Perturbation theory is a relatively inexpensive and size-extensive route towards the exact solution of the Schr\"odinger equation.
However, it rarely works this way in practice as the perturbative series may exhibit quite a large spectrum of behaviors. \cite{Olsen_1996,Christiansen_1996,Cremer_1996,Olsen_2000,Olsen_2019,Stillinger_2000,Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006,Goodson_2011}
For example, in single-reference M{\o}ller-Plesset (MP) perturbation theory, \cite{Moller_1934} erratic, slowly convergent, and divergent behaviors have been observed. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Leininger_2000,Malrieu_2003,Damour_2021}
Systematic improvement is thus difficult to contemplate and it is extremely challenging to predict, \textit{a priori}, the behavior of the series. \cite{Marie_2021a}
This has led to the development, in certain specific contexts, of empirical strategy like MP2.5 where one simply averages the second-order (MP2) and third-order (MP3) total energies. \cite{Pitonak_2009}
Extension of single-reference perturbation theory to electronic excited states is far from being trivial, and the algebraic diagrammatic
construction (ADC) approximation of the polarization propagator is probably the most natural. \cite{Schirmer_1982,Schirmer_1991,Barth_1995,Schirmer_2004,Schirmer_2018,Trofimov_1997,Trofimov_1997b,Trofimov_2002,Trofimov_2005,Trofimov_2006,Harbach_2014,Dreuw_2015}
However, the ADC series naturally inherits from the drawbacks of its MP parent and it has been shown to be not particularly rapidly convergent in the context of vertical excitation energies. \cite{Veril_2021}
This has led some of the authors to recently propose the ADC(2.5) composite approach, where, in the same spirit as MP2.5, one averages the second-order [ADC(2)] and third-order [ADC(3)] vertical transition energies. \cite{Loos_2020d}
Multi-reference perturbation theory is somewhat easier to generalise to excited states as one selects the states of interest to include in the reference space via the so-called complete-active-space self-consistent field (CASSCF) formalism, hence catching efficiently static correlation in the zeroth-order wave function.
The missing dynamical correlation can then be caught via low-order multi-reference 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}
However, the equations are much more involved than their single-reference counterparts and many schemes have been developed.
From the three methods mentioned above, CASPT2 is the most popular approach although it has well-document weaknesses.
In the context of excited states, the most severe one is certainly the intruder state problem which describes a situation where one or several determinants of the outer space, known as perturbers, has 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 common situations, and correcting the second-order energt for the use of this shift.
The use of real-valued level shift \cite{Roos_1995b} or an imaginary level shift \cite{Forsberg_1997} in the case of ``weak'' intruder states have been successfully tested \cite{Roos_1996} and is now routine in excited-state calculations. \cite{Schapiro_2013,Zobel_2017,Sarka_2022}
\titou{The third bottleneck was found in evaluating a large number of chemical problems for which systematic errors were noticed \cite{Andersson_1993,Andersson_1995} and ascribed to the unbalanced description of the zeroth-order Hamiltonian
for the open- and closed-shell electronic configurations. This systematic error can be attenuated by introducing an additional parameter, the so-called ionization-potential-electron-affinity (IPEA) shift, in the zeroth-order
Hamiltonian. \cite{Ghigo_2004}}
Recently, based on the highly-accurate vertical excitation energies of the QUEST dababase, we have reported an exhaustive benchmark of CASPT2 and NEVPT2 for 284 exited states of diverse nature computed in 35 small- to medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarka_2022}
Our main take-home message was that both CASPT2 with IPEA shift and 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.
These values are found to be rather uniform for the various subgroups of transitions.
Here, going one step further in the perturbative expansion, we propose to assess the performances of complete-active-space third-order perturbation theory (CASPT3).
Third-order perturbation theory has a bad reputation especially within MP scheme because of its possible oscillatory behavior.
Because it catches both static (via CASSCF) and dynamic (via PT2) correlation, CASPT2 has been used a lot to compute vertical excitation energies in realistic systems.
Same for NEVPT2 which is an improvement of CASPT2 that does not suffer from the intruder state problem which causes singularities when one or more reference states become (nearly) degenerate with a state in the complementary configuration space.
Example of rhodopsin?
Third-order version have been developed but rarely used and accuracy still need to be assessed. \cite{Grabarek_2016}
Pioneering work along these lines is due to Werner which develops a CASPT3 code in MOLPRO \cite{Werner_2020} based on a hack of the MRCI module. \cite{Werner_1996}
There is also the NEVPT3 method of Angeli and coworkers. \cite{Angeli_2006}
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2021-18005.
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for funding.
\end{acknowledgements}
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\section*{Supporting information available}
\label{sec:SI}
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%\section*{Data availability statement}
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%The data that support the findings of this study are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
