the exhaustive description of the active spaces for each symmetry sector.
Additionally, for the challenging transitions, we have steadily increased the size of the active space to carefully assess the convergence of the vertical excitation energies of interest.
Note that, compared to our previous CASPT2 benchmark study, \cite{Sarka_2022} some of the active spaces has been slightly reduced in order to be able to technically perform the CASPT3 calculations.
In these cases, we have recomputed the CASPT2 values for the same active space.
Although these active space reductions are overall statistically negligible, this explains the small deviations between the statistical quantities reported here and in Ref.~\onlinecite{Sarka_2022}.
Finally, to alleviate the intruder state problem, a level shift of \SI{0.3}{\hartree} has been systematically applied. \cite{Roos_1995b,Roos_1996}
This value has been slightly increased in particularly difficult cases, and is specifically reported in such cases.

The usual statistical indicators are used in the following, namely, the mean signed error (MSE), the mean absolute error (MAE), the root-mean-square error (RMSE), the standard 
deviation of the errors (SDE), as well as largest positive and negative deviations [Max($+$) and Max($-$), respectively].

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% TABLE I %%%
\begin{longtable*}{cllccccccccc}
\caption{Vertical excitation energies (in \si{\eV}) computed with various multi-reference methods.
The reference TBEs of the QUEST database, their percentage of single excitations $\%T_1$ involved in the transition (computed at the CC3 level), their nature 
(V and R stand for valence and Rydberg, respectively) are also reported.
TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute error below \SI{0.05}{\eV}).
[F] indicates a fluorescence transition, \ie, a vertical transition energy computed from an excited-state equilibrium geometry.
\label{tab:BigTab}}
\\
\hline\hline
\#	&Molecule			&Excitation				&Nature	&$\%T_1$	&TBE	&Safe?	&CASSCF	&CASPT2	&CASPT2	&CASPT3	&CASPT3\\
	&					&						&		&			&		&		&		&(IPEA)	&(NOIPEA)&(IPEA)&(NOIPEA)\\
\hline
\endfirsthead
\hline\hline
\#	&Molecule			&Excitation				&Nature	&$\%T_1$	&TBE	&Safe?	&CASSCF	&CASPT2	&CASPT2	&CASPT3	&CASPT3\\
	&					&						&		&			&		&		&		&(IPEA)	&(NOIPEA)&(IPEA)&(NOIPEA)\\
\hline
\endhead
\hline 
\multicolumn{12}{r}{{Continued on next page}} \\
\endfoot
\hline\hline
\endlastfoot
1	&Acetaldehyde		&$^1A''(n,\pis)$			&V	&91.3	&4.31	&\Y	&4.62	&4.35	&4.13	&4.44	&4.41\\
2	&					&$^3A''(n,\pis)$			&V	&97.9	&3.97	&\Y	&4.28	&3.94	&3.71	&4.06	&4.03\\
3	&Acetone			&$^1A_2(n,\pis)$			&V	&91.1	&4.47	&\Y	&4.77	&4.44	&4.19	&4.57	&4.55\\
4	&					&$^1B_2(n,3s)$				&R	&90.5	&6.46	&\Y	&5.50	&6.46	&6.35	&6.64	&6.67\\
5	&					&$^1A_2(n,3p)$				&R	&90.9	&7.47	&\Y	&7.46	&7.80	&7.55	&7.76	&7.68\\
6	&					&$^1A_1(n,3p)$				&R	&90.6	&7.51	&\Y	&7.03	&7.67	&7.46	&7.76	&7.75\\
7	&					&$^1B_2(n,3p)$				&R	&91.2	&7.62	&\Y	&6.44	&7.56	&7.47	&7.73	&7.76\\
8	&					&$^3A_2(n,\pis)$			&V	&97.8	&4.13	&\Y	&4.47	&4.13	&3.89	&4.27	&4.24\\
9	&					&$^3A_1(\pi,\pis)$			&V	&98.7	&6.25	&\Y	&6.22	&6.24	&6.07	&6.26	&6.22\\
10	&Acrolein			&$^1A''(n,\pis)$			&V	&87.6	&3.78	&\Y	&3.48	&3.58	&3.46	&3.66	&3.66\\
11	&					&$^1A'(\pi,\pis)$			&V	&91.2	&6.69	&\Y	&8.84	&6.93	&6.28	&7.18	&7.05\\
12	&					&$^1A''(n,\pis)$			&V	&79.4	&6.72	&\N	&6.76	&6.79	&6.34	&6.88	&6.80\\
13	&					&$^1A'(n,3s)$				&R	&89.4	&7.08	&\Y	&7.20	&7.21	&6.98	&7.20	&7.16\\
14	&					&$^1A'(\pi,\pis)$			&V	&75.0	&7.87	&\Y	&7.01	&8.10	&7.75	&8.02	&7.95\\
15	&					&$^3A''(n,\pis)$			&V	&97.0	&3.51	&\Y	&3.25	&3.28	&3.15	&3.39	&3.40\\
16	&					&$^3A'(\pi,\pis)$			&V	&98.6	&3.94	&\Y	&3.89	&4.01	&3.78	&3.96	&3.91\\
17	&					&$^3A'(\pi,\pis)$			&V	&98.4	&6.18	&\Y	&5.89	&6.20	&5.93	&6.10	&6.02\\
18	&					&$^3A''(n,\pis)$			&V	&92.7	&6.54	&\N	&6.67	&6.65	&6.21	&6.74	&6.66\\
