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Manuscript/ExPerspective.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
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@string{theo = {J. Mol. Struct. (THEOCHEM)}}
@article{Taj18,
Author = {Tajti, Attila and Stanton, John F. and Matthews, Devin A. and Szalay, P{\'e}ter G.},
Date-Added = {2019-11-18 17:57:00 +0100},
Date-Modified = {2019-11-18 17:57:15 +0100},
Doi = {10.1021/acs.jctc.8b00681},
Eprint = {https://doi.org/10.1021/acs.jctc.8b00681},
Journal = {J. Chem. Theory Comput.},
Note = {PMID: 30299948},
Number = {11},
Pages = {5859--5869},
Title = {Accuracy of Coupled Cluster Excited State Potential Energy Surfaces},
Url = {https://doi.org/10.1021/acs.jctc.8b00681},
Volume = {14},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00681}}
@article{Taj19,
Author = {Tajti, Attila and Szalay, P{\'e}ter G.},
Date-Added = {2019-11-18 17:56:17 +0100},
Date-Modified = {2019-11-18 17:56:33 +0100},
Doi = {10.1021/acs.jctc.9b00676},
Eprint = {https://doi.org/10.1021/acs.jctc.9b00676},
Journal = {J. Chem. Theory Comput.},
Note = {PMID: 31433639},
Number = {10},
Pages = {5523--5531},
Title = {Accuracy of Spin-Component-Scaled CC2 Excitation Energies and Potential Energy Surfaces},
Url = {https://doi.org/10.1021/acs.jctc.9b00676},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00676}}
@article{Tun16,
Author = {Tuna, Deniz and Lu, You and Koslowski, Axel and Thiel, Walter},
Date-Added = {2019-11-18 17:53:26 +0100},
Date-Modified = {2019-11-18 17:53:26 +0100},
Doi = {10.1021/acs.jctc.6b00403},
Eprint = {http://dx.doi.org/10.1021/acs.jctc.6b00403},
Journal = {J. Chem. Theory Comput.},
Number = {9},
Pages = {4400--4422},
Title = {Semiempirical Quantum-Chemical Orthogonalization-Corrected Methods: Benchmarks of Electronically Excited States},
Url = {http://dx.doi.org/10.1021/acs.jctc.6b00403},
Volume = {12},
Year = {2016},
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Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jctc.6b00403}}
@article{Bou13,
Author = {Bousquet, Diane and Fukuda, Ryoichi and Maitarad, Phornphimon and Jacquemin, Denis and Ciofini, Ilaria and Adamo, Carlo and Ehara, Masahiro},
Date-Added = {2019-11-18 17:53:14 +0100},
Date-Modified = {2019-11-18 17:53:14 +0100},
Doi = {10.1021/ct400097b},
Eprint = {http://pubs.acs.org/doi/pdf/10.1021/ct400097b},
Journal = {J. Chem. Theory Comput.},
Number = {5},
Pages = {2368-2379},
Title = {Excited-State Geometries of Heteroaromatic Compounds: A Comparative TD-DFT and SAC-CI Study},
Url = {http://pubs.acs.org/doi/abs/10.1021/ct400097b},
Volume = {9},
Year = {2013},
Bdsk-Url-1 = {http://pubs.acs.org/doi/abs/10.1021/ct400097b},
Bdsk-Url-2 = {http://dx.doi.org/10.1021/ct400097b}}
@article{Gua13,
Author = {Riccardo Guareschi and Claudia Filippi},
Date-Added = {2019-11-18 17:52:59 +0100},
Date-Modified = {2019-11-18 17:52:59 +0100},
Doi = {10.1021/ct400876y},
Eprint = {http://dx.doi.org/10.1021/ct400876y},
Journal = {J. Chem. Theory Comput.},
Number = {12},
Pages = {5513--5525},
Title = {Ground- and Excited-State Geometry Optimization of Small Organic Molecules with Quantum Monte Carlo},
Url = {http://dx.doi.org/10.1021/ct400876y},
Volume = {9},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct400876y}}
@article{Bud17,
Author = {Budz{\'a}k, {\v S}. and Scalmani, G. and Jacquemin, D.},
Date-Added = {2019-11-18 17:52:29 +0100},
Date-Modified = {2019-11-18 17:52:29 +0100},
Journal = {J. Chem. Theory Comput.},
Pages = {6237--6252},
Title = {Accurate Excited-State Geometries: a CASPT2 and Coupled-Cluster Reference Database for Small Molecules},
Volume = {13},
Year = {2017}}
@article{Pag03,
Author = {Page, Christopher S. and Olivucci, Massimo},
Date-Added = {2019-11-18 17:51:57 +0100},
Date-Modified = {2019-11-18 17:51:57 +0100},
Doi = {10.1002/jcc.10145},
Issn = {1096-987X},
Journal = {J. Comput. Chem.},
Keywords = {second-order perturbation-theory, electronically excited state, potential energy surface, geometry optimization, conical intersection},
Number = {3},
Pages = {298--309},
Publisher = {Wiley Subscription Services, Inc., A Wiley Company},
Title = {Ground and Excited State CASPT2 Geometry Optimizations of Small Organic Molecules},
Url = {http://dx.doi.org/10.1002/jcc.10145},
Volume = {24},
Year = {2003},
Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.10145}}
@article{Sen11b,
Abstract = { We compile a 109-membered benchmark set of adiabatic excitation energies (AEEs) from high-resolution gas-phase experiments. Our data set includes a variety of organic chromophores with up to 46 atoms, radicals, and inorganic transition metal compounds. Many of the 91 molecules in our set are relevant to atmospheric chemistry, photovoltaics, photochemistry, and biology. The set samples valence, Rydberg, and ionic states of various spin multiplicities. As opposed to vertical excitation energies, AEEs are rigorously defined by energy differences of vibronic states, directly observable, and insensitive to errors in equilibrium structures. We supply optimized ground state and excited state structures, which allows fast and convenient evaluation of AEEs with two single-point energy calculations per system. We apply our benchmark set to assess the performance of time-dependent density functional theory using common semilocal functionals and the configuration interaction singles method. Hybrid functionals such as B3LYP and PBE0 yield the best results, with mean absolute errors around 0.3 eV. We also investigate basis set convergence and correlations between different methods and between the magnitude of the excited state relaxation energy and the AEE error. A smaller, 15-membered subset of AEEs is introduced and used to assess the correlated wave function methods CC2 and ADC(2). These methods improve upon hybrid TDDFT for systems with single-reference ground states but perform less well for radicals and small-gap transition metal compounds. None of the investigated methods reaches ``chemical accuracy'' of 0.05 eV in AEEs. },
