sent to Denis

Manuscript/CASPT3.tex

@@ -82,7 +82,7 @@
 	\email{martial.boggio@irsamc.ups-tlse.fr}
 	\affiliation{\LCPQ}
 \author{Denis \surname{Jacquemin}}
-	\email{denistriou.jacqueminous-aka-gros.lapin@univ-nantes.fr}
+	\email{denistriou.jacqueminous@univ-nantes.fr}
 	\affiliation{\CEISAM}
 \author{Pierre-Fran\c{c}ois \surname{Loos}}
 	\email{loos@irsamc.ups-tlse.fr}
@@ -91,8 +91,8 @@
 % Abstract
 \begin{abstract}
 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 \alert{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.
+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, irrespectively of the type of the transitions and the system size, with a small reduction of the mean absolute error to 0.09 eV when the IPEA shift is switched off.
 %\bigskip
 %\begin{center}
 %        \boxed{\includegraphics[width=0.4\linewidth]{TOC}}
@@ -120,8 +120,8 @@ construction (ADC) approximation of the polarization propagator is probably the
 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}
+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 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 MP2 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-documented weaknesses, CASPT2 is indisputably the most popular of the three approaches mentioned above. 
@@ -132,7 +132,7 @@ One can then introduce a shift in the denominators to avoid such situations, and
 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,Sarkar_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 \alert{infamous} ionization-potential-electron-affinity (IPEA) shift. \cite{Ghigo_2004}
+A cure was quickly proposed via the introduction of an additional parameter in the zeroth-order Hamiltonian, the disputable 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{Sarkar_2022}
@@ -179,13 +179,10 @@ For each system and transition, we report in the {\SupInf} the exhaustive descri
 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{Sarkar_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{Sarkar_2022}.
+Although these active space reductions are overall statistically negligible, this explains the small deviations that one may observe between the data reported here and in Ref.~\onlinecite{Sarkar_2022}.
 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 in such cases.
 
-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].
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
 \label{sec:res}
@@ -196,7 +193,7 @@ deviation of the errors (SDE), as well as the largest positive and negative devi
 \caption{Vertical excitation energies (in \si{\eV}) computed with various multi-reference methods and the aug-cc-pVTZ basis.
 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.
-TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute error below \SI{0.05}{\eV}).
+TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute error below \SI{0.043}{\eV}).
 [F] indicates a fluorescence transition, \ie, a vertical transition energy computed from an excited-state equilibrium geometry.
 \label{tab:BigTab}}
 \\
@@ -565,21 +562,26 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
 A detailed discussion of each individual molecule can be found in Ref.~\onlinecite{Sarkar_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}.)
+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 reported in Table \ref{tab:stat} for various methods considering the 265 ``safe'' TBEs (out of 284), 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}.
+Note that a TBE is listed as ``safe'' if it is assumed to be chemically accurate (i.e., absolute error below \SI{0.043}{\eV}).
 
 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. 
 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 remarkably different (\SI{0.11}{} and \SI{0.27}{\eV}).
-Likewise, the MSEs of CASPT2(IPEA) and CASPT2(NOIPEA), \SI{0.06}{} and \SI{-0.26}{\eV}, clearly highlight the well-known global underestimation of the CASPT2(NOIPEA) excitation energies in molecular systems.
+Likewise, the MSEs of CASPT2(IPEA) and CASPT2(NOIPEA), \SI{0.06}{} and \SI{-0.26}{\eV}, clearly evidence the well-known global underestimation of the CASPT2(NOIPEA) excitation energies in molecular systems.
 For CASPT3, the MSE with IPEA shift is only slightly larger without IPEA (\SI{0.10}{} and \SI{0.05}{\eV}, respectively).
 Importantly, CASPT3 performs slightly better without IPEA shift, which is a nice outcome that holds for each group of transitions and system size (see the MAEs in  Table \ref{tab:stat_subset}).
+
 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}).
 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.
 
 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 is contaminated by some states showing large (positive) deviations at 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 with...}
+\alert{These errors could be certainly alleviated by using a restricted active space (RAS) procedure.}
+
 Comparatively, Liang \textit{et al.} have shown, for a larger set of transitions, that time-dependent density-functional theory with the best exchange-correlation functionals yield RMSEs of the order of \SI{0.3}{\eV}, \cite{Liang_2022} outperforming (more expensive) wave function methods like CIS(D). \cite{Head-Gordon_1994,Head-Gordon_1995} 
 The accuracy of CASPT2(IPEA) and CASPT3 is clearly a step beyond but at a much larger computational cost.
 Although it does not beat the approximate third-order coupled-cluster method CC3 \cite{Christiansen_1995b,Koch_1997} for transitions with a dominant single excitation character (for which CC3 returns a MAEs below the chemical accuracy threshold of \SI{0.043}{\eV} \cite{Sarkar_2022}), it has the undeniable advantage to describe with the same accuracy both single and double excitations.