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Manuscript/QUESTDB.bib
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@ -1,13 +1,133 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-09-04 10:26:50 +0200
%% Created for Pierre-Francois Loos at 2020-09-07 09:29:50 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Goerigk_2010a,
Author = {Goerigk, L. and Grimme, S.},
Date-Added = {2020-09-07 09:29:41 +0200},
Date-Modified = {2020-09-07 09:29:50 +0200},
Journal = JCP,
Pages = {184103},
Title = {Assessment of TD-DFT Methods and of Various Spin Scaled CIS$_n$D and CC2 Versions for the Treatment of Low-Lying Valence Excitations of Large Organic Dyes},
Volume = {132},
Year = 2010}
@article{Hellweg_2008,
Author = {Hellweg, A. and Gr\"un, S. A. and H\"attig, C.},
Date-Added = {2020-09-07 09:29:02 +0200},
Date-Modified = {2020-09-07 09:29:08 +0200},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {4119--4127},
Title = {Benchmarking the Performance of Spin-Component Scaled CC2 in Ground and Electronically Excited States},
Volume = {10},
Year = {2008}}
@article{Nooijen_1997,
Author = {Marcel Nooijen and Rodney J. Bartlett},
Date-Added = {2020-09-07 09:28:19 +0200},
Date-Modified = {2020-09-07 09:28:26 +0200},
Doi = {10.1063/1.474000},
Eprint = {https://doi.org/10.1063/1.474000},
Journal = {J. Chem. Phys.},
Number = {15},
Pages = {6441-6448},
Title = {A New Method for Excited States: Similarity Transformed Equation-Of-Motion Coupled-Cluster Theory},
Url = {https://doi.org/10.1063/1.474000},
Volume = {106},
Year = {1997},
Bdsk-Url-1 = {https://doi.org/10.1063/1.474000}}
@article{Prascjer_2010,
Author = {Prascjer, B. P. and Woon, D. E. and Peterson, K. A. and Dunning, T. H. and Wilson, A. K.},
Date-Added = {2020-09-07 09:26:53 +0200},
Date-Modified = {2020-09-07 09:28:01 +0200},
Journal = {Theor. Chem. Acc.},
Pages = {69--82},
Title = {Gaussian Basis Sets for use in Correlated Molecular Calculations. VII. Valence, Core-valence, and Scalar Relativistic Basis Sets for Li, Be, Na, and Mg},
Volume = {128},
Year = {2010}}
@article{Rolik_2013,
Author = {Zolt{\'a}n Rolik and L{\'o}r{\'a}nt Szegedy and Istv{\'a}n Ladj{\'a}nszki and Bence Lad{\'o}czki and Mih{\'a}ly K{\'a}llay},
Date-Added = {2020-09-07 09:26:23 +0200},
Date-Modified = {2020-09-07 09:26:30 +0200},
Doi = {10.1063/1.4819401},
Eprint = {https://doi.org/10.1063/1.4819401},
Journal = {J. Chem. Phys.},
Number = {9},
Pages = {094105},
Title = {An Efficient Linear-Scaling CCSD(T) Method Based on Local Natural Orbitals},
Url = {https://doi.org/10.1063/1.4819401},
Volume = {139},
Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4819401}}
@misc{Gaussian16,
Author = {M. J. Frisch and G. W. Trucks and H. B. Schlegel and G. E. Scuseria and M. A. Robb and J. R. Cheeseman and G. Scalmani and V. Barone and G. A. Petersson and H. Nakatsuji and X. Li and M. Caricato and A. V. Marenich and J. Bloino and B. G. Janesko and R. Gomperts and B. Mennucci and H. P. Hratchian and J. V. Ortiz and A. F. Izmaylov and J. L. Sonnenberg and D. Williams-Young and F. Ding and F. Lipparini and F. Egidi and J. Goings and B. Peng and A. Petrone and T. Henderson and D. Ranasinghe and V. G. Zakrzewski and J. Gao and N. Rega and G. Zheng and W. Liang and M. Hada and M. Ehara and K. Toyota and R. Fukuda and J. Hasegawa and M. Ishida and T. Nakajima and Y. Honda and O. Kitao and H. Nakai and T. Vreven and K. Throssell and Montgomery, {Jr.}, J. A. and J. E. Peralta and F. Ogliaro and M. J. Bearpark and J. J. Heyd and E. N. Brothers and K. N. Kudin and V. N. Staroverov and T. A. Keith and R. Kobayashi and J. Normand and K. Raghavachari and A. P. Rendell and J. C. Burant and S. S. Iyengar and J. Tomasi and M. Cossi and J. M. Millam and M. Klene and C. Adamo and R. Cammi and J. W. Ochterski and R. L. Martin and K. Morokuma and O. Farkas and J. B. Foresman and D. J. Fox},
