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2
Cover_Letter/CoverLetter.tex
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@ -24,7 +24,7 @@ In particular, by computing vertical excitation energies with and without the in
\end{itemize}
These global trends are also true for specific sets of excitations and various system sizes.
We suggest Stefano Battaglia, Javier Segarra-Marti, Peter Knowles, Hans-Joachim Werner, Leticia Gonz\'alez, Celestino Angeli, and Donald Truhlar as potential referees.
We suggest Stefano Battaglia, Javier Segarra-Marti, Peter Knowles, Hans-Joachim Werner, Leticia Gonz\'alez, and Donald Truhlar as potential referees.
We look forward to hearing from you.
\closing{Sincerely, the authors.}

78
Manuscript/CASPT3.bib
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@ -1,13 +1,68 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-04-08 09:54:20 +0200
%% Created for Pierre-Francois Loos at 2022-04-13 16:47:01 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Olsen_1988,
author = {J. Olsen and B. O. Roos and P. Jorgensen and H. J. A. Jensen},
date-added = {2022-04-13 16:31:23 +0200},
date-modified = {2022-04-13 16:33:11 +0200},
doi = {10.1063/1.455063},
journal = {J. Chem. Phys.},
pages = {2185},
title = {Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces},
volume = {89},
year = {1988}}
@article{Fulscher_1994,
author = {M. P. Fulscher and B. O. Roos},
date-added = {2022-04-13 16:30:07 +0200},
date-modified = {2022-04-13 16:31:10 +0200},
doi = {10.1007/BF01113393},
journal = {Theor. Chim. Acta},
pages = {403},
title = {The excited states of pyrazine: A basis set study},
volume = {87},
year = {1994}}
@article{Tran_2019,
author = {T. Tran and J. Segarra-Marti and M. J. Bearpark and M. A. Robb},
date-added = {2022-04-13 16:28:18 +0200},
date-modified = {2022-04-13 16:33:37 +0200},
doi = {10.1021/acs.jpca.9b03715},
journal = {J. Phys. Chem. A},
pages = {5223},
title = {Molecular Vertical Excitation Energies Studied with First-Order RASSCF (RAS[1,1]): Balancing Covalent and Ionic Excited States},
volume = {123},
year = {2019}}
@article{Boggio-Pasqua_2004,
author = {M. Boggio-Pasqua and M. J. Bearpark and M. Klene and M. A. Robb},
date-added = {2022-04-13 16:27:14 +0200},
date-modified = {2022-04-13 16:29:46 +0200},
doi = {10.1063/1.1690756},
journal = {J. Chem. Phys.},
pages = {7849},
title = {A computational strategy for geometry optimization of ionic and covalent excited states, applied to butadiene and hexatriene},
volume = {120},
year = {2004}}
@article{Borden_1996,
author = {W. T. Borden and E. R. Davidson},
date-added = {2022-04-13 16:25:02 +0200},
date-modified = {2022-04-13 16:29:56 +0200},
doi = {10.1021/ar950134v},
journal = {Acc. Chem. Res.},
pages = {67},
title = {The Importance of Including Dynamic Electron Correlation in ab Initio Calculations},
volume = {29},
year = {1996}}
@article{Boggio-Pasqua_2007,
author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
date-added = {2022-04-04 23:13:51 +0200},
@ -65,9 +120,9 @@
@article{Davidson_1996,
author = {Davidson, Ernest R.},
date-added = {2022-04-04 22:37:02 +0200},
date-modified = {2022-04-04 22:37:02 +0200},
date-modified = {2022-04-13 16:03:23 +0200},
doi = {10.1021/jp952794n},
journal = {J. Phys. Chem},
journal = {J. Phys. Chem.},
number = {15},
pages = {6161-6166},
title = {The Spatial Extent of the V State of Ethylene and Its Relation to Dynamic Correlation in the Cope Rearrangement},
@ -78,9 +133,9 @@
@article{BenAmor_2020,
author = {Ben Amor,Nadia and No{\^u}s,Camille and Trinquier,Georges and Malrieu,Jean-Paul},
date-added = {2022-04-04 22:36:48 +0200},
date-modified = {2022-04-04 22:36:48 +0200},
date-modified = {2022-04-13 16:03:37 +0200},
doi = {10.1063/5.0011582},
journal = {J. Chem. Phys},
journal = {J. Chem. Phys.},
number = {4},
pages = {044118},
title = {Spin polarization as an electronic cooperative effect},
@ -1076,11 +1131,12 @@
@article{Sarkar_2022,
author = {R. Sarkar and P. F. Loos and M. Boggio-Pasqua and D. Jacquemin.},
date-added = {2022-03-16 10:53:25 +0100},
date-modified = {2022-04-05 13:23:50 +0200},
date-modified = {2022-04-13 16:01:34 +0200},
doi = {10.1021/acs.jctc.1c01197},
journal = {J. Chem. Theory Comput.},
pages = {in press},
pages = {2418},
title = {Assessing the performances of CASPT2 and NEVPT2 for vertical excitation energies,},
volume = {18},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c01197}}
@ -2930,12 +2986,14 @@
@article{Garniron_2018,
author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
date-added = {2021-05-06 15:31:25 +0200},
date-modified = {2021-05-06 15:31:25 +0200},
date-modified = {2022-04-13 16:02:40 +0200},
doi = {10.1063/1.5044503},
journal = {J. Chem. Phys.},
pages = {064103},
title = {Selected Configuration Interaction Dressed by Perturbation},
volume = {149},
year = {2018}}
year = {2018},
bdsk-url-1 = {https://doi.org/10.1063/1.5044503}}
@article{Gauss_2006,
author = {Gauss,J{\"u}rgen and Tajti,Attila and K{\'a}llay,Mih{\'a}ly and Stanton,John F. and Szalay,P{\'e}ter G.},

566
Manuscript/CASPT3.tex
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@ -90,7 +90,7 @@
% Abstract
\begin{abstract}
Based on 284 reference vertical transition energies of various natures (singlet, triplet, valence, Rydberg, $n\to\pi^*$, $\pi\to\pi^*$, and double excitations) extracted from the QUEST database, we assess the accuracy of third-order multireference perturbation theory, CASPT3, in the context of molecular excited states.
Based on 280 reference vertical transition energies of various natures (singlet, triplet, valence, Rydberg, $n\to\pi^*$, $\pi\to\pi^*$, and double excitations) 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 disputable 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, irrespective of the transition type and system size, with a small reduction of the mean absolute error to $0.09$ eV when the IPEA shift is switched off.
%\bigskip
@ -134,19 +134,16 @@ A second pitfall was brought to light by Andersson \textit{et al.} \cite{Anderss
A cure was quickly proposed via the introduction of an additional parameter in the zeroth-order Hamiltonian, the 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 that its impact is significantly 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 with a large basis set (aug-cc-pVTZ) in 35 small- and medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarkar_2022}
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 280 excited states of diverse natures (singlet, triplet, valence, Rydberg, $n\to\pis$, $\pi\to\pis$, and double excitations) computed with a large basis set (aug-cc-pVTZ) in 35 small- and medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarkar_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 be crucial as neglecting it increases the mean absolute error to \SI{0.27}{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 the very same set of electronic transitions as the one used in Ref.~\onlinecite{Sarkar_2022}
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 the same set of electronic transitions as the one used in Ref.~\onlinecite{Sarkar_2022}
Although 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, to the best of our knowledge, the first comprehensive benchmark of CASPT3 and allows assessing its accuracy in the framework of electronically excited states.
%DJ: Ce sont des phrases de ccls, dŽjˆ dans l'abstract, pq aussi dans l'Intro ?
%Based on the same 284 highly-accurate vertical excitation energies from the QUEST database, we show that CASPT3 only provides a very slight improvement over CASPT2 as far as accuracy is concerned.
%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.