19	&Benzene			&$^1B_{2u}(\pi,\pis)$		&V	&86.3	&5.06	&\Y	&4.98	&5.14	&4.66	&5.09	&5.01\\
20	&					&$^1B_{1u}(\pi,\pis)$		&V	&92.9	&6.45	&\Y	&7.27	&6.65	&6.23	&6.67	&6.58\\
21	&					&$^1E_{1g}(\pi,3s)$			&R	&92.8	&6.52	&\Y	&5.90	&6.70	&6.57	&6.56	&6.51\\
22	&					&$^1A_{2u}(\pi,3p)$			&R	&93.4	&7.08	&\Y	&6.14	&7.21	&7.07	&7.07	&7.02\\
23	&					&$^1E_{2u}(\pi,3p)$			&R	&92.8	&7.15	&\Y	&6.21	&7.26	&7.12	&7.13	&7.08\\
24	&					&$^1E_{2g}(\pi,\pis)$		&V	&73.0	&8.28	&\Y	&8.10	&8.31	&7.82	&8.26	&8.16\\
25	&					&$^1A_{1g}(\text{double})$	&V	&n.d.	&10.55	&\N	&11.44	&10.24	&9.33	&	&\\
26	&					&$^3B_{1u}(\pi,\pis)$		&V	&98.6	&4.16	&\Y	&3.85	&4.22	&3.92	&4.14	&4.08\\
27	&					&$^3E_{1u}(\pi,\pis)$		&V	&97.1	&4.85	&\Y	&4.85	&4.89	&4.51	&4.87	&4.80\\
28	&					&$^3B_{2u}(\pi,\pis)$		&V	&98.1	&5.81	&\Y	&6.75	&5.85	&5.40	&5.90	&5.81\\
29	&Butadiene			&$^1B_u(\pi,\pis)$			&V	&93.3	&6.22	&\Y	&6.65	&6.76	&6.52	&6.72	&6.65\\
30	&					&$^1B_g(\pi,3s)$			&R	&94.1	&6.33	&\Y	&5.94	&6.49	&6.32	&6.43	&6.38\\
31	&					&$^1A_g(\pi,\pis)$			&V	&75.1	&6.50	&\Y	&6.99	&6.74	&6.30	&6.73	&6.66\\
32	&					&$^1A_u(\pi,3p)$			&R	&94.1	&6.64	&\Y	&5.95	&6.74	&6.64	&6.70	&6.67\\
33	&					&$^1A_u(\pi,3p)$			&R	&94.1	&6.80	&\Y	&6.12	&6.95	&6.84	&6.90	&6.86\\
34	&					&$^1B_u(\pi,3p)$			&R	&93.8	&7.68	&\Y	&7.93	&7.60	&7.30	&7.62	&7.54\\
35	&					&$^3B_u(\pi,\pis)$			&V	&98.4	&3.36	&\Y	&3.55	&3.40	&3.19	&3.40	&3.35\\
36	&					&$^3A_g(\pi,\pis)$			&V	&98.7	&5.20	&\Y	&5.52	&5.32	&4.93	&5.29	&5.19\\
37	&					&$^3B_g(\pi,3s)$			&R	&97.9	&6.29	&\Y	&5.89	&6.44	&6.27	&6.38	&6.33\\
38	&Carbon Trimer		&$^1\Delta_g(\text{double})$&R	&1.0	&5.22	&\Y	&4.98	&5.08	&4.85	&5.20	&5.19\\
39	&					&$^1\Sigma^+_g(\text{double})$&R&1.0	&5.91	&\Y	&5.84	&5.82	&5.58	&5.92	&5.89\\
40	&Cyanoacetylene		&$^1\Sigma^-(\pi,\pis)$		&V	&94.3	&5.80	&\Y	&6.54	&5.85	&5.47	&5.89	&5.81\\
41	&					&$^1\Delta(\pi,\pis)$		&V	&94.0	&6.07	&\Y	&6.80	&6.13	&5.78	&6.17	&6.09\\
42	&					&$^3\Sigma^+(\pi,\pis)$		&V	&98.5	&4.44	&\Y	&4.86	&4.45	&4.04	&4.52	&4.45\\
43	&					&$^3\Delta(\pi,\pis)$		&V	&98.2	&5.21	&\Y	&5.64	&5.21	&4.86	&5.26	&5.19\\
44	&					&$^1A''[F](\pi,\pis)$		&V	&93.6	&3.54	&\Y	&4.30	&3.67	&3.47	&3.64	&3.58\\
45	&Cyanoformaldehyde	&$^1A''(n,\pis)$			&V	&89.8	&3.81	&\Y	&4.02	&3.98	&3.67	&3.94	&3.89\\
46	&					&$^1A''(\pi,\pis)$			&V	&91.9	&6.46	&\Y	&7.61	&6.79	&6.43	&6.77	&6.67\\
47	&					&$^3A''(n,\pis)$			&V	&97.6	&3.44	&\Y	&3.52	&3.46	&3.25	&3.51	&3.50\\
48	&					&$^3A'(\pi,\pis)$			&V	&98.4	&5.01	&\Y	&4.98	&5.25	&5.03	&5.16	&5.12\\
49	&Cyanogen			&$^1\Sigma_u^-(\pi,\pis)$	&V	&94.1	&6.39	&\Y	&7.14	&6.40	&6.03	&6.46	&6.39\\
50	&					&$^1\Delta_u(\pi,\pis)$		&V	&93.4	&6.66	&\Y	&7.46	&6.70	&6.35	&6.75	&6.68\\
51	&					&$^3\Sigma_u^+(\pi,\pis)$	&V	&98.5	&4.91	&\Y	&5.28	&4.85	&4.46	&4.95	&4.89\\
52	&					&$^1\Sigma_u^-[F](\pi,\pis)$&V	&93.4	&5.05	&\Y	&5.68	&5.07	&4.75	&5.11	&5.04\\
53	&Cyclopentadiene	&$^1B_2(\pi,\pis)$			&V	&93.8	&5.56	&\Y	&6.71	&5.96	&5.62	&6.06	&5.99\\
54	&					&$^1A_2(\pi,3s)$			&R	&94.0	&5.78	&\Y	&5.21	&5.88	&5.78	&5.81	&5.77\\
55	&					&$^1B_1(\pi,3p)$			&R	&94.2	&6.41	&\Y	&6.08	&6.59	&6.44	&6.47	&6.41\\
56	&					&$^1A_2(\pi,3p)$			&R	&93.8	&6.46	&\Y	&5.78	&6.55	&6.46	&6.45	&6.41\\
57	&					&$^1B_2(\pi,3p)$			&R	&94.2	&6.56	&\Y	&6.16	&6.72	&6.56	&6.61	&6.54\\
58	&					&$^1A_1(\pi,\pis)$			&V	&78.9	&6.52	&\N	&6.49	&6.63	&6.13	&6.59	&6.50\\
59	&					&$^3B_2(\pi,\pis)$			&V	&98.4	&3.31	&\Y	&3.26	&3.34	&3.09	&3.31	&3.26\\
60	&					&$^3A_1(\pi,\pis)$			&V	&98.6	&5.11	&\Y	&4.92	&5.14	&4.78	&5.10	&5.03\\
61	&					&$^3A_2(\pi,3s)$			&R	&97.9	&5.73	&\Y	&5.53	&5.91	&5.74	&5.81	&5.75\\
62	&					&$^3B_1(\pi,3p)$			&R	&97.9	&6.36	&\Y	&6.05	&6.56	&6.40	&6.43	&6.37\\
63	&Cyclopropene 		&$^1B_1(\sig,\pis)$			&V	&92.8	&6.68	&\Y	&7.48	&6.86	&6.58	&6.85	&6.77\\
64	&					&$^1B_2(\pi,\pis)$			&V	&95.1	&6.79	&\Y	&7.47	&6.89	&6.47	&6.96	&6.87\\
65	&					&$^3B_2(\pi,\pis)$			&V	&98.0	&4.38	&\Y	&4.60	&4.47	&4.27	&4.46	&4.40\\