Author = {Send, Robert and K{\"u}hn, Michael and Furche, Filipp},
Date-Added = {2019-11-18 17:48:34 +0100},
Date-Modified = {2019-11-18 17:48:34 +0100},
Doi = {10.1021/ct200272b},
Journal = {J. Chem. Theory Comput.},
Number = {8},
Pages = {2376-2386},
Title = {Assessing Excited State Methods by Adiabatic Excitation Energies},
Volume = {7},
Year = {2011},
Bdsk-Url-1 = {http://pubs.acs.org/doi/abs/10.1021/ct200272b},
Bdsk-Url-2 = {http://dx.doi.org/10.1021/ct200272b}}
@article{Sue19,
Author = {Suellen, Cinthia and Garcia Freitas, Renato and Loos, Pierre-Francois and Jacquemin, Denis},
Date-Added = {2019-11-18 17:38:43 +0100},
Date-Modified = {2019-11-18 17:38:43 +0100},
Doi = {10.1021/acs.jctc.9b00446},
Eprint = {https://doi.org/10.1021/acs.jctc.9b00446},
Journal = {J. Chem. Theory Comput.},
Pages = {4581--4590},
Title = {Cross Comparisons Between Experiment, TD-DFT, CC, and ADC for Transition Energies},
Url = {https://doi.org/10.1021/acs.jctc.9b00446},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00446}}
@incollection{Hat05c,
Author = {Christof H\"attig},
Booktitle = {Response Theory and Molecular Properties (A Tribute to Jan Linderberg and Poul J{\o}rgensen)},
Date-Added = {2019-11-18 17:38:16 +0100},
Date-Modified = {2019-11-18 17:38:16 +0100},
Doi = {http://dx.doi.org/10.1016/S0065-3276(05)50003-0},
Editor = {H.J. \AA\ Jensen},
Issn = {0065-3276},
Pages = {37--60},
Publisher = {Academic Press},
Series = {Advances in Quantum Chemistry},
Title = {Structure Optimizations for Excited States with Correlated Second-Order Methods: CC2 and ADC(2)},
Url = {http://www.sciencedirect.com/science/article/pii/S0065327605500030},
Volume = {50},
Year = {2005},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0065327605500030},
Bdsk-Url-2 = {http://dx.doi.org/10.1016/S0065-3276(05)50003-0}}
@article{Kuc01,
Author = {Stanis{\l}aw A. Kucharski and Marta W{\l}och and Monika Musia{\l} and Rodney J. Bartlett},
Date-Added = {2019-11-18 16:38:35 +0100},
Date-Modified = {2019-11-18 16:38:35 +0100},
Doi = {10.1063/1.1416173},
Eprint = {https://doi.org/10.1063/1.1416173},
Journal = {J. Chem. Phys.},
Number = {18},
Pages = {8263-8266},
Title = {Coupled-Cluster Theory for Excited Electronic States: The Full Equation-Of-Motion Coupled-Cluster Single, Double, and Triple Excitation Method},
Url = {https://doi.org/10.1063/1.1416173},
Volume = {115},
Year = {2001},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1416173}}
@article{Goe10a,
Author = {Goerigk, L. and Grimme, S.},
Date-Added = {2019-11-18 16:34:32 +0100},
Date-Modified = {2019-11-18 16:34:32 +0100},
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{Bre16,
Author = {Br{\'e}mond, Eric and Ciofini, Ilaria and Sancho-Garc{\'\i}a, Juan Carlos and Adamo, Carlo},
Date-Added = {2019-11-18 16:33:11 +0100},
Date-Modified = {2019-11-18 16:33:11 +0100},
Doi = {10.1021/acs.accounts.6b00232},
Eprint = {http://dx.doi.org/10.1021/acs.accounts.6b00232},
Journal = {Acc. Chem. Res.},
Number = {8},
Pages = {1503--1513},
Title = {Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists},
Url = {http://dx.doi.org/10.1021/acs.accounts.6b00232},
Volume = {49},
Year = {2016},
Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.accounts.6b00232}}
@article{Sch17,
Author = {Schwabe, Tobias and Goerigk, Lars},
Date-Added = {2019-11-18 16:32:36 +0100},
Date-Modified = {2019-11-18 16:32:36 +0100},
Doi = {10.1021/acs.jctc.7b00386},
Eprint = {https://doi.org/10.1021/acs.jctc.7b00386},
Journal = {J. Chem. Theory Comput.},
Number = {9},
Pages = {4307--4323},
Title = {Time-Dependent Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling},
Url = {https://doi.org/10.1021/acs.jctc.7b00386},
Volume = {13},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00386}}
@article{Lau13,
Author = {Laurent, Ad{\`e}le D. and Jacquemin, Denis},
Date-Added = {2019-11-18 16:30:05 +0100},
Date-Modified = {2019-11-18 16:30:05 +0100},
Journal = {Int. J. Quantum Chem.},
Owner = {chibani-s},
Pages = {2019--2039},
Timestamp = {2013.04.29},
Title = {TD-DFT Benchmarks: A Review},
Volume = {113},
Year = {2013}}
@misc{Bai19,
Author = {Alberto Baiardi and Markus Reiher},
Date-Added = {2019-11-18 16:26:39 +0100},
Date-Modified = {2019-11-18 16:27:55 +0100},
Note = {arXiv, 1910.00137},
Title = {The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges},
Year = {2019}}
@article{Win13,
