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Cyclopentadiene "^1 B_2 (\pi \rightarrow \pi^*)" 5.79 5.80 5.80 0.02 5.79 0.02
Cyclopentadiene "^3 B_2 (\pi \rightarrow \pi^*)" 3.33 3.33 3.32 0.04 3.29 0.07
Furan "^1A_2(\pi \rightarrow 3s)" 6.26 6.28 6.31 0.05 6.37 0.01
Furan "^3B_2(\pi \rightarrow \pi^*)" 4.28 4.28 4.26 0.04 4.22 0.07
Imidazole "^1A''(\pi \rightarrow 3s)" 5.77 5.77 5.78 0.05 5.96 0.14
Imidazole "^3A'(\pi \rightarrow \pi^*)" 4.83 4.81 4.82 0.07 4.65 0.22
Pyrrole "^1A_2(\pi \rightarrow 3s)" 5.25 5.25 5.23 0.07 5.31 0.01
Pyrrole "^3B_2(\pi \rightarrow \pi^*)" 4.59 4.58 4.54 0.07 4.37 0.23
Thiophene "^1A_1(\pi \rightarrow \pi^*)" 5.79 5.77 5.75 0.08 5.73 0.09
Thiophene "^3B_2(\pi \rightarrow \pi^*)" 3.95 3.94 3.98 0.01 3.99 0.02
Benzene "^1B_{2u}(\pi \rightarrow \pi^*)" 5.13 5.10 5.06 0.09 5.21 0.07
Benzene "^3B_{1u}(\pi \rightarrow \pi^*)" 4.18 4.16 4.28 0.06 4.17 0.07
Cyclopentadienone "^1A_2(n \rightarrow \pi^*)" 3.03 3.03 3.08 0.02 3.13 0.03
Cyclopentadienone "^3B_2(\pi \rightarrow \pi^*)" 2.30 2.32 2.37 0.05 2.10 0.25
Pyrazine "^1B_{3u}(n \rightarrow \pi^*)" 4.28 4.28 4.26 0.09 4.10 0.25
Pyrazine "^3B_{3u}(n \rightarrow \pi^*)" 3.68 3.68 3.70 0.03 3.70 0.01
Tetrazine "^1B_{3u}(n \rightarrow \pi^*)" 2.53 2.54 2.56 0.05 5.07 0.16
Tetrazine "^3B_{3u}(n \rightarrow \pi^*)" 1.87 1.88 1.91 0.03 4.04 0.49
Pyridazine "^1B_1(n \rightarrow \pi^*)" 3.95 3.95 3.97 0.10 3.60 0.43
Pyridazine "^3B_1(n \rightarrow \pi^*)" 3.27 3.26 3.27 0.15 3.46 0.14
Pyridine "^1B_1(n \rightarrow \pi^*)" 5.12 5.10 5.15 0.12 4.90 0.24
Pyridine "^3A_1(\pi \rightarrow \pi^*)" 4.33 4.31 4.42 0.85 3.68 1.05
Pyrimidine "^1B_1(n \rightarrow \pi^*)" 4.58 4.57 4.64 0.11 2.54 0.05
Pyrimidine "^3B_1(n \rightarrow \pi^*)" 4.20 4.20 4.55 0.37 2.18 0.27
Triazine "^1A_1''(n \rightarrow \pi^*)" 4.85 4.84 4.77 0.13 5.12 0.51
Triazine "^3A_2''(n \rightarrow \pi^*)" 4.40 4.40 4.45 0.39 4.73 0.06

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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-09-08 20:46:59 +0200
%% Created for Pierre-Francois Loos at 2020-10-26 13:47:35 +0100
%% Saved with string encoding Unicode (UTF-8)
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Date-Modified = {2020-10-26 13:04:17 +0100},
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Issn = {0009-2614},
Journal = {Chemical Physics Letters},
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Date-Added = {2020-10-26 09:04:40 +0100},
Date-Modified = {2020-10-26 09:04:46 +0100},
Doi = {10.1063/1.455827},
Eprint = {https://doi.org/10.1063/1.455827},
Journal = {The Journal of Chemical Physics},
Number = {7},
Pages = {3700-3703},
Title = {Is coupled cluster singles and doubles (CCSD) more computationally intensive than quadratic configuration interaction (QCISD)?},
Url = {https://doi.org/10.1063/1.455827},
Volume = {90},
Year = {1989},
Bdsk-Url-1 = {https://doi.org/10.1063/1.455827}}
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Eprint = {https://doi.org/10.1063/1.455269},
Journal = {The Journal of Chemical Physics},
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Title = {An efficient reformulation of the closed‐shell coupled cluster single and double excitation (CCSD) equations},
Url = {https://doi.org/10.1063/1.455269},
Volume = {89},
Year = {1988},
Bdsk-Url-1 = {https://doi.org/10.1063/1.455269}}
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Date-Added = {2020-10-26 08:59:18 +0100},
Date-Modified = {2020-10-26 09:03:32 +0100},
Doi = {10.1002/9780470143599},
Editor = {P. C. Hariharan},
Pages = {35},
Publisher = {Wiley Interscience, New York},
Title = {Advances in Chemical Physics},
Volume = {14},
Year = {1969},
Bdsk-Url-1 = {https://doi.org/10.1002/9780470143599}}
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Author = {C. J. Cramer},
Date-Added = {2020-10-26 08:50:25 +0100},