66	&					&$^3B_1(\sig,\pis)$			&V	&98.9	&6.45	&\Y	&7.08	&6.56	&6.32	&6.55	&6.47\\
67	&Cyclopropenone		&$^1B_1(n,\pis)$			&V	&87.7	&4.26	&\Y	&4.92	&4.12	&3.75	&4.40	&4.38\\
68	&					&$^1A_2(n,\pis)$			&V	&91.0	&5.55	&\Y	&5.64	&5.62	&5.31	&5.67	&5.64\\
69	&					&$^1B_2(n,3s)$				&R	&90.8	&6.34	&\Y	&5.68	&6.28	&6.21	&6.41	&6.44\\
70	&					&$^1B_2(\pi,\pis)$			&V	&86.5	&6.54	&\Y	&6.40	&6.54	&6.20	&6.63	&6.62\\
71	&					&$^1B_2(n,3p)$				&R	&91.1	&6.98	&\Y	&6.35	&6.84	&6.70	&6.99	&7.01\\
72	&					&$^1A_1(n,3p)$				&R	&91.2	&7.02	&\Y	&6.84	&7.27	&7.03	&7.26	&7.24\\
73	&					&$^1A_1(\pi,\pis)$			&V	&90.8	&8.28	&\Y	&10.42	&8.96	&8.11	&9.21	&9.07\\
74	&					&$^3B_1(n,\pis)$			&V	&96.0	&3.93	&\Y	&4.72	&3.65	&3.28	&4.00	&3.98\\
75	&					&$^3B_2(\pi,\pis)$			&V	&97.9	&4.88	&\Y	&4.39	&4.76	&4.60	&4.76	&4.74\\
76	&					&$^3A_2(n,\pis)$			&V	&97.5	&5.35	&\Y	&5.40	&5.36	&5.06	&5.44	&5.42\\
77	&					&$^3A_1(\pi,\pis)$			&V	&98.1	&6.79	&\Y	&6.59	&6.93	&6.61	&6.86	&6.82\\
78	&Cyclopropenethione	&$^1A_2(n,\pis)$			&V	&89.6	&3.41	&\Y	&3.44	&3.43	&3.14	&3.46	&3.40\\
79	&					&$^1B_1(n,\pis)$			&V	&84.8	&3.45	&\Y	&3.57	&3.45	&3.17	&3.52	&3.46\\
80	&					&$^1B_2(\pi,\pis)$			&V	&83.0	&4.60	&\Y	&4.51	&4.64	&4.35	&4.66	&4.61\\
81	&					&$^1B_2(n,3s)$				&R	&91.8	&5.34	&\Y	&4.59	&5.25	&5.15	&5.25	&5.22\\
82	&					&$^1A_1(\pi,\pis)$			&V	&89.0	&5.46	&\Y	&6.46	&5.84	&5.32	&5.88	&5.75\\
83	&					&$^1B_2(n,3p)$				&R	&91.3	&5.92	&\Y	&5.27	&5.93	&5.86	&5.92	&5.90\\
84	&					&$^3A_2(n,\pis)$			&V	&97.2	&3.28	&\Y	&3.26	&3.28	&3.00	&3.33	&3.28\\
85	&					&$^3B_1(n,\pis)$			&V	&94.5	&3.32	&\Y	&3.51	&3.35	&3.07	&3.42	&3.36\\
86	&					&$^3B_2(\pi,\pis)$			&V	&96.5	&4.01	&\Y	&3.80	&3.97	&3.75	&3.99	&3.95\\
87	&					&$^3A_1(\pi,\pis)$			&V	&98.2	&4.01	&\Y	&3.83	&4.01	&3.77	&4.00	&3.95\\
88	&Diacetylene		&$^1\Sigma_u^-(\pi,\pis)$	&V	&94.4	&5.33	&\Y	&6.13	&5.42	&5.01	&5.45	&5.36\\
89	&					&$^1\Delta_u(\pi,\pis)$		&V	&94.1	&5.61	&\Y	&6.39	&5.68	&5.30	&5.72	&5.63\\
90	&					&$^3\Sigma_u^+(\pi,\pis)$	&V	&98.5	&4.10	&\Y	&4.54	&4.11	&3.67	&4.17	&4.09\\
91	&					&$^3\Delta_u(\pi,\pis)$		&V	&98.2	&4.78	&\Y	&5.28	&4.82	&4.45	&4.86	&4.78\\
92	&Diazomethane 		&$^1A_2(\pi,\pis)$			&V	&90.1	&3.14	&\Y	&3.27	&3.13	&2.92	&3.09	&3.04\\
93	&					&$^1B_1(\pi,3s)$			&R	&93.8	&5.54	&\Y	&4.59	&5.50	&5.30	&5.48	&5.45\\
94	&					&$^1A_1(\pi,\pis)$			&V	&91.4	&5.90	&\Y	&5.65	&6.21	&5.92	&6.18	&6.13\\
95	&					&$^3A_2(\pi,\pis)$			&V	&97.7	&2.79	&\Y	&3.02	&2.87	&2.67	&2.84	&2.79\\
96	&					&$^3A_1(\pi,\pis)$			&V	&98.6	&4.05	&\Y	&4.27	&4.10	&3.88	&4.06	&4.01\\
97	&					&$^3B_1(\pi,3s )$			&R	&98.0	&5.35	&\Y	&4.45	&5.34	&5.15	&5.33	&5.30\\
98	&					&$^3A_1(\pi,3p)$			&R	&98.5	&6.82	&\Y	&6.34	&7.00	&6.76	&6.96	&6.91\\
99	&					&$^1A''[F](\pi,\pis)$		&V	&87.4	&0.71	&\Y	&0.72	&0.69	&0.52	&0.66	&0.62\\
100	&Formamide 			&$^1A''(n,\pis)$			&V	&90.8	&5.65	&\Y	&5.95	&5.66	&5.45	&5.71	&5.67\\
101	&					&$^1A'(n,3s)$				&R	&88.6	&6.77	&\Y	&6.17	&6.80	&6.64	&6.82	&6.81\\
102	&					&$^1A'(n,3p)$				&R	&89.6	&7.38	&\N	&6.74	&7.45	&7.32	&7.46	&7.46\\
103	&					&$^1A'(\pi,\pis)$			&V	&89.3	&7.63	&\N	&8.80	&7.88	&7.13	&7.95	&7.78\\
104	&					&$^3A''(n,\pis)$			&V	&97.7	&5.38	&\Y	&5.89	&5.36	&5.16	&5.41	&5.37\\
105	&					&$^3A'(\pi,\pis)$			&V	&98.2	&5.81	&\Y	&6.10	&5.88	&5.62	&5.91	&5.87\\
106	&Furan				&$^1A_2(\pi,3s)$			&R	&93.8	&6.09	&\Y	&5.26	&6.16	&6.04	&6.06	&6.02\\
107	&					&$^1B_2(\pi,\pis)$			&V	&93.0	&6.37	&\Y	&7.78	&6.59	&6.02	&6.80	&6.71\\
108	&					&$^1A_1(\pi,\pis)$			&V	&92.4	&6.56	&\Y	&6.73	&6.66	&6.10	&6.69	&6.62\\
109	&					&$^1B_1(\pi,3p)$			&R	&93.9	&6.64	&\Y	&6.07	&6.79	&6.63	&6.65	&6.60\\
110	&					&$^1A_2(\pi,3p)$			&R	&93.6	&6.81	&\Y	&5.87	&6.87	&6.77	&6.76	&6.72\\
111	&					&$^1B_2(\pi,3p)$			&R	&93.5	&7.24	&\Y	&6.54	&7.11	&6.84	&6.96	&6.88\\
112	&					&$^3B_2(\pi,\pis)$			&V	&98.4	&4.20	&\Y	&3.94	&4.26	&4.01	&4.17	&4.12\\