Abstract = {In the present study a benchmark set of medium-sized and large aromatic organic molecules with 10--78 atoms is presented. For this test set 0--0 transition energies measured in supersonic jets are compared to those calculated with DFT and the B3LYP functional{,} ADC(2){,} CC2 and the spin-scaled CC2 variants SOS-CC2 and SCS-CC2. Geometries of the ground and excited states have been optimized with these methods in polarized triple zeta basis sets. Zero-point vibrational corrections have been calculated with the same methods and basis sets. In addition the energies have been corrected by single point calculations with a triple zeta basis augmented with diffuse functions{,} aug-cc-pVTZ. The deviations of the theoretical results from experimental electronic origins{,} which have all been measured in the gas phase with high-resolution techniques{,} were evaluated. The accuracy of SOS-CC2 is comparable to that of unscaled CC2{,} whereas ADC(2) has slightly larger errors. The lowest errors were found for SCS-CC2. All correlated wave function methods provide significantly better results than DFT with the B3LYP functional. The effects of the energy corrections from the augmented basis set and the method-consistent calculation of the zero-point vibrational corrections are small. With this benchmark set reliable reference data for 0--0 transition energies for larger organic chromophores are available that can be used to benchmark the accuracy of other quantum chemical methods such as new DFT functionals or semi-empirical methods for excitation energies and structures and thereby augments available benchmark sets augments present benchmark sets which include mainly smaller molec},
Author = {Winter, Nina O. C. and Graf, Nora K. and Leutwyler, Samuel and H{\"a}ttig, Christof},
Date-Added = {2019-11-18 16:13:25 +0100},
Date-Modified = {2019-11-18 16:13:25 +0100},
Doi = {10.1039/C2CP42694C},
Issue = {18},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {6623-6630},
Publisher = {The Royal Society of Chemistry},
Title = {Benchmarks for 0--0 Transitions of Aromatic Organic Molecules: DFT/B3LYP{,} ADC(2){,} CC2{,} SOS-CC2 and SCS-CC2 Compared to High-resolution Gas-Phase Data},
Url = {http://dx.doi.org/10.1039/C2CP42694C},
Volume = {15},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1039/C2CP42694C}}
@article{Fan14b,
Author = {Fang, Changfeng and Oruganti, Baswanth and Durbeej, Bo},
Date-Added = {2019-11-18 16:13:08 +0100},
Date-Modified = {2019-11-18 16:13:08 +0100},
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{Die04b,
Author = {Dierksen, M. and Grimme, S.},
Date-Added = {2019-11-18 16:12:51 +0100},
Date-Modified = {2019-11-18 16:12:51 +0100},
Journal = JPCA,
Pages = {10225-10237},
Title = {The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of \emph{Exact} Hartree-Fock Exchange},
Volume = 108,
Year = 2004}
@article{Gho18,
Author = {Ghosh, Soumen and Verma, Pragya and Cramer, Christopher J. and Gagliardi, Laura and Truhlar, Donald G.},
Date-Added = {2019-11-18 16:10:53 +0100},
Date-Modified = {2019-11-18 16:10:53 +0100},
Doi = {10.1021/acs.chemrev.8b00193},
Eprint = {https://doi.org/10.1021/acs.chemrev.8b00193},
Journal = {Chem. Rev.},
Number = {15},
Pages = {7249--7292},
Title = {Combining Wave Function Methods with Density Functional Theory for Excited States},
Url = {https://doi.org/10.1021/acs.chemrev.8b00193},
Volume = {118},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.chemrev.8b00193}}
@article{Dre05,
Author = {Dreuw, A. and Head-Gordon, M.},
Date-Added = {2019-11-18 16:10:24 +0100},
Date-Modified = {2019-11-18 16:10:24 +0100},
Journal = CR,
Pages = {4009--4037},
Title = {Single-Reference \emph{ab initio} Methods for the Calculation of Excited States of Large Molecules},
Volume = {105},
Year = 2005}
@article{Shu17,
Author = {Shu, Yinan and Truhlar, Donald G.},
Date-Added = {2019-11-16 13:38:18 +0100},
@ -1284,16 +1579,6 @@
Year = {2015},
Bdsk-Url-1 = {http://dx.doi.org/10.1002/wcms.1206}}
@article{Dre05,
Author = {Dreuw, A. and Head-Gordon, M.},
Date-Added = {2019-10-30 13:41:22 +0100},
Date-Modified = {2019-10-30 13:41:22 +0100},
Journal = CR,
Pages = {4009--4037},
Title = {Single-Reference \emph{ab initio} Methods for the Calculation of Excited States of Large Molecules},
Volume = {105},
Year = 2005}
@article{Dre04,
Author = {Dreuw, A. and Head-Gordon, M.},
Date-Added = {2019-10-30 13:37:19 +0100},
@ -1338,19 +1623,6 @@
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5041327}}
@article{Sue19,
Author = {Suellen, Cinthia and Garcia Freitas, Renato and Loos, Pierre-Francois and Jacquemin, Denis},
Date-Added = {2019-10-30 13:35:26 +0100},
Date-Modified = {2019-10-30 13:35:26 +0100},
Doi = {10.1021/acs.jctc.9b00446},
Eprint = {https://doi.org/10.1021/acs.jctc.9b00446},
Journal = {J. Chem. Theory Comput.},
Note = {doi: 10.1021/acs.jctc.9b00446},
Title = {Cross Comparisons Between Experiment, TD-DFT, CC, and ADC for Transition Energies},
Url = {https://doi.org/10.1021/acs.jctc.9b00446},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00446}}
@article{Gar17b,
Author = {Garniron, Yann and Scemama, Anthony and Loos, Pierre-Fran{\c c}ois and Caffarel, Michel},
Date-Added = {2019-10-30 13:35:06 +0100},
@ -1786,10 +2058,11 @@
@article{Sce13b,
Author = {Scemama, Anthony and Giner, Emmanuel},
Date-Modified = {2019-11-18 17:43:02 +0100},
Eprint = {1311.6244},
Journal = {arXiv},
Month = {Nov},
Title = {{An efficient implementation of Slater-Condon rules}},
Title = {An efficient implementation of Slater-Condon rules},
Url = {https://arxiv.org/abs/1311.6244},
Year = {2013},
Bdsk-Url-1 = {https://arxiv.org/abs/1311.6244}}

192
Manuscript/ExPerspective.tex
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@ -70,12 +70,13 @@
\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 with computational chemistry tools, eventually leading to chemically accurate vertical transition energies for small- and medium-size molecules.