Date-Added = {2020-09-07 09:26:11 +0200},
Date-Modified = {2020-09-07 09:26:11 +0200},
Note = {Gaussian Inc. Wallingford CT},
Title = {Gaussian 16 {R}evision {A}.03},
Year = {2016}}
@misc{cfour,
Date-Added = {2020-09-07 09:25:53 +0200},
Date-Modified = {2020-09-07 09:25:53 +0200},
Note = {CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package by J.F. Stanton, J. Gauss, L. Cheng, M.E. Harding, D.A. Matthews, P.G. Szalay with contributions from A.A. Auer, R.J. Bartlett, U. Benedikt, C. Berger, D.E. Bernholdt, Y.J. Bomble, O. Christiansen, F. Engel, R. Faber, M. Heckert, O. Heun, M. Hilgenberg, C. Huber, T.-C. Jagau, D. Jonsson, J. Jus{\'e}lius, T. Kirsch, K. Klein, W.J. Lauderdale, F. Lipparini, T. Metzroth, L.A. M{\"u}ck, D.P. O'Neill, D.R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, C. Simmons, S. Stopkowicz, A. Tajti, J. V{\'a}zquez, F. Wang, J.D. Watts and the integral packages MOLECULE (J. Alml{\"o}f and P.R. Taylor), PROPS (P.R. Taylor), ABACUS (T. Helgaker, H.J. Aa. Jensen, P. J{\o}rgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van W{\"u}llen. For the current version, see http://www.cfour.de.}}
@article{Fang_2014,
Author = {Fang, Changfeng and Oruganti, Baswanth and Durbeej, Bo},
Date-Added = {2020-09-07 09:09:06 +0200},
Date-Modified = {2020-09-07 09:09:17 +0200},
Doi = {10.1021/jp501974p},
Eprint = {http://pubs.acs.org/doi/pdf/10.1021/jp501974p},
Journal = {J. Phys. Chem. A},
Pages = {4157--4171},
Title = {How Method-Dependent Are Calculated Differences Between Vertical, Adiabatic and 0-0 Excitation Energies?},
Url = {http://pubs.acs.org/doi/abs/10.1021/jp501974p},
Volume = {118},
Year = {2014},
Bdsk-Url-1 = {http://pubs.acs.org/doi/abs/10.1021/jp501974p},
Bdsk-Url-2 = {http://dx.doi.org/10.1021/jp501974p}}
@article{Goerigk_2017,
Abstract = {We present the GMTKN55 benchmark database for general main group thermochemistry{,} kinetics and noncovalent interactions. Compared to its popular predecessor GMTKN30 [Goerigk and Grimme J. Chem. Theory Comput.{,} 2011{,} 7{,} 291]{,} it allows assessment across a larger variety of chemical problems---with 13 new benchmark sets being presented for the first time---and it also provides reference values of significantly higher quality for most sets. GMTKN55 comprises 1505 relative energies based on 2462 single-point calculations and it is accessible to the user community via a dedicated website. Herein{,} we demonstrate the importance of better reference values{,} and we re-emphasise the need for London-dispersion corrections in density functional theory (DFT) treatments of thermochemical problems{,} including Minnesota methods. We assessed 217 variations of dispersion-corrected and -uncorrected density functional approximations{,} and carried out a detailed analysis of 83 of them to identify robust and reliable approaches. Double-hybrid functionals are the most reliable approaches for thermochemistry and noncovalent interactions{,} and they should be used whenever technically feasible. These are{,} in particular{,} DSD-BLYP-D3(BJ){,} DSD-PBEP86-D3(BJ){,} and B2GPPLYP-D3(BJ). The best hybrids are ωB97X-V{,} M052X-D3(0){,} and ωB97X-D3{,} but we also recommend PW6B95-D3(BJ) as the best conventional global hybrid. At the meta-generalised-gradient (meta-GGA) level{,} the SCAN-D3(BJ) method can be recommended. Other meta-GGAs are outperformed by the GGA functionals revPBE-D3(BJ){,} B97-D3(BJ){,} and OLYP-D3(BJ). We note that many popular methods{,} such as B3LYP{,} are not part of our recommendations. In fact{,} with our results we hope to inspire a change in the user community{'}s perception of common DFT methods. We also encourage method developers to use GMTKN55 for cross-validation studies of new methodologies.},
