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@ -1,13 +1,93 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-23 22:35:50 +0100
%% Created for Pierre-Francois Loos at 2022-04-04 23:18:23 +0200
%% Saved with string encoding Unicode (UTF-8)
@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},
date-modified = {2022-04-04 23:13:51 +0200},
doi = {10.1021/jo070452v},
issn = {0022-3263, 1520-6904},
journal = {J. Org. Chem.},
language = {en},
month = jun,
number = {12},
pages = {4497-4503},
title = {Toward a {{Mechanistic Understanding}} of the {{Photochromism}} of {{Dimethyldihydropyrenes}}},
volume = {72},
year = {2007},
bdsk-url-1 = {https://doi.org/10.1021/jo070452v}}
@article{Head-Gordon_1994,
author = {M. Head-Gordon and R. J. Rico and M. Oumi and T. J. Lee},
date-added = {2022-04-04 22:56:32 +0200},
date-modified = {2022-04-04 22:56:32 +0200},
doi = {10.1016/0009-2614(94)00070-0},
journal = {Chem. Phys. Lett.},
pages = {21--29},
title = {A Doubles Correction To Electronic Excited States From Configuration Interaction In The Space Of Single Substitutions},
volume = {219},
year = {1994},
bdsk-url-1 = {https://doi.org/10.1016/0009-2614(94)00070-0}}
@article{Head-Gordon_1995,
author = {Head-Gordon, M. and Maurice, D. and Oumi, M.},
date-added = {2022-04-04 22:56:32 +0200},
date-modified = {2022-04-04 22:56:32 +0200},
doi = {10.1016/0009-2614(95)01111-L},
journal = {Chem. Phys. Lett.},
pages = {114--121},
title = {A Perturbative Correction to Restricted Open-Shell Configuration-Interaction with Single Substitutions for Excited-States of Radicals},
volume = {246},
year = {1995},
bdsk-url-1 = {https://doi.org/10.1016/0009-2614(95)01111-L}}
@misc{Liang_2022,
author = {Liang, Jiashu and Feng, Xintian and Hait, Diptarka and Head-Gordon, Martin},
copyright = {arXiv.org perpetual, non-exclusive license},
date-added = {2022-04-04 22:47:24 +0200},
date-modified = {2022-04-04 22:47:30 +0200},
doi = {10.48550/ARXIV.2202.13208},
keywords = {Chemical Physics (physics.chem-ph), Other Condensed Matter (cond-mat.other), Computational Physics (physics.comp-ph), Quantum Physics (quant-ph), FOS: Physical sciences, FOS: Physical sciences},
publisher = {arXiv},
title = {Revisiting the performance of time-dependent density functional theory for electronic excitations: Assessment of 43 popular and recently developed functionals from rungs one to four},
url = {https://arxiv.org/abs/2202.13208},
year = {2022},
bdsk-url-1 = {https://arxiv.org/abs/2202.13208},
bdsk-url-2 = {https://doi.org/10.48550/ARXIV.2202.13208}}
@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},
doi = {10.1021/jp952794n},
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},
volume = {100},
year = {1996},
bdsk-url-1 = {https://doi.org/10.1021/jp952794n}}
@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},
doi = {10.1063/5.0011582},
journal = {J. Chem. Phys},
number = {4},
pages = {044118},
title = {Spin polarization as an electronic cooperative effect},
volume = {153},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0011582}}
@article{Bittererova_2001,
author = {Bittererova, M and Brinck, T and Ostmark, H},
date-added = {2022-03-17 21:11:59 +0100},
@ -993,10 +1073,10 @@
year = {2013},
bdsk-url-1 = {https://doi.org/10.1021/ct400136y}}
@article{Sarka_2022,
author = {R. Sarka and P. F. Loos and M. Boggio-Pasqua and D. Jacquemin.},
@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 = {2022-03-16 10:54:12 +0100},
date-modified = {2022-03-24 16:39:50 +0100},
journal = {J. Chem. Theory Comput.},
pages = {in press},
title = {Assessing the performances of CASPT2 and NEVPT2 for vertical excitation energies,},

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Manuscript/CASPT3.tex
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@ -91,7 +91,7 @@
% Abstract
\begin{abstract}
Based on 284 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 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.
