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@ -84,7 +84,7 @@ We provide a global overview of the successive steps that made possible to obtai
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. 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. 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. 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 performances of yet accurate, lower-order methods (\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 performances of computationally lighter models (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of excited states.
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. 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} \end{abstract}
@ -93,8 +93,8 @@ We conclude this \textit{Perspective} by discussing the current potentiality of
%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%
%%% INTRODUCTION %%% %%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%
The accurate modeling of excited-state properties from \textit{ab initio} quantum chemistry methods is a challenging yet self-proclaimed ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come (see, for example, Ref.~\onlinecite{Gon12} and references therein). The accurate modeling of excited-state properties with \textit{ab initio} quantum chemistry methods is a challenging yet self-proclaimed ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come (see, for example, Ref.~\onlinecite{Gon12} and references therein).
Of particular interest is the access to precise excitation energies, \ie, the energy difference between ground and excited states, and their intimate link with photochemical processes and photochemistry in general. Of particular interest is the access to precise excitation energies, \ie, the energy difference between ground and excited electronic states, and their intimate link with photochemical processes and photochemistry in general.
The factors that makes this quest for high accuracy particularly delicate are very diverse. The factors that makes this quest for high accuracy particularly delicate are very diverse.
First of all (and maybe surprisingly), it is, in most cases, tricky to obtain reliable and accurate experimental data that one can straightforwardly compare to theoretical values. First of all (and maybe surprisingly), it is, in most cases, tricky to obtain reliable and accurate experimental data that one can straightforwardly compare to theoretical values.
@ -110,7 +110,7 @@ What are the requirement of the ``perfect'' theoretical model?
As mentioned above, a balanced treatment of excited states with different character is highly desirable. As mentioned above, a balanced treatment of excited states with different character is highly desirable.
Moreover, chemically accurate excitation energies (\ie, with error smaller than $1$ kcal/mol or $0.043$ eV) would be also beneficial in order to provide a quantitative chemical picture. Moreover, chemically accurate excitation energies (\ie, with error smaller than $1$ kcal/mol or $0.043$ eV) would be also beneficial in order to provide a quantitative chemical picture.
The access to other properties, such as oscillator strengths, dipole moments and analytical energy gradients, is also an asset if one wants to compare with experimental data. The access to other properties, such as oscillator strengths, dipole moments and analytical energy gradients, is also an asset if one wants to compare with experimental data.
Let us not forget about the requirements of minimal user input and chemical intuition (\ie, black box method preferable) in order to minimize the potential bias brought by the user appreciation of the problem complexity. Let us not forget about the requirements of minimal user input and minimal chemical intuition (\ie, black box method preferable) in order to minimize the potential bias brought by the user appreciation of the problem complexity.
Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded. Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
Although the simultaneous fulfillment of all these requirements seems elusive, it is essential to keep these criteria in mind. Although the simultaneous fulfillment of all these requirements seems elusive, it is essential to keep these criteria in mind.
Table \ref{tab:method} is here for fulfill such a purpose. Table \ref{tab:method} is here for fulfill such a purpose.
@ -118,22 +118,23 @@ Table \ref{tab:method} is here for fulfill such a purpose.
