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@ -1,13 +1,42 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-11-25 13:44:50 +0100
%% Created for Pierre-Francois Loos at 2020-11-27 22:23:32 +0100
%% Saved with string encoding Unicode (UTF-8)
@book{Robb_2018,
author = {Robb, Michael A},
date-added = {2020-11-27 22:23:19 +0100},
date-modified = {2020-11-27 22:23:27 +0100},
doi = {10.1039/9781788013642},
isbn = {978-1-78262-864-4},
pages = {P001-225},
publisher = {The Royal Society of Chemistry},
series = {Theoretical and Computational Chemistry Series},
title = {Theoretical Chemistry for Electronic Excited States},
url = {http://dx.doi.org/10.1039/9781788013642},
year = {2018},
Bdsk-Url-1 = {http://dx.doi.org/10.1039/9781788013642}}
@article{Mai_2020,
abstract = {Abstract Photochemistry is a fascinating branch of chemistry that is concerned with molecules and light. However, the importance of simulating light-induced processes is reflected also in fields as diverse as biology, material science, and medicine. This Minireview highlights recent progress achieved in theoretical chemistry to calculate electronically excited states of molecules and simulate their photoinduced dynamics, with the aim of reaching experimental accuracy. We focus on emergent methods and give selected examples that illustrate the progress in recent years towards predicting complex electronic structures with strong correlation, calculations on large molecules, describing multichromophoric systems, and simulating non-adiabatic molecular dynamics over long time scales, for molecules in the gas phase or in complex biological environments.},
author = {Mai, Sebastian and Gonz{\'a}lez, Leticia},
date-added = {2020-11-27 22:21:08 +0100},
date-modified = {2020-11-27 22:21:51 +0100},
doi = {https://doi.org/10.1002/anie.201916381},
journal = {Angew. Chem. Int. Ed.},
keywords = {excited states, molecular chemistry, non-adiabatic dynamics, photochemistry, quantum chemistry},
pages = {16832-16846},
title = {Molecular Photochemistry: Recent Developments in Theory},
volume = {59},
year = {2020},
Bdsk-Url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/anie.201916381},
Bdsk-Url-2 = {https://doi.org/10.1002/anie.201916381}}
@article{Lee_2020,
author = {Joonho Lee and Fionn D. Malone and David R. Reichman},
date-added = {2020-11-25 13:44:34 +0100},

60
Manuscript/QUEST_WIREs.tex
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@ -107,7 +107,7 @@ absolute and/or relative energies of electronic states in molecular systems \cit
methods is their ability to access the energies of electronic excited states, i.e., states that have higher total energies than the so-called ground (that is, lowest-energy) state
\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020a}.
The faithful description of excited states is particularly challenging from a theoretical point of view but is key to a deeper understanding of photochemical and photophysical processes
like absorption, fluorescence, phosphorescence or even chemoluminescence \cite{Bernardi_1996,Olivucci_2010,Robb_2007,Navizet_2011,Crespo_2018}.
like absorption, fluorescence, phosphorescence, chemoluminescence, and others \cite{Bernardi_1996,Olivucci_2010,Robb_2007,Navizet_2011,Crespo_2018,Robb_2018,Mai_2020}.
For a given level of theory, ground-state methods are usually more accurate than their excited-state analogs.
The reasons behind this are (at least) threefold: i) one might lack a proper variational principle for excited-state energies and one may have to rely on response theory
\cite{Monkhorst_1977,Helgaker_1989,Koch_1990,Koch_1990b,Christiansen_1995b,Christiansen_1998b,Hattig_2003,Kallay_2004,Hattig_2005c} formalisms which inherently introduce a
@ -118,7 +118,7 @@ The reasons behind this are (at least) threefold: i) one might lack a proper var
articles on this particular subject \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a}.
