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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-06 15:17:19 +0100
%% Created for Pierre-Francois Loos at 2019-11-06 20:54:53 +0100
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@ -82,6 +82,71 @@
@string{theo = {J. Mol. Struct. (THEOCHEM)}}
@inbook{Mai12,
Address = {Berlin, Heidelberg},
Author = {Maitra, Neepa T.},
Booktitle = {Fundamentals of Time-Dependent Density Functional Theory},
Date-Added = {2019-11-06 20:54:06 +0100},
Date-Modified = {2019-11-06 20:54:10 +0100},
Doi = {10.1007/978-3-642-23518-4_8},
Editor = {Marques, Miguel A.L. and Maitra, Neepa T. and Nogueira, Fernando M.S. and Gross, E.K.U. and Rubio, Angel},
File = {/Users/loos/Zotero/storage/MAFNZHIQ/Maitra - 2012 - Memory History , Initial-State Dependence , and D.pdf},
Isbn = {978-3-642-23517-7 978-3-642-23518-4},
Pages = {167-184},
Publisher = {Springer Berlin Heidelberg},
Title = {Memory: History , Initial-State Dependence , and Double-Excitations},
Volume = {837},
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1007/978-3-642-23518-4_8}}
@article{Mai04,
Author = {Maitra, Neepa T. and Zhang, Fan and Cave, Robert J. and Burke, Kieron},
Date-Added = {2019-11-06 20:53:34 +0100},
Date-Modified = {2019-11-06 20:53:39 +0100},
Doi = {10.1063/1.1651060},
File = {/Users/loos/Zotero/storage/KQFDU7KL/Maitra et al. - 2004 - Double excitations within time-dependent density f.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {J. Chem. Phys.},
Language = {en},
Month = apr,
Number = {13},
Pages = {5932-5937},
Title = {Double Excitations within Time-Dependent Density Functional Theory Linear Response},
Volume = {120},
Year = {2004},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1651060}}
@article{Eli11,
Abstract = {The adiabatic approximation in time-dependent density functional theory (TDDFT) yields reliable excitation spectra with great efficiency in many cases, but fundamentally fails for states of double-excitation character. We discuss how double-excitations are at the root of some of the most challenging problems for \{TDDFT\} today. We then present new results for (i) the calculation of autoionizing resonances in the helium atom, (ii) understanding the nature of the double excitations appearing in the quadratic response function, and (iii) retrieving double-excitations through a real-time semiclassical approach to correlation in a model quantum dot. },
Author = {Peter Elliott and Sharma Goldson and Chris Canahui and Neepa T. Maitra},
Date-Added = {2019-11-06 20:52:35 +0100},
Date-Modified = {2019-11-06 20:52:35 +0100},
Doi = {http://dx.doi.org/10.1016/j.chemphys.2011.03.020},
Issn = {0301-0104},
Journal = {Chem. Phys.},
Keywords = {Adiabatic approximation},
Number = {1},
Pages = {110--119},
Title = {Perspectives on double-excitations in \{TDDFT\}},
Url = {http://www.sciencedirect.com/science/article/pii/S0301010411000966},
Volume = {391},
Year = {2011},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0301010411000966},
Bdsk-Url-2 = {http://dx.doi.org/10.1016/j.chemphys.2011.03.020}}
@inbook{Cas95,
Address = {Singapore},
Author = {Casida, M. E.},
Date-Added = {2019-11-06 20:50:07 +0100},
Date-Modified = {2019-11-06 20:50:07 +0100},
Editor = {D. P. Chong},
Pages = {155--192},
Publisher = {World Scientific},
Series = {Recent Advances in Density Functional Methods},
Title = {Time-Dependent Density-Functional Response Theory for Molecules},
Volume = {1},
Year = 1995}
@article{Wat96,
Author = {John D. Watts and Rodney J. Bartlett},
Date-Added = {2019-11-03 21:54:08 +0100},

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Manuscript/ExPerspective.tex
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@ -74,11 +74,11 @@
\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, 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, broad 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.
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.
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}
@ -87,7 +87,7 @@ We conclude this \textit{Perspective} by discussing the current potentiality of
%%%%%%%%%%%%%%%%%%%%
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
The accurate modelling 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 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).
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.
The factors that makes this quest for high accuracy particularly delicate are very diverse.
@ -102,15 +102,15 @@ And let's be honest, none of the existing methods does provide this at an afford
What are the requirement of the ``perfect'' theoretical model?
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.