Date-Modified = {2020-10-26 08:50:25 +0100},
Keywords = {qmech},
Publisher = {Wiley},
Title = {Essentials of Computational Chemistry: Theories and Models},
Year = {2004}}
@article{Pitonak_2009,
Abstract = {Abstract Scaled MP3 interaction energies calculated as a sum of MP2/CBS (complete basis set limit) interaction energies and scaled third-order energy contributions obtained in small or medium size basis sets agree very closely with the estimated CCSD(T)/CBS interaction energies for the 22 H-bonded, dispersion-controlled and mixed non-covalent complexes from the S22 data set. Performance of this so-called MP2.5 (third-order scaling factor of 0.5) method has also been tested for 33 nucleic acid base pairs and two stacked conformers of porphine dimer. In all the test cases, performance of the MP2.5 method was shown to be superior to the scaled spin-component MP2 based methods, e.g. SCS--MP2, SCSN--MP2 and SCS(MI)--MP2. In particular, a very balanced treatment of hydrogen-bonded compared to stacked complexes is achieved with MP2.5. The main advantage of the approach is that it employs only a single empirical parameter and is thus biased by two rigorously defined, asymptotically correct ab-initio methods, MP2 and MP3. The method is proposed as an accurate but computationally feasible alternative to CCSD(T) for the computation of the properties of various kinds of non-covalently bound systems.},
Author = {Pito{\v n}{\'a}k, Michal and Neogr{\'a}dy, Pavel and {\v C}ern{\'y}, Ji{\v r}{\'\i} and Grimme, Stefan and Hobza, Pavel},
@ -539,10 +1069,10 @@
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Date-Added = {2020-09-04 09:47:52 +0200},
Date-Modified = {2020-09-04 09:48:02 +0200},
Date-Modified = {2020-10-26 10:05:30 +0100},
Eprint = {2008.11145},
Primaryclass = {physics.chem-ph},
Title = {Note: The performance of CIPSI on the ground state electronic energy of benzene},
Title = {The performance of CIPSI on the ground state electronic energy of benzene},
Year = {2020}}
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Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00476}}
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Archiveprefix = {arXiv},
Author = {Janus J. Eriksen and Tyler A. Anderson and J. Emiliano Deustua and Khaldoon Ghanem and Diptarka Hait and Mark R. Hoffmann and Seunghoon Lee and Daniel S. Levine and Ilias Magoulas and Jun Shen and Norman M. Tubman and K. Birgitta Whaley and Enhua Xu and Yuan Yao and Ning Zhang and Ali Alavi and Garnet Kin-Lic Chan and Martin Head-Gordon and Wenjian Liu and Piotr Piecuch and Sandeep Sharma and Seiichiro L. Ten-no and C. J. Umrigar and J{\"u}rgen Gauss},
Date-Added = {2020-09-03 21:52:47 +0200},
Date-Modified = {2020-09-03 21:52:47 +0200},
Eprint = {2008.02678},
Primaryclass = {physics.chem-ph},
Title = {The Ground State Electronic Energy of Benzene},
Year = {2020}}
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Author = {Tubman, Norm M. and Lee, Joonho and Takeshita, Tyler Y. and {Head-Gordon}, Martin and Whaley, K. Birgitta},
Date-Added = {2020-09-03 21:52:47 +0200},
@ -2705,24 +3225,6 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.37.8351},
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.37.8351}}
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Author = {Liu, Peitao and Kaltak, Merzuk and Klime\ifmmode \check{s}\else \v{s}\fi{}, Ji\ifmmode \check{r}\else \v{r}\fi{}\'{\i} and Kresse, Georg},
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Month = {Oct},
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Pages = {165109},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Editor = {D. P. Chong},
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Pages = {164112},

528
Manuscript/QUEST_WIREs.tex
View File

@ -11,11 +11,18 @@
% Use "num-refs" option for numerical citation and references style.