113	&					&$^3A_1(\pi,\pis)$			&V	&98.1	&5.46	&\Y	&5.41	&5.50	&5.09	&5.47	&5.40\\
114	&					&$^3A_2(\pi,3s)$			&R	&97.9	&6.02	&\Y	&5.57	&6.16	&5.99	&6.05	&5.99\\
115	&					&$^3B_1(\pi,3p)$			&R	&97.9	&6.59	&\Y	&6.04	&6.76	&6.60	&6.62	&6.56\\
116	&Glyoxal			&$^1A_u(n,\pis)$			&V	&91.0	&2.88	&\Y	&3.42	&2.82	&2.51	&2.97	&2.94\\
117	&					&$^1B_g(n,\pis)$			&V	&88.3	&4.24	&\Y	&4.68	&4.21	&3.89	&4.36	&4.31\\
118	&					&$^1A_g(\text{double})$		&V	&0.5	&5.61	&\Y	&5.92	&5.37	&5.21	&5.53	&5.55\\
119	&					&$^1B_g(n,\pis)$			&V	&83.9	&6.57	&\Y	&7.35	&6.52	&5.98	&6.76	&6.72\\
120	&					&$^1B_u(n,3p)$				&R	&91.7	&7.71	&\Y	&7.04	&7.61	&7.34	&7.78	&7.81\\
121	&					&$^3A_u(n,\pis)$			&V	&97.6	&2.49	&\Y	&3.06	&2.41	&2.12	&2.57	&2.55\\
122	&					&$^3B_g(n,\pis)$			&V	&97.4	&3.89	&\Y	&4.61	&3.90	&3.53	&4.04	&4.01\\
123	&					&$^3B_u(\pi,\pis)$			&V	&98.5	&5.15	&\Y	&5.46	&5.14	&4.91	&5.17	&5.14\\
124	&					&$^3A_g(\pi,\pis)$			&V	&98.8	&6.30	&\Y	&6.69	&6.32	&6.02	&6.33	&6.27\\
125	&Imidazole			&$^1A''(\pi,3s)$			&R	&93.0	&5.70	&\Y	&5.04	&5.88	&5.66	&5.74	&5.68\\
126	&					&$^1A'(\pi,3p)$				&R	&90.0	&6.41	&\Y	&6.18	&6.69	&6.45	&6.61	&6.56\\
127	&					&$^1A''(\pi,3p)$			&R	&93.6	&6.50	&\Y	&5.43	&6.57	&6.47	&6.47	&6.44\\
128	&					&$^1A''(n,\pis)$			&V	&89.0	&6.71	&\Y	&7.13	&6.94	&6.57	&6.92	&6.85\\
129	&					&$^1A'(\pi,\pis)$			&V	&88.9	&6.86	&\Y	&6.73	&6.88	&6.46	&6.89	&6.83\\
130	&					&$^1A'(n,3s)$				&R	&89.0	&7.00	&\Y	&6.36	&7.10	&6.91	&7.09	&7.07\\
131	&					&$^3A'(\pi,\pis)$			&V	&98.3	&4.74	&\Y	&4.55	&4.78	&4.52	&4.73	&4.68\\
132	&					&$^3A''(\pi,3s)$			&R	&97.6	&5.66	&\Y	&5.03	&5.86	&5.63	&5.72	&5.66\\
133	&					&$^3A'(\pi,\pis)$			&V	&97.9	&5.74	&\Y	&5.69	&5.85	&5.48	&5.80	&5.72\\
134	&					&$^3A''(n,\pis)$			&V	&97.3	&6.31	&\Y	&6.58	&6.44	&6.10	&6.43	&6.37\\
135	&Isobutene			&$^1B_1(\pi,3s)$			&R	&94.1	&6.46	&\Y	&6.21	&6.74	&6.59	&6.64	&6.57\\
136	&					&$^1A_1(\pi,3p)$			&R	&94.2	&7.01	&\Y	&6.90	&7.32	&7.14	&7.24	&7.18\\
137	&					&$^3A_1(\pi,\pis)$			&V	&98.9	&4.53	&\Y	&4.66	&4.59	&4.41	&4.58	&4.53\\
138	&Ketene 			&$^1A_2(\pi,\pis)$			&V	&91.0	&3.86	&\Y	&3.98	&3.92	&3.70	&3.90	&3.85\\
139	&					&$^1B_1(\pi,3s)$			&R	&93.9	&6.01	&\Y	&5.22	&5.99	&5.79	&6.00	&5.97\\
140	&					&$^1A_1(\pi,\pis)$			&V	&92.4	&7.25	&\Y	&		&		&&&\\
141	&					&$^1A_2(\pi,3p)$			&R	&94.4	&7.18	&\Y	&6.38	&7.25	&7.05	&7.19	&7.15\\
142	&					&$^3A_2(\pi,\pis)$			&V	&91.0	&3.77	&\Y	&3.92	&3.81	&3.59	&3.79	&3.74\\
143	&					&$^3A_1(\pi,\pis)$			&V	&98.6	&5.61	&\Y	&5.79	&5.65	&5.43	&5.63	&5.59\\
144	&					&$^3B_1(\pi,3s)$			&R	&98.1	&5.79	&\Y	&5.05	&5.79	&5.60	&5.80	&5.77\\
145	&					&$^3A_2(\pi,3p)$			&R	&94.4	&7.12	&\Y	&6.35	&7.22	&7.01	&7.15	&7.11\\
146	&					&$^1A''[F](\pi,\pis)$		&V	&87.9	&1.00	&\Y	&0.95	&1.05	&0.88	&1.00	&0.95\\
147	&Methylenecycloprope&ne$^1B_2(\pi,\pis)$		&V	&85.4	&4.28	&\Y	&4.47	&4.40	&4.12	&4.39	&4.33\\
148	&					&$^1B_1(\pi,3s)$			&R	&93.6	&5.44	&\Y	&4.92	&5.57	&5.44	&5.46	&5.41\\
149	&					&$^1A_2(\pi,3p)$			&R	&93.3	&5.96	&\Y	&5.37	&6.09	&5.97	&5.97	&5.92\\
150	&					&$^1A_1(\pi,\pis)$			&V	&92.8	&6.12	&\N	&5.37	&6.26	&6.16	&6.17	&6.13\\
151	&					&$^3B_2(\pi,\pis)$			&V	&97.2	&3.49	&\Y	&3.44	&3.57	&3.34	&3.55	&3.49\\
152	&					&$^3A_1(\pi,\pis)$			&V	&98.6	&4.74	&\Y	&4.60	&4.82	&4.58	&4.77	&4.72\\
153	&Nitrosomethane 	&$^1A''(n,\pis)$			&V	&93.0	&1.96	&\Y	&2.12	&1.84	&1.60	&1.94	&1.91\\
154	&					&$^1A'(\text{double})$		&V	&2.5	&4.76	&\Y	&4.74	&4.69	&4.67	&4.71	&4.71\\
155	&					&$^1A'(\text{n.d.})$		&R	&90.8	&6.29	&\Y	&5.87	&6.32	&6.07	&6.34	&6.31\\
156	&					&$^3A''(n,\pis)$			&V	&98.4	&1.16	&\Y	&1.31	&1.00	&0.75	&1.12	&1.09\\
157	&					&$^3A'(\pi,\pis)$			&V	&98.9	&5.60	&\Y	&5.52	&5.52	&5.37	&5.54	&5.50\\
158	&					&$^1A''[F](n,\pis)$			&V	&92.7	&1.67	&\Y	&1.83	&1.55	&1.32	&1.66	&1.62\\
159	&Propynal			&$^1A''(n,\pis)$			&V	&89.0	&3.80	&\Y	&4.00	&3.92	&3.64	&3.90	&3.86\\
160	&					&$^1A''(\pi,\pis)$			&V	&92.9	&5.54	&\Y	&6.62	&5.82	&5.49	&5.81	&5.72\\