First, we describe the evolution of \textit{ab initio} state-of-the-art methods, with originally Roos' CASPT2 method, and then third-order coupled cluster methods as in the renowned Thiel set of excitation energies described in a remarkable series of papers in the 2000's.
More recently, this quest for highly accurate excitation energies was reinitiated thanks to the resurgence of selected configuration interaction (SCI) methods and their efficient parallel implementation.
These methods have been able to routinely deliver highly accurate excitation energies for small molecules as well as medium-size molecules in compact basis sets for single and double excitations.
Second, we describe how these high-level methods and the creation of large, diverse and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performance of computationally lighter theoretical models (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of excited states ($\pi \ra \pi^*$, $n \ra \pi^*$, valence, Rydberg, singlet, triplet, double excitation, etc).
We conclude this \textit{Perspective} by discussing the current potentiality of these methods from both an expert and a non-expert point of view, and what we believe could be the future theoretical and technological developments in the field.
We provide a personal overview of the successive steps that made possible to obtain increasingly accurate excitation energies and properties with computational chemistry tools, eventually leading to chemically accurate vertical transition energies for small- and
medium-size molecules. First, we describe the evolution of \textit{ab initio} state-of-the-art methods use to define benchmark values, 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 with a compact basis sets for both single and double excitations. Second, we describe how these high-level methods
and the creation of large, diverse and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performance of computationally lighter theoretical models (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of
excited states ($\pi \ra \pi^*$, $n \ra \pi^*$, valence, Rydberg, singlet, triplet, double excitation, etc). We conclude this \textit{Perspective} by discussing the current potentiality of these methods from both an expert and a non-expert points of view, and what we
believe could be the future theoretical and technological developments in the field.
\end{abstract}
\maketitle
@ -83,33 +84,31 @@ We conclude this \textit{Perspective} by discussing the current potentiality of
%%%%%%%%%%%%%%%%%%%%
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
The accurate modeling of excited-state properties with \textit{ab initio} quantum chemistry methods is a challenging yet self-proclaimed ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come (see, for example, Ref.~\onlinecite{Gon12} and references therein).
Of particular interest is the access to precise excitation energies, \ie, the energy difference between ground and excited electronic states, and their intimate link with photochemical processes and photochemistry in general.
The factors that makes this quest for high accuracy particularly delicate are very diverse.
The accurate modeling of excited-state properties with \textit{ab initio} quantum chemistry methods is a challenging yet self-proclaimed ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come
(see, for example, Refs.~\onlinecite{Dre05,Gon12,Gho18} and references therein). Of particular interest is the access to precise excitation energies, \ie, the energy difference between ground and excited electronic states, and their intimate link with photophysical and
photochemical processes. The factors that makes this quest for high accuracy particularly delicate are very diverse.
First of all (and maybe surprisingly), it is, in most cases, tricky to obtain reliable and accurate experimental data that one can straightforwardly compare to theoretical values.
In the case of vertical excitation energies, \ie, excitation energies at a fixed geometry, band maxima do not usually match theoretical values as one needs to take into account geometric relaxation and zero-point vibrational energy motion.
Even more problematic, experimental spectra might not be available in gas phase, and, in the worst-case scenario, wrong assignments may have occurred.
For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Loo18b,Loo19a,Loo19b}
First of all (and maybe surprisingly), it is, in most cases, tricky to obtain reliable and accurate experimental data that one can straightforwardly compare to theoretical values. In the case of vertical excitation energies, \ie, excitation energies at a fixed geometry, band maxima
do not usually match theoretical values as one needs to take into account both geometric relaxation and zero-point vibrational energy motion. Even more problematic, experimental spectra might not be available in gas phase, and, in the worst-case scenario, no clear
assignments could be made. For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Die04b,Win13,Fan14b,Loo19b}
Another feature that makes excited states particularly fascinating and challenging is that they can be both extremely close in energy from each other and have very different natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet, triplet, etc).
Therefore, it would be highly desirable to possess a computational method (or protocol) that provides a balanced treatment of the entire ``spectrum'' of excited states.
And let's be honest, none of the existing methods does provide this at an affordable cost.
Second, developing theories suite for excited states is more complex because some fundamental safety nets, \eg, bounding from below, are not available in contrast to the ground electronic states, which additionally makes the results often less accurate for excited states
than for ground states.
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.
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 minimal chemical intuition (\ie, black box method preferable) in order to minimize the potential bias brought by the user appreciation of the problem complexity.
Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
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.
Another feature that makes excited states particularly fascinating and challenging is that they can be both extremely close in energy from each other and have very different natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet,
triplet, etc). Therefore, it would be highly desirable to possess a computational method (or protocol) that provides a balanced treatment of the entire ``spectrum'' of excited states. And let's be honest, none of the existing methods does provide such feat for large compounds
at an affordable cost.
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. 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 minimal chemical intuition (\ie, black box models are 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 fulfillment of all these requirements seems elusive, it is useful to keep these criteria in mind. Table \ref{tab:method} is here for fulfill such a purpose.
%%% TABLE I %%%
%\begin{squeezetable}
\begin{table}
\caption{Formal computational scaling of various excited-state methods with respect to the number of one-electron basis functions $N$ and the availability of various key properties.
The typical error range or estimate for single excitations is also provided as a rough indicator of the method accuracy.}
\caption{Formal computational scaling of various excited-state methods with respect to the number of one-electron basis functions $N$ and the accessibility of various key properties in widely available codes.
