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%% This BibTeX bibliography file was created using BibDesk.
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@string{theo = {J. Mol. Struct. (THEOCHEM)}}
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Date-Added = {2019-11-16 13:38:18 +0100},
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Shorttitle = {Excitation {{Number}}},
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publisher = {IOP Publishing},
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Author = {Kim, Jeongnim and Baczewski, Andrew D. and Beaudet, Todd D. and Benali, Anouar and Bennett, M. Chandler and Berrill, Mark A. and Blunt, Nick S. and Borda, Edgar Josu{\ifmmode\acute{e}\else\'{e}\fi} Landinez and Casula, Michele and Ceperley, David M. and Chiesa, Simone and Clark, Bryan K. and Iii, Raymond C. Clay and Delaney, Kris T. and Dewing, Mark and Esler, Kenneth P. and Hao, Hongxia and Heinonen, Olle and Kent, Paul R. C. and Krogel, Jaron T. and Kyl{\ifmmode\ddot{a}\else\"{a}\fi}np{\ifmmode\ddot{a}\else\"{a}\fi}{\ifmmode\ddot{a}\else\"{a}\fi}, Ilkka and Li, Ying Wai and Lopez, M. Graham and Luo, Ye and Malone, Fionn D. and Martin, Richard M. and Mathuriya, Amrita and McMinis, Jeremy and Melton, Cody A. and Mitas, Lubos and Morales, Miguel A. and Neuscamman, Eric and Parker, William D. and Flores, Sergio D. Pineda and Romero, Nichols A. and Rubenstein, Brenda M. and Shea, Jacqueline A. R. and Shin, Hyeondeok and Shulenburger, Luke and Tillack, Andreas F. and Townsend, Joshua P. and Tubman, Norm M. and Van Der Goetz, Brett and Vincent, Jordan E. and Yang, D. ChangMo and Yang, Yubo and Zhang, Shuai and Zhao, Luning},
Doi = {10.1088/1361-648x/aab9c3},
Issn = {0953-8984},
Journal = {J. Phys.: Condens. Matter},
Month = {Apr},
Number = {19},
Pages = {195901},
Publisher = {IOP Publishing},
Title = {{QMCPACK: an open source ab initio quantum Monte Carlo package for the electronic structure of atoms, molecules and solids}},
Volume = {30},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1088/1361-648x/aab9c3}}
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Issn = {0021-9606},
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Number = {15},
Pages = {154107},
Publisher = {American Institute of Physics},
Title = {{An atomic orbital-based formulation of analytical gradients and nonadiabatic coupling vector elements for the state-averaged complete active space self-consistent field method on graphical processing units}},
Volume = {143},
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Journal = {arXiv},
Month = {Nov},
Title = {{An efficient implementation of Slater-Condon rules}},
Url = {https://arxiv.org/abs/1311.6244},
Year = {2013},
Bdsk-Url-1 = {https://arxiv.org/abs/1311.6244}}

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Manuscript/ExPerspective.tex
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\begin{document}
\title{The Quest For Highly Accurate Excitation Energies}
\title{The Quest For Highly Accurate Excitation Energies: A Computational Perspective}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
@ -83,11 +70,11 @@
\centering
\includegraphics[width=\linewidth]{TOC}
\end{wrapfigure}
We provide a global overview of the successive steps that made possible to obtain increasingly accurate excitation energies and properties, eventually leading to chemically accurate vertical transition energies for small- and medium-size molecules.
We provide a global overview of the successive steps that made possible to obtain increasingly accurate excitation energies and properties with computational chemistry tools, eventually leading to chemically accurate vertical transition energies for small- and medium-size molecules.
First, we describe the evolution of \textit{ab initio} state-of-the-art methods, with originally Roos' CASPT2 method, and then third-order coupled cluster methods as in the renowned Thiel set of excitation energies described in a remarkable series of papers in the 2000's.
More recently, this quest for highly accurate excitation energies was reinitiated thanks to the resurgence of selected configuration interaction (SCI) methods and their efficient parallel implementation.
These methods have been able to routinely deliver highly accurate excitation energies for small molecules as well as medium-size molecules in compact basis sets for single and double excitations.
Second, we describe how these high-level methods and the creation of large, diverse and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performance of computationally lighter theoretical models (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of excited states.