We underline that, although a third-order version of NEVPT has been developed \cite{Angeli_2006} and has been used in some 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, no NEVPT3 implementation are publicly available.
We underline that, although a third-order version of NEVPT has been developed \cite{Angeli_2006} and has been used in some 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, no NEVPT3 implementation is publicly available.
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -191,7 +188,7 @@ A detailed discussion of each individual molecule can be found in Ref.~\onlineci
We therefore decided to focus on global trends here.
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}.
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 the largest positive and negative deviations [Max($+$) and Max($-$), respectively].
These are given in Table \ref{tab:stat} considering the 265 ``safe'' TBEs (out of 284) for which chemical accuracy is assumed (absolute error below \SI{0.043}{\eV}).
These are given in Table \ref{tab:stat} considering the 265 ``safe'' TBEs (out of 280) for which chemical accuracy is assumed (absolute error below \SI{0.043}{\eV}).
The MAEs determined for 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) can be found in Table \ref{tab:stat_subset}.
Error patterns for selected subsets are reported in {\SupMat}.
@ -243,272 +240,268 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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 &$^1\Delta_g(\text{double})$&V &1.0 &5.22 &\Y &4.98 &5.08 &4.85 &5.20 &5.19\\
39 & &$^1\Sigma^+_g(\text{double})$&V&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 &Cyclopropenethione &$^1A_2(n,\pis)$ &V &89.6 &3.41 &\Y &3.44 &3.43 &3.14 &3.46 &3.40\\
68 & &$^1B_1(n,\pis)$ &V &84.8 &3.45 &\Y &3.57 &3.45 &3.17 &3.52 &3.46\\
69 & &$^1B_2(\pi,\pis)$ &V &83.0 &4.60 &\Y &4.51 &4.64 &4.35 &4.66 &4.61\\
70 & &$^1B_2(n,3s)$ &R &91.8 &5.34 &\Y &4.59 &5.25 &5.15 &5.25 &5.22\\
71 & &$^1A_1(\pi,\pis)$ &V &89.0 &5.46 &\Y &6.46 &5.84 &5.32 &5.88 &5.75\\
72 & &$^1B_2(n,3p)$ &R &91.3 &5.92 &\Y &5.27 &5.93 &5.86 &5.92 &5.90\\
73 & &$^3A_2(n,\pis)$ &V &97.2 &3.28 &\Y &3.26 &3.28 &3.00 &3.33 &3.28\\
74 & &$^3B_1(n,\pis)$ &V &94.5 &3.32 &\Y &3.51 &3.35 &3.07 &3.42 &3.36\\
75 & &$^3B_2(\pi,\pis)$ &V &96.5 &4.01 &\Y &3.80 &3.97 &3.75 &3.99 &3.95\\
76 & &$^3A_1(\pi,\pis)$ &V &98.2 &4.01 &\Y &3.83 &4.01 &3.77 &4.00 &3.95\\
77 &Cyclopropenone &$^1B_1(n,\pis)$ &V &87.7 &4.26 &\Y &4.92 &4.12 &3.75 &4.40 &4.38\\
78 & &$^1A_2(n,\pis)$ &V &91.0 &5.55 &\Y &5.64 &5.62 &5.31 &5.67 &5.64\\
79 & &$^1B_2(n,3s)$ &R &90.8 &6.34 &\Y &5.68 &6.28 &6.21 &6.41 &6.44\\
80 & &$^1B_2(\pi,\pis)$ &V &86.5 &6.54 &\Y &6.40 &6.54 &6.20 &6.63 &6.62\\
81 & &$^1B_2(n,3p)$ &R &91.1 &6.98 &\Y &6.35 &6.84 &6.70 &6.99 &7.01\\
82 & &$^1A_1(n,3p)$ &R &91.2 &7.02 &\Y &6.84 &7.27 &7.03 &7.26 &7.24\\
83 & &$^1A_1(\pi,\pis)$ &V &90.8 &8.28 &\Y &10.42 &8.96 &8.11 &9.21 &9.07\\
84 & &$^3B_1(n,\pis)$ &V &96.0 &3.93 &\Y &4.72 &3.65 &3.28 &4.00 &3.98\\
85 & &$^3B_2(\pi,\pis)$ &V &97.9 &4.88 &\Y &4.39 &4.76 &4.60 &4.76 &4.74\\
86 & &$^3A_2(n,\pis)$ &V &97.5 &5.35 &\Y &5.40 &5.36 &5.06 &5.44 &5.42\\
87 & &$^3A_1(\pi,\pis)$ &V &98.1 &6.79 &\Y &6.59 &6.93 &6.61 &6.86 &6.82\\
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 &4.73 &\Y &4.55 &4.78 &4.53 &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 & &$^1A_2(\pi,3p)$ &R &94.4 &7.18 &\Y &6.38 &7.25 &7.05 &7.19 &7.15\\
141 & &$^1A_1(\pi,\pis)$ &V &92.4 &7.25 &\Y & & &&&\\
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 &Methylenecyclopropene&$^1B_2(\pi,\pis)$ &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 & &$^1A'(n,3s)$ &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 &4.59 &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 &3.93 &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-C1 &$^1B_2(\pi,\pis)$ &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 &8.83 &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\\
25 & &$^3B_{1u}(\pi,\pis)$ &V &98.6 &4.16 &\Y &3.85 &4.22 &3.92 &4.14 &4.08\\
26 & &$^3E_{1u}(\pi,\pis)$ &V &97.1 &4.85 &\Y &4.85 &4.89 &4.51 &4.87 &4.80\\