% Use "alpha-refs" option for author-year citation and references style.
\documentclass[num-refs]{wiley-article}
\documentclass[num-refs,sort&compress]{wiley-article}
% \documentclass[blind,alpha-refs]{wiley-article}
% Add additional packages here if required
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,mhchem,siunitx}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,mhchem,siunitx,rotating}
\usepackage[
colorlinks=true,
citecolor=blue,
breaklinks=true
]{hyperref}
\urlstyle{same}
% macros
\newcommand{\ra}{\rightarrow}
@ -41,7 +48,7 @@
\title{QUESTDB: a database of highly-accurate excitation energies for the electronic structure community}
% List abbreviations here, if any. Please note that it is preferred that abbreviations be defined at the first instance they appear in the text, rather than creating an abbreviations list.
\abbrevs{ABC, a black cat; DEF, doesn't ever fret; GHI, goes home immediately.}
%\abbrevs{ABC, a black cat; DEF, doesn't ever fret; GHI, goes home immediately.}
% Include full author names and degrees, when required by the journal.
% Use the \authfn to add symbols for additional footnotes and present addresses, if any. Usually start with 1 for notes about author contributions; then continuing with 2 etc if any author has a different present address.
@ -75,12 +82,12 @@
\maketitle
\begin{abstract}
We describe our efforts of the past few years to create a mega set of more than \alert{470} highly-accurate vertical excitation energies of various natures ($\pi \to \pis$, $n \to \pis$, double excitation, Rydberg, singlet, doublet, triplet, etc) for small- and medium-sized molecules.
We describe our efforts of the past few years to create a large set of more than \alert{470} highly-accurate vertical excitation energies of various natures ($\pi \to \pis$, $n \to \pis$, double excitation, Rydberg, singlet, doublet, triplet, etc) for small- and medium-sized molecules.
These values have been obtained using a combination of high-order coupled cluster and selected configuration interaction calculations using increasingly large diffuse basis sets.
One of the key aspect of the so-called QUEST dataset of vertical excitations is that it does not rely on any experimental values, avoiding potential biases inherently linked to experiments and facilitating in the process theoretical cross comparisons.
Following this composite protocol, we have been able to produce theoretical best estimate (TBEs) with the aug-cc-pVTZ basis set, as well as basis set corrected TBEs (i.e., near the complete basis set limit) for each of these transitions.
These TBEs have been employed to benchmark a large number of (lower-order) wave function methods such as CIS(D), ADC(2), STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, ADC(3), CC3, CASPT2, NEVPT2, and others.
In order to gather the huge number of data produced during the QUEST project, we have created a website where one can easily test and compare the accuracy of a given method with respect to various variables such as the molecule size or its family, the nature of the excited states, the size of the basis set, and many others.
In order to gather the huge number of data produced during the QUEST project, we have created a website where one can easily test and compare the accuracy of a given method with respect to various variables such as the molecule size or its family, the nature of the excited states, the size of the basis set, etc.
We hope that the present review will provide a useful summary of our work so far and foster new developments around excited-state methods.
% Please include a maximum of seven keywords
\keywords{Excited states, full configuration interaction, excitation energies}
@ -91,15 +98,15 @@ 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).
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{SzaboBook,JensenBook,CramerBook,HelgakerBook}.
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 (that is, lowest-energy) state.
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 analogs.
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 able to 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 with well-defined setup (same geometry, basis set, etc).
These values can be absolute or relative energies, geometrical parameters, physical or chemical properties, extracted from experiments, high-level theoretical calculations, or a combination of both.