Author = {Goerigk, Lars and Hansen, Andreas and Bauer, Christoph and Ehrlich, Stephan and Najibi, Asim and Grimme, Stefan},
Date-Added = {2020-09-07 08:57:06 +0200},
Date-Modified = {2020-09-07 08:57:12 +0200},
Doi = {10.1039/C7CP04913G},
Issue = {48},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {32184-32215},
Publisher = {The Royal Society of Chemistry},
Title = {A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry{,} kinetics and noncovalent interactions},
Url = {http://dx.doi.org/10.1039/C7CP04913G},
Volume = {19},
Year = {2017},
Bdsk-Url-1 = {http://dx.doi.org/10.1039/C7CP04913G}}
@article{Gould_2018b,
Abstract = {The GMTKN55 benchmarking protocol introduced by [Goerigk et al.{,} Phys. Chem. Chem. Phys.{,} 2017{,} 19{,} 32184] allows comprehensive analysis and ranking of density functional approximations with diverse chemical behaviours. But this comprehensiveness comes at a cost: GMTKN55{'}s 1500 benchmarking values require energies for around 2500 systems to be calculated{,} making it a costly exercise. This manuscript introduces three subsets of GMTKN55{,} consisting of 30{,} 100 and 150 systems{,} as `diet' substitutes for the full database. The subsets are chosen via a stochastic genetic approach{,} and consequently can reproduce key results of the full GMTKN55 database{,} including ranking of approximations. Some results are also included for the recent MGCDB84 database.},
Author = {Gould, Tim},
Date-Added = {2020-09-07 08:52:32 +0200},
Date-Modified = {2020-09-07 08:52:52 +0200},
Doi = {10.1039/C8CP05554H},
Issue = {44},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {27735-27739},
Publisher = {The Royal Society of Chemistry},
Title = {`Diet GMTKN55' offers accelerated benchmarking through a representative subset approach},
Url = {http://dx.doi.org/10.1039/C8CP05554H},
Volume = {20},
Year = {2018},
Bdsk-Url-1 = {http://dx.doi.org/10.1039/C8CP05554H}}
@misc{Benali_2020,
Archiveprefix = {arXiv},
Author = {Anouar Benali and Kevin Gasperich and Kenneth D. Jordan and Thomas Applencourt and Ye Luo and M. Chandler Bennett and Jaron T. Krogel and Luke Shulenburger and Paul R. C. Kent and Pierre-Fran{\c c}ois Loos and Anthony Scemama and Michel Caffarel},
@ -6298,10 +6418,10 @@
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.119.243001}}
@article{Gould_2018,
@article{Gould_2018a,
Author = {Gould, Tim and Kronik, Leeor and Pittalis, Stefano},
Date-Added = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-01-01 21:36:52 +0100},
Date-Modified = {2020-09-07 08:52:48 +0200},
Doi = {10.1063/1.5022832},
File = {/Users/loos/Zotero/storage/C5DEDGG2/Gould et al. - 2018 - Charge transfer excitations from exact and approxi.pdf},
Issn = {0021-9606, 1089-7690},

65
Manuscript/QUEST_WIREs.tex
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@ -92,10 +92,10 @@ We hope that the present review will provide a useful summary of our work so far
Nowadays, there exists a very large number of electronic structure computational approaches, more or less expensive depending on their overall accuracy, able to quantitatively predict the absolute and/or relative energies of electronic states in molecular systems \cite{JensenBook}.