27 & &$^3B_{2u}(\pi,\pis)$ &V &98.1 &5.81 &\Y &6.75 &5.85 &5.40 &5.90 &5.81\\
28 &Butadiene &$^1B_u(\pi,\pis)$ &V &93.3 &6.22 &\Y &6.65 &6.76 &6.52 &6.72 &6.65\\
29 & &$^1B_g(\pi,3s)$ &R &94.1 &6.33 &\Y &5.94 &6.49 &6.32 &6.43 &6.38\\
30 & &$^1A_g(\pi,\pis)$ &V &75.1 &6.50 &\Y &6.99 &6.74 &6.30 &6.73 &6.66\\
31 & &$^1A_u(\pi,3p)$ &R &94.1 &6.64 &\Y &5.95 &6.74 &6.64 &6.70 &6.67\\
32 & &$^1A_u(\pi,3p)$ &R &94.1 &6.80 &\Y &6.12 &6.95 &6.84 &6.90 &6.86\\
33 & &$^1B_u(\pi,3p)$ &R &93.8 &7.68 &\Y &7.93 &7.60 &7.30 &7.62 &7.54\\
34 & &$^3B_u(\pi,\pis)$ &V &98.4 &3.36 &\Y &3.55 &3.40 &3.19 &3.40 &3.35\\
35 & &$^3A_g(\pi,\pis)$ &V &98.7 &5.20 &\Y &5.52 &5.32 &4.93 &5.29 &5.19\\
36 & &$^3B_g(\pi,3s)$ &R &97.9 &6.29 &\Y &5.89 &6.44 &6.27 &6.38 &6.33\\
37 &Carbon trimer &$^1\Delta_g(\text{double})$&V &1.0 &5.22 &\Y &4.98 &5.08 &4.85 &5.20 &5.19\\
38 & &$^1\Sigma^+_g(\text{double})$&V&1.0 &5.91 &\Y &5.84 &5.82 &5.58 &5.92 &5.89\\
39 &Cyanoacetylene &$^1\Sigma^-(\pi,\pis)$ &V &94.3 &5.80 &\Y &6.54 &5.85 &5.47 &5.89 &5.81\\
40 & &$^1\Delta(\pi,\pis)$ &V &94.0 &6.07 &\Y &6.80 &6.13 &5.78 &6.17 &6.09\\
41 & &$^3\Sigma^+(\pi,\pis)$ &V &98.5 &4.44 &\Y &4.86 &4.45 &4.04 &4.52 &4.45\\
42 & &$^3\Delta(\pi,\pis)$ &V &98.2 &5.21 &\Y &5.64 &5.21 &4.86 &5.26 &5.19\\
43 & &$^1A''[F](\pi,\pis)$ &V &93.6 &3.54 &\Y &4.30 &3.67 &3.47 &3.64 &3.58\\
44 &Cyanoformaldehyde &$^1A''(n,\pis)$ &V &89.8 &3.81 &\Y &4.02 &3.98 &3.67 &3.94 &3.89\\
45 & &$^1A''(\pi,\pis)$ &V &91.9 &6.46 &\Y &7.61 &6.79 &6.43 &6.77 &6.67\\
46 & &$^3A''(n,\pis)$ &V &97.6 &3.44 &\Y &3.52 &3.46 &3.25 &3.51 &3.50\\
47 & &$^3A'(\pi,\pis)$ &V &98.4 &5.01 &\Y &4.98 &5.25 &5.03 &5.16 &5.12\\
48 &Cyanogen &$^1\Sigma_u^-(\pi,\pis)$ &V &94.1 &6.39 &\Y &7.14 &6.40 &6.03 &6.46 &6.39\\
49 & &$^1\Delta_u(\pi,\pis)$ &V &93.4 &6.66 &\Y &7.46 &6.70 &6.35 &6.75 &6.68\\
50 & &$^3\Sigma_u^+(\pi,\pis)$ &V &98.5 &4.91 &\Y &5.28 &4.85 &4.46 &4.95 &4.89\\
51 & &$^1\Sigma_u^-[F](\pi,\pis)$&V &93.4 &5.05 &\Y &5.68 &5.07 &4.75 &5.11 &5.04\\
52 &Cyclopentadiene &$^1B_2(\pi,\pis)$ &V &93.8 &5.56 &\Y &6.71 &5.96 &5.62 &6.06 &5.99\\
53 & &$^1A_2(\pi,3s)$ &R &94.0 &5.78 &\Y &5.21 &5.88 &5.78 &5.81 &5.77\\
54 & &$^1B_1(\pi,3p)$ &R &94.2 &6.41 &\Y &6.08 &6.59 &6.44 &6.47 &6.41\\
55 & &$^1A_2(\pi,3p)$ &R &93.8 &6.46 &\Y &5.78 &6.55 &6.46 &6.45 &6.41\\
56 & &$^1B_2(\pi,3p)$ &R &94.2 &6.56 &\Y &6.16 &6.72 &6.56 &6.61 &6.54\\
57 & &$^1A_1(\pi,\pis)$ &V &78.9 &6.52 &\N &6.49 &6.63 &6.13 &6.59 &6.50\\
58 & &$^3B_2(\pi,\pis)$ &V &98.4 &3.31 &\Y &3.26 &3.34 &3.09 &3.31 &3.26\\
59 & &$^3A_1(\pi,\pis)$ &V &98.6 &5.11 &\Y &4.92 &5.14 &4.78 &5.10 &5.03\\
60 & &$^3A_2(\pi,3s)$ &R &97.9 &5.73 &\Y &5.53 &5.91 &5.74 &5.81 &5.75\\
61 & &$^3B_1(\pi,3p)$ &R &97.9 &6.36 &\Y &6.05 &6.56 &6.40 &6.43 &6.37\\
62 &Cyclopropene &$^1B_1(\sig,\pis)$ &V &92.8 &6.68 &\Y &7.48 &6.86 &6.58 &6.85 &6.77\\
63 & &$^1B_2(\pi,\pis)$ &V &95.1 &6.79 &\Y &7.47 &6.89 &6.47 &6.96 &6.87\\
64 & &$^3B_2(\pi,\pis)$ &V &98.0 &4.38 &\Y &4.60 &4.47 &4.27 &4.46 &4.40\\
65 & &$^3B_1(\sig,\pis)$ &V &98.9 &6.45 &\Y &7.08 &6.56 &6.32 &6.55 &6.47\\
66 &Cyclopropenethione &$^1A_2(n,\pis)$ &V &89.6 &3.41 &\Y &3.44 &3.43 &3.14 &3.46 &3.40\\
67 & &$^1B_1(n,\pis)$ &V &84.8 &3.45 &\Y &3.57 &3.45 &3.17 &3.52 &3.46\\
68 & &$^1B_2(\pi,\pis)$ &V &83.0 &4.60 &\Y &4.51 &4.64 &4.35 &4.66 &4.61\\
69 & &$^1B_2(n,3s)$ &R &91.8 &5.34 &\Y &4.59 &5.25 &5.15 &5.25 &5.22\\
70 & &$^1A_1(\pi,\pis)$ &V &89.0 &5.46 &\Y &6.46 &5.84 &5.32 &5.88 &5.75\\
71 & &$^1B_2(n,3p)$ &R &91.3 &5.92 &\Y &5.27 &5.93 &5.86 &5.92 &5.90\\
72 & &$^3A_2(n,\pis)$ &V &97.2 &3.28 &\Y &3.26 &3.28 &3.00 &3.33 &3.28\\
73 & &$^3B_1(n,\pis)$ &V &94.5 &3.32 &\Y &3.51 &3.35 &3.07 &3.42 &3.36\\
74 & &$^3B_2(\pi,\pis)$ &V &96.5 &4.01 &\Y &3.80 &3.97 &3.75 &3.99 &3.95\\
75 & &$^3A_1(\pi,\pis)$ &V &98.2 &4.01 &\Y &3.83 &4.01 &3.77 &4.00 &3.95\\
76 &Cyclopropenone &$^1B_1(n,\pis)$ &V &87.7 &4.26 &\Y &4.92 &4.12 &3.75 &4.40 &4.38\\
77 & &$^1A_2(n,\pis)$ &V &91.0 &5.55 &\Y &5.64 &5.62 &5.31 &5.67 &5.64\\
78 & &$^1B_2(n,3s)$ &R &90.8 &6.34 &\Y &5.68 &6.28 &6.21 &6.41 &6.44\\
79 & &$^1B_2(\pi,\pis)$ &V &86.5 &6.54 &\Y &6.40 &6.54 &6.20 &6.63 &6.62\\
80 & &$^1B_2(n,3p)$ &R &91.1 &6.98 &\Y &6.35 &6.84 &6.70 &6.99 &7.01\\