%%% TABLE I %%% %%% TABLE I %%%
%\begin{squeezetable} %\begin{squeezetable}
\begin{table} \begin{table}
\caption{Scaling of various excited state methods with the number of basis functions $N$ and the availability of various key properties.} \caption{Scaling of various excited-state methods with the number of 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} \label{tab:method}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccc} \begin{tabular}{lcccc}
\mr{2}{*}{Method} & Formal & Oscillator & Analytic & Typical \\ \mr{2}{*}{Method} & Formal & Oscillator & Analytic & Typical \\
& scaling & strength & gradients & error (eV) \\ & scaling & strength & gradients & error (eV) \\
\hline \hline
TD-DFT & $N^4$ & \cmark & \cmark & $0.2$-$0.4$ \\ TD-DFT & $N^4$ & \cmark & \cmark & $0.2$--$0.4$ \\
BSE@GW & $N^4$ & \cmark & \xmark & $0.1$-$0.3$ \\ BSE@GW & $N^4$ & \cmark & \xmark & $0.1$--$0.3$ \\
\\ \\
CIS & $N^5$ & \cmark & \cmark & $1.0$-$1.5$ \\ CIS & $N^5$ & \cmark & \cmark & $1.0$--$1.5$ \\
CIS(D) & $N^5$ & \xmark & \cmark & $0.2$-$0.3$ \\ CIS(D) & $N^5$ & \xmark & \cmark & $0.2$--$0.3$ \\
ADC(2) & $N^5$ & \cmark & \cmark & $0.1$-$0.2$ \\ ADC(2) & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
CC2 & $N^5$ & \cmark & \cmark & $0.1$-$0.2$ \\ CC2 & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
\\ \\
ADC(3) & $N^6$ & \cmark & \xmark & $0.1$-$0.2$ \\ ADC(3) & $N^6$ & \cmark & \xmark & $0.1$--$0.2$ \\
EOM-CCSD & $N^6$ & \cmark & \cmark & $\sim 0.10$ \\ EOM-CCSD & $N^6$ & \cmark & \cmark & $\sim 0.10$ \\
\\ \\
CC3 & $N^7$ & \cmark & \xmark & $\sim 0.04$ \\ CC3 & $N^7$ & \cmark & \xmark & $\sim 0.04$ \\
@ -141,8 +142,8 @@ Table \ref{tab:method} is here for fulfill such a purpose.
EOM-CCSDT & $N^8$ & \xmark & \xmark & $\sim 0.03$ \\ EOM-CCSDT & $N^8$ & \xmark & \xmark & $\sim 0.03$ \\
EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark & $\sim 0.01$ \\ EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark & $\sim 0.01$ \\
\\ \\
CASPT2 or NEVPT2 & $N!$ & \cmark & \cmark & $0.1$-$0.2$ \\ CASPT2 or NEVPT2 & $N!$ & \cmark & \cmark & $0.1$--$0.2$ \\
SCI & $N!$ & \cmark & \cmark & $\sim 0.03$ \\ SCI & $N!$ & \xmark & \xmark & $\sim 0.03$ \\
FCI & $N!$ & \cmark & \xmark & $0.0$ \\ FCI & $N!$ & \cmark & \xmark & $0.0$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
@ -153,7 +154,7 @@ Table \ref{tab:method} is here for fulfill such a purpose.
%** HISTORY **% %** HISTORY **%
%************** %**************
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. 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). Interestingly, for pretty much every single methods, 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.
%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
@ -163,7 +164,7 @@ The first mainstream \textit{ab initio} method for excited states was probably C
CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states. CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states.
It is not unusual to have errors of the order of $1$ eV which precludes the usage of CIS as a quantitative quantum chemistry method. It is not unusual to have errors of the order of $1$ eV which precludes the usage of CIS as a quantitative quantum chemistry method.
Twenty years later, CIS(D) which adds a second-order perturbative correction to CIS was developed and implemented thanks to the efforts of Head-Gordon and coworkers. \cite{Hea94} Twenty years later, CIS(D) which adds a second-order perturbative correction to CIS was developed and implemented thanks to the efforts of Head-Gordon and coworkers. \cite{Hea94}
This second-order correction significantly reduces the magnitude of the error compared to CIS, with a typical error range of $0.2$-$0.3$ eV. This second-order correction significantly reduces the magnitude of the error compared to CIS, with a typical error range of $0.2$--$0.3$ eV.
Unfortunately, to the best of our knowledge, analytic nuclear gradients are not available for CIS(D). Unfortunately, to the best of our knowledge, analytic nuclear gradients are not available for CIS(D).
%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%
@ -172,16 +173,17 @@ Unfortunately, to the best of our knowledge, analytic nuclear gradients are not
In the early 1990's, the complete-active-space self-consistent field (CASSCF) method \cite{And90} and its second-order perturbation-corrected variant CASPT2 \cite{And92} (both originally developed in Roos' group) appeared. In the early 1990's, the complete-active-space self-consistent field (CASSCF) method \cite{And90} and its second-order perturbation-corrected variant CASPT2 \cite{And92} (both originally developed in Roos' group) appeared.