161	&					&$^3A''(n,\pis)$			&V	&97.4	&3.47	&\Y	&3.52	&3.48	&3.26	&3.52	&3.50\\
162	&					&$^3A'(\pi,\pis)$			&V	&98.3	&4.47	&\Y	&4.69	&4.59	&4.30	&4.54	&4.54\\
163	&Pyrazine			&$^1B_{3u}(n,\pis)$			&V	&90.1	&4.15	&\Y	&4.76	&4.09	&3.66	&4.31	&4.30\\
164	&					&$^1A_u(n,\pis)$			&V	&88.6	&4.98	&\Y	&5.90	&4.76	&4.26	&5.10	&5.10\\
165	&					&$^1B_{2u}(\pi,\pis)$		&V	&86.9	&5.02	&\Y	&4.97	&5.13	&4.65	&5.09	&5.03\\
166	&					&$^1B_{2g}(n,\pis)$			&V	&85.6	&5.71	&\Y	&5.80	&5.68	&5.27	&5.73	&5.70\\
167	&					&$^1A_g(n,3s)$				&R	&91.1	&6.65	&\Y	&6.69	&6.66	&6.27	&6.81	&6.80\\
168	&					&$^1B_{1g}(n,\pis)$			&V	&84.2	&6.74	&\Y	&7.16	&6.61	&6.07	&6.78	&6.76\\
169	&					&$^1B_{1u}(\pi,\pis)$		&V	&92.8	&6.88	&\Y	&8.04	&7.14	&6.72	&7.20	&7.12\\
170	&					&$^1B_{1g}(\pi,3s)$			&R	&93.8	&7.21	&\Y	&6.73	&7.41	&7.27	&7.24	&7.18\\
171	&					&$^1B_{2u}(n,3p)$			&R	&90.8	&7.24	&\Y	&7.49	&7.34	&6.93	&7.43	&7.40\\
172	&					&$^1B_{1u}(n,3p)$			&R	&91.4	&7.44	&\Y	&7.83	&7.55	&7.08	&7.64	&7.59\\
173	&					&$^1B_{1u}(\pi,\pis)$		&V	&90.5	&7.98	&\N	&9.65	&8.59	&7.96	&8.68	&8.57\\
174	&					&$^1A_g(\text{double})$		&V	&12.0	&8.04	&\N	&		&		&&&\\
175	&					&$^1A_g(\pi,\pis)$			&V	&71.0	&8.69	&\N	&		&		&&&\\
176	&					&$^3B_{3u}(n,\pis)$			&V	&97.3	&3.59	&\Y	&4.16	&3.49	&3.08	&3.72	&3.71\\
177	&					&$^3B_{1u}(\pi,\pis)$		&V	&98.5	&4.35	&\Y	&3.98	&4.44	&4.15	&4.34	&4.28\\
178	&					&$^3B_{2u}(\pi,\pis)$		&V	&97.6	&4.39	&\Y	&4.62	&4.44	&4.09	&4.47	&4.41\\
179	&					&$^3A_u(n,\pis)$			&V	&96.1	&4.93	&\Y	&5.85	&4.73	&4.21	&5.07	&5.07\\
180	&					&$^3B_{2g}(n,\pis)$			&V	&97.0	&5.08	&\Y	&5.25	&5.04	&4.66	&5.14	&5.11\\
181	&					&$^3B_{1u}(\pi,\pis)$		&V	&97.0	&5.28	&\Y	&5.15	&5.29	&4.92	&5.25	&5.19\\
182	&Pyridazine			&$^1B_1(n,\pis)$			&V	&89.0	&3.83	&\Y	&4.29	&3.74	&3.36	&3.94	&3.92\\
183	&					&$^1A_2(n,\pis)$			&V	&86.9	&4.37	&\Y	&4.83	&4.29	&3.87	&4.49	&4.48\\
184	&					&$^1A_1(\pi,\pis)$			&V	&85.8	&5.26	&\Y	&5.12	&5.34	&4.87	&5.30	&5.25\\
185	&					&$^1A_2(n,\pis)$			&V	&86.2	&5.72	&\Y	&6.26	&5.73	&5.19	&5.93	&5.89\\
186	&					&$^1B_2(n,3s)$				&R	&88.5	&6.17	&\Y	&5.99	&6.18	&5.90	&6.28	&6.27\\
187	&					&$^1B_1(n,\pis)$			&V	&87.0	&6.37	&\Y	&7.16	&6.50	&5.94	&6.72	&6.67\\
188	&					&$^1B_2(\pi,\pis)$			&V	&90.6	&6.75	&\Y	&7.54	&7.26	&6.82	&7.25	&7.17\\
189	&					&$^3B_1(n,\pis)$			&V	&97.1	&3.19	&\Y	&3.60	&3.08	&2.72	&3.29	&3.28\\
190	&					&$^3A_2(n,\pis)$			&V	&96.1	&4.11	&\Y	&4.49	&4.01	&3.59	&4.20	&4.18\\
191	&					&$^3B_2(\pi,\pis)$			&V	&98.5	&4.34	&\N	&3.92	&4.44	&4.13	&4.30	&4.24\\
192	&					&$^3A_1(\pi,\pis)$			&V	&97.3	&4.82	&\Y	&4.93	&4.87	&4.48	&4.89	&4.83\\
193	&Pyridine			&$^1B_1(n,\pis)$			&V	&88.4	&4.95	&\Y	&5.43	&5.15	&4.81	&5.18	&5.13\\
194	&					&$^1B_2(\pi,\pis)$			&V	&86.5	&5.14	&\Y	&5.03	&5.18	&4.76	&5.15	&5.09\\
195	&					&$^1A_2(n,\pis)$			&V	&87.9	&5.40	&\Y	&6.30	&5.46	&5.03	&5.63	&5.59\\
196	&					&$^1A_1(\pi,\pis)$			&V	&92.1	&6.62	&\Y	&7.90	&6.92	&6.27	&7.04	&6.93\\
197	&					&$^1A_1(n,3s)$				&R	&89.7	&6.76	&\Y	&6.40	&6.90	&6.67	&6.97	&6.96\\
198	&					&$^1A_2(\pi,3s)$			&R	&93.2	&6.82	&\Y	&6.60	&7.08	&6.87	&6.88	&6.80\\
199	&					&$^1B_1(\pi,3p)$			&R	&93.6	&7.38	&\Y	&7.12	&7.70	&7.51	&7.48	&7.40\\
200	&					&$^1A_1(\pi,\pis)$			&V	&90.5	&7.39	&\Y	&9.49	&7.66	&6.63	&7.87	&7.70\\
201	&					&$^1B_2(\pi,\pis)$			&V	&90.0	&7.40	&\N	&7.45	&7.92	&7.67	&7.80	&7.73\\
202	&					&$^3A_1(\pi,\pis)$			&V	&98.5	&4.30	&\Y	&3.98	&4.40	&4.06	&4.29	&4.22\\
203	&					&$^3B_1(n,\pis)$			&V	&97.0	&4.46	&\Y	&4.65	&4.48	&4.21	&4.57	&4.55\\
204	&					&$^3B_2(\pi,\pis)$			&V	&97.3	&4.79	&\Y	&4.83	&4.86	&4.53	&4.81	&4.74\\
205	&					&$^3A_1(\pi,\pis)$			&V	&97.1	&5.04	&\Y	&5.11	&5.09	&4.63	&5.09	&5.02\\
206	&					&$^3A_2(n,\pis)$			&V	&95.8	&5.36	&\Y	&5.94	&5.33	&4.96	&5.53	&5.51\\
207	&					&$^3B_2(\pi,\pis)$			&V	&97.7	&6.24	&\Y	&6.93	&6.40	&5.99	&6.43	&6.35\\
208	&Pyrimidine			&$^1B_1(n,\pis)$			&V	&88.6	&4.44	&\Y	&4.85	&4.44	&4.07	&4.58	&4.55\\