81 & &$^1A_1(n,3p)$ &R &91.2 &7.02 &\Y &6.84 &7.27 &7.03 &7.26 &7.24\\
82 & &$^1A_1(\pi,\pis)$ &V &90.8 &8.28 &\Y &10.42 &8.96 &8.11 &9.21 &9.07\\
83 & &$^3B_1(n,\pis)$ &V &96.0 &3.93 &\Y &4.72 &3.65 &3.28 &4.00 &3.98\\
84 & &$^3B_2(\pi,\pis)$ &V &97.9 &4.88 &\Y &4.39 &4.76 &4.60 &4.76 &4.74\\
85 & &$^3A_2(n,\pis)$ &V &97.5 &5.35 &\Y &5.40 &5.36 &5.06 &5.44 &5.42\\
86 & &$^3A_1(\pi,\pis)$ &V &98.1 &6.79 &\Y &6.59 &6.93 &6.61 &6.86 &6.82\\
87 &Diacetylene &$^1\Sigma_u^-(\pi,\pis)$ &V &94.4 &5.33 &\Y &6.13 &5.42 &5.01 &5.45 &5.36\\
88 & &$^1\Delta_u(\pi,\pis)$ &V &94.1 &5.61 &\Y &6.39 &5.68 &5.30 &5.72 &5.63\\
89 & &$^3\Sigma_u^+(\pi,\pis)$ &V &98.5 &4.10 &\Y &4.54 &4.11 &3.67 &4.17 &4.09\\
90 & &$^3\Delta_u(\pi,\pis)$ &V &98.2 &4.78 &\Y &5.28 &4.82 &4.45 &4.86 &4.78\\
91 &Diazomethane &$^1A_2(\pi,\pis)$ &V &90.1 &3.14 &\Y &3.27 &3.13 &2.92 &3.09 &3.04\\
92 & &$^1B_1(\pi,3s)$ &R &93.8 &5.54 &\Y &4.59 &5.50 &5.30 &5.48 &5.45\\
93 & &$^1A_1(\pi,\pis)$ &V &91.4 &5.90 &\Y &5.65 &6.21 &5.92 &6.18 &6.13\\
94 & &$^3A_2(\pi,\pis)$ &V &97.7 &2.79 &\Y &3.02 &2.87 &2.67 &2.84 &2.79\\
95 & &$^3A_1(\pi,\pis)$ &V &98.6 &4.05 &\Y &4.27 &4.10 &3.88 &4.06 &4.01\\
96 & &$^3B_1(\pi,3s)$ &R &98.0 &5.35 &\Y &4.45 &5.34 &5.15 &5.33 &5.30\\
97 & &$^3A_1(\pi,3p)$ &R &98.5 &6.82 &\Y &6.34 &7.00 &6.76 &6.96 &6.91\\
98 & &$^1A''[F](\pi,\pis)$ &V &87.4 &0.71 &\Y &0.72 &0.69 &0.52 &0.66 &0.62\\
99 &Formamide &$^1A''(n,\pis)$ &V &90.8 &5.65 &\Y &5.95 &5.66 &5.45 &5.71 &5.67\\
100 & &$^1A'(n,3s)$ &R &88.6 &6.77 &\Y &6.17 &6.80 &6.64 &6.82 &6.81\\
101 & &$^1A'(n,3p)$ &R &89.6 &7.38 &\N &6.74 &7.45 &7.32 &7.46 &7.46\\
102 & &$^1A'(\pi,\pis)$ &V &89.3 &7.63 &\N &8.80 &7.88 &7.13 &7.95 &7.78\\
103 & &$^3A''(n,\pis)$ &V &97.7 &5.38 &\Y &5.89 &5.36 &5.16 &5.41 &5.37\\
104 & &$^3A'(\pi,\pis)$ &V &98.2 &5.81 &\Y &6.10 &5.88 &5.62 &5.91 &5.87\\
105 &Furan &$^1A_2(\pi,3s)$ &R &93.8 &6.09 &\Y &5.26 &6.16 &6.04 &6.06 &6.02\\
106 & &$^1B_2(\pi,\pis)$ &V &93.0 &6.37 &\Y &7.78 &6.59 &6.02 &6.80 &6.71\\
107 & &$^1A_1(\pi,\pis)$ &V &92.4 &6.56 &\Y &6.73 &6.66 &6.10 &6.69 &6.62\\
108 & &$^1B_1(\pi,3p)$ &R &93.9 &6.64 &\Y &6.07 &6.79 &6.63 &6.65 &6.60\\
109 & &$^1A_2(\pi,3p)$ &R &93.6 &6.81 &\Y &5.87 &6.87 &6.77 &6.76 &6.72\\
110 & &$^1B_2(\pi,3p)$ &R &93.5 &7.24 &\Y &6.54 &7.11 &6.84 &6.96 &6.88\\
111 & &$^3B_2(\pi,\pis)$ &V &98.4 &4.20 &\Y &3.94 &4.26 &4.01 &4.17 &4.12\\
112 & &$^3A_1(\pi,\pis)$ &V &98.1 &5.46 &\Y &5.41 &5.50 &5.09 &5.47 &5.40\\
113 & &$^3A_2(\pi,3s)$ &R &97.9 &6.02 &\Y &5.57 &6.16 &5.99 &6.05 &5.99\\
114 & &$^3B_1(\pi,3p)$ &R &97.9 &6.59 &\Y &6.04 &6.76 &6.60 &6.62 &6.56\\
115 &Glyoxal &$^1A_u(n,\pis)$ &V &91.0 &2.88 &\Y &3.42 &2.82 &2.51 &2.97 &2.94\\
116 & &$^1B_g(n,\pis)$ &V &88.3 &4.24 &\Y &4.68 &4.21 &3.89 &4.36 &4.31\\
117 & &$^1A_g(\text{double})$ &V &0.5 &5.61 &\Y &5.92 &5.37 &5.21 &5.53 &5.55\\
118 & &$^1B_g(n,\pis)$ &V &83.9 &6.57 &\Y &7.35 &6.52 &5.98 &6.76 &6.72\\
119 & &$^1B_u(n,3p)$ &R &91.7 &7.71 &\Y &7.04 &7.61 &7.34 &7.78 &7.81\\
120 & &$^3A_u(n,\pis)$ &V &97.6 &2.49 &\Y &3.06 &2.41 &2.12 &2.57 &2.55\\
121 & &$^3B_g(n,\pis)$ &V &97.4 &3.89 &\Y &4.61 &3.90 &3.53 &4.04 &4.01\\
122 & &$^3B_u(\pi,\pis)$ &V &98.5 &5.15 &\Y &5.46 &5.14 &4.91 &5.17 &5.14\\
123 & &$^3A_g(\pi,\pis)$ &V &98.8 &6.30 &\Y &6.69 &6.32 &6.02 &6.33 &6.27\\
124 &Imidazole &$^1A''(\pi,3s)$ &R &93.0 &5.70 &\Y &5.04 &5.88 &5.66 &5.74 &5.68\\
125 & &$^1A'(\pi,3p)$ &R &90.0 &6.41 &\Y &6.18 &6.69 &6.45 &6.61 &6.56\\
126 & &$^1A''(\pi,3p)$ &R &93.6 &6.50 &\Y &5.43 &6.57 &6.47 &6.47 &6.44\\
127 & &$^1A''(n,\pis)$ &V &89.0 &6.71 &\Y &7.13 &6.94 &6.57 &6.92 &6.85\\
128 & &$^1A'(\pi,\pis)$ &V &88.9 &6.86 &\Y &6.73 &6.88 &6.46 &6.89 &6.83\\
129 & &$^1A'(n,3s)$ &R &89.0 &7.00 &\Y &6.36 &7.10 &6.91 &7.09 &7.07\\
130 & &$^3A'(\pi,\pis)$ &V &98.3 &4.73 &\Y &4.55 &4.78 &4.53 &4.73 &4.68\\
131 & &$^3A''(\pi,3s)$ &R &97.6 &5.66 &\Y &5.03 &5.86 &5.63 &5.72 &5.66\\
132 & &$^3A'(\pi,\pis)$ &V &97.9 &5.74 &\Y &5.69 &5.85 &5.48 &5.80 &5.72\\
133 & &$^3A''(n,\pis)$ &V &97.3 &6.31 &\Y &6.58 &6.44 &6.10 &6.43 &6.37\\
134 &Isobutene &$^1B_1(\pi,3s)$ &R &94.1 &6.46 &\Y &6.21 &6.74 &6.59 &6.64 &6.57\\