These values can be absolute or relative energies, geometrical parameters, physical or chemical properties extracted from experiments, high-level theoretical calculations, or a combination of both.
To do so, the electronic structure community has designed along the years benchmark sets, i.e., sets of molecules for which one could (very) accurately compute theoretical estimates and/or access solid experimental data for given properties.
Regarding ground-states properties, two of the oldest and most employed sets are probably the Gaussian-1 and Gaussian-2 benchmark sets \cite{Pople_1989,Curtiss_1991,Curtiss_1997} developed by the group of Pople in the 1990's.
For example, the Gaussian-2 set gathers atomization energies, ionization energies, electron affinities, proton affinities, bond dissociation energies, and reaction barriers.
@ -107,18 +114,18 @@ This set was subsequently extended and refined \cite{Curtiss_1998,Curtiss_2007}.
Another very useful set for the design of methods able to catch dispersion effects is the S22 benchmark set \cite{Jureka_2006} (and its extended S66 version \cite{Rezac_2011}) of Hobza and collaborators which provides benchmark interaction energies for weakly-interacting (non covalent) systems.
One could also mentioned the $GW$100 set \cite{vanSetten_2015,Krause_2015,Maggio_2016} (and its $GW$5000 extension \cite{Stuke_2020}) of ionization energies which has helped enormously the community to settle on the implementation of $GW$-type methods for molecular systems \cite{vanSetten_2013,Bruneval_2016,Caruso_2016,Govoni_2018}.
The extrapolated ab initio thermochemistry (HEAT) set designed to achieve high accuracy for enthalpies of formation of atoms and small molecules (without experimental data) is yet another successful example of benchmark set \cite{Tajti_2004,Bomble_2006,Harding_2008}.
More recently, the benchmark datasets provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable to the community by providing, for example, highly-accurate ground state energies for hydrogen chains \cite{Motta_2017} and transition metal atoms and their ions and monoxides \cite{Williams_2020}.
Let us also mention the set of Zhao and Truhlar for small transition metal complexes employed to compare the accuracy density-functional methods for $3d$ transition-metal chemistry \cite{Zhao_2006}.
More recently, the benchmark datasets provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable to the community by providing, for example, highly-accurate ground state energies for hydrogen chains \cite{Motta_2017} as well as transition metal atoms and their ions and monoxides \cite{Williams_2020}.
Let us also mention the set of Zhao and Truhlar for small transition metal complexes employed to compare the accuracy density-functional methods \cite{ParrBook} for $3d$ transition-metal chemistry \cite{Zhao_2006}.
The examples presented above are all designed for ground-state properties, and there exists now specific protocols designed to accurately model excited-state energies and properties.
The examples of benchmark sets presented above are all designed for ground-state properties, and there now exists specific protocols taylored to accurately model excited-state energies and properties.
Benchmark datasets of excited-state energies and/or properties are less numerous than their ground-state counterparts but their number have been growing at a consistent pace in the past few years.
Below, we provide a short description for some of them.
One of the most characteristic example is the benchmark set of vertical excitations proposed by Thiel and coworkers \cite{Schreiber_2008,Silva-Junior_2008,Silva-Junior_2010,Silva-Junior_2010b,Silva-Junior_2010c}.
The so-called Thiel (or M\"ulheim) set of excitation energies gathers a large number of excitation energies determined in 28 medium-size organic molecules with a total of 223 valence excited states (152 singlet and 71 triplet states) for which theoretical best estimates (TBEs) were defined.
In their first study, Thiel and collaborators performed CC2 \cite{Christiansen_1995a,Hattig_2000}, CCSD \cite{Purvis_1982}, CC3 \cite{Christiansen_1995b,Koch_1997}, and CASPT2 \cite{Andersson_1990,Andersson_1992,Roos,Roos_1996} calculations (with the TZVP basis) on MP2/6-31G(d) geometries in order to provide (based on additional high-quality literature data) TBEs for these transitions \cite{Silva-Junior_2010b}.
These TBEs were quickly refined with the larger aug-cc-pVTZ basis set, highlighting the importance of diffuse functions in the faithful description of excited states (especially for Rydberg states).