Let us not forget about the requirements of minimal user input and chemical intuition (\ie, black box method preferable) in order to minimise the potential bias brought by the user appreciation of the problem complexity.
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.
Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
Although the simultaneous fulfilment of all these requirements seems elusive, it is essential to keep these criteria in mind.
Table \ref{tab:method} is here for fulfil such a purpose.
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 I %%%
\begin{squeezetable}
%\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.}
\label{tab:method}
@ -141,12 +141,12 @@ Table \ref{tab:method} is here for fulfil such a purpose.
\end{tabular}
\end{ruledtabular}
\end{table}
\end{squeezetable}
%\end{squeezetable}
%**************
%** 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).
Here, we only mention methods that, we think, ended up becoming mainstream.
@ -157,7 +157,8 @@ 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.
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}
This second-order correction significantly reduces the magnitude of the error associated with CIS excitation energies, 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).
%%%%%%%%%%%%%%%%%%%
%%% ROOS' GROUP %%%
@ -168,15 +169,15 @@ Although it took more than ten years to obtain analytic nuclear gradients, \cite
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.
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 range of CASPT2 or NEVPT2 is $0.1$-$0.2$ eV.
With a typical minimal valence active space, the typical error in CASPT2 or NEVPT2 calculations is $0.1$-$0.2$ eV.
%%%%%%%%%%%%%
%%% TDDFT %%%
%%%%%%%%%%%%%
The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Dre05} 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.
However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Dre05,Lev06}
In the present context, one of the most annoying feature of TD-DFT --- in its most standard form --- is its inability to describe, even qualitatively, charge-transfer states, \cite{Toz99,Dre04} Rydberg states, \cite{Toz98} and double excitations. \cite{Lev06}
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 issue 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}
Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community (and beyond).
@ -188,45 +189,46 @@ Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90}
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.
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}
As a $N^7$ method, CC3 has a particularly interesting accuracy/cost ratio with errors usually below chemical accuracy.
The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively, where $N$ is the number of basis functions.
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 study.
%%%%%%%%%%%%%%%%%%%
%%% ADC METHODS %%%
%%%%%%%%%%%%%%%%%%%
It is also important to mention the recent advent 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.
Moreover, Dreuw's group has put an enormous amount of work to provide a fast and efficient implementation of these methods as well as other interesting variants. \cite{Dre15}
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}
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) might generally overcorrect 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 %%%
%%%%%%%%%%%%%%
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-state energetics (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}
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.
%%%%%%%%%%%%%%%%%%%
%%% SCI METHODS %%%
%%%%%%%%%%%%%%%%%%%
Importantly in the context of the present \textit{Perspective} article, we have witnessed a resurgence of selected CI (SCI) methods \cite{Ben69,Whi69,Hur73} in the past five years \cite{Gin13,Gin15} thanks to the development and implementation of new, fast and efficient algorithms to select cleverly determinants in the full CI (FCI) space (see Refs.~\onlinecite{Gar18,Gar19} and references therein).
SCI methods rely on the same principle as the usual CI approach, except that determinants are not chosen a priori based on occupation or excitation criteria but selected among the entire set of determinants based on their estimated contribution to the FCI wave function or energy.
In the past five years, \cite{Gin13,Gin15} we have witnessed a resurgence of the so-called selected CI (SCI) methods \cite{Ben69,Whi69,Hur73} thanks to the development and implementation of new, fast and efficient algorithms to select cleverly determinants in the full CI (FCI) space (see Refs.~\onlinecite{Gar18,Gar19} and references therein).
SCI methods rely on the same principle as the usual CI approach, except that determinants are not chosen \textit{a priori} based on occupation or excitation criteria but selected among the entire set of determinants based on their estimated contribution to the FCI wave function or energy.
Indeed, it has been noticed long ago that, even inside a predefined subspace of determinants, only a small number of them significantly contributes.
The main advantage of SCI methods is that no a priori assumption is made on the type of electron correlation.
Therefore, at the price of a brute force calculation, a SCI calculation is less biased by the user appreciation of the problem complexity.
One of the strength of our implementation (based on the CIPSI algorithm developed by Huron, Rancurel, and Malrieu in 1973 \cite{Hur73}) is its parallel efficiency which makes possible to run on thousands of cores.
