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\begin{document}

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

\title{Benchmarking CASPT3 Vertical Excitation Energies}

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

% Abstract
\begin{abstract}
The present study assesses the performance of the third-order multireference perturbation theory, CASPT3, in the context of molecular excited states.
Based on 284 vertical excitation energies of various natures extracted from the QUEST database, we show that...
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% Title
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\section{Introduction}
\label{sec:intro}
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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 achieve and it is extremely challenging to predict, \textit{a priori}, the behavior of the series. \cite{Marie_2021a} 
This has led, in certain specific contexts, to the development of empirical strategy like MP2.5 where one simply averages the second-order (MP2) and third-order (MP3) total energies. \cite{Pitonak_2009}

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

Multi-reference perturbation theory is somewhat easier to generalize to excited states as one selects the states of interest to include in the reference (zeroth-order) space via the so-called complete-active-space self-consistent field (CASSCF) formalism, hence catching effectively static correlation in the zeroth-order wave function.
The missing dynamical correlation can then be recovered 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, its most severe drawback is certainly the intruder state problem which describes a situation where one or several determinants of the outer (first-order) 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 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 shift has been successfully tested and is now routine in excited-state calculations. \cite{Schapiro_2013,Zobel_2017,Sarka_2022}
%NEVPT2 which is an improvement of CASPT2 that does not suffer from the intruder state problem.

\titou{The second drawback 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 database, we have reported an exhaustive benchmark of CASPT2 and NEVPT2 for 284 excited states of diverse nature computed in 35 small- and medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarka_2022}
Our main take-home message was that both CASPT2 with IPEA shift and 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 were 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). 
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} which has been used in several applications, \cite{Pastore_2006a,Pastore_2006b,Pastore_2007,Angeli_2007,Camacho_2010,Angeli_2011,Angeli_2012} but, as far as we are aware of, only standalone implementation of NEVPT3 exists.
Third-order perturbation theory has a bad reputation especially within MP perturbation theory because of it does not always yield to an significant improvement compared to its cheaper second-order version. \cite{Rettig_2020}
Third-order version have been developed but rarely used and accuracy still need to be assessed.
except in the case of rhodopsin. \cite{Grabarek_2016}

\begin{figure}
	\includegraphics[width=\linewidth,viewport=1.cm 10cm 18cm 27cm,clip]{mol.pdf}
	\caption{Various molecular systems considered in this study.
	\label{fig:mol}}
\end{figure}

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\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All the CASPT2 and CASPT3 calculations have been carried out with MOLPRO within the RS2 and RS3 schemes. \cite{Werner_2020}
Both methods have been tested with and without IPEA (labeled as NOIPEA).

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\section{Results and discussion}
\label{sec:res}
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\section{Conclusion}
\label{sec:ccl}
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\begin{acknowledgements}
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}.

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\bibliography{CASPT3}
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