In the same spirit, it is also worth mentioning Gordon's set of vertical transitions (based on experimental values) used to benchmark the performance of time-dependent density-functional theory (TD-DFT) \cite{Leang_2012}, as well as its extended version by Goerigk and coworkers who decided to replace the experimental reference values by CC3 excitation energies instead \cite{Schwabe_2017,Casanova-Paez_2019,Casanova_Paes_2020}.
Let us also mention the new benchmark set of charge-transfer excited states recently introduced by Szalay and coworkers [based on coupled cluster (CC) methods] \cite{Kozma_2020} as well as the Gagliardi-Truhlar set employed to compare the accuracy of multiconfiguration pair-density functional theory \cite{Ghosh_2018} against the well-established CASPT2 method \cite{Hoyer_2016}.
In their first study, Thiel and collaborators performed CC2 \cite{Christiansen_1995a,Hattig_2000}, EOM-CCSD \cite{Koch_1990,Stanton_1993,Koch_1994}, CC3 \cite{Christiansen_1995b,Koch_1997}, and CASPT2 \cite{Andersson_1990,Andersson_1992,Roos,Roos_1996} calculations (with the TZVP basis) on MP2/6-31G(d) geometries in order to provide (based on additional high-quality literature data) TBEs for these transitions.
These TBEs were quickly refined with the larger aug-cc-pVTZ basis set, highlighting the importance of diffuse functions in the faithful description of excited states (especially for Rydberg states) \cite{Silva-Junior_2010b,Silva-Junior_2010c}.
In the same spirit, it is also worth mentioning Gordon's set of vertical transitions (based on experimental values) \cite{Leang_2012} used to benchmark the performance of time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Casida_2012,Ulrich_2012}, as well as its extended version by Goerigk and coworkers who decided to replace the experimental reference values by CC3 excitation energies instead \cite{Schwabe_2017,Casanova-Paez_2019,Casanova_Paes_2020}.
Let us also mention the new benchmark set of charge-transfer excited states recently introduced by Szalay and coworkers [based on equation-of-motion coupled cluster (EOM-CC) methods] \cite{Kozma_2020} as well as the Gagliardi-Truhlar set employed to compare the accuracy of multiconfiguration pair-density functional theory \cite{Ghosh_2018} against the well-established CASPT2 method \cite{Hoyer_2016}.
Following a similar philosophy and striving for chemical accuracy, we have recently reported in several studies highly-accurate vertical excitations for small- and medium-sized molecules \cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}.
@ -130,19 +137,19 @@ In the same vein, we have also produced chemically-accurate theoretical 0-0 ener
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 past two decades.
%%% FIGURE 1 %%%
\begin{figure}[ht]
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{fig1/fig1}
\includegraphics[width=0.6\linewidth]{fig1/fig1}
\caption{Composition of each of the five subsets making up the present QUEST dataset of highly-accurate vertical excitation energies.}
\label{fig:scheme}
\end{figure}
The QUEST dataset has the particularity to be based in a large proportion on selected configuration interaction (SCI) reference excitation energies as well as high-order CC methods such as CCSDT and CCSDTQ \cite{Oliphant_1991,Kucharski_1992}.
Recently, SCI methods have been a force to reckon with for the computation of highly-accurate energies in small- and medium-sized molecules as they yield near full configuration interaction (FCI) quality energies for only a fraction of the computational cost of a genuine FCI calculation \cite{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 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 equation-of-motion (EOM) CC methods such as EOM-CCSDT and EOM-CCSDTQ \cite{Hirata_2000}.
Recently, SCI methods have been a force to reckon with for the computation of highly-accurate energies in small- and medium-sized molecules as they yield near full configuration interaction (FCI) quality energies for only a fraction of the computational cost of a genuine FCI calculation \cite{Booth_2009,Booth_2010,Cleland_2010,Booth_2011,Daday_2012,Blunt_2015,Ghanem_2019,Deustua_2017,Deustua_2018,Holmes_2017,Chien_2018,Li_2018,Yao_2020,Li_2020,Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b,Xu_2018,Xu_2020,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020a,Loos_2020e}.