135 & &$^1A_1(\pi,3p)$ &R &94.2 &7.01 &\Y &6.90 &7.32 &7.14 &7.24 &7.18\\
136 & &$^3A_1(\pi,\pis)$ &V &98.9 &4.53 &\Y &4.66 &4.59 &4.41 &4.58 &4.53\\
137 &Ketene &$^1A_2(\pi,\pis)$ &V &91.0 &3.86 &\Y &3.98 &3.92 &3.70 &3.90 &3.85\\
138 & &$^1B_1(\pi,3s)$ &R &93.9 &6.01 &\Y &5.22 &5.99 &5.79 &6.00 &5.97\\
139 & &$^1A_2(\pi,3p)$ &R &94.4 &7.18 &\Y &6.38 &7.25 &7.05 &7.19 &7.15\\
140 & &$^3A_2(\pi,\pis)$ &V &91.0 &3.77 &\Y &3.92 &3.81 &3.59 &3.79 &3.74\\
141 & &$^3A_1(\pi,\pis)$ &V &98.6 &5.61 &\Y &5.79 &5.65 &5.43 &5.63 &5.59\\
142 & &$^3B_1(\pi,3s)$ &R &98.1 &5.79 &\Y &5.05 &5.79 &5.60 &5.80 &5.77\\
143 & &$^3A_2(\pi,3p)$ &R &94.4 &7.12 &\Y &6.35 &7.22 &7.01 &7.15 &7.11\\
144 & &$^1A''[F](\pi,\pis)$ &V &87.9 &1.00 &\Y &0.95 &1.05 &0.88 &1.00 &0.95\\
145 &Methylenecyclopropene&$^1B_2(\pi,\pis)$ &V &85.4 &4.28 &\Y &4.47 &4.40 &4.12 &4.39 &4.33\\
146 & &$^1B_1(\pi,3s)$ &R &93.6 &5.44 &\Y &4.92 &5.57 &5.44 &5.46 &5.41\\
147 & &$^1A_2(\pi,3p)$ &R &93.3 &5.96 &\Y &5.37 &6.09 &5.97 &5.97 &5.92\\
148 & &$^1A_1(\pi,\pis)$ &V &92.8 &6.12 &\N &5.37 &6.26 &6.16 &6.17 &6.13\\
149 & &$^3B_2(\pi,\pis)$ &V &97.2 &3.49 &\Y &3.44 &3.57 &3.34 &3.55 &3.49\\
150 & &$^3A_1(\pi,\pis)$ &V &98.6 &4.74 &\Y &4.60 &4.82 &4.58 &4.77 &4.72\\
151 &Nitrosomethane &$^1A''(n,\pis)$ &V &93.0 &1.96 &\Y &2.12 &1.84 &1.60 &1.94 &1.91\\
152 & &$^1A'(\text{double})$ &V &2.5 &4.76 &\Y &4.74 &4.69 &4.67 &4.71 &4.71\\
153 & &$^1A'(n,3s)$ &R &90.8 &6.29 &\Y &5.87 &6.32 &6.07 &6.34 &6.31\\
154 & &$^3A''(n,\pis)$ &V &98.4 &1.16 &\Y &1.31 &1.00 &0.75 &1.12 &1.09\\
155 & &$^3A'(\pi,\pis)$ &V &98.9 &5.60 &\Y &5.52 &5.52 &5.37 &5.54 &5.50\\
156 & &$^1A''[F](n,\pis)$ &V &92.7 &1.67 &\Y &1.83 &1.55 &1.32 &1.66 &1.62\\
157 &Propynal &$^1A''(n,\pis)$ &V &89.0 &3.80 &\Y &4.00 &3.92 &3.64 &3.90 &3.86\\
158 & &$^1A''(\pi,\pis)$ &V &92.9 &5.54 &\Y &6.62 &5.82 &5.49 &5.81 &5.72\\
159 & &$^3A''(n,\pis)$ &V &97.4 &3.47 &\Y &3.52 &3.48 &3.26 &3.52 &3.50\\
160 & &$^3A'(\pi,\pis)$ &V &98.3 &4.47 &\Y &4.69 &4.59 &4.30 &4.59 &4.54\\
161 &Pyrazine &$^1B_{3u}(n,\pis)$ &V &90.1 &4.15 &\Y &4.76 &4.09 &3.66 &4.31 &4.30\\
162 & &$^1A_u(n,\pis)$ &V &88.6 &4.98 &\Y &5.90 &4.76 &4.26 &5.10 &5.10\\
163 & &$^1B_{2u}(\pi,\pis)$ &V &86.9 &5.02 &\Y &4.97 &5.13 &4.65 &5.09 &5.03\\
164 & &$^1B_{2g}(n,\pis)$ &V &85.6 &5.71 &\Y &5.80 &5.68 &5.27 &5.73 &5.70\\
165 & &$^1A_g(n,3s)$ &R &91.1 &6.65 &\Y &6.69 &6.66 &6.27 &6.81 &6.80\\
166 & &$^1B_{1g}(n,\pis)$ &V &84.2 &6.74 &\Y &7.16 &6.61 &6.07 &6.78 &6.76\\
167 & &$^1B_{1u}(\pi,\pis)$ &V &92.8 &6.88 &\Y &8.04 &7.14 &6.72 &7.20 &7.12\\
168 & &$^1B_{1g}(\pi,3s)$ &R &93.8 &7.21 &\Y &6.73 &7.41 &7.27 &7.24 &7.18\\
169 & &$^1B_{2u}(n,3p)$ &R &90.8 &7.24 &\Y &7.49 &7.34 &6.93 &7.43 &7.40\\
170 & &$^1B_{1u}(n,3p)$ &R &91.4 &7.44 &\Y &7.83 &7.55 &7.08 &7.64 &7.59\\
171 & &$^1B_{1u}(\pi,\pis)$ &V &90.5 &7.98 &\N &9.65 &8.59 &7.96 &8.68 &8.57\\
172 & &$^3B_{3u}(n,\pis)$ &V &97.3 &3.59 &\Y &4.16 &3.49 &3.08 &3.72 &3.71\\
173 & &$^3B_{1u}(\pi,\pis)$ &V &98.5 &4.35 &\Y &3.98 &4.44 &4.15 &4.34 &4.28\\
174 & &$^3B_{2u}(\pi,\pis)$ &V &97.6 &4.39 &\Y &4.62 &4.44 &4.09 &4.47 &4.41\\
175 & &$^3A_u(n,\pis)$ &V &96.1 &4.93 &\Y &5.85 &4.73 &4.21 &5.07 &5.07\\
176 & &$^3B_{2g}(n,\pis)$ &V &97.0 &5.08 &\Y &5.25 &5.04 &4.66 &5.14 &5.11\\
177 & &$^3B_{1u}(\pi,\pis)$ &V &97.0 &5.28 &\Y &5.15 &5.29 &4.92 &5.25 &5.19\\
178 &Pyridazine &$^1B_1(n,\pis)$ &V &89.0 &3.83 &\Y &4.29 &3.74 &3.36 &3.94 &3.92\\
179 & &$^1A_2(n,\pis)$ &V &86.9 &4.37 &\Y &4.83 &4.29 &3.87 &4.49 &4.48\\
180 & &$^1A_1(\pi,\pis)$ &V &85.8 &5.26 &\Y &5.12 &5.34 &4.87 &5.30 &5.25\\
181 & &$^1A_2(n,\pis)$ &V &86.2 &5.72 &\Y &6.26 &5.73 &5.19 &5.93 &5.89\\
182 & &$^1B_2(n,3s)$ &R &88.5 &6.17 &\Y &5.99 &6.18 &5.90 &6.28 &6.27\\
183 & &$^1B_1(n,\pis)$ &V &87.0 &6.37 &\Y &7.16 &6.50 &5.94 &6.72 &6.67\\
184 & &$^1B_2(\pi,\pis)$ &V &90.6 &6.75 &\Y &7.54 &7.26 &6.82 &7.25 &7.17\\
185 & &$^3B_1(n,\pis)$ &V &97.1 &3.19 &\Y &3.60 &3.08 &2.72 &3.29 &3.28\\
186 & &$^3A_2(n,\pis)$ &V &96.1 &4.11 &\Y &4.49 &4.01 &3.59 &4.20 &4.18\\
187 & &$^3B_2(\pi,\pis)$ &V &98.5 &4.34 &\N &3.93 &4.44 &4.13 &4.30 &4.24\\
188 & &$^3A_1(\pi,\pis)$ &V &97.3 &4.82 &\Y &4.93 &4.87 &4.48 &4.89 &4.83\\