When designing a new theoretical model, the first feature that one might want to test is its overall accuracy, i.e., its ability to reproduce reference (or benchmark) values for a given system with well-defined
When designing a new theoretical model, the first feature that one might want to test is its overall accuracy, i.e., its ability to reproduce reference (or benchmark) values for a given system with a well-defined
setup (same geometry, basis set, etc). These values can be absolute and/or relative energies, geometrical parameters, physical or chemical spectroscopic properties extracted from experiments,
high-level theoretical calculations, or any combination of these. To this end, the electronic structure community has designed along the years benchmark sets, i.e., sets of molecules for which one
could (very) accurately compute theoretical estimates and/or access solid experimental data for given properties. Regarding ground-states properties, two of the oldest and most employed sets are
@ -129,17 +129,17 @@ of Hobza and collaborators which provides benchmark interaction energies for wea
(and its $GW$5000 extension \cite{Stuke_2020}) of ionization energies which has helped enormously the community to compare the implementation of $GW$-type methods for molecular
systems \cite{vanSetten_2013,Bruneval_2016,Caruso_2016,Govoni_2018}. The extrapolated ab initio thermochemistry (HEAT) set designed to achieve high accuracy for enthalpies of formation
of atoms and small molecules (without experimental data) is yet another successful example of benchmark set \cite{Tajti_2004,Bomble_2006,Harding_2008}. More recently, the benchmark datasets
provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable to the community by providing, for example, highly-accurate ground state energies for
provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable to the community by providing, for example, highly-accurate ground-state energies for
hydrogen chains \cite{Motta_2017} as well as transition metal atoms and their ions and monoxides \cite{Williams_2020}. Let us also mention the set of Zhao and Truhlar for small transition metal complexes
employed to compare the accuracy density-functional methods \cite{ParrBook} for $3d$ transition-metal chemistry \cite{Zhao_2006}, and finally the popular GMTKN24 \cite{Goerigk_2010},
employed to compare the accuracy of density-functional methods \cite{ParrBook} for $3d$ transition-metal chemistry \cite{Zhao_2006}, and finally the popular GMTKN24 \cite{Goerigk_2010},
GMTKN30 \cite{Goerigk_2011a,Goerigk_2011b} and GMTKN55 \cite{Goerigk_2017} databases for general main group thermochemistry, kinetics, and non-covalent interactions developed by Goerigk, Grimme and
their coworkers.
The examples of benchmark sets presented above are all designed for ground-state properties, and there exists specific protocols taylored to accurately model excited-state energies and properties as well.
Indeed, benchmark datasets of excited-state energies and/or properties are less numerous than their ground-state counterparts but their number have been growing at a consistent pace in the past few years.
Indeed, benchmark datasets of excited-state energies and/or properties are less numerous than their ground-state counterparts but their number has been growing at a consistent pace in the past few years.
Below, we provide a short description for some of them. One of the most characteristic example is the benchmark set of vertical excitation energies proposed by Thiel and coworkers
\cite{Schreiber_2008,Silva-Junior_2008,Silva-Junior_2010,Silva-Junior_2010b,Silva-Junior_2010c}. The so-called Thiel (or M\"ulheim) set of excitation energies gathers a large number of excitation energies
determined in 28 medium-size organic CNOH molecules with a total of 223 valence excited states (152 singlet and 71 triplet states) for which theoretical best estimates (TBEs) were defined.
determined in 28 medium-sized organic CNOH molecules with a total of 223 valence excited states (152 singlet and 71 triplet states) for which theoretical best estimates (TBEs) were defined.
In their first study, Thiel and collaborators performed CC2 \cite{Christiansen_1995a,Hattig_2000}, CCSD \cite{Rowe_1968,Koch_1990,Stanton_1993,Koch_1994}, CC3 \cite{Christiansen_1995b,Koch_1997}, and
CASPT2 \cite{Andersson_1990,Andersson_1992,Roos,Roos_1996} calculations (with the TZVP basis) on MP2/6-31G(d) geometries in order to provide (based on additional high-quality literature data) TBEs for these
transitions. These TBEs were quickly refined with the larger aug-cc-pVTZ basis set \cite{Silva-Junior_2010b,Silva-Junior_2010c}. In the same spirit, it is also worth mentioning Gordon's set of vertical transitions
@ -151,15 +151,15 @@ Let us also mention the new benchmark set of charge-transfer excited states rece
as well as the Gagliardi-Truhlar set employed to compare the accuracy of multiconfiguration pair-density functional theory \cite{Ghosh_2018} against the well-established CASPT2 method \cite{Hoyer_2016}.