Due to the fairly natural idea underlying these methods, the SCI family is composed by numerous members \cite{Bender_1969,Whitten_1969,Huron_1973,Abrams_2005,Bunge_2006,Bytautas_2009,Giner_2013,Caffarel_2014,Giner_2015,Garniron_2017b,Caffarel_2016a,Caffarel_2016b,Holmes_2016,Sharma_2017,Holmes_2017,Chien_2018,Scemama_2018,Scemama_2018b,Garniron_2018,Evangelista_2014,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.
Their fundamental philosophy consists, roughly speaking, in retaining only the most energetically relevant determinants of the FCI space following a given criterion to slow down the exponential increase of the size of the CI expansion.
Originally developed in the late 1960's by Bender and Davidson \cite{Bender_1969} as well as Whitten and Hackmeyer \cite{Whitten_1969}, new efficient SCI algorithms have resurfaced recently.
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,Giner_2013,Giner_2015,Garniron_2019}.
Four examples are adaptive sampling CI (ASCI) \cite{Tubman_2016,Tubman_2018,Tubman_2020}, iCI \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020}, semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017,Li_2018,Li_2020,Yao_2020}), and \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2019}.
These four flavors of SCI include a second-order perturbative (PT2) correction which is key to estimate the ``distance'' to the FCI solution (see below).
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}).
@ -151,9 +158,9 @@ CIPSI is also frequently employed to provide accurate trial wave functions for q
We refer the interested reader to Ref.~\cite{Garniron_2019} where one can find additional details regarding the implementation of the CIPSI algorithm.
The present article is organized as follows.
In Sec.~\ref{sec:tools}, we detail the specificities of our protocol by providing computational details regarding geometries, basis sets, (reference and benchmarked) computational methods, and a new way of estimating rigorously the extrapolation error in SCI calculations.
In Sec.~\ref{sec:tools}, we detail the specificities of our protocol by providing computational details regarding geometries, basis sets, (reference and benchmarked) computational methods, and a new way of estimating rigorously the extrapolation error in SCI calculations which is tested by computing additional FCI values for five- and six-membered rings.
We then describe in Sec.~\ref{sec:QUEST} the content of our five QUEST subsets providing for each of them the number of reference excitation energies, the nature and size of the molecules, the list of benchmarked methods, as well as other specificities.
A special emphasis is placed on our latest add-on, QUEST\#5, specifically designed for the present manuscript where we have considered, in particular but not only, larger molecules as well as additional FCI values for five- and six-membered rings.
A special emphasis is placed on our latest add-on, QUEST\#5, specifically designed for the present manuscript where we have considered, in particular but not only, larger molecules.
Section \ref{sec:TBE} discusses the generation of the TBEs, while Sec.~\ref{sec:bench} proposes a comprehensive benchmark of various methods on the entire QUEST set which is composed by more than \alert{470} excitations with, in addition, a specific analysis for each type of excited states.
Section \ref{sec:website} describes the feature of the website that we have specifically designed to gather the entire data generated during these last few years.
Thanks to this website, one can easily test and compare the accuracy of a given method with respect to various variables such as the molecule size or its family, the nature of the excited states, the size of the basis set, etc.
@ -170,15 +177,16 @@ Finally, we draw our conclusions in Sec.~\ref{sec:ccl} where we discuss, in part
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 \cite{Loos_2020c}, the geometries are optimized at the UCCSD(T)/aug-cc-pVTZ level using the GAUSSIAN16 program \cite{Gaussian16} and applying the ``tight'' convergence threshold.
For the open-shell derivatives beloging to QUEST\#4 \cite{Loos_2020c}, the geometries are optimized at the UCCSD(T)/aug-cc-pVTZ level using the GAUSSIAN16 program \cite{Gaussian16} and applying the ``tight'' convergence threshold.
For the purpose of the present review article, we have gathered all the geometries in the {\SupInf}.
%\footnote{These geometries can be found at...}
%=======================
\subsection{Basis sets}
%=======================
For the entire set, we rely on the 6-31+G(d) Pople basis set, the augmented family of Dunning basis sets aug-cc-pVXZ (where X $=$ D, T, Q, and 5), and sometimes its doubly- and triply-augmented variants, d-aug-cc-pVXZ and t-aug-cc-pVXZ respectively.
