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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 CASPT3 provides a significant improvement compared to its second-order counterpart, CASPT2.
As already reported, we have also observed that the accuracy of CASPT3 is much less sensitive to the infamous ionization-potential-electron-affinity (IPEA) shift.
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% Title
\maketitle


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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{Loos_2018a,Loos_2020a,Veril_2021}
This has led some of the authors to recently propose the ADC(2.5) composite approach, where, in the same spirit as MP2.5, one averages the second-order [ADC(2)] and third-order [ADC(3)] vertical transition energies. \cite{Loos_2020d}

Multi-reference perturbation theory is somewhat easier to generalize to excited states as one selects the states of interest to include in the reference (zeroth-order) space via the so-called complete-active-space self-consistent field (CASSCF) formalism, hence catching effectively static correlation in the zeroth-order wave function.
The missing dynamical correlation can then be recovered via low-order perturbation theory, as performed in the complete-active-space second-order perturbation theory (CASPT2) of Roos and 
coworkers, \cite{Andersson_1990,Andersson_1992,Roos_1995a} Hirao's multireference second-order M{\o}llet-Plesset (MRMP2) approach, \cite{Hirao_1992} or the $N$-electron valence state second-order perturbation theory (NEVPT2) developed by Angeli, Malrieu, and coworkers. \cite{Angeli_2001a,Angeli_2001b,Angeli_2002,Angeli_2006}
However, these multi-reference formalisms and their implementation are much more involved and costly than their single-reference counterparts.
Although it has well-document weaknesses, CASPT2 is indisputably the most popular of the three approaches mentioned above. 
As such, it has been employed in countless computational studies involving electronic excited states. \cite{Serrano-Andres_1993a,Serrano-Andres_1993b,Serrano-Andres_1993c,Serrano-Andres_1995,Roos_1996,Serrano-Andres_1996a,Serrano-Andres_1996b,Serrano-Andres_1998b,Roos_1999,Merchan_1999,Roos_2002,Serrano-Andres_2002,Serrano-Andres_2005,Tozer_1999,Burcl0_2002,Peach_2008,Faber_2013,Schreiber_2008,Silva-Junior_2008,Sauer_2009,Silva-Junior_2010a,Silva-Junior_2010b,Silva-Junior_2010c}

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

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

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 (singlets, triplets, valence, Rydberg, $n\to\pis$, $\pi\to\pis$, and double excitations) computed in 35 small- and medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarka_2022}
Our main take-home message was that both CASPT2 with IPEA shift and the partially-contracted version of NEVPT2 provide fairly reliable vertical transition energy estimates, with slight overestimations and mean absolute errors of \SI{0.11}{} and \SI{0.13}{\eV}, respectively.
Importantly, the introduction of the IPEA shift in CASPT2 was found to lower the mean absolute errors from \SI{0.27}{} to \SI{0.11}{eV}.

In the electronic structure community, third-order perturbation theory has a fairly bad reputation especially within MP perturbation theory where it is rarely worth its extra cost. \cite{Rettig_2020} 
Nonetheless, going against popular believes 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} 
Very few CASPT3 calculations have been reported in the literature \cite{Angeli_2006,Yanai_2007,Grabarek_2016,Li_2017,Li_2018,Li_2021} with only a single work reporting CASPT3 vertical excitation energies. \cite{Grabarek_2016}
Based on the same 284 vertical excitation energies from the QUEST database, we show that CASPT3 provides a significant improvement compared to CASPT2.
Moreover, as already reported, \cite{Grabarek_2016} we also observe that the accuracy of CASPT3 is much less sensitive to the IPEA shift.
Note that, although a third-order version of NEVPT has been developed \cite{Angeli_2006} and has been used in several applications \cite{Pastore_2006a,Pastore_2006b,Pastore_2007,Angeli_2007,Camacho_2010,Angeli_2011,Angeli_2012} by Angeli and coworkers, as far as we are aware of, only standalone implementation of NEVPT3 exists.

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

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

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

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

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

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

A detailed discussion of each individual molecule can be found in Ref.~\onlinecite{Sarka_2022} where we 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 are represented in Fig.~\ref{fig:}.

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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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\end{document}
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