Second, we describe how these high-level methods and the creation of large, diverse and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performance of computationally lighter theoretical models (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of excited states ($\pi \ra \pi^*$, $n \ra \pi^*$, valence, Rydberg, singlet, triplet, double excitation, etc).
We conclude this \textit{Perspective} by discussing the current potentiality of these methods from both an expert and a non-expert point of view, and what we believe could be the future theoretical and technological developments in the field.
\end{abstract}
@ -106,7 +93,7 @@ Even more problematic, experimental spectra might not be available in gas phase,
For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Loo18b,Loo19a,Loo19b}
Another feature that makes excited states particularly fascinating and challenging is that they can be both extremely close in energy from each other and have very different natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet, triplet, etc).
Therefore, it would be highly desirable to possess a computational method or protocol providing a balanced treatment of the entire ``spectrum'' of excited states.
Therefore, it would be highly desirable to possess a computational method (or protocol) that provides a balanced treatment of the entire ``spectrum'' of excited states.
And let's be honest, none of the existing methods does provide this at an affordable cost.
What are the requirement of the ``perfect'' theoretical model?
@ -121,7 +108,7 @@ Table \ref{tab:method} is here for fulfill such a purpose.
%%% TABLE I %%%
%\begin{squeezetable}
\begin{table}
\caption{Scaling of various excited-state methods with the number of basis functions $N$ and the availability of various key properties.
\caption{Formal computational scaling of various excited-state methods with respect to the number of one-electron basis functions $N$ and the availability of various key properties.
The typical error range or estimate for single excitations is also provided as a rough indicator of the method accuracy.}
\label{tab:method}
\begin{ruledtabular}
@ -132,7 +119,7 @@ The typical error range or estimate for single excitations is also provided as a
TD-DFT & $N^4$ & \cmark & \cmark & $0.2$--$0.4$ \\
BSE@GW & $N^4$ & \cmark & \xmark & $0.1$--$0.3$ \\
\\
CIS & $N^5$ & \cmark & \cmark & $1.0$--$1.5$ \\
CIS & $N^5$ & \cmark & \cmark & $\sim 1.0$ \\
CIS(D) & $N^5$ & \xmark & \cmark & $0.2$--$0.3$ \\
ADC(2) & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
CC2 & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
@ -158,7 +145,7 @@ The typical error range or estimate for single excitations is also provided as a
%**************
Before detailing some key past and present contributions towards the obtention of highly accurate excitation energies, we start by giving a historical overview of the various excited-state \textit{ab initio} methods that have emerged in the last fifty years.
Interestingly, for pretty much every single method, the theory was derived much earlier than their actual implementation in electronic structure software packages (same applies to the analytic gradients when available).
Here, we only mention methods that, we think, ended up becoming mainstream.
%Here, we only mention methods that, we think, ended up becoming mainstream.
%%%%%%%%%%%%%%%%%%%%%
%%% POPLE'S GROUP %%%
@ -178,7 +165,7 @@ Although it took more than ten years to obtain analytic nuclear gradients, \cite
Nonetheless, it is common knowledge that CASPT2 has the strong tendency of underestimating vertical excitation energies in organic molecules.
Driven by Angeli and Malrieu, \cite{Ang01} the creation of the second-order $n$-electron valence state perturbation theory (NEVPT2) method several years later was able to cure some of the main theoretical deficiencies of CASPT2.
For example, NEVPT2 is known to be intruder state free and size consistent.
The limited applicability of these so-called multiconfigurational methods is mainly due to the necessity of defining an active space based on the desired transition(s), as well as their factorial computational growth with the number of active electrons and orbitals.
The limited applicability of these so-called multiconfigurational methods is mainly due to the necessity of carefully defining an active space based on the desired transition(s) in order to obtain meaningful results, as well as their factorial computational growth with the number of active electrons and orbitals.
With a typical minimal valence active space tailored for the desired transitions, the usual error in CASPT2 or NEVPT2 calculations is $0.1$--$0.2$ eV.