This was a real breakthrough. This was a real breakthrough.
Although it took more than ten years to obtain analytic nuclear gradients, \cite{Cel03} CASPT2 was probably the first method that could provide quantitative results for molecular excited states of genuine photochemical interest. \cite{Roo96} Although it took more than ten years to obtain analytic nuclear gradients, \cite{Cel03} CASPT2 was probably the first method that could provide quantitative results for molecular excited states of genuine photochemical interest. \cite{Roo96}
Nonetheless, it is common knowledge that CASPT2 has the strong tendency of underestimating vertical excitation energies in organic molecules.
Driven by Celestino 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. Driven by Celestino 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. 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 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.
With a typical minimal valence active space, the typical error in CASPT2 or NEVPT2 calculations is $0.1$-$0.2$ eV. With a typical minimal valence active space tailored for the desired transitions, the typical error in CASPT2 or NEVPT2 calculations is $0.1$--$0.2$ eV.
%%%%%%%%%%%%% %%%%%%%%%%%%%
%%% TDDFT %%% %%% TDDFT %%%
%%%%%%%%%%%%% %%%%%%%%%%%%%
The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Cas95} was a real shock for the community as TD-DFT was able to provide accurate excitation energies at a much lower cost than its predecessors in a very black-box way. The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Cas95} was a real shock for the community as TD-DFT was able to provide accurate excitation energies at a much lower cost than its predecessors in a very black-box way.
For low-lying excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$-$0.4$ eV. For low-lying excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$--$0.4$ eV.
However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Mai04,Dre05,Lev06,Eli11} 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} 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. Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
@ -193,14 +195,15 @@ Despite all of this, TD-DFT is still nowadays the most employed excited-state me
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the huge growth of computational ressources, equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) \cite{Sta93} became mainstream in the 2000's. Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the huge growth of computational ressources, equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) \cite{Sta93} became mainstream in the 2000's.
EOM-CCSD gradients were also quickly available. \cite{Sta95} EOM-CCSD gradients were also quickly available. \cite{Sta95}
With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV. With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV, and a typical overestimation of the vertical transition energies.
Its third-order version, EOM-CCSDT, was also implemented and provides, at a significant higher cost, high accuracy for single excitations. \cite{Nog87} Its third-order version, EOM-CCSDT, was also implemented and provides, at a significant higher cost, high accuracy for single excitations. \cite{Nog87}
Although extremely expensive and tedious to implement, higher orders are also technically possible for small systems thanks to automatically generated code. \cite{Kuc91,Hir04} Although extremely expensive and tedious to implement, higher orders are also technically possible for small systems thanks to automatically generated code. \cite{Kuc91,Hir04}
The EOM-CC family of methods was quickly followed by an approximated and computationally lighter family with, in front line, the second-order CC2 model \cite{Chr95} and its third-order extension, CC3. \cite{Chr95b} The EOM-CC family of methods was quickly followed by an approximated and computationally lighter family with, in front line, the second-order CC2 model \cite{Chr95} and its third-order extension, CC3. \cite{Chr95b}
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. 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. 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} 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 study. 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.
%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%
%%% ADC METHODS %%% %%% ADC METHODS %%%
@ -208,14 +211,14 @@ For the sake of brevity, we drop the EOM acronym in the rest of this study.
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. 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 [including the ADC(2) analytical gradients] as well as other interesting variants. \cite{Dre15} 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 [including the ADC(2) analytical gradients] as well as other interesting variants. \cite{Dre15}
These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents. 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. 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} However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies. \cite{Loo18a,Loo20}
%%%%%%%%%%%%%% %%%%%%%%%%%%%%
%%% BSE@GW %%% %%% 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}). 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 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}
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} 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. However, due to the adiabatic (\ie, static) approximation, doubly excited states are completely absent from the BSE spectrum.