Following a similar philosophy and striving for chemical accuracy, we have recently reported in several studies highly-accurate vertical excitations for small- and medium-sized molecules
\cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}. The so-called QUEST dataset of vertical excitations which we will describe in details in the present review article is composed by 5
\cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}. The so-called QUEST dataset of vertical excitations which we will describe in detail in the present review article is composed by 5
subsets (see Fig.~\ref{fig:scheme}): i) a subset of excitations in small molecules containing from 1 to 3 non-hydrogen atoms known as QUEST\#1, ii) a subset of double excitations for molecules of small and
medium sizes known as QUEST\#2, iii) a subset of excitation energies for medium-sized molecules containing from 4 to 6 non-hydrogen atoms known as QUEST\#3, iv) a subset composed by more ``exotic''
molecules and radicals labeled as QUEST\#4, and v) a subset known as QUEST\#5, specifically designed for the present article, gathering excitation energies in larger molecules as well as additional smaller molecules.
One of the key aspect of the QUEST dataset is that it does not rely on any experimental values, avoiding potential biases inherently linked to experiments and facilitating in the process theoretical comparisons.
Moreover, our protocol has been designed to be as uniform as possible, which means that we have designed a very systematic procedure for all excited states in order to make cross-comparison as straightforward as possible.
Importantly, it allowed us to benchmark, in a very systematic and balanced way, a series of popular excited-state wave function methods partially or fully accounting for double and triple excitations as well as multiconfigurational methods (see below).
In the same vein, as evoked above, we have also produced chemically-accurate theoretical 0-0 energies \cite{Loos_2018,Loos_2019a,Loos_2019b} which can be more straightforwardly compare to experimental data \cite{Furche_2002,Kohn_2003,Dierksen_2004,Goerigk_2010a,Send_2011a,Jacquemin_2012,Winter_2013,Fang_2014,Jacquemin_2015b,Oruganti_2016}. We refer the interested reader to Ref.~\cite{Loos_2019b} for a
review the generic benchmark studies devoted to adiabatic and 0-0 energies performed in the past two decades.
In the same vein, as evoked above, we have also produced chemically-accurate theoretical 0-0 energies \cite{Loos_2018,Loos_2019a,Loos_2019b} which can be more straightforwardly compared to experimental data \cite{Furche_2002,Kohn_2003,Dierksen_2004,Goerigk_2010a,Send_2011a,Jacquemin_2012,Winter_2013,Fang_2014,Jacquemin_2015b,Oruganti_2016}. We refer the interested reader to Ref.~\cite{Loos_2019b} for a
review of the generic benchmark studies devoted to adiabatic and 0-0 energies performed in the past two decades.
%%% FIGURE 1 %%%
\begin{figure}
@ -169,10 +169,10 @@ review the generic benchmark studies devoted to adiabatic and 0-0 energies perfo
\label{fig:scheme}
\end{figure}
The QUEST dataset has the particularity to be based in a large proportion on selected configuration interaction (SCI) reference excitation energies as well as high-order linear-response (LR) CC methods such as LR-CCSDT and
The QUEST dataset has the particularity to be based to a large extent on selected configuration interaction (SCI) reference excitation energies as well as high-order linear-response (LR) CC methods such as LR-CCSDT and
LR-CCSDTQ \cite{Noga_1987,Koch_1990,Kucharski_1991,Christiansen_1998b,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}. Recently, SCI methods have been a force to reckon with for
the computation of highly-accurate energies in small- and medium-sized molecules as they yield near full configuration interaction (FCI) quality energies for only a fraction of the computational cost of a genuine FCI calculation \cite{Booth_2009,Booth_2010,Cleland_2010,Booth_2011,Daday_2012,Blunt_2015,Ghanem_2019,Deustua_2017,Deustua_2018,Holmes_2017,Chien_2018,Li_2018,Yao_2020,Li_2020,Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b,Xu_2018,Xu_2020,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020a,Loos_2020e,Eriksen_2021}.
Due to the fairly natural idea underlying these methods, the SCI family is composed by numerous members \cite{Bender_1969,Whitten_1969,Huron_1973,Abrams_2005,Bunge_2006,Bytautas_2009,Giner_2013,Caffarel_2014,Giner_2015,Garniron_2017b,Caffarel_2016a,Caffarel_2016b,Holmes_2016,Sharma_2017,Holmes_2017,Chien_2018,Scemama_2018,Scemama_2018b,Garniron_2018,Evangelista_2014,Tubman_2016,Tubman_2020,Schriber_2016,Schriber_2017,Liu_2016,Per_2017,Ohtsuka_2017,Zimmerman_2017,Li_2018,Ohtsuka_2017,Coe_2018,Loos_2019}.