189 &Pyridine &$^1B_1(n,\pis)$ &V &88.4 &4.95 &\Y &5.43 &5.15 &4.81 &5.18 &5.13\\
190 & &$^1B_2(\pi,\pis)$ &V &86.5 &5.14 &\Y &5.03 &5.18 &4.76 &5.15 &5.09\\
191 & &$^1A_2(n,\pis)$ &V &87.9 &5.40 &\Y &6.30 &5.46 &5.03 &5.63 &5.59\\
192 & &$^1A_1(\pi,\pis)$ &V &92.1 &6.62 &\Y &7.90 &6.92 &6.27 &7.04 &6.93\\
193 & &$^1A_1(n,3s)$ &R &89.7 &6.76 &\Y &6.40 &6.90 &6.67 &6.97 &6.96\\
194 & &$^1A_2(\pi,3s)$ &R &93.2 &6.82 &\Y &6.60 &7.08 &6.87 &6.88 &6.80\\
195 & &$^1B_1(\pi,3p)$ &R &93.6 &7.38 &\Y &7.12 &7.70 &7.51 &7.48 &7.40\\
196 & &$^1A_1(\pi,\pis)$ &V &90.5 &7.39 &\Y &9.49 &7.66 &6.63 &7.87 &7.70\\
197 & &$^1B_2(\pi,\pis)$ &V &90.0 &7.40 &\N &7.45 &7.92 &7.67 &7.80 &7.73\\
198 & &$^3A_1(\pi,\pis)$ &V &98.5 &4.30 &\Y &3.98 &4.40 &4.06 &4.29 &4.22\\
199 & &$^3B_1(n,\pis)$ &V &97.0 &4.46 &\Y &4.65 &4.48 &4.21 &4.57 &4.55\\
200 & &$^3B_2(\pi,\pis)$ &V &97.3 &4.79 &\Y &4.83 &4.86 &4.53 &4.81 &4.74\\
201 & &$^3A_1(\pi,\pis)$ &V &97.1 &5.04 &\Y &5.11 &5.09 &4.63 &5.09 &5.02\\
202 & &$^3A_2(n,\pis)$ &V &95.8 &5.36 &\Y &5.94 &5.33 &4.96 &5.53 &5.51\\
203 & &$^3B_2(\pi,\pis)$ &V &97.7 &6.24 &\Y &6.93 &6.40 &5.99 &6.43 &6.35\\
204 &Pyrimidine &$^1B_1(n,\pis)$ &V &88.6 &4.44 &\Y &4.85 &4.44 &4.07 &4.58 &4.55\\
205 & &$^1A_2(n,\pis)$ &V &88.5 &4.85 &\Y &5.52 &4.80 &4.36 &5.02 &5.00\\
206 & &$^1B_2(\pi,\pis)$ &V &86.3 &5.38 &\Y &5.28 &5.42 &4.98 &5.41 &5.36\\
207 & &$^1A_2(n,\pis)$ &V &86.7 &5.92 &\Y &6.70 &5.92 &5.32 &6.16 &6.10\\
208 & &$^1B_1(n,\pis)$ &V &86.7 &6.26 &\Y &7.20 &6.31 &5.65 &6.58 &6.53\\
209 & &$^1B_2(n,3s)$ &R &90.3 &6.70 &\Y &6.86 &6.85 &6.50 &6.89 &6.86\\
210 & &$^1A_1(\pi,\pis)$ &V &91.5 &6.88 &\Y &7.69 &7.31 &6.94 &7.29 &7.22\\
211 & &$^3B_1(n,\pis)$ &V &96.8 &4.09 &\Y &4.45 &4.05 &3.67 &4.20 &4.18\\
212 & &$^3A_1(\pi,\pis)$ &V &98.3 &4.51 &\N &4.22 &4.57 &4.25 &4.51 &4.44\\
213 & &$^3A_2(n,\pis)$ &V &96.5 &4.66 &\Y &5.20 &4.63 &4.16 &4.81 &4.78\\
214 & &$^3B_2(\pi,\pis)$ &V &97.4 &4.96 &\Y &5.10 &5.01 &4.60 &5.03 &4.97\\
215 &Pyrrole &$^1A_2(\pi,3s)$ &R &92.9 &5.24 &\Y &4.49 &5.44 &5.23 &5.28 &5.23\\
216 & &$^1B_1(\pi,3p)$ &R &92.4 &6.00 &\Y &5.22 &6.26 &6.07 &6.08 &6.02\\
217 & &$^1A_2(\pi,3p)$ &R &93.0 &6.00 &\Y &4.89 &6.16 &6.02 &6.01 &5.97\\
218 & &$^1B_2(\pi,\pis)$ &V &92.5 &6.26 &\Y &7.73 &6.62 &6.36 &6.45 &6.38\\
219 & &$^1A_1(\pi,\pis)$ &V &86.3 &6.30 &\Y &6.47 &6.41 &5.84 &6.43 &6.34\\
220 & &$^1B_2(\pi,3p)$ &R &92.6 &6.83 &\Y &5.82 &6.75 &6.11 &6.92 &6.82\\
221 & &$^3B_2(\pi,\pis)$ &V &98.3 &4.51 &\Y &4.24 &4.57 &4.30 &4.49 &4.44\\
222 & &$^3A_2(\pi,3s)$ &R &97.6 &5.21 &\Y &4.47 &5.41 &5.21 &5.26 &5.20\\
223 & &$^3A_1(\pi,\pis)$ &V &97.8 &5.45 &\Y &5.52 &5.50 &5.04 &5.49 &5.40\\
224 & &$^3B_1(\pi,3p)$ &R &97.4 &5.91 &\Y &5.18 &6.22 &6.03 &6.04 &5.98\\
225 &Streptocyanine-C1 &$^1B_2(\pi,\pis)$ &V &88.7 &7.13 &\Y &7.82 &7.17 &6.76 &7.28 &7.21\\
226 & &$^3B_2(\pi,\pis)$ &V &98.3 &5.52 &\Y &5.86 &5.49 &5.22 &5.54 &5.49\\
227 &Tetrazine &$^1B_{3u}(n,\pis)$ &V &89.8 &2.47 &\Y &2.99 &2.31 &1.91 &2.54 &2.53\\
228 & &$^1A_u(n,\pis)$ &V &87.9 &3.69 &\Y &4.37 &3.49 &3.00 &3.77 &3.78\\
229 & &$^1A_g(\text{double})$ &V &0.7 &4.61 &\N &5.42 &4.69 &4.48 &4.85 &4.87\\
230 & &$^1B_{1g}(n,\pis)$ &V &83.1 &4.93 &\Y &5.41 &4.83 &4.33 &5.02 &5.00\\
231 & &$^1B_{2u}(\pi,\pis)$ &V &85.4 &5.21 &\Y &5.04 &5.31 &4.84 &5.26 &5.23\\
232 & &$^1B_{2g}(n,\pis)$ &V &81.7 &5.45 &\Y &5.43 &5.38 &4.90 &5.42 &5.38\\
233 & &$^1A_u(n,\pis)$ &V &87.7 &5.53 &\Y &6.37 &5.51 &4.92 &5.80 &5.80\\
234 & &$^1B_{3g}(\text{double})$ &V &0.7 &6.15 &\N &6.59 &5.85 &5.22 &6.20 &6.22\\
235 & &$^1B_{2g}(n,\pis)$ &V &80.2 &6.12 &\Y &6.79 &5.96 &5.18 &6.27 &6.28\\
236 & &$^1B_{1g}(n,\pis)$ &V &85.1 &6.91 &\Y &7.18 &6.59 &5.89 &6.79 &6.72\\
237 & &$^3B_{3u}(n,\pis)$ &V &97.1 &1.85 &\Y &2.38 &1.70 &1.31 &1.94 &1.93\\
238 & &$^3A_u(n,\pis)$ &V &96.3 &3.45 &\Y &4.06 &3.26 &2.78 &3.52 &3.52\\
239 & &$^3B_{1g}(n,\pis)$ &V &97.0 &4.20 &\Y &4.66 &4.10 &3.62 &4.32 &4.30\\
240 & &$^1B_{1u}(\pi,\pis)$ &V &98.5 &4.49 &\N &3.90 &4.55 &4.29 &4.39 &4.34\\
241 & &$^3B_{2u}(\pi,\pis)$ &V &97.5 &4.52 &\Y &4.68 &4.55 &4.20 &4.60 &4.55\\
242 & &$^3B_{2g}(n,\pis)$ &V &96.4 &5.04 &\Y &5.17 &5.02 &4.53 &5.10 &5.07\\