One important aspect of some of these theoretical 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 state (that is, the lowest-energy state).
The faithful description of excited states are particularly challenging from a theoretical point of view and is key to a deeper understanding of photochemical and photophysical processes like absorption, fluorescence, or even chemoluminescence \cite{Bernardi_1996,Olivucci_2010,Robb_2007,Navizet_2011}.
The faithful description of excited states is particularly challenging from a theoretical point of view \cite{Gonzales_2012,Ghosh_2018,Loos_2020a} and is key to a deeper understanding of photochemical and photophysical processes like absorption, fluorescence, or even chemoluminescence \cite{Bernardi_1996,Olivucci_2010,Robb_2007,Navizet_2011}.
For a given level of theory, ground-state methods are usually more accurate than their excited-state analog.
The reasons behind this are (at least) twofold: i) one might lack a proper variational principle for excited-state energies, and ii) excited states are often very close in energy from each other but they can have very different natures ($\pi \to \pis$, $n \to \pis$, charge transfer, double excitation, valence, Rydberg, singlet, doublet, triplet, etc).
Designing excited-state methods which can tackle on the same footing all these types of excited states at an affordable cost remain an open challenge in theoretical computational chemistry \cite{Gonzales_2012, Loos_2020a}.
Designing excited-state methods which can tackle on the same footing all these types of excited states at an affordable cost remain an open challenge in theoretical computational chemistry \cite{Gonzales_2012,Ghosh_2018,Loos_2020a}.
When one designs 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 in a well-defined setup (same geometry, same basis set, etc).
These values can be absolute or relative energies, geometrical parameters, physical or chemical properties, etc, extracted from experiments, high-level theoretical calculations, or a combination of both.
@ -121,16 +121,16 @@ Following a similar philosophy, we have recently reported in several studies hig
One of the key aspect of the so-called QUEST dataset of vertical excitations which we will describe in details in the present review article 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 use a very systematic procedure for all excited states in order to make cross-comparison as straightforward as possible.
Importantly, it allowed us to benchmark a series of popular excited-state wave function methods partially or fully accounting for double and triple excitations as well as multiconfigurational methods such as CASPT2 and NEVPT2.
In the same vein, 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{Kohn_2003,Send_2011a,Winter_2013}.
In the same vein, 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{Kohn_2003,Dierksen_2004,Goerigk_2010a,Send_2011a,Winter_2013,Fang_2014}.
We refer the interested reader to Ref.~\cite{Loos_2019b} where we review the generic benchmark studies devoted to adiabatic and 0-0 energies performed in the last two decades.
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 CC methods such as CCSDT and CCSDTQ.
Recently, SCI methods have been a force to reckon with for the computation of highly-accurate energies in small- and medium-sized molecules \cite{Holmes_2017,Chien_2018,Loos_2018a,Li_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020a,Li_2020,Eriksen_2020,Loos_2020e,Yao_2020}.
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,Schriber_2016,Schriber_2017,Liu_2016,Per_2017,Ohtsuka_2017,Zimmerman_2017,Li_2018,Ohtsuka_2017,Coe_2018,Loos_2019} and their fundamental philosophy consists, roughly speaking, in retaining only the most energetically relevant determinants of the FCI space following a given criterion to avoid the exponential increase of the size of the CI expansion.