Due to the fairly natural idea underlying these methods, the SCI family is composed of numerous members \cite{Bender_1969,Whitten_1969,Huron_1973,Abrams_2005,Bunge_2006,Bytautas_2009,Giner_2013,Caffarel_2014,Giner_2015,Garniron_2017b,Caffarel_2016a,Caffarel_2016b,Holmes_2016,Sharma_2017,Holmes_2017,Chien_2018,Scemama_2018,Scemama_2018b,Garniron_2018,Evangelista_2014,Tubman_2016,Tubman_2020,Schriber_2016,Schriber_2017,Liu_2016,Per_2017,Ohtsuka_2017,Zimmerman_2017,Li_2018,Ohtsuka_2017,Coe_2018,Loos_2019}.
Their fundamental philosophy consists, roughly speaking, in retaining only the most relevant determinants of the FCI space following a given criterion to slow down the exponential increase of the size of the CI expansion.
Originally developed in the late 1960's by Bender and Davidson \cite{Bender_1969} as well as Whitten and Hackmeyer \cite{Whitten_1969}, new efficient SCI algorithms have resurfaced recently.
Three examples are iCI \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020}, semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017,Li_2018,Li_2020,Yao_2020}, and \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2019}.
@ -202,8 +202,7 @@ of the excited states, the size of the basis set, etc. Finally, we draw our conc
%=======================
The ground-state structures of the molecules included in the QUEST dataset have been systematically optimized at the CC3/aug-cc-pVTZ level of theory, except for a very few cases.
As shown in Refs.~\cite{Hattig_2005c,Budzak_2017}, CC3 provides extremely accurate ground- and excited-state geometries. These optimizations have been performed using DALTON 2017
\cite{dalton} and CFOUR 2.1 \cite{cfour} applying default parameters. For the open-shell derivatives belonging to QUEST\#4 \cite{Loos_2020c}, the geometries are optimized at the UCCSD(T)/aug-cc-pVTZ
level using the GAUSSIAN16 program \cite{Gaussian16} and applying the ``tight'' convergence threshold. For the purpose of the present review article, we have gathered all the geometries in the {\SupInf}.
\cite{dalton} and CFOUR 2.1 \cite{cfour} applying default parameters. For the open-shell derivatives belonging to QUEST\#4 \cite{Loos_2020c}, the geometries are optimized at the UCCSD(T)/aug-cc-pVTZ level using the GAUSSIAN16 program \cite{Gaussian16} and applying the ``tight'' convergence threshold. For the purpose of the present review article, we have gathered all the geometries in the {\SupInf}.
%=======================
\subsection{Basis sets}
@ -248,7 +247,7 @@ Below, we provide a much cleaner way of estimating the extrapolation error in SC
The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019}.
Moreover, a renormalized version of the PT2 correction (dubbed rPT2) has been recently implemented for a more efficient extrapolation to the FCI limit \cite{Garniron_2019}.
We refer the interested reader to Ref.~\cite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm.
Note that, all our SCI wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator which is, unlike ground-state calculations, paramount in the case of excited states \cite{Applencourt_2018}.
Note that all our SCI wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator which is, unlike ground-state calculations, paramount in the case of excited states \cite{Applencourt_2018}.
%------------------------------------------------
\subsubsection{Benchmarked computational methods}
@ -458,15 +457,15 @@ by overcorrecting ADC(2) excitation energies.
%=======================
\subsection{QUEST\#2}
%=======================
The QUEST\#2 benchmark set \cite{Loos_2019} reports reference energies for double excitations. This set gathers 20 vertical transitions from 14 small- and medium-size molecules (acrolein, benzene, beryllium atom,
The QUEST\#2 benchmark set \cite{Loos_2019} reports reference energies for double excitations. This 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). The TBEs of the QUEST\#2 set are obtained with SCI and/or multiconfigurational
[CASSCF, CASPT2, (X)MS-CASPT2, and NEVPT2] calculations depending on the size of the molecules and the level of theory that we could afford. An important addition to this second study was also the inclusion of
various flavors of multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including, at least, perturbative triples (CC3, CCSDT, CCSDTQ, etc).