When one applies the \alert{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 and the system size, with a small reduction of the mean absolute errors to 0.09 eV when the IPEA shift is switched off.
%\bigskip
%\begin{center}
@ -124,18 +124,18 @@ Multi-reference perturbation theory is somewhat easier to generalize to excited
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.
Although it has well-documented 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}
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,Sarkar_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}
A cure was quickly proposed via the introduction of an additional parameter in the zeroth-order Hamiltonian, the \alert{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}
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{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 lower the mean absolute errors from \SI{0.27}{} to \SI{0.11}{eV}.
@ -145,9 +145,10 @@ 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.
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.
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}
@ -174,11 +175,11 @@ For the sake of computational efficiency, the doubly-excited external configurat
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 added 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.
For each system and transition, we report in the {\SupInf} the exhaustive description of the active spaces for each symmetry representation.
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.
Note that, compared to our previous CASPT2 benchmark study, \cite{Sarkar_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}.
Although these active space reductions are overall statistically negligible, this explains the small deviations between the statistical quantities reported here and in Ref.~\onlinecite{Sarkar_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.
@ -227,7 +228,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
14 & &$^1A'(\pi,\pis)$ &V &75.0 &7.87 &\Y &7.91 &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\\
@ -251,8 +252,8 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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\\
@ -280,27 +281,27 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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\\
@ -344,7 +345,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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\\
@ -353,14 +354,14 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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 &Methylenecycloprope&ne$^1B_2(\pi,\pis)$ &V &85.4 &4.28 &\Y &4.47 &4.40 &4.12 &4.39 &4.33\\
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\\
@ -368,14 +369,14 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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 &4.54 &4.54\\
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\\
@ -404,7 +405,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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\\
@ -442,7 +443,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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\\
@ -491,7 +492,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
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\\
@ -520,16 +521,16 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
&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 &$0.58$ &$0.14$ &$0.21$ &$0.13$ &$0.13$ &$0.14$ &$0.14$\\
MSE &$0.12$ &$0.06$ &$-0.26$ &$0.10$ &$0.05$ &$0.13$ &$0.09$\\
SDE &$0.58$ &$0.14$ &$0.21$ &$0.14$ &$0.13$ &$0.14$ &$0.14$\\
RMSE &$0.61$ &$0.16$ &$0.33$ &$0.17$ &$0.14$ &$0.19$ &$0.17$\\
MAE &$0.48$ &$0.11$ &$0.27$ &$0.11$ &$0.09$ &$0.15$ &$0.13$\\
MAE &$0.47$ &$0.11$ &$0.27$ &$0.11$ &$0.09$ &$0.15$ &$0.13$\\
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}.}
\fnt[2]{Values taken from Ref.~\onlinecite{Sarkar_2022}.}
\end{table*}
%%% %%% %%% %%%
@ -544,40 +545,45 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
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\\
Singlet &174 &0.56 &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\\
Valence &208 &0.44 &0.11 &0.33 &0.13 &0.10 &0.15 &0.12\\
Rydberg &76 &0.55 &0.13 &0.13 &0.08 &0.07 &0.15 &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\\
$\pi \to \pis$ &119 &0.45 &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\\
4 non-H atoms &94 &0.45 &0.11 &0.22 &0.12 &0.09 &0.14 &0.13\\
5-6 non-H atoms &151 &0.51 &0.12 &0.33 &0.13 &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}.}
\fnt[2]{Values taken from Ref.~\onlinecite{Sarkar_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.
A detailed discussion of each individual molecule can be found in Ref.~\onlinecite{Sarkar_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}. (The error distributions for some of these subsets are reported in {\SupInf}.)
From the different statistical quantities reported in Table \ref{tab:stat}, one can highlight the two following trends.
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.
First, as previously reported, \cite{Werner_1996,Grabarek_2016} CASPT3 vertical excitation energies are much less sensitive to the IPEA shift, which drastically alters 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 remarkably different (\SI{0.11}{} and \SI{0.27}{\eV}).