The typical error range of estimate for single excitations is also provided as a very rough indicator of the method accuracy.}
\label{tab:method}
\begin{ruledtabular}
\begin{tabular}{lcccc}
@ -117,25 +116,25 @@ The typical error range or estimate for single excitations is also provided as a
& scaling & strength & gradients & error (eV) \\
\hline
TD-DFT & $N^4$ & \cmark & \cmark & $0.2$--$0.4$ \\
BSE@GW & $N^4$ & \cmark & \xmark & $0.1$--$0.3$ \\
BSE@\emph{GW} & $N^4$ & \cmark & \xmark & $0.1$--$0.3$ \\
\\
CIS & $N^5$ & \cmark & \cmark & $\sim 1.0$ \\
CIS(D) & $N^5$ & \xmark & \cmark & $0.2$--$0.3$ \\
ADC(2) & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
CC2 & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
\\
ADC(3) & $N^6$ & \cmark & \cmark & $0.1$--$0.2$ \\
EOM-CCSD & $N^6$ & \cmark & \cmark & $\sim 0.10$ \\
ADC(3) & $N^6$ & \cmark & \xmark & $0.1$--$0.2$ \\
EOM-CCSD & $N^6$ & \cmark & \cmark & $0.1$--$0.2$ \\
\\
CC3 & $N^7$ & \cmark & \cmark & $\sim 0.04$ \\
CC3 & $N^7$ & \cmark & \xmark & $\sim 0.04$ \\
\\
EOM-CCSDT & $N^8$ & \xmark & \xmark & $\sim 0.03$ \\
EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark & $\sim 0.01$ \\
EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark & $\sim 0.01$ \\
\\
CASPT2 or NEVPT2 & $N!$ & \cmark & \cmark & $0.1$--$0.2$ \\
SCI & $N!$ & \xmark & \xmark & $\sim 0.03$ \\
CASPT2 or NEVPT2 & $N!$ & \cmark & \cmark & $0.1$--$0.2$ \\
SCI & $N!$ & \xmark & \xmark & $\sim 0.03$ \\
FCI & $N!$ & \cmark & \cmark & $0.0$ \\
\end{tabular}
\end{tabular}
\end{ruledtabular}
\end{table}
%\end{squeezetable}
@ -143,8 +142,8 @@ The typical error range or estimate for single excitations is also provided as a
%**************
%** 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.
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).
Before detailing some key past and present contributions aiming to obtain 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 and the same applies to the analytic gradients when they are available.
%%%%%%%%%%%%%%%%%%%%%
%%% POPLE'S GROUP %%%
@ -153,7 +152,7 @@ The first mainstream \textit{ab initio} method for excited states was probably C
CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states.
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,Ish95}
This second-order correction significantly reduces the magnitude of the error compared to CIS, with a typical error range of $0.2$--$0.3$ eV.
This second-order correction greatly reduces the magnitude of the error compared to CIS, with a typical error range of $0.2$--$0.3$ eV.
%%%%%%%%%%%%%%%%%%%
%%% ROOS' GROUP %%%
@ -161,38 +160,40 @@ This second-order correction significantly reduces the magnitude of the error co
In the early 1990's, the complete-active-space self-consistent field (CASSCF) method \cite{And90} and its second-order perturbation-corrected variant CASPT2 \cite{And92} (both originally developed in Roos' group) appeared.
This was a real breakthrough.
Although it took more than ten years to obtain analytic nuclear gradients, \cite{Cel03} CASPT2 was probably the first method that could provide quantitative results for molecular excited states of genuine photochemical interest. \cite{Roo96}
Nonetheless, it is common knowledge that CASPT2 has the strong tendency of underestimating vertical excitation energies in organic molecules.
Nonetheless, it is common knowledge that CASPT2 has the clear tendency of underestimating vertical excitation energies in organic molecules.
Driven by Angeli and Malrieu, \cite{Ang01} the creation of the second-order $n$-electron valence state perturbation theory (NEVPT2) method several years later was able to cure some of the main theoretical deficiencies of CASPT2.
For example, NEVPT2 is known to be intruder state free and size consistent.
The limited applicability of these so-called multiconfigurational methods is mainly due to the necessity of carefully defining an active space based on the desired transition(s) in order to obtain meaningful results, as well as their factorial computational growth with the number of active electrons and orbitals.
The limited applicability of these multiconfigurational methods is mainly due to the need of carefully defining an active space based on the desired transition(s) in order to obtain meaningful results, as well as their factorial computational growth with the number of active electrons and orbitals.
We also point out that some emergent approaches, like DMRG (density matrix renormalization group), \cite{Bai19} also offer a new path of development for these multiconfigurational theories.
With a typical minimal valence active space tailored for the desired transitions, the usual error in CASPT2 or NEVPT2 calculations is $0.1$--$0.2$ eV.
%%%%%%%%%%%%%
%%% TDDFT %%%
%%%%%%%%%%%%%
The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Cas95} was a real shock for the community as TD-DFT was able to provide accurate excitation energies at a much lower cost than its predecessors in a very black-box way.
For low-lying excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$--$0.4$ eV.
For low-lying valence excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$--$0.4$ eV.
However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Mai04,Dre05,Lev06,Eli11}
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}
One of the main related issues is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals: \cite{Goe19,Sue19} it is difficult to
select a functional adequate for all families of transitions, \cite{Lau13} despite the development of new more robust approaches, including the so-called double-hybrids. \cite{Goe10a,Bre16,Sch17}
Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
One of the main issues is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals. \cite{Goe19,Sue19}
Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community (and beyond).
Despite all of this, TD-DFT remains nowadays the most employed excited-state method in the electronic structure community (and beyond).
%%%%%%%%%%%%%%%%%%
%%% CC METHODS %%%
%%%%%%%%%%%%%%%%%%
Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the huge growth of computational resources, equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) \cite{Sta93} became mainstream in the 2000's.
EOM-CCSD gradients were also quickly available. \cite{Sta95}
With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV, and a typical overestimation of the vertical transition energies.
Its third-order version, EOM-CCSDT, was also implemented and provides, at a significantly higher cost, high accuracy for single excitations. \cite{Nog87}
With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV for small compounds and generally $0.2$ eV for larger ones, with a typical overestimation of the vertical transition energies.