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-FCI quality energies for only a fraction of the computational cost of a genuine FCI calculation \cite{Holmes_2017,Chien_2018,Loos_2018a,Li_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020a,Li_2020,Eriksen_2020,Loos_2020e,Yao_2020}.
Due to the fairly natural idea underlying SCI 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,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 energetically relevant determinants of the FCI space following a given criterion to avoid 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.
Four examples are adaptive sampling CI (ASCI) \cite{Tubman_2016,Tubman_2018,Tubman_2020}, iCI \cite{Liu_2016}, semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017,Li_2018}), and \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973}
These four flavors of SCI includes a second-order perturbative (PT2) correction which is key to estimate the ``distance'' to the genuine FCI solution.
Four examples are adaptive sampling CI (ASCI) \cite{Tubman_2016,Tubman_2018,Tubman_2020}, iCI \cite{Liu_2016}, semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017,Li_2018}), and \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) \cite{Huron_1973}.
These four flavors of SCI include a second-order perturbative (PT2) correction which is key to estimate the ``distance'' to the FCI solution.
The QUEST set of excitation energies relies on the CIPSI algorithm, which is, from a historical point of view, one of the oldest SCI algorithm.
It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see also Refs.~\cite{Evangelisti_1983,Cimiraglia_1985,Cimiraglia_1987,Illas_1988,Povill_1992}).
Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by our group enabling to perform massively parallel computations \cite{Garniron_2017,Garniron_2018,Garniron_2019,Loos_2020e}.
@ -156,6 +156,8 @@ Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
%=======================
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, the geometries are optimized at the UCCSD(T)/aug-cc-pVTZ level using the GAUSSIAN16 program \cite{Gaussian16} and applying the \textsc{tight}convergence threshold.
For the present review article, we have gathered all the geometries in the {\SupInf}.
\footnote{These geometries can be found at...}
@ -169,6 +171,9 @@ Doubly- and triply-augmented basis sets are usually employed for Rydberg states
\subsection{Computational methods}
%==================================
%------------------------------------------------
\subsubsection{Reference computational methods}
%------------------------------------------------
@ -177,13 +182,40 @@ For small systems (typically 1--3 non-hydrogen atoms), we resort to SCI methods
Obviously, the smaller the molecule, the larger the basis we can afford.
For larger systems (\ie, 4--6 non-hydrogen atom), one cannot afford SCI calculations anymore expect in a few exceptions, and we then rely on CC theory (CCSDT and CCSDTQ typically) to obtain accurate transition energies.
The CC calculations are performed with several codes. For closed-shell molecules, CC3 \cite{Christiansen_1995b,Koch_1997} calculations are achieved with DALTON \cite{dalton} and CFOUR; \cite{cfour} CCSDT calculations are performed
with CFOUR \cite{cfour} and MRCC 2017;\cite{Rolik_2013,mrcc} the latter code being also used for CCSDTQ and CCSDTQP. Note that all our excited-state CC calculations are performed within the equation-of-motion (EOM)
or linear-response (LR) formalism that yield equivalent excited-state energies. The reported oscillator strengths have been computed in the LR-CC3 formalism only. For open-shell molecules, the CCSDT, CCSDTQ, and
CCSDTQP calculations performed with MRCC \cite{Rolik_2013,mrcc} do consider an unrestricted Hartree-Fock (UHF) wave function as reference. All excited-state calculations are performed, except when explicitly mentioned, in
the FC approximation using large cores for the third-row atoms. All electrons are correlated for the \ce{Be} atom, for which we systematically applied the basis set as included in MRCC. \cite{Prascjer_2010} (We have noted
differences in the definition of the Dunning bases for this particular atom depending on the software that one considers.)