%%%%%%%%%%%%%
@ -189,7 +176,7 @@ For low-lying excited states, TD-DFT calculations relying on hybrid exchange-cor
However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Mai04,Dre05,Lev06,Eli11}
In the present context, one of the most annoying feature of TD-DFT --- in its most standard (adiabatic) approximation --- is its inability to describe, even qualitatively, charge-transfer states, \cite{Toz99,Dre04} Rydberg states, \cite{Toz98} and double excitations. \cite{Mai04,Lev06,Eli11}
Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
One of the main issues is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals. \cite{Sue19}
One of the main issues is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals. \cite{Goe19,Sue19}
Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community (and beyond).
%%%%%%%%%%%%%%%%%%
@ -205,7 +192,7 @@ The EOM-CC family of methods was quickly followed by an approximated and computa
As a $N^7$ method (where $N$ is the number of basis functions), CC3 has a particularly interesting accuracy/cost ratio with errors usually below the chemical accuracy threshold.
The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively.
Thanks to the introduction of triples, EOM-CCSDT and CC3 also provide qualitative results for double excitations, a feature that is completely absent from EOM-CCSD and CC2. \cite{Loo19c}
For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Perspective} keeping in mind that these CC methods are applied to excited states.
For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Perspective} keeping in mind that these CC methods are applied to excited states in the present context.
%%%%%%%%%%%%%%%%%%%
%%% ADC METHODS %%%
@ -213,14 +200,14 @@ For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Per
It is also important to mention the recent rejuvenation of the second- and third-order algebraic diagrammatic construction [ADC(2) \cite{Sch82} and ADC(3) \cite{Tro99,Har14}] which scale as $N^5$ and $N^6$, respectively.
This renaissance was certainly initiated by the enormous amount of work invested by Dreuw's group in order to provide a fast and efficient implementation of these methods, \cite{Dre15} including the analytical gradients, \cite{Reh19} as well as other interesting variants.
These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
In that regard, ADC(2) is particularly attractive with an error generally around $0.1$--$0.2$ eV.
However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies. \cite{Loo18a,Loo20}
In that regard, ADC(2) is particularly attractive with an error around $0.1$--$0.2$ eV.
However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies \cite{Loo18a,Loo20} (see also Ref.~\onlinecite{Tro02}).
%%%%%%%%%%%%%%
%%% BSE@GW %%%
%%%%%%%%%%%%%%
Finally, let us mention the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Str88} (which is usually performed on top of a GW calculation \cite{Hed65}).
BSE has gained momentum in the past few years and is a good candidate as a computationally inexpensive electronic structure theory method that can model accurately excited states (with a typical error of $0.1$--$0.3$ eV) and their corresponding properties. \cite{Jac17b,Bla18}
BSE has gained momentum in the past few years and is a serious candidate as a computationally inexpensive electronic structure theory method that can model accurately excited states (with a typical error of $0.1$--$0.3$ eV) and their corresponding properties. \cite{Jac17b,Bla18}
One of the main advantage of BSE compared to TD-DFT (with a similar computational cost) is that it allows a faithful description of charge-transfer states and, when performed on top of a (partially) self-consistently GW calculation, BSE@GW has been shown to be weakly dependent on its starting point. \cite{Jac16,Gui18}
However, due to the adiabatic (\ie, static) approximation, doubly excited states are completely absent from the BSE spectrum.
@ -244,7 +231,7 @@ In summary, each method has its own strengths and weaknesses, and none of them i
%%% FIG 1 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{Set1}
\caption{Mean absolute error (in eV) with respect to the TBE/aug-cc-pVTZ values from the {\SetA} set (as described in Ref.~\onlinecite{Loo18a}) for various methods and types of excited states.}
\caption{Mean absolute error (in eV) with respect to the TBE/\emph{aug}-cc-pVTZ values from the {\SetA} set (as described in Ref.~\onlinecite{Loo18a}) for various methods and types of excited states.}
\label{fig:Set1}
\end{figure*}
%%% %%% %%%
@ -253,7 +240,7 @@ In summary, each method has its own strengths and weaknesses, and none of them i
\begin{figure}
\includegraphics[width=\linewidth]{Set2}
\caption{Mean absolute error (in eV) (with respect to exFCI excitation energies) for the doubly excited states reported in Ref.~\onlinecite{Loo19c} for various methods taking into account at least triple excitations.