Its third-order version, EOM-CCSDT, was also implemented and provides, at a significantly higher cost, high accuracy for single excitations. \cite{Kuc01}
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 (where $N$ is the number of basis functions), CC3 has a particularly interesting accuracy/cost ratio with errors usually below the chemical accuracy threshold.
The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively.
Thanks to the introduction of triples, EOM-CCSDT and CC3 also provide qualitative results for double excitations, a feature that is completely absent from EOM-CCSD and CC2. \cite{Loo19c}
For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Perspective} keeping in mind that these CC methods are applied to excited states in the present context.
The CC family of methods was quickly followed by an approximated and computationally lighter family with, in front line, the second-order CC2 model \cite{Chr95} and its third-order extension, CC3. \cite{Chr95b}
As a $N^7$ method (where $N$ is the number of basis functions), CC3 has a particularly interesting accuracy/cost ratio with errors usually below the chemical accuracy threshold. \cite{Hat05c,Loo18a,Loo18b,Loo19a}
The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively.
It is also noteworthy that CCSDT and CC3 are also able to pinpoint the presence of double excitations, a feature that is absent from both CCSD and CC2. \cite{Loo19c}
%%%%%%%%%%%%%%%%%%%
%%% ADC METHODS %%%
%%%%%%%%%%%%%%%%%%%
@ -200,14 +201,14 @@ It is also important to mention the recent rejuvenation of the second- and third
This renaissance was certainly initiated by the enormous amount of work invested by Dreuw's group in order to provide a fast and efficient implementation of these methods, \cite{Dre15} including the analytical gradients, \cite{Reh19} as well as other interesting variants.
These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
In that regard, ADC(2) is particularly attractive with an error around $0.1$--$0.2$ eV.
However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies \cite{Loo18a,Loo20} (see also Ref.~\onlinecite{Tro02}).
However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies and is less accurate than CC3 \cite{Tro02,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 serious candidate as a computationally inexpensive electronic structure theory method that can model accurately excited states (with a typical error of $0.1$--$0.3$ eV) and their corresponding properties. \cite{Jac17b,Bla18}
One of the main advantage of BSE compared to TD-DFT (with a similar computational cost) is that it allows a faithful description of charge-transfer states and, when performed on top of a (partially) self-consistently GW calculation, BSE@GW has been shown to be weakly dependent on its starting point. \cite{Jac16,Gui18}
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 \emph{GW} calculation \cite{Hed65}).
BSE has gained momentum in the past few years and is a serious candidate as a computationally inexpensive electronic structure theory method that can model accurately excited states (with a typical error of $0.1$--$0.3$ eV) and some of the 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 \emph{GW} calculation, BSE@\emph{GW} has been shown to be weakly dependent on its starting point, that is on the functional selected for the underlying DFT calculation. \cite{Jac16,Gui18}
However, due to the adiabatic (\ie, static) approximation, doubly excited states are completely absent from the BSE spectrum.
%%%%%%%%%%%%%%%%%%%
@ -218,19 +219,22 @@ SCI methods rely on the same principle as the usual CI approach, except that det
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's complexity.
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 deliver 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 method is only applicable to molecules with a small number of heavy atoms with relatively compact basis sets.
One of the strength of one of the 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 deliver 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 originating from the extrapolation procedure. \cite{Gar19}
However, although the \textit{``exponential wall''} is pushed back, this type of method is only applicable to molecules with a small number of heavy atoms and/or relatively compact basis sets.
%DJ to T2: \hl{ordonner tjrs les methodes pareil + ajouter ADC(3). J'ai les valeurs dispos, je te renverrais cela}
%%%%%%%%%%%%%%%
%%% SUMMARY %%%
%%% 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 excited states at an affordable cost, 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, balanced, and reliable excitation energies for all classes of electronic excited states at an affordable cost.
%%% FIG 1 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{Set1}
\caption{Mean absolute error (in eV) with respect to the TBE/\emph{aug}-cc-pVTZ values from the {\SetA} set (as described in Ref.~\onlinecite{Loo18a}) for various methods and types of excited states.}
\caption{Mean absolute error (in eV) with respect to the TBE/\emph{aug}-cc-pVTZ values from the {\SetA} set (as described in Ref.~\onlinecite{Loo18a}) for various methods and types of excited states.
}
\label{fig:Set1}
\end{figure*}
%%% %%% %%%
@ -238,8 +242,8 @@ 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) 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 (calculated at the CC3 level) in the transition.}
\caption{Mean absolute error (in eV) (with respect to FCI 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 single excitation percentage in the transition calculated at the CC3 level.}
\label{fig:Set2}
\end{figure}
%%% %%% %%%
@ -257,18 +261,18 @@ In summary, each method has its own strengths and weaknesses, and none of them i
%*****************
Although sometimes decried, benchmark sets of molecules and their corresponding reference data are definitely essential for the validation of existing theoretical models and to bring to light and subsequently understand their strengths and, more importantly, their limitations.
These sets have started to emerge at the end of the 1990's for ground-state properties with the acclaimed G2 test set designed by the Pople group. \cite{Cur97}
For excited states, things started moving a little later but some major contributions were able to put things back on track pretty quickly.
For excited states, things started moving a little later but some major contributions were able to put things back on track.
%%%%%%%%%%%%%%%%%%%
%%% 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, diverse and accurate enough to be used as a proper benchmarking set for excited-state methods.
More specifically, it gathers a large number of excitation energies consisting of 28 medium-size organic molecules with a total of 223 excited states (152 singlet and 71 triplet states).
More specifically, it gathers a large number of excitation energies consisting of 28 medium-size organic molecules with a total of 223 valence excited states (152 singlet and 71 triplet states).
In their first study they performed CC2, CCSD, CC3 and CASPT2 calculations (with the TZVP basis) in order to provide (based on additional high-level literature data) a list of best theoretical estimates (TBEs) for all these transitions.
Their main conclusion was that \textit{``CC3 and CASPT2 excitation energies are in excellent agreement for states which are dominated by single excitations''}.
