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almost done with CASPT3 paper

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@ -8,7 +8,7 @@
\newcommand{\ie}{\textit{i.e.}} \newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}} \newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{black}{#1}} \newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
@ -166,13 +166,13 @@ For each compound represented in Fig.~\ref{fig:mol}, we have computed the CASPT2
basis set. \cite{Kendall_1992} basis set. \cite{Kendall_1992}
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}. 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 with MOLPRO within the RS2 and RS3 contraction schemes as described in Refs.~\onlinecite{Werner_1996} and \onlinecite{Werner_2020}. 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}.
Both methods have been tested with and without IPEA (labeled as NOIPEA). Both methods have been tested with and without IPEA (labeled as NOIPEA).
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}.
The MOLPRO implementation of CASPT3 is based on a modification of the multi-reference configuration interaction (MRCI) module. \cite{Werner_1988,Knowles_1988} 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} 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. 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. In several occasions, we have added 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. 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. 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.
@ -183,7 +183,7 @@ Finally, to alleviate the intruder state problem, a level shift of \SI{0.3}{\har
This value has been slightly increased in particularly difficult cases, and is specifically reported in such cases. 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 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]. deviation of the errors (SDE), as well as the largest positive and negative deviations [Max($+$) and Max($-$), respectively].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion} \section{Results and discussion}
@ -192,7 +192,7 @@ deviation of the errors (SDE), as well as largest positive and negative deviatio
%%% TABLE I %%% %%% TABLE I %%%
\begin{longtable*}{cllccccccccc} \begin{longtable*}{cllccccccccc}
\caption{Vertical excitation energies (in \si{\eV}) computed with various multi-reference methods. \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 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. (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{0.05}{\eV}).
@ -505,13 +505,15 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{PT2_vs_PT3.pdf} \includegraphics[width=\linewidth]{PT2_vs_PT3.pdf}
\caption{Histograms of the errors (in \si{\eV}) obtained for CASPT2 and CASPT3 with and without IPEA shift. \caption{Histograms of the errors (in \si{\eV}) obtained for CASPT2 and CASPT3 with and without IPEA shift.
\label{fig:PT2_vs_PT3}} Raw data are given in Table \ref{tab:BigTab}.}
\label{fig:PT2_vs_PT3}
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE II %%% %%% TABLE II %%%
\begin{table*} \begin{table*}
\caption{Statistical quantities (in eV), considering the 265 ``safe'' TBEs (out of 284) as reference, for various multi-reference methods.} \caption{Statistical quantities (in eV), considering the 265 ``safe'' TBEs (out of 284) as reference, for various multi-reference methods.
Raw data are given in Table \ref{tab:BigTab}.}
\label{tab:stat} \label{tab:stat}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrrrrrrr} \begin{tabular}{lrrrrrrr}
@ -534,7 +536,8 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
%%% TABLE II %%% %%% TABLE II %%%
\begin{table*} \begin{table*}
\caption{MAEs determined for several subsets of transitions and system sizes computed with various multi-reference methods. \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.} Count is the number of excited states considered in each subset.
Raw data are given in Table \ref{tab:BigTab}.}
\label{tab:stat_subset} \label{tab:stat_subset}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrrrrrrrr} \begin{tabular}{lrrrrrrrr}
@ -563,38 +566,61 @@ 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}. 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}. 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}.
From the different statistical quantities reported in Table \ref{tab:stat}, one can highlight the two following observations. 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 alter the accuracy of CASPT2. 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}). 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. 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). 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}). 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. 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.
\titou{Here, discuss difficult case where we have a large (positive) error in CASPT2 and CASPT3. \alert{Here, discuss difficult case where we have a large (positive) error in CASPT2 and CASPT3.
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} 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.} These errors could be alleviated by using a RAS space.}
%%% TABLE III %%% %%% TABLE III %%%
\begin{table*} \begin{table}
\caption{CASPT2 and CASPT3 timings (in seconds) for a selection of systems and transitions.} \caption{Wall times (in seconds) for the computation of the (ground-state) second-order (PT2) and third-order (PT3) energies of benzene.