209	&					&$^1A_2(n,\pis)$			&V	&88.5	&4.85	&\Y	&5.52	&4.80	&4.36	&5.02	&5.00\\
210	&					&$^1B_2(\pi,\pis)$			&V	&86.3	&5.38	&\Y	&5.28	&5.42	&4.98	&5.41	&5.36\\
211	&					&$^1A_2(n,\pis)$			&V	&86.7	&5.92	&\Y	&6.70	&5.92	&5.32	&6.16	&6.10\\
212	&					&$^1B_1(n,\pis)$			&V	&86.7	&6.26	&\Y	&7.20	&6.31	&5.65	&6.58	&6.53\\
213	&					&$^1B_2(n,3s)$				&R	&90.3	&6.70	&\Y	&6.86	&6.85	&6.50	&6.89	&6.86\\
214	&					&$^1A_1(\pi,\pis)$			&V	&91.5	&6.88	&\Y	&7.69	&7.31	&6.94	&7.29	&7.22\\
215	&					&$^3B_1(n,\pis)$			&V	&96.8	&4.09	&\Y	&4.45	&4.05	&3.67	&4.20	&4.18\\
216	&					&$^3A_1(\pi,\pis)$			&V	&98.3	&4.51	&\N	&4.22	&4.57	&4.25	&4.51	&4.44\\
217	&					&$^3A_2(n,\pis)$			&V	&96.5	&4.66	&\Y	&5.20	&4.63	&4.16	&4.81	&4.78\\
218	&					&$^3B_2(\pi,\pis)$			&V	&97.4	&4.96	&\Y	&5.10	&5.01	&4.60	&5.03	&4.97\\
219	&Pyrrole			&$^1A_2(\pi,3s)$			&R	&92.9	&5.24	&\Y	&4.49	&5.44	&5.23	&5.28	&5.23\\
220	&					&$^1B_1(\pi,3p)$			&R	&92.4	&6.00	&\Y	&5.22	&6.26	&6.07	&6.08	&6.02\\
221	&					&$^1A_2(\pi,3p)$			&R	&93.0	&6.00	&\Y	&4.89	&6.16	&6.02	&6.01	&5.97\\
222	&					&$^1B_2(\pi,\pis)$			&V	&92.5	&6.26	&\Y	&7.73	&6.62	&6.36	&6.45	&6.38\\
223	&					&$^1A_1(\pi,\pis)$			&V	&86.3	&6.30	&\Y	&6.47	&6.41	&5.84	&6.43	&6.34\\
224	&					&$^1B_2(\pi,3p)$			&R	&92.6	&6.83	&\Y	&5.82	&6.75	&6.11	&6.92	&6.82\\
225	&					&$^3B_2(\pi,\pis)$			&V	&98.3	&4.51	&\Y	&4.24	&4.57	&4.30	&4.49	&4.44\\
226	&					&$^3A_2(\pi,3s)$			&R	&97.6	&5.21	&\Y	&4.47	&5.41	&5.21	&5.26	&5.20\\
227	&					&$^3A_1(\pi,\pis)$			&V	&97.8	&5.45	&\Y	&5.52	&5.50	&5.04	&5.49	&5.40\\
228	&					&$^3B_1(\pi,3p)$			&R	&97.4	&5.91	&\Y	&5.18	&6.22	&6.03	&6.04	&5.98\\
229	&Streptocyanine-1	&$^1B_2(\pi,\pis)$			&V	&88.7	&7.13	&\Y	&7.82	&7.17	&6.76	&7.28	&7.21\\
230	&					&$^3B_2(\pi,\pis)$			&V	&98.3	&5.52	&\Y	&5.86	&5.49	&5.22	&5.54	&5.49\\
231	&Tetrazine			&$^1B_{3u}(n,\pis)$			&V	&89.8	&2.47	&\Y	&2.99	&2.31	&1.91	&2.54	&2.53\\
232	&					&$^1A_u(n,\pis)$			&V	&87.9	&3.69	&\Y	&4.37	&3.49	&3.00	&3.77	&3.78\\
233	&					&$^1A_g(\text{double})$		&V	&0.7	&4.61	&\N	&5.42	&4.69	&4.48	&4.85	&4.87\\
234	&					&$^1B_{1g}(n,\pis)$			&V	&83.1	&4.93	&\Y	&5.41	&4.83	&4.33	&5.02	&5.00\\
235	&					&$^1B_{2u}(\pi,\pis)$		&V	&85.4	&5.21	&\Y	&5.04	&5.31	&4.84	&5.26	&5.23\\
236	&					&$^1B_{2g}(n,\pis)$			&V	&81.7	&5.45	&\Y	&5.43	&5.38	&4.90	&5.42	&5.38\\
237	&					&$^1A_u(n,\pis)$			&V	&87.7	&5.53	&\Y	&6.37	&5.51	&4.92	&5.80	&5.80\\
238	&					&$^1B_{3g}(\text{double})$	&V	&0.7	&6.15	&\N	&6.59	&5.85	&5.22	&6.20	&6.22\\
239	&					&$^1B_{2g}(n,\pis)$			&V	&80.2	&6.12	&\Y	&6.79	&5.96	&5.18	&6.27	&6.28\\
240	&					&$^1B_{1g}(n,\pis)$			&V	&85.1	&6.91	&\Y	&7.18	&6.59	&5.89	&6.79	&6.72\\
241	&					&$^3B_{3u}(n,\pis)$			&V	&97.1	&1.85	&\Y	&2.38	&1.70	&1.31	&1.94	&1.93\\
242	&					&$^3A_u(n,\pis)$			&V	&96.3	&3.45	&\Y	&4.06	&3.26	&2.78	&3.52	&3.52\\
243	&					&$^3B_{1g}(n,\pis)$			&V	&97.0	&4.20	&\Y	&4.66	&4.10	&3.62	&4.32	&4.30\\
244	&					&$^1B_{1u}(\pi,\pis)$		&V	&98.5	&4.49	&\N	&3.90	&4.55	&4.29	&4.39	&4.34\\
245	&					&$^3B_{2u}(\pi,\pis)$		&V	&97.5	&4.52	&\Y	&4.68	&4.55	&4.20	&4.60	&4.55\\
246	&					&$^3B_{2g}(n,\pis)$			&V	&96.4	&5.04	&\Y	&5.17	&5.02	&4.53	&5.10	&5.07\\
247	&					&$^3A_u(n,\pis)$			&V	&96.6	&5.11	&\Y	&6.12	&5.07	&4.44	&5.41	&5.41\\
248	&					&$^3B_{3g}(\text{double})$	&V	&5.7	&5.51	&\N	&6.56	&5.39	&4.86	&5.83	&5.85\\
249	&					&$^3B_{1u}(\pi,\pis)$		&V	&96.6	&5.42	&\Y	&5.32	&5.46	&5.08	&5.44	&5.39\\
250	&Thioacetone		&$^1A_2(n,\pis)$			&V	&88.9	&2.53	&\Y	&2.72	&2.58	&2.33	&2.60	&2.53\\
251	&					&$^1B_2(n,3s)$				&R	&91.3	&5.56	&\Y	&4.80	&5.60	&5.48	&5.64	&5.61\\
252	&					&$^1A_1(\pi,\pis)$			&V	&90.6	&5.88	&\Y	&6.94	&6.42	&5.98	&6.40	&6.26\\
253	&					&$^1B_2(n,3p)$				&R	&92.4	&6.51	&\Y	&5.57	&6.51	&6.40	&6.53	&6.49\\
254	&					&$^1A_1(n,3p)$				&R	&91.6	&6.61	&\Y	&6.24	&6.66	&6.41	&6.59	&6.50\\
255	&					&$^3A_2(n,\pis)$			&V	&97.4	&2.33	&\Y	&2.52	&2.34	&2.09	&2.38	&2.31\\