These TBEs were quickly refined with the larger \emph{aug}-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions (especially for Rydberg states).
As a direct evidence of the actual value of reference data, these TBEs were quickly picked up to benchmark various computationally effective methods from semi-empirical to state-of-the-art \textit{ab initio} methods (see Ref.~\onlinecite{Loo18a} and references therein).
These TBEs were quickly refined with the larger \emph{aug}-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions.
As a direct evidence of the actual value of reference data, these TBEs were quickly picked up to benchmark various computationally effective methods from semi-empirical to state-of-the-art \textit{ab initio} methods (see the Introduction of Ref.~\onlinecite{Loo18a} and references therein).
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.
@ -282,37 +286,20 @@ These two studies clearly demonstrate and motivate the need for higher accuracy
%%% JACQUEMIN'S SET %%%
%%%%%%%%%%%%%%%%%%%%%%%
Recently, we made, what we think, is a significant contribution to this quest for highly accurate excitation energies. \cite{Loo18a}
More specifically, we studied 18 small molecules with sizes ranging from one to three non-hydrogen atoms.
More specifically, we studied 18 small molecules (1 to 3 non-hydrogen atoms) with sizes ranging from one to three non-hydrogen atoms.
For such systems, using a combination of high-order CC methods, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various natures (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly excited) based on accurate CC3/\emph{aug}-cc-pVTZ geometries.
In the following, we label this set of TBEs as {\SetA}.
Importantly, it allowed us to benchmark a series of 12 popular excited-state wave function methods accounting for double and triple excitations (see Fig.~\ref{fig:Set1}): CIS(D), ADC(2), CC2, STEOM-CCSD, \cite{Noo97} CCSD, CCSDR(3), \cite{Chr77} and CCSDT-3. \cite{Wat96}
Our main conclusion was that CC3 is extremely accurate (with a mean absolute error of only $\sim 0.03$ eV), and that, although less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies.
Quite surprisingly, ADC(3) was found to deliver quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version ADC(2).
Quite surprisingly, ADC(3) was found to have a clear tendency to overcorrect its second-order version ADC(2).
In a second study, \cite{Loo19c} using a similar combination of theoretical models (but mostly extrapolated SCI energies), we provided accurate reference excitation energies for transitions involving a substantial amount of double excitations using a series of increasingly large diffuse-containing atomic basis sets (up to \emph{aug}-cc-pVQZ when technically feasible).
Our set gathers 20 vertical transitions from 14 small- and medium-sized molecules (acrolein, benzene, beryllium atom, butadiene, carbon dimer and trimer, ethylene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine), a set we label as {\SetB} in the remaining of this paper.
This set gathers 20 vertical transitions from 14 small- and medium-sized molecules, a set we label as {\SetB} in the remaining of this \emph{Perspective}.
An important addition to this second study was the computation of double excitations with various flavors of multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative or iterative triple corrections (see Fig.~\ref{fig:Set2}).
Our results clearly evidence that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition.
For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and CCSDT can easily reach $1$ and $0.5$ eV, respectively, while it goes down to a few tenths of an electronvolt for more common transitions (such as in \emph{trans}-butadiene and benzene) involving a significant amount of singles.\cite{Shu17,Bar18b,Bar18a}
The quality of the excitation energies obtained with multiconfigurational methods was harder to predict as the overall accuracy of these methods is highly dependent on both the system and the selected active space.
Interestingly, CASPT2 and NEVPT2 were found to be more accurate for transition with a small percentage of single excitations (error usually below $0.1$ eV) than for excitations dominated by single excitations where the error is more around $0.1$--$0.2$ eV (see Fig.~\ref{fig:Set2}).
In our latest study, \cite{Loo20} in order to provide more general conclusions, we generated highly accurate vertical transition energies for larger compounds with a set composed by 27 molecules encompassing from four to six non-hydrogen atoms for a total of \alert{238} vertical transition energies of various natures.
This set, labeled as {\SetC} and still based on CC3/\emph{aug}-cc-pVTZ geometries, is constituted by a reasonably good balance of singlet, triplet, valence, and Rydberg states.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with up to hundred million determinants), as well as the most robust multiconfigurational method, NEVPT2.
Each approach was applied in combination with diffuse-containing atomic basis sets.
Because the SCI energy converges obviously slower for these larger systems, the extrapolated SCI values were employed as a``safely net'' to demonstrate the overall consistency of our CC-based protocol rather than straight out-of-the-box reference values.
For all the transitions of the {\SetC} set, we reported at least CC3/\emph{aug}-cc-pVQZ transition energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition.
Pursuing our previous benchmarking efforts, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a dominant double excitation character (see Fig.~\ref{fig:Set3}).
These findings were in perfect agreement with our two previous studies. \cite{Loo18a,Loo19c}
This definitely settles down the debate by demonstrating the superiority of CC3 (in terms of accuracy) compared to methods like CCSDT-3 or ADC(3).
Moreover, thanks to the exhaustive and detailed comparisons made in Ref.~\onlinecite{Loo20}, we could safely conclude that CC3 also regularly outperforms CASPT2 and NEVPT2 as long as the corresponding transition does not show any strong multiple excitation character.
Our current efforts are now focussing on expanding and merging these sets to create an \textit{ultimate} test set of highly accurate excitations energies.
In particular, we are currently generating reference excitations energies for radicals as well as more ``exotic'' molecules containing heavier atoms (such as \ce{Cl}, \ce{F}, \ce{P}, and \ce{Si}). \cite{Loo20b}
The combination of these various sets would potentially create a mega-set of more than 400 vertical transition energies for small- and medium-size molecules based on accurate ground-state geometries.
Such a set would definitely be a terrific asset for the entire electronic structure community.
It would surely stimulate further theoretical developments in excited-state methods and provide a fair ground for the assessments of the currently available and under development excited-state models.
Nevertheless, 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}).