All the SCI calculations are performed within the FC approximation using QUANTUM PACKAGE \cite{Garniron_2019} where the CIPSI algorithm \cite{Huron_1973} is implemented. Details regarding this specific CIPSI implementation
can be found in Refs.~\citenum{Gar19} and \citenum{Sce19}. We use a state-averaged formalism which means that the ground and excited states are described with the same number and same set of determinants, but
different CI coefficients. The SCI energy is defined as the sum of the variational energy (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction which estimates the
contribution of the determinants not included in the CI space. \cite{Garniron_2017b} By extrapolating this second-order correction to zero, one can efficiently estimate the FCI limit for the total energies, and hence, compute the
corresponding transition energies. We estimate the extrapolation error by the difference between the transition energies obtained with the largest SCI wave function and the FCI extrapolated value. These errors are
systematically reported in the Tables below. Although this cannot be viewed as a true error bar, it provides a rough idea of the quality of the FCI extrapolation and estimate.
%------------------------------------------------
\subsubsection{Benchmarked computational methods}
%------------------------------------------------
%------------------------------------------------
\subsubsection{Electronic structure software}
%------------------------------------------------
Our benchmark effort consists in evaluating the accuracy of vertical transition energies obtained at lower levels of theory.
These calculations are performed with a variety of codes. For the exotic set, we rely on: GAUSSIAN \cite{Gaussian16} and TURBOMOLE 7.3 \cite{Turbomole} for CIS(D); \cite{Hea94,Hea95} Q-CHEM 5.2 \cite{Kry13} for EOM-MP2 [CCSD(2)] \cite{Sta95c} and ADC(3); \cite{Tro02,Har14,Dre15}
Q-CHEM \cite{Kry13} and TURBOMOLE \cite{Turbomole} for ADC(2); \cite{Tro97,Dre15} DALTON \cite{dalton} and TURBOMOLE \cite{Turbomole} for CC2; \cite{Chr95,Hat00} DALTON \cite{dalton} and GAUSSIAN
for CCSD;\cite{Pur82} DALTON \cite{dalton} for CCSDR(3); \cite{Chr96b} CFOUR \cite{cfour} for CCSDT-3; \cite{Wat96,Pro10} and ORCA \cite{Nee12} for similarity-transformed EOM-CCSD (STEOM-CCSD). \cite{Nooijen_1997,Dut18}
In addition, we evaluate the spin-opposite scaling (SOS) variants of ADC(2), SOS-ADC(2), as implemented in both Q-CHEM, \cite{Kra13} and TURBOMOLE. \cite{Hellweg_2008} Note that these two codes have distinct SOS implementations, as explained in Ref.~\citenum{Kra13}. We also test the SOS and spin-component scaled (SCS) versions of CC2, as implemented in TURBOMOLE. \cite{Hellweg_2008,Turbomole} Discussion of various spin-scaling
schemes can be found elsewhere. \cite{Goerigk_2010a} When available, we take advantage of the resolution-of-the-identity (RI) approximation in TURBOMOLE and Q-CHEM. For the STEOM-CCSD calculations, it was checked that the
active character percentage was, at least, $98\%$. When comparisons between various codes/implementations were possible, we could not detect variations in the transition energies larger than $0.01$ eV. For the radical set
molecules, we applied both the U (unrestricted) and RO (restricted open-shell) versions of CCSD and CC3 as implemented in the PSI4 code, \cite{Psi4} to perform our benchmarks.
State-averaged (SA) CASSCF and CASPT2 \cite{Roos,Andersson_1990} have been performed with MOLPRO (RS2 contraction level). \cite{molpro}
Concerning the NEVPT2 calculations, the partially-contracted (PC) and strongly-contracted (SC) variants have been systematically tested. \cite{Angeli_2001a, Angeli_2001b, Angeli_2002}
From a strict theoretical point of view, we point out that PC-NEVPT2 is supposed to be more accurate than SC-NEVPT2 given that it has a larger number of perturbers and greater flexibility.