256	&					&$^3A_1(\pi,\pis)$			&V	&98.7	&3.45	&\Y	&3.52	&3.48	&3.29	&3.48	&3.43\\
257	&Thiophene			&$^1A_1(\pi,\pis)$			&V	&87.6	&5.64	&\Y	&6.11	&5.84	&5.21	&5.89	&5.79\\
258	&					&$^1B_2(\pi,\pis)$			&V	&91.5	&5.98	&\Y	&6.94	&6.35	&5.89	&6.44	&6.35\\
259	&					&$^1A_2(\pi,3s)$			&R	&92.6	&6.14	&\Y	&5.70	&6.28	&6.07	&6.16	&6.10\\
260	&					&$^1B_1(\pi,3p)$			&R	&90.1	&6.14	&\Y	&6.02	&6.21	&5.90	&6.16	&6.10\\
261	&					&$^1A_2(\pi,3p)$			&R	&91.8	&6.21	&\Y	&6.05	&6.32	&5.98	&6.28	&6.21\\
262	&					&$^1B_1(\pi,3s)$			&R	&92.8	&6.49	&\Y	&5.78	&6.57	&6.28	&6.51	&6.44\\
263	&					&$^1B_2(\pi,3p)$			&R	&92.4	&7.29	&\Y	&6.80	&7.29	&7.03	&7.20	&7.13\\
264	&					&$^1A_1(\pi,\pis)$			&V	&86.5	&7.31	&\N	&8.29	&7.62	&6.85	&7.71	&7.56\\
265	&					&$^3B_2(\pi,\pis)$			&V	&98.2	&3.92	&\Y	&3.68	&3.98	&3.71	&3.90	&3.84\\
266	&					&$^3A_1(\pi,\pis)$			&V	&97.7	&4.76	&\Y	&4.97	&4.85	&4.39	&4.87	&4.79\\
267	&					&$^3B_1(\pi,3p)$			&R	&96.6	&5.93	&\Y	&5.86	&5.97	&5.64	&5.94	&5.88\\
268	&					&$^3A_2(\pi,3s)$			&R	&97.5	&6.08	&\Y	&5.65	&6.22	&6.01	&6.11	&6.04\\
269	&Thiopropynal		&$^1A''(n,\pis)$			&V	&87.5	&2.03	&\Y	&2.06	&2.05	&1.84	&2.05	&2.00\\
270	&					&$^3A''(n,\pis)$			&V	&97.2	&1.80	&\Y	&1.85	&1.81	&1.60	&1.84	&1.79\\
271	&Triazine			&$^1A_1''(n,\pis)$			&V	&88.3	&4.72	&\Y	&5.88	&4.62	&3.90	&5.00	&4.99\\
272	&					&$^1A_2''(n,\pis)$			&V	&88.3	&4.75	&\Y	&5.14	&4.77	&4.39	&4.90	&4.87\\
273	&					&$^1E''(n,\pis)$			&V	&88.3	&4.78	&\Y	&5.51	&4.76	&4.14	&5.01	&4.98\\
274	&					&$^1A_2'(\pi,\pis)$			&V	&85.7	&5.75	&\Y	&5.55	&5.76	&5.32	&5.75	&5.72\\
275	&					&$^1A_1'(\pi,\pis)$			&V	&90.4	&7.24	&\Y	&8.20	&7.43	&6.89	&7.50	&7.41\\
276	&					&$^1E'(n,3s)$				&R	&90.9	&7.32	&\Y	&7.40	&7.48	&7.15	&7.53	&7.49\\
277	&					&$^1E''(n,\pis)$			&V	&82.6	&7.78	&\Y	&8.26	&7.75	&7.04	&7.92	&7.90\\
278	&					&$^1E'(\pi,\pis)$			&V	&90.0	&7.94	&\Y	&10.03	&8.65	&7.70	&8.63	&8.72\\
279	&					&$^3A_2''(n,\pis)$			&V	&96.7	&4.33	&\Y	&4.74	&4.37	&3.99	&4.51	&4.49\\
280	&					&$^3E''(n,\pis)$			&V	&96.6	&4.51	&\Y	&5.14	&4.47	&3.88	&4.71	&4.68\\
281	&					&$^3A_1''(n,\pis)$			&V	&96.2	&4.73	&\Y	&5.88	&4.70	&3.94	&5.06	&5.04\\
282	&					&$^3A_1'(\pi,\pis)$			&V	&98.2	&4.85	&\Y	&4.46	&4.88	&4.55	&4.81	&4.75\\
283	&					&$^3E'(\pi,\pis)$			&V	&96.9	&5.59	&\Y	&5.57	&5.62	&5.20	&5.62	&5.57\\
284	&					&$^3A_2'(\pi,\pis)$			&V	&97.6	&6.62	&\Y	&7.70	&6.62	&6.12	&6.76	&6.68\\
\end{longtable*}
%%% %%% %%% %%%

%%% FIGURE 2 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{PT2_vs_PT3.pdf}
	\caption{Histograms of the errors (in \si{\eV}) obtained for CASPT2 and CASPT3 with and without IPEA shift.
	\label{fig:PT2_vs_PT3}}
\end{figure}
%%% %%% %%% %%%

%%% TABLE II %%%
\begin{table*}
	\caption{Statistical quantities (in eV), considering the 265 ``safe'' TBEs (out of 284) as reference, for various multi-reference methods.} 
	\label{tab:stat}
	\begin{ruledtabular}
        \begin{tabular}{lrrrrrrr}
			&CASSCF\fnm[1]	&CASPT2\fnm[1]	&CASPT2\fnm[1]	&CASPT3\fnm[1]	&CASPT3\fnm[1]	&SC-NEVPT2\fnm[2]	&PC-NEVPT2\fnm[2]\\
			&				&(IPEA)			&(NOIPEA)		&(IPEA)			&(NOIPEA)\\
			\hline
            MSE			&$0.11$		&$0.06$		&$-0.26$	&$0.10$		&$0.05$		&$0.13$		&$0.09$\\
            SDE			&$0.58$		&$0.14$		&$0.21$		&$0.13$		&$0.13$		&$0.14$		&$0.14$\\
            RMSE		&$0.61$		&$0.16$		&$0.33$		&$0.17$		&$0.14$		&$0.19$		&$0.17$\\
            MAE			&$0.48$		&$0.11$		&$0.27$		&$0.11$		&$0.09$		&$0.15$		&$0.13$\\
            Max($+$)	&$2.15$		&$0.71$		&$0.30$		&$0.93$		&$0.79$		&$0.65$		&$0.46$\\
            Max($-$)	&$-1.18$	&$-0.32$	&$-1.02$	&$-0.28$	&$-0.36$	&$-0.38$	&$-0.57$\\
		\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Values from the present study.}
	\fnt[2]{Values taken from Ref.~\onlinecite{Sarka_2022}.}
\end{table*}
%%% %%% %%% %%%

%%% TABLE II %%%
\begin{table*}
	\caption{MAEs determined for several subsets of transitions and system sizes computed with various multi-reference methods.
