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.
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.
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_2000a,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 model space.
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}
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 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}
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}
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 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 a significant set of electronic transitions.
Although few CASPT3 calculations have been reported in the literature,
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.
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.
Geometries and reference theoretical best estimates (TBEs) for the vertical excitation energies 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 in the frozen-core approximation and with MOLPRO within the RS2 and RS3 contraction schemes as described in Refs.~\onlinecite{Werner_1996} and \onlinecite{Werner_2020}.
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}
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.
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.
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.
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}.
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
The reference TBEs of the QUEST database, their percentage of single excitations $\%T_1$ involved in the transition (computed at the CC3 level), their nature
(V and R stand for valence and Rydberg, respectively) are also reported.
A detailed discussion of each individual molecule can be found in Ref.~\onlinecite{Sarka_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}.)
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.
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}).
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.
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.}
\caption{Ratio $t_\text{PT3}/t_\text{PT2}$ of the wall times associated with the computation of the third- and second-order energies as a function of the total number of contracted and uncontracted external configurations for benzene (see Table \ref{tab:timings} for raw data).
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.
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.
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.
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.
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting information available}
\label{sec:SI}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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}.
