loos/CASPT3
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

OK for T2

Browse Source
This commit is contained in:
Pierre-Francois Loos 2022-04-11 22:38:40 +02:00
parent 8fab3b3d12
commit 325f3ab06b
1 changed files with 23 additions and 23 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

46
Manuscript/CASPT3.tex
View File

@ -17,7 +17,7 @@
\newcommand{\trashAS}[1]{\textcolor{green}{\sout{#1}}}
\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}}
\newcommand{\ant}[1]{\textcolor{orange}{#1}}
\newcommand{\SupInf}{\textcolor{blue}{Supporting Information}}
\newcommand{\SupMat}{\textcolor{blue}{supplementary material}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
@ -91,8 +91,8 @@
% Abstract
\begin{abstract}
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.
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 transition type and 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}}
@ -112,7 +112,7 @@ 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 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 of the series when increasing the perturbation order. \cite{Marie_2021a}
Systematic improvement is thus difficult to achieve and it is extremely challenging to predict, \textit{a priori}, the evolution of the series when ramping up 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
@ -130,7 +130,7 @@ In the context of excited states, its most severe drawback is certainly the intr
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.
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 second pitfall was brought to light 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 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}
@ -156,7 +156,7 @@ We underline that, although a third-order version of NEVPT has been developed \c
%%% FIGURE 1 %%%
\begin{figure}
\includegraphics[width=0.8\linewidth]{mol.pdf}
\includegraphics[width=\linewidth]{mol.pdf}
\caption{Various molecular systems considered in this study.
\label{fig:mol}}
\end{figure}
@ -166,21 +166,21 @@ For each compound represented in Fig.~\ref{fig:mol}, we have computed the CASPT2
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}.
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}.
All the CASPT2 and CASPT3 calculations have been carried out in the frozen-core approximation and within the RS2 and RS3 contraction schemes as implemented in MOLPRO and described in Refs.~\onlinecite{Werner_1996} and \onlinecite{Werner_2020}.
Both methods have been tested with and without IPEA (labeled as NOIPEA).
When an IPEA shift is applied, its value is set to the \SI{0.25}{\hartree} default, as discussed in Ref.~\onlinecite{Ghigo_2004}.
When an IPEA shift is applied, it 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}
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.
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 representation.
For each system and transition, we report in the {\SupMat} the exhaustive description of the active spaces for each symmetry representation.
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} the active spaces of acrolein, pyrimidine, and pyridazine have been slightly reduced in order to make the CASPT3 calculations technically achievable.
In these cases, we have recomputed the CASPT2 values for the same active space for the sake of consistency.
In these cases, for the sake of consistency, 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 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 such cases are detailed in {\SupInf}.
This value has been slightly increased in particularly difficult cases, and such cases are detailed in {\SupMat}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion}
@ -193,7 +193,7 @@ The exhaustive list of CASPT2 and CASPT3 transitions can be found in Table \ref{
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 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}.
Error patterns for selected subsets are reported in {\SupMat}.
%%% TABLE I %%%
\begin{longtable*}{cllccccccccc}
@ -582,7 +582,7 @@ It is worth mentioning that CASPT3(NOIPEA) yields MAEs for each subset that is a
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.}
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}
Comparatively, Liang \textit{et al.} have recently 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, these two methods do 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{Veril_2021} it has the undisputable advantage to describe with the same accuracy both single and double excitations.
This feature is crucial in the description of some photochemistry mechanisms. \cite{Boggio-Pasqua_2007}
@ -627,7 +627,7 @@ This feature is crucial in the description of some photochemistry mechanisms. \c
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 the frozen-core approximation (42 electrons and 414 basis functions) for increasingly large active spaces.
All these calculations have been performed on a single core of an Intel Xeon E5-2670 2.6Ghz.
It is particularly instructive to study the wall time ratio as the number of (contracted and uncontracted) external configurations grows (Fig.~\ref{fig:timings}).
It is particularly instructive to study the wall time ratio as the number of (contracted and uncontracted) external configurations grows (see 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, and remains thus typically affordable for these kinds of calculations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -637,10 +637,16 @@ Overall, the PT3 step takes between 5 and 10 times longer than the PT2 step for
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 principal take-home messages of this study are that:
(i) CASPT3 transition energies are almost independent of the IPEA shift;
(ii) CASPT2(IPEA) and CASPT3 deliver a very similar accuracy.
(ii) CASPT2(IPEA) and CASPT3 have a very similar accuracy.
These global trends are also true for specific sets of excitations and various system sizes.
Therefore, if one can afford the extra computation cost associated with the third-order energy (which is only several times more than its second-order counterpart), one can eschew the delicate choice of the IPEA value in CASPT2, and rely solely on the CASPT3(NOIPEA) excitation energies.
%%%%%%%%%%%%%%%%%%%%%%
\section*{Supplementary Material}
\label{sec:supmat}
%%%%%%%%%%%%%%%%%%%%%%
Included in the {\SupMat} are the error distributions obtained for CASPT2 and CASPT3 with and without IPEA shift for various subsets of transitions, as well as the description and specification of the active space for each molecule.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2021-18005.
@ -649,16 +655,10 @@ PFL thanks the European Research Council (ERC) under the European Union's Horizo
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting information available}
\label{sec:SI}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement}
\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}.
The data that supports the findings of this study are available within the article and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{CASPT3}