Our results demonstrated that the error of CC methods is intimately linked to the amount of double-excitation character in the vertical transition. For ``pure'' double excitations (i.e., 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 eV for more common transitions involving a significant amount of single excitations
(such as the well-known $A_g$ transition in butadiene or the $E_{2g}$ excitation in benzene). The quality of the excitation energies obtained with CASPT2 and NEVPT2 was harder to predict as the overall accuracy of
these methods is highly dependent on both the system and the selected active space. Nevertheless, these two methods were found to be more accurate for transition with a very small percentage of single excitations
(error usually below $0.1$ eV) than for excitations dominated by single excitations where the error is closer from $0.1$--$0.2$ eV.
these methods is highly dependent on both the system and the selected active space. Nevertheless, these two methods were found to be more accurate for transitions with a very small percentage of single excitations
(error usually below $0.1$ eV) than for excitations dominated by single excitations where the error is closer to $0.1$--$0.2$ eV.
%=======================
\subsection{QUEST\#3}
@ -499,9 +498,9 @@ are much less satisfying for open-shell derivatives (MAE of $0.20$ eV with UCCSD
\subsection{QUEST\#5}
%=======================
The QUEST\#5 subset is composed by additional accurate excitation energies that we have produced for the present article. This new set gathers 13 new systems composed by small molecules as well as larger molecules
The QUEST\#5 subset is composed of additional accurate excitation energies that we have produced for the present article. This new set gathers 13 new systems composed by small molecules as well as larger molecules
(see blue molecules in Fig.~\ref{fig:molecules}): aza-naphthalene, benzoquinone, cyclopentadienone, cyclopentadienethione, diazirine, hexatriene, maleimide, naphthalene, nitroxyl, octatetraene, streptocyanine-C3, streptocyanine-C5,
and thioacrolein. For these new transitions, we report again quality vertical energies, the vast majority being of CCSDT quality, and we consider that, out of these \alert{80} new transitions, \alert{55} of them can be labeled
and thioacrolein. For these new transitions, we report again quality vertical transition energies, the vast majority being of CCSDT quality, and we consider that, out of these 80 new transitions, 55 of them can be labeled
as ``safe'', \ie, considered as chemically accurate or within 0.05 eV of the FCI limit for the given geometry and basis set. We refer the interested reader to the {\SupInf} for a detailed discussion of each molecule for which comparisons
are made with literature data.
@ -510,13 +509,13 @@ are made with literature data.
\label{sec:TBE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We discuss in this section the generation of the TBEs obtained with the aug-cc-pVTZ basis.
For the closed-shell compounds, the exhaustive list of TBEs can be found in Table \ref{tab:TBE} alongside various specifications: the molecule's name, the excitation, its nature (valence, Rydberg, or charge transfer), its oscillator strength (when spatially- and spin-allowed),
For the closed-shell compounds, the exhaustive list of TBEs can be found in Table \ref{tab:TBE} alongside various specifications: the molecule's name, the excitation, its nature (valence, Rydberg, or charge transfer), its oscillator strength (when symmetry- and spin-allowed),
and its percentage of single excitations $\%T_1$ (computed at the LR-CC3 level). All these quantities are computed with the same aug-cc-pVTZ basis.
Importantly, we also report the composite approach considered to compute the TBEs (see column ``Method'').
Following an ONIOM-like strategy \cite{Svensson_1996a,Svensson_1996b}, the TBEs are computed as ``A/SB + [B/TB - B/SB]'', where A/SB is the excitation energy computed with a method A in a smaller basis (SB), and B/SB and B/TB are excitation energies computed with a method B in the small basis and target basis TB, respectively.
Table \ref{tab:rad} reports the TBEs for the open-shell molecules belonging to the QUEST\#4 subset.
Talking about numbers, the QUEST database is composed by 551 excitation energies, including 302 singlet, 197 triplet, 51 doublet, 412 valence, and 176 Rydberg excited states.