243 & &$^3A_u(n,\pis)$ &V &96.6 &5.11 &\Y &6.12 &5.07 &4.44 &5.41 &5.41\\
244 & &$^3B_{3g}(\text{double})$ &V &5.7 &5.51 &\N &6.56 &5.39 &4.86 &5.83 &5.85\\
245 & &$^3B_{1u}(\pi,\pis)$ &V &96.6 &5.42 &\Y &5.32 &5.46 &5.08 &5.44 &5.39\\
246 &Thioacetone &$^1A_2(n,\pis)$ &V &88.9 &2.53 &\Y &2.72 &2.58 &2.33 &2.60 &2.53\\
247 & &$^1B_2(n,3s)$ &R &91.3 &5.56 &\Y &4.80 &5.60 &5.48 &5.64 &5.61\\
248 & &$^1A_1(\pi,\pis)$ &V &90.6 &5.88 &\Y &6.94 &6.42 &5.98 &6.40 &6.26\\
249 & &$^1B_2(n,3p)$ &R &92.4 &6.51 &\Y &5.57 &6.51 &6.40 &6.53 &6.49\\
250 & &$^1A_1(n,3p)$ &R &91.6 &6.61 &\Y &6.24 &6.66 &6.41 &6.59 &6.50\\
251 & &$^3A_2(n,\pis)$ &V &97.4 &2.33 &\Y &2.52 &2.34 &2.09 &2.38 &2.31\\
252 & &$^3A_1(\pi,\pis)$ &V &98.7 &3.45 &\Y &3.52 &3.48 &3.29 &3.48 &3.43\\
253 &Thiophene &$^1A_1(\pi,\pis)$ &V &87.6 &5.64 &\Y &6.11 &5.84 &5.21 &5.89 &5.79\\
254 & &$^1B_2(\pi,\pis)$ &V &91.5 &5.98 &\Y &6.94 &6.35 &5.89 &6.44 &6.35\\
255 & &$^1A_2(\pi,3s)$ &R &92.6 &6.14 &\Y &5.70 &6.28 &6.07 &6.16 &6.10\\
256 & &$^1B_1(\pi,3p)$ &R &90.1 &6.14 &\Y &6.02 &6.21 &5.90 &6.16 &6.10\\
257 & &$^1A_2(\pi,3p)$ &R &91.8 &6.21 &\Y &6.05 &6.32 &5.98 &6.28 &6.21\\
258 & &$^1B_1(\pi,3s)$ &R &92.8 &6.49 &\Y &5.78 &6.57 &6.28 &6.51 &6.44\\
259 & &$^1B_2(\pi,3p)$ &R &92.4 &7.29 &\Y &6.80 &7.29 &7.03 &7.20 &7.13\\
260 & &$^1A_1(\pi,\pis)$ &V &86.5 &7.31 &\N &8.29 &7.62 &6.85 &7.71 &7.56\\
261 & &$^3B_2(\pi,\pis)$ &V &98.2 &3.92 &\Y &3.68 &3.98 &3.71 &3.90 &3.84\\
262 & &$^3A_1(\pi,\pis)$ &V &97.7 &4.76 &\Y &4.97 &4.85 &4.39 &4.87 &4.79\\
263 & &$^3B_1(\pi,3p)$ &R &96.6 &5.93 &\Y &5.86 &5.97 &5.64 &5.94 &5.88\\
264 & &$^3A_2(\pi,3s)$ &R &97.5 &6.08 &\Y &5.65 &6.22 &6.01 &6.11 &6.04\\
265 &Thiopropynal &$^1A''(n,\pis)$ &V &87.5 &2.03 &\Y &2.06 &2.05 &1.84 &2.05 &2.00\\
266 & &$^3A''(n,\pis)$ &V &97.2 &1.80 &\Y &1.85 &1.81 &1.60 &1.84 &1.79\\
267 &Triazine &$^1A_1''(n,\pis)$ &V &88.3 &4.72 &\Y &5.88 &4.62 &3.90 &5.00 &4.99\\
268 & &$^1A_2''(n,\pis)$ &V &88.3 &4.75 &\Y &5.14 &4.77 &4.39 &4.90 &4.87\\
269 & &$^1E''(n,\pis)$ &V &88.3 &4.78 &\Y &5.51 &4.76 &4.14 &5.01 &4.98\\
270 & &$^1A_2'(\pi,\pis)$ &V &85.7 &5.75 &\Y &5.55 &5.76 &5.32 &5.75 &5.72\\
271 & &$^1A_1'(\pi,\pis)$ &V &90.4 &7.24 &\Y &8.20 &7.43 &6.89 &7.50 &7.41\\
272 & &$^1E'(n,3s)$ &R &90.9 &7.32 &\Y &7.40 &7.48 &7.15 &7.53 &7.49\\
273 & &$^1E''(n,\pis)$ &V &82.6 &7.78 &\Y &8.26 &7.75 &7.04 &7.92 &7.90\\
274 & &$^1E'(\pi,\pis)$ &V &90.0 &7.94 &\Y &10.03 &8.65 &7.70 &8.83 &8.72\\
275 & &$^3A_2''(n,\pis)$ &V &96.7 &4.33 &\Y &4.74 &4.37 &3.99 &4.51 &4.49\\
276 & &$^3E''(n,\pis)$ &V &96.6 &4.51 &\Y &5.14 &4.47 &3.88 &4.71 &4.68\\
277 & &$^3A_1''(n,\pis)$ &V &96.2 &4.73 &\Y &5.88 &4.70 &3.94 &5.06 &5.04\\
278 & &$^3A_1'(\pi,\pis)$ &V &98.2 &4.85 &\Y &4.46 &4.88 &4.55 &4.81 &4.75\\
279 & &$^3E'(\pi,\pis)$ &V &96.9 &5.59 &\Y &5.57 &5.62 &5.20 &5.62 &5.57\\
280 & &$^3A_2'(\pi,\pis)$ &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]{fig2.pdf}
\includegraphics[width=\linewidth]{fig2}
\caption{Histograms of the errors (in \si{\eV}) obtained for CASPT2 and CASPT3 with and without IPEA shift.
Raw data are given in Table \ref{tab:BigTab}.}
\label{fig:PT2_vs_PT3}
@ -517,7 +510,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
%%% TABLE II %%%
\begin{table*}
\caption{Statistical quantities (in eV), considering the 265 ``safe'' TBEs (out of 284) as reference, for various multi-reference methods.
\caption{Statistical quantities (in eV), considering the 265 ``safe'' TBEs (out of 280) as reference, for various multi-reference methods.
Raw data are given in Table \ref{tab:BigTab}.}
\label{tab:stat}
\begin{ruledtabular}
@ -579,8 +572,33 @@ Because the relative size of the active space naturally decreases as the number
Note that combining CASPT2 and CASPT3 via an hybrid protocol such as CASPT2.5, as proposed by Zhang and Truhlar in the context of spin splitting energies in transition metals, \cite{Zhang_2020} is not beneficial in the present situation.
It is worth mentioning that CASPT3(NOIPEA) yields MAEs for each subset that is almost systematically below \SI{0.1}{\eV}, except for the singlet subset which contains some states showing large (positive) deviations at both the CASPT2 and CASPT3 levels.