Calculations have been performed in the frozen-core approximation and with the aug-cc-pVTZ basis set on an AMD Zen3 node (see main text).}
\label{tab:timings} \label{tab:timings}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llcccccccc} \begin{tabular}{cccccc}
System & Transition &Active &\# electrons &\# basis &\# CAS &\# contracted &\# uncontracted &CPU &CPU\\ Active &\# CAS &\# contracted &\# uncontracted &$t_\text{PT2}$ &$t_\text{PT3}$\\
& &Space & &functions &det. &config. &config. &CASPT2 &CASPT3 \\ space &det. &config. &config. & & \\
\hline \hline
Acetone &$^1A_2(n,\pis)$ &(6e,6o) &32 &322 &104 &$3.86 \times 10^6$ &$1.49 \times 10^8$ &12.50 &33.25\\ % Acetone &$^1A_2(n,\pis)$ &32 &322 &(6e,6o) &104 &$3.86 \times 10^6$ &$1.49 \times 10^8$ &12.50 &33.25\\
Pyrrole &$^1A_2(\pi,3s)$ &(6e,6o) &36 &345 &96 &$4.79 \times 10^6$ &$2.04 \times 10^8$ &13.24 &49.36\\ % Pyrrole &$^1A_2(\pi,3s)$ &36 &345 &(6e,6o) &96 &$4.79 \times 10^6$ &$2.04 \times 10^8$ &13.24 &49.36\\
Imidazole &$^1A''(\pi,3s)$ &(8e,7o) &36 &322 &600 &$1.46 \times 10^7$ &$1.82 \times 10^9$ &193.93 &282.62\\ % Imidazole &$^1A''(\pi,3s)$ &36 &322 &(8e,7o) &600 &$1.46 \times 10^7$ &$1.82 \times 10^9$ &193.93 &282.62\\
Pyrazine &$^1B_{3u}(n,\pis)$ &(10e,8o) &42 &368 &392 &$6.95 \times 10^6$ &$6.38 \times 10^8$ &29.58 &174.96\\ % Pyrazine &$^1B_{3u}(n,\pis)$ &42 &368 &(10e,8o) &392 &$6.95 \times 10^6$ &$6.38 \times 10^8$ &29.58 &174.96\\
% \mc{9}{l}{Methylenecyclopropene}\\
% $^1A_1(S_0)$ &28 &276 &(4e,4o) &18 &$1.32 \times 10^6$ &$1.52 \times 10^7$ &1.86 &5.61\\
% $^1A_1(S_0)$ &28 &276 &(4e,5o) &28 &$1.73 \times 10^6$ &$2.42 \times 10^7$ &5.52 &6.98\\
% $^1A_1(S_0)$ &28 &276 & (4e,6o) &125 &$2.45 \times 10^6$ &$9.71 \times 10^7$ &8.56 &21.48\\
% $^1A_1(S_0)$ &28 &276 & (4e,7o) &261 &$3.72 \times 10^6$ &$1.97 \times 10^8$ &23.26 &52.92\\
% \mc{9}{l}{Benzene}\\
(6e,6o) &104 &$4.50 \times 10^6$ &$2.29 \times 10^8$ &10.64 &59.76\\
(6e,7o) &165 &$7.27 \times 10^6$ &$3.69 \times 10^8$ &38.82 &249.01\\
(6e,8o) &412 &$1.59 \times 10^7$ &$8.98 \times 10^8$ &158.74 &1332.66\\
(6e,9o) &1800 &$3.96 \times 10^7$ &$3.53 \times 10^9$ &578.49 &6332.44\\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
\titou{Table \ref{tab:timings} reports the evolution of the CPU timings for CASPT2 and CASPT3 as a function of the size of the active space. %%% FIGURE 3 %%%
It is particularly instructive to study the increase in CPU times as the number of external configuration grows.} \begin{figure}
\includegraphics[width=\linewidth]{timings}
\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).
Calculations have been performed in the frozen-core approximation and with the aug-cc-pVTZ basis set on an AMD Zen3 node (see main text).}
\label{fig:timings}
\end{figure}
%%% %%% %%% %%%
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.
All these calculations have been performed on an AMD Zen3 node \alert{with...}
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
@ -605,6 +631,7 @@ The two take-home messages are that:
i) CASPT3 transition energies are almost independent of the IPEA shift; i) CASPT3 transition energies are almost independent of the IPEA shift;
ii) CASPT2(IPEA) and CASPT3 have very similar accuracy. ii) CASPT2(IPEA) and CASPT3 have very similar accuracy.
The global trends are also true for specific sets of excitations and various system size. 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements} \begin{acknowledgements}

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