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Manuscript/CASPT3.bib
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Manuscript/CASPT3.tex
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% Abstract
\begin{abstract}
Using 284 reference 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.
Based on 284 reference 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 disputable 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, irrespective of the considered type of the transition and system size, with a small reduction of the mean absolute error to 0.09 eV when the IPEA shift is switched off.
%\bigskip
@ -110,22 +110,21 @@ However, as already reported, we also observe that the accuracy of CASPT3 is alm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Perturbation theory is a relatively inexpensive route towards the exact solution of the Schr\"odinger equation.
However, it rarely works this way in real-life as the perturbative series may exhibit quite a large spectrum of (non-optimal) 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}
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 evolution when increasing the perturbation order. \cite{Marie_2021a}
This has led, in certain specific contexts, to the development of empirical strategy like MP2.5 where one averages the second-order (MP2) and third-order (MP3) total energies, to obtain improved values. \cite{Pitonak_2009}
Systematic improvement is thus difficult to achieve and it is extremely challenging to predict, \textit{a priori}, the evolution of the series when increasing the perturbation order. \cite{Marie_2021a}
This has led, in certain specific contexts, to the development of empirical strategy like MP2.5 where one averages the second-order (MP2) and third-order (MP3) total energies, to obtain more accurate values. \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 rather slowly convergent in the context of vertical excitation energies. \cite{Loos_2018a,Loos_2020a,Veril_2021}
This has led some of us 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 has the freedom to select 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 effectively catching static correlation in the zeroth-order model space.
Multi-reference perturbation theory is somewhat easier to generalize to excited states as one has the freedom to select 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 effectively catching 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} the multireference MP2 approach of Hirao, \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-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} \hl{Que des vieiilles refs, utiles ?}
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 introduce a shift in the denominators to avoid such situations, and correcting afterwards the second-order energy for the use of this shift.
@ -135,12 +134,12 @@ A second pitfall was revealed by Andersson \textit{et al.} \cite{Andersson_1993,
A cure was quickly proposed via the introduction of an additional parameter in the zeroth-order Hamiltonian, the 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 that its impact is significantly 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 with a large basis set ({aug}-cc-pVTZ) in 35 small- and medium-sized organic molecules containing from three to six non-hydrogen atoms. \cite{Sarkar_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 with a large basis set (aug-cc-pVTZ) 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 be crucial as it neglect yields a mean absolute error of \SI{0.27}{eV}.
Importantly, the introduction of the IPEA shift in CASPT2 was found to be crucial as neglecting it increases the mean absolute error to \SI{0.27}{eV}.
In the electronic structure community, third-order perturbation theory has not a great reputation especially within MP perturbation theory where it is seen as rarely worth its 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 the very same set of electronic transitions as the one used in Ref. \citenum{Sarkar_2022}
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 the very same set of electronic transitions as the one used in Ref.~\onlinecite{Sarkar_2022}
Although 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, to the best of our knowledge, the first comprehensive benchmark of CASPT3 and allows assessing its accuracy in the framework of electronically excited states.
@ -192,7 +191,8 @@ A detailed discussion of each individual molecule can be found in Ref.~\onlineci
We therefore decided to focus on global trends here.
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}.
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 the largest positive and negative deviations [Max($+$) and Max($-$), respectively].
These are given in Table \ref{tab:stat} considering the 265 ``safe'' TBEs (out of 284) for which chemical accurate is assumed (absolute error below \SI{0.043}{\eV}). The MAEs determined for 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) can be found in Table \ref{tab:stat_subset}.
These are given in Table \ref{tab:stat} considering the 265 ``safe'' TBEs (out of 284) for which chemical accuracy is assumed (absolute error below \SI{0.043}{\eV}).
The MAEs determined for 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) can be found in Table \ref{tab:stat_subset}.
Error patterns for selected subsets are reported in {\SupInf}.
%%% TABLE I %%%
@ -541,7 +541,7 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
%%% TABLE II %%%
\begin{table*}
\caption{MAEs determined for several subsets of transitions and system sizes computed with various multi-reference methods.
Count is the number of excited states considered in each subset. \hl{definir double clairement ?}
Count is the number of excited states considered in each subset.
Raw data are given in Table \ref{tab:BigTab}.}
\label{tab:stat_subset}
\begin{ruledtabular}
@ -566,7 +566,6 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
\end{table*}
%%% %%% %%% %%%
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 alters the accuracy of CASPT2.
\hl{Ce sont deux choses differentes: je separerais l'effet de l'IPEA et la precision finale. Donc, quel est le MAD avec/sans IPEA en PT2 et en PT3 sans tenir compte des TBEs, puis aller vers les TBEs}
@ -577,9 +576,10 @@ Importantly, CASPT3 performs slightly better without IPEA shift, which is a nice
Second, CASPT3 (with or without IPEA) has a similar accuracy as CASPT2(IPEA).
Again, this observation stands for each subset of excitations and irrespectively of the system size (see Table \ref{tab:stat_subset}).
Because the relative size of the active space naturally decreases as the number of electrons and orbitals get larger, we observe that the MAEs of each subset increase with the size of the molecules.
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 in transition metals, \cite{Zhang_2020} is not beneficial in the present situation.
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 contains some states showing large (positive) deviations at both CASPT2 and CASPT3 levels.
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 contains some states showing large (positive) deviations at both 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.}