This can be tracked down to the relatively small active spaces that we have considered here and, more precisely, to the lack of direct $\sig$-$\pi$ coupling in the active space which are known to be important in ionic states for example. \cite{Davidson_1996,Angeli_2009,Garniron_2018,BenAmor_2020}
\alert{These errors could be certainly alleviated by using a restricted active space (RAS) procedure.}
This is most notably the case for the $^1 B_u(\pi,\pis)$ state of butadiene, the $^1B_2(\pi,\pis)$ state of cyclopentadiene, the $^1A_1(\pi,\pis)$ state of cyclopropenone, the second $^1B_{1u}(\pi,\pis)$ state of pyrazine, the $^1B_2(\pi,\pis)$ state of pyridazine, and the $^1E'(\pi,\pis)$ state of triazine, for which both CASPT2(IPEA) and CASPT3(NOIPEA) overestimate the corresponding vertical transition energies by at least \SI{0.4}{\eV} with respect to the TBEs.
This can be tracked down to the relatively small active spaces that we have considered here and, more precisely, to the lack of direct $\sig$-$\pi$ coupling in the active space that is known to be important in ionic states, for example. \cite{Davidson_1996,Borden_1996,Boggio-Pasqua_2004,Angeli_2009,Garniron_2018,Tran_2019,BenAmor_2020}
For this family of states, it is particularly important to describe the dynamic response of the $\sig$-electron framework to the field of the $\pi$-electron system, a phenomenon known as dynamic $\sig$ polarization.
Because the dynamic $\sig$ polarization is generally more important for the ionic excited state than for the ground state, its contribution is expected to lower the vertical transition energy.
Furthermore, this part of the dynamic $\sig$-$\pi$ correlation needs to be included at the orbital optimization stage, otherwise the orbitals become too diffuse, resulting in artificial valence-Rydberg mixing which cannot be disentangled using non-degenerate perturbation theory such as the version of CASPT2 and CASPT3 considered here. \cite{Angeli_2009}
As an illustration of this problematic, we have chosen to address the specific case of the second $^1B_{1u}(\pi,\pis)$ state of pyrazine, which is known to exhibit a strong ionic character. \cite{Fulscher_1994}
As shown in Table \ref{tab:BigTab} (\#171), the TBE for the vertical transition energy to this state is \SI{7.98}{\eV}.
CASPT2(IPEA) and CASPT3(NOIPEA) locate this state at \SI{8.59}{} and \SI{8.57}{\eV}, respectively, providing a large overestimation of \SI{0.6}{\eV}.
This state was computed using a reference CASSCF wave function averaged over four states [the ground state, two valence $B_{1u}(\pi,\pis)$ states and one Rydberg $B_{1u}(\pi,3p_x)$ state] with an active space comprising the $\pi$ valence and three $3p_x$ orbitals.
(The $3p_x$ orbitals were included to recover part of the radial correlation.)
However, this strategy leads to a valence-Rydberg mixing due to the fact that the dynamic correlation is not sufficiently described at the CASSCF level.
The ionic $B_{1u}(\pi,\pis)$ state lies \SI{9.65}{\eV} vertically above the ground state, while the Rydberg $B_{1u}(\pi,3p_x)$ state is \SI{0.2}{\eV} below at the CASSCF level.
For this reason, the two states are mixed and both CASPT2 and CASPT3 fails to predict accurate transition energies for the ionic state.
The Rydberg character of the ionic $B_{1u}(\pi,\pis)$ state is evident from the inspection of the CASSCF wave function and also from its value of $\expval*{x^2}$, which measures the spatial extent of the wave function out of the molecular plane (hence characteristic of the size of the $\pi$ orbitals in the considered state).
The $\expval*{x^2}$ value is \SI{31.9}{\bohr^2} for the ionic $B_{1u}(\pi,\pis)$ state and \SI{51.1}{\bohr^2} for the $B_{1u}(\pi,3p_x)$ Rydberg state, while it is only \SI{26.6}{\bohr^2} for the ground state.
To remove the artificial valence-Rydberg mixing in the reference CASSCF wave function, we included the dynamic $\sig$ polarization at the orbital optimization stage using a restricted active space self-consistent field (RASSCF) approach. \cite{Olsen_1988}
We selected the bonding $\sigCC$ and $\sigCN$ orbitals in the RAS1 partition and the corresponding anti-bonding $\sigsCC$ and $\sigsCN$ orbitals in RAS3 allowing a single hole in RAS1 and a single electron in RAS3.
The six valence $\pi$ orbitals were kept in RAS2 (full CI space).
In this way, the contraction of the $\pi$ orbitals as a result of the dynamic $\sig$ polarization is ensured and the interference of the Rydberg state is removed allowing to compute the two valence $B_{1u}(\pi,\pis)$ states without including the Rydberg $B_{1u}(\pi,3p_x)$ state in the state-averaging procedure.
The $\expval*{x^2}$ value associated with the ionic $B_{1u}(\pi,\pis)$ state is reduced to \SI{26.9}{\bohr^2}, providing a spatial extent similar to that of the ground state ($\expval*{x^2} = \SI{27.0}{\bohr^2}$ at the RASSCF level).
Using the RASSCF orbitals to perform the CASPT2(IPEA) and CASPT3(NOIPEA) calculations using a CAS-CI(6,6) reference, we obtain vertical transition energies of \SI{7.92}{} and \SI{8.10}{\eV}, respectively.
The agreement with the TBE is now within the expected accuracy of the method with an error of about \SI{0.1}{\eV}.
To be complete the vertical transition energy to the first $B_{1u}(\pi,\pis)$ state, which also possesses a significant ionic character, is improved too with respect to the TBE at \SI{6.88}{\eV} with transition energies of \SI{6.83}{\eV} and \SI{6.87}{\eV} at the CASPT2(IPEA) and CASPT3(NOIPEA) levels, respectively.
This represents a significant improvement compared to the \SI{7.14}{} and \SI{7.12}{\eV} values obtained at the same level of theory but using a reference SA4-CASSCF wave function.
We thus believe that the difficult cases listed above can be handled more rigorously provided that more suitable active spaces are used to describe the reference (zeroth-order) wave function prior to the CASPT2/CASPT3 calculations.
Comparatively, Liang \textit{et al.} have recently shown, for a larger set of transitions, that time-dependent density-functional theory with the best exchange-correlation functionals yield RMSEs of the order of \SI{0.3}{\eV}, \cite{Liang_2022} outperforming (more expensive) wave function methods like CIS(D). \cite{Head-Gordon_1994,Head-Gordon_1995}
The accuracy of CASPT2(IPEA) and CASPT3 is clearly a step beyond but at a much larger computational cost.
@ -626,7 +644,7 @@ This feature is crucial in the description of some photochemistry mechanisms. \c
%%% %%% %%% %%%
Table \ref{tab:timings} reports the evolution of the wall times associated with the computation of the second- and third-order energies in benzene with the aug-cc-pVTZ basis and the frozen-core approximation (42 electrons and 414 basis functions) for increasingly large active spaces.
All these calculations have been performed on a single core of an Intel Xeon E5-2670 2.6Ghz.
All these calculations have been performed on a single core of an Intel Xeon E5-2670 2.6 Ghz.
It is particularly instructive to study the wall time ratio as the number of (contracted and uncontracted) external configurations grows (see Fig.~\ref{fig:timings}).
Overall, the PT3 step takes between 5 and 10 times longer than the PT2 step for the active spaces that we have considered here, and remains thus typically affordable for these kinds of calculations.
@ -634,7 +652,7 @@ Overall, the PT3 step takes between 5 and 10 times longer than the PT2 step for
\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.
In the present study, we have benchmarked, using 280 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 principal take-home messages of this study are that:
(i) CASPT3 transition energies are almost independent of the IPEA shift;
(ii) CASPT2(IPEA) and CASPT3 have a very similar accuracy.