Likewise, the MSEs of CASPT2(IPEA) and CASPT2(NOIPEA), \SI{0.06}{} and \SI{-0.26}{\eV}, clearly highlight the well-known global underestimation of the CASPT2(NOIPEA) excitation energies in molecular systems.
For CASPT3, the MSE with IPEA shift is only slightly larger without IPEA (\SI{0.10}{} and \SI{0.05}{\eV}, respectively).
Importantly, CASPT3 performs slightly better without IPEA shift, which is a nice outcome that holds for each group of transitions and system size (see the MAEs in Table \ref{tab:stat_subset}).
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}).
Again, this observation stands for each subset of excitations and irrespectively of the system size (see Table \ref{tab:stat_subset}).
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 of transition metals, \cite{Zhang_2020} is not beneficial in the present situation.
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.
\alert{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.}
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 is contaminated by some states showing large (positive) deviations at 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 with...}
Comparatively, Liang \textit{et al.} have 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.
Although it does not beat the approximate third-order coupled-cluster method CC3 \cite{Christiansen_1995b,Koch_1997} for transitions with a dominant single excitation character (for which CC3 returns a MAEs below the chemical accuracy threshold of \SI{0.043}{\eV} \cite{Sarkar_2022}), it has the undeniable advantage to describe with the same accuracy both single and double excitations.
This feature is crucial in the description of some photochemistry mechanisms. \cite{Boggio-Pasqua_2007}
%%% TABLE III %%%
\begin{table}
@ -599,10 +605,10 @@ These errors could be alleviated by using a RAS space.}
% $^1A_1(S_0)$ &28 &276 & (4e,6o) &125 &$2.45 \times 10^6$ &$9.71 \times 10^7$ &8.56 &21.48\\
% $^1A_1(S_0)$ &28 &276 & (4e,7o) &261 &$3.72 \times 10^6$ &$1.97 \times 10^8$ &23.26 &52.92\\
% \mc{9}{l}{Benzene}\\
(6e,6o) &104 &$4.50 \times 10^6$ &$2.29 \times 10^8$ &10.64 &59.76\\
(6e,7o) &165 &$7.27 \times 10^6$ &$3.69 \times 10^8$ &38.82 &249.01\\
(6e,8o) &412 &$1.59 \times 10^7$ &$8.98 \times 10^8$ &158.74 &1332.66\\
(6e,9o) &1800 &$3.96 \times 10^7$ &$3.53 \times 10^9$ &578.49 &6332.44\\
(6e,6o) &104 &$4.50 \times 10^6$ &$2.29 \times 10^8$ &11 &60\\
(6e,7o) &165 &$7.27 \times 10^6$ &$3.69 \times 10^8$ &39 &249\\
(6e,8o) &412 &$1.59 \times 10^7$ &$8.98 \times 10^8$ &159&1333\\
(6e,9o) &1800 &$3.96 \times 10^7$ &$3.53 \times 10^9$ &578&6332\\
\end{tabular}
\end{ruledtabular}
\end{table}
@ -618,19 +624,19 @@ These errors could be alleviated by using a RAS space.}
%%% %%% %%% %%%
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 within the frozen-core approximation (42 electrons and 414 basis functions) for increasingly large active spaces.
All these calculations have been performed on an Intel Xeon E5-2670 node with 8 physical cores at 2.6Ghz node and 64GB of memory.
It is particularly instructive to study the wall time ratio as the number of (contracted and uncontracted) external configuration grows (see also 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, which usually affordable for these kinds of calculations.
All these calculations have been performed on a single core of an Intel Xeon E5-2670 2.6Ghz.
It is particularly instructive to study the wall time ratio as the number of (contracted and uncontracted) external configurations grows (see also 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 is thus usually affordable for these kinds of calculations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.
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.
These global trends are also true for specific sets of excitations and various system sizes.
Therefore, if one can afford the additional computation of the third-order energy (which is only several times longer to compute than its second-order counterpart), one can eschew the delicate choice of the IPEA value in CASPT2, and rely solely on the CASPT3(NOIPEA) energy.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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