Talking about numbers, the QUEST database is composed of 551 excitation energies, including 302 singlet, 197 triplet, 51 doublet, 412 valence, and 176 Rydberg excited states.
Amongst the valence transitions in closed-shell compounds, 135 transitions correspond to $n \ra \pis$ excitations, 200 to $\pi \ra \pis$ excitations, and 23 are doubly-excited states. In terms of molecular sizes, 146 excitations are obtained
in molecules having in-between 1 and 3 non-hydrogen atoms, 97 excitations from 4 non-hydrogen atom compounds, 177 from molecules composed by 5 and 6 non-hydrogen atoms, and, finally, 68 excitations are obtained from systems with 7 to 10 non-hydrogen atoms.
In addition, QUEST is composed by 24 open-shell molecules with a single unpaired electron.
@ -530,7 +529,7 @@ Besides this energetic criterion, we consider as ``safe'' transitions that are e
\caption{Theoretical best estimates TBEs (in eV), oscillator strengths $f$, percentage of single excitations $\%T_1$ involved in the transition (computed at the CC3 level) for the full set of closed-shell compounds of the QUEST database.
``Method'' provides the protocol employed to compute the TBEs.
The nature of the excitation is also provided: V, R, and CT stands for valence, Rydberg, and charge transfer, respectively.
[F] indicates a fluorescence transition, \ie, a transition energy computed from an excited-state geometry.
[F] indicates a fluorescence transition, \ie, a vertical transition energy computed from an excited-state geometry.
AVXZ stands for aug-cc-pVXZ.
\label{tab:TBE}}
\\
@ -1192,9 +1191,9 @@ MAE & & 0.22 & 0.16 & 0.22 & 0.11 & 0.12 & 0.05 & 0.04 & 0.02 & 0.20 & 0.22
The most striking feature from the statistical indicators gathered in Table \ref{tab:stat} is the overall accuracy of CC3 with MAEs and MSEs systematically below the chemical accuracy threshold (errors $<$ 0.043 eV or 1 kcal/mol), irrespective of the nature of the transition and the size of the molecule.
CCSDR(3) are CCCSDT-3 can also be regarded as excellent performers with overall MAEs below $0.05$ eV, though one would notice a slight degradation of their performances for the $n \ra \pis$ excitations and the largest molecules of the database.
The other third-order method, ADC(3), which enjoys a lower computational cost, is significantly less accuracy and does not really improve upon its second-order analog, even for the largest systems considered here, observation in line with a previous analysis by some of the authors \cite{Loos_2020d}.
The other third-order method, ADC(3), which enjoys a lower computational cost, is significantly less accurate and does not really improve upon its second-order analog, even for the largest systems considered here, an observation in line with a previous analysis by some of the authors \cite{Loos_2020d}.
Nonetheless, ADC(3)'s accuracy improves in larger compounds, with a MAE of 0.24 eV (0.16 eV) for the subsets of the most compact (extended) compounds considered herein. The ADC(2.5) composite method introduced in Ref.~\cite{Loos_2020d}, which corresponds to grossly average the ADC(2) and ADC(3)
values, yield an appreciable accuracy improvement, as shown in Fig.~\ref{fig:QUEST_stat}. Indeed, we note that the MAE of 0.07 eV obtained for ``large'' compounds is comparable to the one obtained with CCSDR(3) and CCSDT-3 for these molecules. All these third-order methods
values, yields an appreciable accuracy improvement, as shown in Fig.~\ref{fig:QUEST_stat}. Indeed, we note that the MAE of 0.07 eV obtained for ``large'' compounds is comparable to the one obtained with CCSDR(3) and CCSDT-3 for these molecules. All these third-order methods
are rather equally efficient for valence and Rydberg transitions.
Concerning the second-order methods (which have the indisputable advantage to be applicable to larger molecules than the ones considered here), we have the following ranking in terms of MAEs: EOM-MP2 $\approx$ CIS(D) $<$ CC2 $\approx$ ADC(2) $<$ CCSD $\approx$ STEOM-CCSD, which fits our previous conclusions on the specific subsets \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020d}.
@ -1243,10 +1242,10 @@ via GitHub pull requests.