$\%T_1$ corresponds to the percentage of single excitations in the transition calculated at the CC3 level.}
$\%T_1$ corresponds to the percentage of single excitations (calculated at the CC3 level) in the transition.}
\label{fig:Set2}
\end{figure}
%%% %%% %%%
@ -261,7 +248,7 @@ In summary, each method has its own strengths and weaknesses, and none of them i
%%% FIG 3 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{Set3}
\caption{Mean absolute error (in eV) with respect to the TBE/aug-cc-pVTZ values from the {\SetC} set (as described in Ref.~\onlinecite{Loo20}) for various methods and types of excited states.}
\caption{Mean absolute error (in eV) with respect to the TBE/\emph{aug}-cc-pVTZ values from the {\SetC} set (as described in Ref.~\onlinecite{Loo20}) for various methods and types of excited states.}
\label{fig:Set3}
\end{figure*}
%%% %%% %%%
@ -279,9 +266,9 @@ For excited states, things started moving a little later but some major contribu
One of these major contributions was provided by the group of Walter Thiel \cite{Sch08,Sil08,Sau09,Sil10b,Sil10c} with the introduction of the so-called Thiel set of excitation energies. \cite{Sch08}
For the first time, this set was large, diverse and accurate enough to be used as a proper benchmarking set for excited-state methods.
More specifically, it gathers a large number of excitation energies consisting of 28 medium-size organic molecules with a total of 223 excited states (152 singlet and 71 triplet states).
In their first study they performed CC2, CCSD, CC3 and MS-CASPT2 calculations (with the TZVP basis) in order to provide (based on additional high-level literature data) a list of best theoretical estimates (TBEs) for all these transitions.
In their first study they performed CC2, CCSD, CC3 and CASPT2 calculations (with the TZVP basis) in order to provide (based on additional high-level literature data) a list of best theoretical estimates (TBEs) for all these transitions.
Their main conclusion was that \textit{``CC3 and CASPT2 excitation energies are in excellent agreement for states which are dominated by single excitations''}.
These TBEs were quickly refined with the larger aug-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions (especially for Rydberg states).
These TBEs were quickly refined with the larger \emph{aug}-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions (especially for Rydberg states).
As a direct evidence of the actual value of reference data, these TBEs were quickly picked up to benchmark various computationally effective methods from semi-empirical to state-of-the-art \textit{ab initio} methods (see Ref.~\onlinecite{Loo18a} and references therein).
Theoretical improvements of Thiel's set were slow but steady, highlighting further its quality.
@ -297,32 +284,32 @@ These two studies clearly demonstrate and motivate the need for higher accuracy
%%%%%%%%%%%%%%%%%%%%%%%
Recently, we made, what we think, is a significant contribution to this quest for highly accurate excitation energies. \cite{Loo18a}
More specifically, we studied 18 small molecules with sizes ranging from one to three non-hydrogen atoms.
For such systems, using a combination of high-order CC methods, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various natures (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly excited) based on accurate CC3/aug-cc-pVTZ geometries.
For such systems, using a combination of high-order CC methods, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various natures (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly excited) based on accurate CC3/\emph{aug}-cc-pVTZ geometries.
In the following, we label this set of TBEs as {\SetA}.
Importantly, it allowed us to benchmark a series of 12 popular excited-state wave function methods accounting for double and triple excitations (see Fig.~\ref{fig:Set1}): CIS(D), ADC(2), CC2, STEOM-CCSD, \cite{Noo97} CCSD, CCSDR(3), \cite{Chr77} and CCSDT-3. \cite{Wat96}
Our main conclusion was that CC3 was extremely accurate (with a mean absolute error of only $\sim 0.03$ eV), and that, although less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies.
Our main conclusion was that CC3 is extremely accurate (with a mean absolute error of only $\sim 0.03$ eV), and that, although less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies.
Quite surprisingly, ADC(3) was found to deliver quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version ADC(2).
In a second study, \cite{Loo19c} using a similar combination of theoretical models (but mostly extrapolated SCI energies), we provided accurate reference excitation energies for transitions involving a substantial amount of double excitations using a series of increasingly large diffuse-containing atomic basis sets (up to aug-cc-pVQZ when technically feasible).
In a second study, \cite{Loo19c} using a similar combination of theoretical models (but mostly extrapolated SCI energies), we provided accurate reference excitation energies for transitions involving a substantial amount of double excitations using a series of increasingly large diffuse-containing atomic basis sets (up to \emph{aug}-cc-pVQZ when technically feasible).