@ -239,7 +242,7 @@ In summary, each method has its own strengths and weaknesses, and none of them i
%%% FIG 1 %%% %%% FIG 1 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{Set1} \includegraphics[width=\linewidth]{Set1}
\caption{Mean absolute error (in eV) with respect to the TBE/aug-cc-pVTZ values from the Jacquemin set \#1 (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/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} \label{fig:Set1}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%%
@ -256,7 +259,7 @@ In summary, each method has its own strengths and weaknesses, and none of them i
%%% FIG 3 %%% %%% FIG 3 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{Set3} \includegraphics[width=\linewidth]{Set3}
\caption{Mean absolute error (in eV) with respect to the TBE/aug-cc-pVTZ values from the Jacquemin set \#3 (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/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} \label{fig:Set3}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%%
@ -274,14 +277,14 @@ 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} 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. 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). 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 (in 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 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.
Their main conclusion was that \textit{``CC3 and CASPT2 excitation energies are in excellent agreement for states which are dominated by single excitations''}. 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 sooner 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 aug-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions (especially for Rydberg states).
As a direct evidence of the 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). 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. Theoretical improvements of Thiel's set were slow but steady, highlighting further its quality.
In 2013, Watson et al.\cite{Wat13} computed CCSDT-3/TZVP (an iterative approximation of the triples of CCSDT \cite{Wat96}) excitation energies for the Thiel set. In 2013, Watson et al.\cite{Wat13} computed CCSDT-3/TZVP (an iterative approximation of the triples of CCSDT \cite{Wat96}) excitation energies for the Thiel set.
Their quality were very similar to the CC3 values reported in Ref.~\onlinecite{Sau09} and the authors could not appreciate which model was more accurate. Their quality were very similar to the CC3 values reported in Ref.~\onlinecite{Sau09} and the authors could not appreciate which model was the most accurate.
Similarly, Dreuw and coworkers performed ADC(3) calculations on Thiel's set and arrived at the same kind of conclusion: \cite{Har14} Similarly, Dreuw and coworkers performed ADC(3) calculations on Thiel's set and arrived at the same kind of conclusion: \cite{Har14}
\textit{``based on the quality of the existing benchmark set it is practically not possible to judge whether ADC(3) or CC3 is more accurate''}. \textit{``based on the quality of the existing benchmark set it is practically not possible to judge whether ADC(3) or CC3 is more accurate''}.
%Finally, let us mention the work of Kannar and Szalay who reported CCSDT excitation energies \cite{Kan14,Kan17} for a subset of the original Thiel set. %Finally, let us mention the work of Kannar and Szalay who reported CCSDT excitation energies \cite{Kan14,Kan17} for a subset of the original Thiel set.
@ -293,9 +296,10 @@ 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} 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. 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/aug-cc-pVTZ geometries.
In the following, we label this set a {\SetA}. 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} 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, although less accurate than CC3, CCSDT-3 can be used as a reliable reference for benchmark studies, and that ADC(3) delivers quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version ADC(2). 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.
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 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. 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.
@ -303,11 +307,11 @@ An important addition to this second study was the computation of double excitat
Our results clearly evidence that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition. 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 trans-butadiene) involving a significant amount of singles.
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. 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}). 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. 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. 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 these energies, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with tens of millions of determinants), as well as the most robust multiconfigurational method, NEVPT2. 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 tens of millions of determinants), as well as the most robust multiconfigurational method, NEVPT2.
Each approach was applied in combination with diffuse-containing atomic basis sets. 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. 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/aug-cc-pVQZ transition energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition.
@ -316,11 +320,11 @@ These findings were in perfect agreement with our two previous studies. \cite{Lo
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). 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. 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 focussing on expanding and combining these sets to create an \textit{ultimate} test set of highly-accurate excitations energies. Our current efforts are now focussing on expanding and combining 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} 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 single mega set of more than 400 vertical transition energies for small- and medium-size molecules based on accurate ground-state geometries. 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. Such a set would definitely be a terrific asset for the entire electronic structure community.
It would surely stimulate further theoretical developments in excited-state methods and provide a fair ground for the assessements of the currently available or under development excited-state models. It would surely stimulate further theoretical developments in excited-state methods and provide a fair ground for the assessments of the currently available and under development excited-state models.
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