\section{Concluding remarks}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the present review article, we have presented and extended the QUEST database of highly-accurate excitation energies for molecules systems \cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c} that we started building
In the present review article, we have presented and extended the QUEST database of highly-accurate excitation energies for molecular systems \cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c} that we started building
in 2018 and that is now composed by more than 500 vertical excitations, many of which can be reasonably considered as within 1 kcal/mol (or less) of the FCI limit for the considered CC3/aug-cc-pVTZ geometry and basis set (\emph{aug}-cc-pVTZ).
In particular, we have detailed the specificities of our protocol by providing computational details regarding geometries, basis sets, as well as reference and benchmarked computational methods. The content of our five QUEST subsets has
been presented in details, and for each of them, we have provided the number of reference excitation energies, the nature and size of the molecules, the list of benchmarked methods, as well as other useful specificities. Importantly, we have
been presented in detail, and for each of them, we have provided the number of reference excitation energies, the nature and size of the molecules, the list of benchmarked methods, as well as other useful specificities. Importantly, we have
proposed a new method to faithfully estimate the extrapolation error in SCI calculations. This new method based on Gaussian random variables has been tested by computing additional FCI values for five- and six-membered rings.
After having discussed the generation of our TBEs, we have reported a comprehensive benchmark for a significant number of methods on the entire QUEST set with, in addition, a specific analysis for each type of excited states.
Finally, the main features of the website specifically designed to gather the entire data generated during these past few years have been presented and discussed.
@ -1255,10 +1254,10 @@ Paraphrasing Thiel's conclusions \cite{Schreiber_2008}, we hope that not only th
(inevitable in such a large data set), but more importantly extensions with both improved estimates for some compounds and states, or new molecules.
In this framework, we provide in the {\SupInf} a file with all our benchmark data.
Regarding future improvements and extensions, we would like to mention that although our present goal is to produce chemically accurate vertical excitation energies, we are currently devoting great efforts to obtain highly-accurate excited-state properties \cite{Hodecker_2019,Eriksen_2020b} as such dipoles and oscillator strengths for molecules of small and medium sizes \cite{Chrayteh_2021,Sarkar_2021}, so as to complete previous efforts aiming at determining accurate excited-state geometries \cite{Budzak_2017,Jacquemin_2018}.
Regarding future improvements and extensions, we would like to mention that although our present goal is to produce chemically accurate vertical excitation energies, we are currently devoting great efforts to obtain highly-accurate excited-state properties \cite{Hodecker_2019,Eriksen_2020b} such as dipoles and oscillator strengths for molecules of small and medium sizes \cite{Chrayteh_2021,Sarkar_2021}, so as to complete previous efforts aiming at determining accurate excited-state geometries \cite{Budzak_2017,Jacquemin_2018}.
Reference ground-state properties (such as correlation energies and atomization energies) are also being currently produced \cite{Scemama_2020,Loos_2020e}.
Besides this, because computing 500 (or so) excitation energies can be a costly exercise even with cheap computational methods, we are planning on developing a ``diet set'' following the philosophy of the ``diet GMTKN55'' set proposed recently by Gould \cite{Gould_2018b}.
We hope to report on this in the new future.
We hope to report on this in the near future.
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\section*{acknowledgements}
@ -1318,7 +1317,10 @@ got his Ph.D. in Chemistry from Scuola Normale Superiore, Pisa in 2013. He worke
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\begin{biography}[MBoggioPasqua]{M.~Boggio-Pasqua}
is a CNRS researcher at the Laboratoire de Chimie et Physique Quantiques at the University of Toulouse III - Paul Sabatier. His main research interests are focused on the theoretical studies of photochemical processes in complex molecular systems including the description of excited-state reaction mechanisms based on static exploration of potential energy surfaces and simulations of the nonadiabatic dynamics.
received his PhD in Physical Chemistry from the Universit\'e Bordeaux 1 in 1999.
He then worked as a post-doctoral research associate at King's College London (2000-2003) and at Imperial College London (2004-2007) with M.~Robb and M.~Bearpark.
He was then appointed as a CNRS researcher at the Laboratoire de Chimie et Physique Quantiques at the Universit\'e Paul Sabatier (Toulouse).
His main research interests are focused on the theoretical studies of photochemical processes in complex molecular systems including the description of excited-state reaction mechanisms based on static explorations of potential energy surfaces and simulations of nonadiabatic dynamics.
\end{biography}
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