Thanks to these tremendous features, SCI methods delivers near FCI quality excitation energies for both singly- and doubly-excited states, \cite{Hol17,Chi18,Loo18a,Loo19c} with an error of roughly $0.03$ eV (mostly due to the extrapolation procedure). \cite{Gar19}
One of the strength of our implementation, based on the CIPSI (configuration interaction using a perturbative selection made iteratively) algorithm developed by Huron, Rancurel, and Malrieu in 1973 \cite{Hur73}) is its parallel efficiency which makes possible to run on thousands of cores. \cite{Gar19}
Thanks to these tremendous features, SCI methods delivers near FCI quality excitation energies for both singly and doubly excited states, \cite{Hol17,Chi18,Loo18a,Loo19c} with an error of roughly $0.03$ eV (mostly due to the extrapolation procedure \cite{Gar19}).
However, although the \textit{``exponential wall''} is pushed back, this type of methods is only applicable to molecules with a small number of heavy atoms with relatively compact basis sets.
%%%%%%%%%%%%%%%
%%% SUMMARY %%%
%%%%%%%%%%%%%%%
In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate and reliable excitation energies for all classes of electronic states, ultimately leading to an unbalanced description of excited states with distinct natures.
In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate and reliable excitation energies for all classes of electronic excited states at an affordable cost, ultimately leading to an unbalanced description of excited states with distinct natures.
%%% FIG 1 %%%
\begin{figure*}
@ -239,8 +241,8 @@ In summary, each method has its own strengths and weaknesses, and none of them i
%%% FIG 2 %%%
\begin{figure}
\includegraphics[width=\linewidth]{Set2}
\caption{Mean absolute error (in eV) (with respect to exFCI) excitation energies from the Jacquemin set \#2 (as described 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.}
\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.}
\label{fig:Set2}
\end{figure}
%%% %%% %%%
@ -264,40 +266,44 @@ For excited states, things started moving a little later but some major contribu
%%% THIEL'S SET %%%
%%%%%%%%%%%%%%%%%%%
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, broad and accurate enough to be used as a proper benchmarking set for excited states.
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, EOM-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 (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.
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).
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).
Theoretical improvements of Thiel's set were slow but steady, highlighting further their quality.
In 2013, Watson et al.\cite{Wat13} computed EOM-CCSDT-3/TZVP (an iterative approximation of the triples of EOM-CCSDT \cite{Wat96}) excitation energies for the Thiel set.
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.
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.
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''}.
Finally, let us mention the work of Kannar and Szalay who reported EOM-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.
These two studies clearly demonstrate and motivate the need for higher accuracy benchmark excited-states energies.
%%%%%%%%%%%%%%%%%%%%%%%
%%% JACQUEMIN'S SET %%%
%%%%%%%%%%%%%%%%%%%%%%%
Recently, we also made, what we think, is a significant contribution to the 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.
For such systems, using a combination of high-order CC, 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 characters (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly-excited) based on accurate CC3/aug-cc-pVTZ geometries (see Fig.~\ref{fig:Set1}).
Importantly, it allowed us to benchmark a series of 12 popular excited-state wave function methods accounting for double and triple excitations [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, EOM-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).
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.
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).
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 excitation using a series of increasingly large diffuse-containing atomic basis sets (up to aug-cc-pVTZ 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).
An important addition to this second study was the computation of excitation energies with multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative and iterative triple corrections (see Fig.~\ref{fig:Set2}).
Our results clearly evidenced 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 EOM-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.
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}).
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.
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.
To obtain these energies, we used EOM-CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), SCI calculations up to several million determinants, as well as the most robust multiconfigurational method, NEVPT2, each approach being combined with diffuse-containing atomic basis sets.
For all these transitions, we reported at least CC3/aug-cc-pVQZ transition energies and as well as CC3/aug-cc-pVTZ oscillator strengths for all dipole-allowed transitions.
In agreement with our two previous studies, \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 predominant double excitation character (see Fig.~\ref{fig:Set3}).
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.
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.
Each approach was applied in combination with diffuse-containing atomic basis sets.
For all these transitions, we reported at least CC3/aug-cc-pVQZ transition energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition.
In agreement with our two previous studies, \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}).
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).
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@ -308,7 +314,7 @@ In agreement with our two previous studies, \cite{Loo18a,Loo19c} we confirmed th
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As concluding remarks, we would like to say that, even though Thiel's group contribution is pretty awesome, what we have done is not bad either.
Thanks to new technological advances, we hope to be able to push further our quest to highly accurate excitation energies.
Thanks to new technological advances, we hope to be able to push further our quest to highly accurate excitation energies in years to come.
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%%% ACKNOWLEDGEMENTS %%%

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