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%%% COMPUTERS %%%
@ -320,7 +307,7 @@ It would surely stimulate further theoretical developments in excited-state meth
For someone who has never worked with SCI methods, it might be surprising to see that one is able to compute near-FCI excitation energies for molecules as big as benzene. \cite{Chi18,Loo19c,Loo20}
This is mainly due to some specific choices in terms of implementation as explained below.
Indeed, to keep up with Moore's ``Law'' in the early 2000's, the processor designers had no other choice than to propose multi-core chips to avoid an explosion of the energy requirements.
Increasing the number of floating-point operations per second (flops/s) by doubling the number of CPU cores only requires to double the required energy, while doubling the frequency multiplies the required energy by a factor close to 8.
Increasing the number of floating-point operations per second by doubling the number of CPU cores only requires to double the required energy, while doubling the frequency multiplies the required energy by a factor of $\sim$ 8.
This bifurcation in hardware design implied a \emph{change of paradigm} \cite{Sut05} in the implementation and design of computational algorithms. A large degree of parallelism is now required to benefit from a significant acceleration.
Fifteen years later, the community has made a significant effort to redesign the methods with parallel-friendly algorithms. \cite{Val10,Cle10,Gar17b,Pen16,Kri13,Sce13}
In particular, the change of paradigm to reach FCI accuracy with SCI methods came
@ -333,20 +320,43 @@ Block-Davidson methods can require a large amount of memory, and the recent intr
The next generation of supercomputers is going to generalize the presence of accelerators (graphical processing units, GPUs), leading to a new software crisis.
Fortunately, some authors have already prepared this transition. \cite{Dep11,Kim18,Sny15,Ufi08,Kal17}
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 organic molecules encompassing from four to six non-hydrogen atoms for a total of 223 vertical transition energies of various natures.
This set, labeled as {\SetC} and still based on CC3/\emph{aug}-cc-pVTZ geometries, is constituted by a reasonably good balance of singlet, triplet, valence, and Rydberg states.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with up to hundred million determinants), as well as the most robust multiconfigurational method, NEVPT2.
Each approach was applied in combination with diffuse-containing atomic basis sets.
For all the transitions of the {\SetC} set, we reported at least CC3/\emph{aug}-cc-pVQZ transition energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition.
Pursuing our previous benchmarking efforts, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a dominant double excitation character (see Fig.~\ref{fig:Set3}).
This settles down, at least for now, the debate by demonstrating the superiority of CC3 (in terms of accuracy) compared to methods like CCSDT-3 or ADC(3).
Moreover, thanks to the exhaustive and detailed comparisons made in Ref.~\onlinecite{Loo20}, we could safely conclude that CC3 also regularly outperforms CASPT2 (that underestimates) and NEVPT2 (that overestimates) as long as the corresponding transition does not show any strong multiple excitation character.
Our current efforts are now focussing on expanding and merging these sets to create an complete test set of highly accurate excitations energies.
In particular, we are currently generating reference excitations energies for radicals as well as more ``exotic'' molecules containing heavier atoms (such as \ce{Cl}, \ce{F}, \ce{P}, and \ce{Si}). \cite{Loo20b}
The combination of these various sets would potentially create an ensemble of more than 400 vertical transition energies for small- and medium-size molecules based on accurate ground-state geometries.
Such a set would definitely be a valuable asset for the entire electronic structure community.
It would likely stimulate further theoretical developments in excited-state methods and provide a fair ground for the assessments of the currently available and under development excited-state models.
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%%% Properties
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Besides all previously described works aiming at reaching chemically accurate vertical transition energies, it should be pointed out that an increasing effort is now set to determine highly-trustable excited-state properties as well. This includes first the 0-0 energies, \cite{Die04b,Hat05c,Goe10a,Sen11b,Win13,Fan14b,Loo18b,Loo19a,Loo19b}
that, as mentioned above, as they offer well-grounded comparisons with experiment. However, as these 0-0 energies are not very sensitive to the used geometry, \cite{Sen11b,Win13,Loo19a} they are not very indicative of the quality of the underlying structures. This is why, one can find several sets of excited-state geometries
determined with various wavefunction approaches, \cite{Pag03,Gua13,Bou13,Tun16,Bud17} a few using very refined models, \cite{Gua13,Bud17} as well as evaluations of the accuracy of the gradients at the FC point. \cite{Taj18,Taj19} The interested researchers can also find several investigations proposing sets of
reference oscillator strengths, \cite{Sil10c,Har14,Kan14,Loo18a,Loo20b} but other more complex properties, such as two-photon cross-sections and vibrations, have been mostly determined at lower levels of theory hinting that the story is far from its end.
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%%% CONCLUSION %%%
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As concluding remarks, we would like to highlight once again the major contribution brought by Thiel's group in an effort to define broad and accurate benchmark sets for excited states. \cite{Sch08,Sil08,Sau09,Sil10b,Sil10c}
Following their footsteps, we have recently proposed a larger, even more accurate set of transitions energies for various types of excited states (including double excitations). \cite{Loo18a,Loo19c,Loo20}
As concluding remarks, we would like to highlight once again the major contribution brought by Roos' and Thiel's groups in an effort to define benchmark values for excited states.
Following their footsteps, we have recently proposed a larger, even more accurate set of vertical transitions energies for various types of excited states (including double excitations). \cite{Loo18a,Loo19c,Loo20}
This was made possible thanks to a technological renaissance of SCI methods which can now routinely produce near-FCI excitation energies for small- and medium-size organic molecules. \cite{Chi18,Gar18,Gar19}
We hope that new technological advances will enable us to push further our quest to highly accurate excitation energies in years to come.
We hope that new technological advances will enable us to push further our quest to highly accurate excitation energies, and, importantly, of excited-state properties as well, in years to come.
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%%% ACKNOWLEDGEMENTS %%%
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PFL would like to thank Peter Gill for useful discussions.
He also acknowledges funding from the \textit{``Centre National de la Recherche Scientifique''}.
DJ acknowledges the R\'egion des Pays de la Loire for financial support.
DJ acknowledges the \emph{R\'egion des Pays de la Loire} for financial support.
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%%% BIBLIOGRAPHY %%%