	Count is the number of excited states considered in each subset.}
	\label{tab:stat_subset}
	\begin{ruledtabular}
		\begin{tabular}{lrrrrrrrr}
			Transitions		&	Count	&CASSCF\fnm[1]	&CASPT2\fnm[1]	&CASPT2\fnm[1]	&CASPT3\fnm[1]	&CASPT3\fnm[1]	&SC-NEVPT2\fnm[2]	&PC-NEVPT2\fnm[2]\\
							&			&				&(IPEA)			&(NOIPEA)		&(IPEA)			&(NOIPEA)\\
			\hline
 			Singlet			&174	&0.57	&0.14	&0.27	&0.14	&0.12	&0.16	&0.14\\
			Triplet			&110	&0.34	&0.07	&0.29	&0.07	&0.06	&0.13	&0.11\\
			Valence 		&206	&0.45	&0.11	&0.33	&0.13	&0.10	&0.15	&0.12\\
			Rydberg			&78		&0.54	&0.13	&0.13	&0.08	&0.07	&0.14	&0.15\\
			$n \to \pis$	&78		&0.44	&0.08	&0.44	&0.13	&0.10	&0.12	&0.10\\
			$\pi \to \pis$ 	&119	&0.46	&0.12	&0.27	&0.13	&0.10	&0.18	&0.14\\
			Double			&9		&0.46	&0.11	&0.22	&0.12	&0.09	&0.14	&0.13\\
			3 non-H atoms	&39		&0.38	&0.07	&0.21	&0.06	&0.05	&0.10	&0.08\\
			4 non-H	atoms	&94		&0.46	&0.11	&0.22	&0.12	&0.09	&0.14	&0.13\\
			5-6	non-H atoms	&151	&0.51	&0.12	&0.33	&0.12	&0.11	&0.17	&0.15\\	
		\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Values from the present study.}
	\fnt[2]{Values taken from Ref.~\onlinecite{Sarka_2022}.}
\end{table*}
%%% %%% %%% %%%

A detailed discussion of each individual molecule can be found in Ref.~\onlinecite{Sarka_2022} where we also report relevant values from the literature.
Here, we focus on global trends.
The exhaustive list of CASPT2 and CASPT3 transitions can be found in Table \ref{tab:BigTab} and the distribution of the errors are represented in Fig.~\ref{fig:PT2_vs_PT3}.
Various statistical indictors are given in Table \ref{tab:stat} while MAEs determined for several subsets of transitions (singlet, triplet, valence, Rydberg, $n\to\pis$, $\pi\to\pis$, and double excitations) and system sizes (3 non-H atoms, 4 non-H atoms, and 5-6 non-H atoms) are reported in Table \ref{tab:stat_subset}.

From the different statistical quantities reported in Table \ref{tab:stat}, one can highlight the two following observations.
First, as previously reported, \cite{Werner_1996,Grabarek_2016} CASPT3 vertical excitation energies are much less sensitive to the IPEA shift, which drastically alter the accuracy of CASPT2. 
For example, the MAEs of CASPT3(IPEA) and CASPT3(NOIPEA) are amazingly close (\SI{0.11}{} and \SI{0.09}{\eV}), while the MAEs of CASPT2(IPEA) and CASPT2(NOIPEA) are drastically different (\SI{0.27}{} and \SI{0.11}{\eV}).
Importantly, CASPT3 seems to perform slightly better without IPEA shift, which is a great outcome.
Second, CASPT3 (with or without IPEA) has a similar accuracy as CASPT2(IPEA).
All these observations stand for each subset of excitations and irrespectively of the system size (see Table \ref{tab:stat_subset}).

Interestingly, CASPT3(NOIPEA) yields MAEs for each subset that is almost systematically below \SI{0.1}{\eV}, except for the singlet subsets which is polluted by some states showing larger deviations at the CASPT2 and CASPT3 levels.
\titou{Here, discuss difficult case where we have a large (positive) error in CASPT2 and CASPT3.
This is due to the relative small size of the active space and, more precisely, to the lack of direct $\sig$-$\pi$ coupling in the active space which are known to be important in such ionic states. \cite{Garniron_2018}
These errors could be alleviated by using a RAS space.}

%%% TABLE III %%%
\begin{table*}
	\caption{CASPT2 and CASPT3 timings (in seconds) for a selection of systems and transitions.}
	\label{tab:timings}
    \begin{ruledtabular}
        \begin{tabular}{llcccccccc}
        	System	&	Transition	&Active	&\# electrons	&\# basis 	&\# CAS 	&\# contracted	&\# uncontracted 	&CPU 	&CPU\\
        			&			&Space	&			&functions	&det.		&config. 		&config.			&CASPT2	&CASPT3	\\
        	\hline
        	Acetone		&$^1A_2(n,\pis)$		&(6e,6o)	&32	&322	&104	&$3.86 \times 10^6$	&$1.49 \times 10^8$	&12.50		&33.25\\
        	Pyrrole		&$^1A_2(\pi,3s)$		&(6e,6o)	&36	&345	&96		&$4.79 \times 10^6$	&$2.04 \times 10^8$	&13.24		&49.36\\
        	Imidazole	&$^1A''(\pi,3s)$		&(8e,7o)	&36	&322	&600	&$1.46 \times 10^7$	&$1.82 \times 10^9$	&193.93		&282.62\\
        	Pyrazine	&$^1B_{3u}(n,\pis)$		&(10e,8o)	&42	&368	&392	&$6.95 \times 10^6$	&$6.38 \times 10^8$	&29.58		&174.96\\
        \end{tabular}
    \end{ruledtabular}
\end{table*}
%%% %%% %%% %%%

\titou{Table \ref{tab:timings} reports the evolution of the CPU timings for CASPT2 and CASPT3 as a function of the size of the active space.
It is particularly instructive to study the increase in CPU times as the number of external configuration grows.}

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\section{Conclusion}
\label{sec:ccl}
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In the present study, we have benchmarked, using 284 highly-accurate electronic transitions extracted from the QUEST database, \cite{Veril_2021} the third-order multi-reference perturbation theory method, CASPT3, by computing vertical excitation energies with and without IPEA shift.
The two take-home messages are that: 
i) CASPT3 transition energies are almost independent of the IPEA shift; 
ii) CASPT2(IPEA) and CASPT3 have very similar accuracy.
The global trends are also true for specific sets of excitations and various system size.

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\begin{acknowledgements}
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2021-18005.
DJ is indebted to the CCIPL computational center installed in Nantes for a generous allocation of computational time.
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for funding.
\end{acknowledgements}
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\section*{Supporting information available}
\label{sec:SI}
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%\section*{Data availability statement}
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%The data that support the findings of this study are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.

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\bibliography{CASPT3}
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