Our set gathers 20 vertical transitions from 14 small- and medium-sized molecules (acrolein, benzene, beryllium atom, butadiene, carbon dimer and trimer, ethylene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine), a set we label as {\SetB} in the remaining of this paper.
An important addition to this second study was the computation of double excitations with multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative or iterative triple corrections (see Fig.~\ref{fig:Set2}).
An important addition to this second study was the computation of double excitations with various flavors of multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative or iterative triple corrections (see Fig.~\ref{fig:Set2}).
Our results clearly evidence that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition.
For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and CCSDT can easily reach $1$ and $0.5$ eV, respectively, while it goes down to a few tenths of an electronvolt for more common transitions (such as in trans-butadiene) involving a significant amount of singles.
For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and CCSDT can easily reach $1$ and $0.5$ eV, respectively, while it goes down to a few tenths of an electronvolt for more common transitions (such as in \emph{trans}-butadiene and benzene) involving a significant amount of singles.\cite{Shu17,Bar18b,Bar18a}
The quality of the excitation energies obtained with multiconfigurational methods was harder to predict as the overall accuracy of these methods is highly dependent on both the system and the selected active space.
Interestingly, CASPT2 and NEVPT2 were found to be more accurate for transition with a small percentage of single excitations (error usually below $0.1$ eV) than for excitations dominated by single excitations where the error is more around $0.1$--$0.2$ eV (see Fig.~\ref{fig:Set2}).
In our latest study, \cite{Loo20} in order to provide more general conclusions, we generated highly accurate vertical transition energies for larger compounds with a set composed by 27 molecules encompassing from 4 to 6 non-hydrogen atoms for a total of \alert{238} vertical transition energies of various natures.
This set, labeled as {\SetC} and still based on CC3/aug-cc-pVTZ geometries, is constituted by a reasonably good balance of singlet, triplet, valence, and Rydberg states.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with a hundred million of determinants), as well as the most robust multiconfigurational method, NEVPT2.
In our latest study, \cite{Loo20} in order to provide more general conclusions, we generated highly accurate vertical transition energies for larger compounds with a set composed by 27 molecules encompassing from four to six non-hydrogen atoms for a total of \alert{238} vertical transition energies of various natures.
This set, labeled as {\SetC} and still based on CC3/\emph{aug}-cc-pVTZ geometries, is constituted by a reasonably good balance of singlet, triplet, valence, and Rydberg states.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with up to hundred million determinants), as well as the most robust multiconfigurational method, NEVPT2.
Each approach was applied in combination with diffuse-containing atomic basis sets.
Because the SCI energy converges obviously slower for these larger systems, the extrapolated SCI values were employed as a``safely net'' to demonstrate the overall consistency of our CC-based protocol rather than straight out-of-the-box reference values.
For all the transitions of the {\SetC} set, we reported at least CC3/aug-cc-pVQZ transition energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition.
For all the transitions of the {\SetC} set, we reported at least CC3/\emph{aug}-cc-pVQZ transition energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition.
Pursuing our previous benchmarking efforts, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a dominant double excitation character (see Fig.~\ref{fig:Set3}).
These findings were in perfect agreement with our two previous studies. \cite{Loo18a,Loo19c}
This definitely settles down the debate by demonstrating the superiority of CC3 (in terms of accuracy) compared to methods like CCSDT-3 or ADC(3).
Moreover, thanks to the exhaustive and detailed comparisons made in Ref.~\onlinecite{Loo20}, we could safely conclude that CC3 also regularly outperforms CASPT2 and NEVPT2 as long as the corresponding transition does not show any strong multiple excitation character.
Our current efforts are now focussing on expanding and merging these sets to create an \textit{ultimate} test set of highly-accurate excitations energies.
Our current efforts are now focussing on expanding and merging these sets to create an \textit{ultimate} test set of highly accurate excitations energies.
In particular, we are currently generating reference excitations energies for radicals as well as more ``exotic'' molecules containing heavier atoms (such as \ce{Cl}, \ce{F}, \ce{P}, and \ce{Si}). \cite{Loo20b}
The combination of these various sets would potentially create a mega-set of more than 400 vertical transition energies for small- and medium-size molecules based on accurate ground-state geometries.
Such a set would definitely be a terrific asset for the entire electronic structure community.

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