When there is a strong mixing between states with same spin and spatial symmetries, we have also performed calculations with multi-state (MS) CASPT2 (MS-MR formalism), \cite{Finley_1998} and its extended variant (XMS-CASPT2). \cite{Shiozaki_2011}
Unless otherwise stated, all CASPT2 calculations have been performed with level shift and IPEA parameters set to the standard values of $0.3$ and $0.25$ a.u., respectively.
Large active spaces carefully chosen and tailored for the desired transitions have been selected.
The definition of the active space considered for each system as well as the number of states in the state-averaged calculation is provided in their corresponding publication.
%------------------------------------------------
\subsubsection{Estimating the extrapolation error}
@ -207,7 +239,7 @@ The QUEST\#1 benchmark set \cite{Loos_2018a} consists of 110 vertical excitation
streptocyanine). For this set, we provided two sets of TBEs: i) one obtained within the frozen-core approximation and the aug-cc-pVTZ basis set, and ii) another one including further corrections for basis set incompleteness and ``all electron'' effects.
For the former set, we systematically selected FCI/aug-cc-pVTZ values to define our TBEs except in very few cases.
For the latter set, both the ``all electron'' correlation and the basis set corrections were systematically obtained at the CC3 level of theory and with the d-aug-cc-pV5Z basis for the nine smallest molecules, and slightly more compact basis sets for the larger compounds.
Our TBE/aug-cc-pVTZ reference excitation energies were employed to benchmark a series of popular excited-state wave function methods partially or fully accounting for double and triple excitations, namely CIS(D), CC2, CCSD, STEOM-CCSD, CCSDR(3), CCSDT- CC3, ADC(2), and ADC(3).
Our TBE/aug-cc-pVTZ reference excitation energies were employed to benchmark a series of popular excited-state wave function methods partially or fully accounting for double and triple excitations, namely CIS(D), CC2, CCSD, STEOM-CCSD, CCSDR(3), CCSDT-3, CC3, ADC(2), and ADC(3).
Our main conclusions were that i) ADC(2) and CC2 show strong similarities in terms of accuracy, ii) STEOM-CCSD is, on average, as accurate as CCSD, the latter overestimating transition energies, iii) CC3 is extremely accurate (with a mean absolute error of only $\sim 0.03$ eV) and that although slightly less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies, and iv) ADC(3) was found to be significantly less accurate than CC3 by overcorrecting ADC(2) excitation energies.
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@ -462,6 +494,15 @@ Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.769(132) \\
\section{Concluding remarks}
\label{sec:ccl}
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Because computing 450 excitation energies can be a costly exercise, we are planning on developing a ``diet set'' following the philosophy of the ``diet GMTKN55'' set \cite{Goerigk_2017} proposed recently by Gould \cite{Gould_2018b}.
Besides all the studies described above aiming at reaching chemically accurate vertical transition energies, it should be pointed out that an increasing amount of effort is currently devoted to the obtention of highly-trustable excited-state properties.
This includes, first, 0-0 energies, \cite{Die04b,Hat05c,Goerigk_2010a,Sen11b,Win13,Fan14b,Loo18b,Loo19a,Loo19b} which, as mentioned above, offer well-grounded comparisons with experiment.
However, because 0-0 energies are fairly insensitive to the underlying molecular geometries, \cite{Sen11b,Win13,Loo19a} they are not a good indicator of their overall quality.
Consequently, one can find in the literature several sets of excited-state geometries obtained at various levels of theory, \cite{Pag03,Gua13,Bou13,Tun16,Bud17} some of them being determined using state-of-the-art models. \cite{Gua13,Bud17}
There are also investigations of the accuracy of the nuclear gradients at the Franck-Condon point. \cite{Taj18,Taj19}
The interested reader may find useful several investigations reporting sets of reference oscillator strengths. \cite{Sil10c,Har14,Kan14,Loo18a,Loo20a}
More complex properties, such as two-photon cross-sections and vibrations, have been mostly determined at lower levels of theory, hinting at future studies on this particular subject.
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\section*{acknowledgements}