For the entire set, we rely on the 6-31+G(d) Pople basis set \cite{Binkley_1977a,Clark_1983a,Dill_1975a,Ditchfield_1971a,Francl_1982a,Gordon_1982a,Hehre_1972a}, the augmented family of Dunning basis sets aug-cc-pVXZ (where X $=$ D, T, Q, and 5) \cite{Dunning_1989a,Kendall_1992a,Prascher_2011a,Woon_1993a,Woon_1994a}, and sometimes its doubly- and triply-augmented variants, d-aug-cc-pVXZ and t-aug-cc-pVXZ respectively.
Doubly- and triply-augmented basis sets are usually employed for Rydberg states where it is not uncommon to observe a strong basis set dependence due to the very diffuse nature of these excited states.
These basis sets have been downloaded from the \href{https://www.basissetexchange.org}{basis set exchange} website \cite{Feller_1996a,Pritchard_2019a,Schuchardt_2007a}.
%==================================
\subsection{Computational methods}
@ -190,17 +198,18 @@ Doubly- and triply-augmented basis sets are usually employed for Rydberg states
In order to compute reference vertical energies, we have designed different strategies depending on the actual nature of the transition and the size of the system.
For small molecules (typically 1--3 non-hydrogen atoms), we resort to SCI methods which can provide near-FCI excitation energies for compact basis sets.
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.
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 EOM-CC theory (EOM-CCSDT and EOM-CCSDTQ typically \cite{Hirata_2000}) to obtain accurate transition energies.
In the following, we will omit the prefix EOM for the sake of clarity.
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 the same excited-state energies.
The reported oscillator strengths have been computed in the LR-CC3 formalism only.
%Note that all our excited-state CC calculations are performed within the equation-of-motion (EOM) or linear-response (LR) formalism that yield the same excited-state energies.
The reported oscillator strengths have been computed in the linear-response (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 wave function as reference.
All excited-state calculations are performed, except when explicitly mentioned, in the frozen-core (FC) approximation using large cores for the third-row atoms.
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.~\cite{Garniron_2019} and \cite{Scemama_2019}.
All the SCI calculations are performed within the frozen-core 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.~\cite{Garniron_2019} and \cite{Scemama_2019}.
A state-averaged formalism is employed, i.e., the ground and excited states are described with the same set of determinants, but different CI coefficients.
Our usual protocol \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} consists of performing a preliminary CIPSI calculation using Hartree-Fock orbitals in order to generate a CIPSI wave function with at least $10^7$ determinants.
Natural orbitals are then computed based on this wave function, and a new, larger CIPSI calculation is performed with this new set of orbitals.
@ -217,7 +226,6 @@ Moreover, a renormalized version of the PT2 correction (dubbed rPT2) has been re
We refer the interested reader to Ref.~\cite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm.
Note that, all our SCI wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator which is, unlike ground-state calculations, paramount in the case of excited states \cite{Applencourt_2018}.
%------------------------------------------------
\subsubsection{Benchmarked computational methods}
%------------------------------------------------
@ -227,7 +235,7 @@ For example, we rely on GAUSSIAN \cite{Gaussian16} and TURBOMOLE 7.3 \cite{Turbo
Q-CHEM 5.2 \cite{Krylov_2013} for EOM-MP2 [CCSD(2)] \cite{Stanton_1995c} and ADC(3) \cite{Trofimov_2002,Harbach_2014,Dreuw_2015};
Q-CHEM \cite{Krylov_2013} and TURBOMOLE \cite{Turbomole} for ADC(2) \cite{Trofimov_1997,Dreuw_2015};
DALTON \cite{dalton} and TURBOMOLE \cite{Turbomole} for CC2 \cite{Christiansen_1995a,Hattig_2000};
DALTON \cite{dalton} and GAUSSIAN \cite{Gaussian16} for CCSD \cite{Purvis_1982};
DALTON \cite{dalton} and GAUSSIAN \cite{Gaussian16} for CCSD \cite{Koch_1990,Stanton_1993,Koch_1994};
DALTON \cite{dalton} for CCSDR(3) \cite{Christiansen_1996b};
CFOUR \cite{cfour} for CCSDT-3 \cite{Watts_1996b,Prochnow_2010};
and ORCA \cite{Neese_2012} for similarity-transformed EOM-CCSD (STEOM-CCSD) \cite{Nooijen_1997,Dutta_2018}.
@ -235,17 +243,17 @@ In addition, we evaluate the spin-opposite scaling (SOS) variants of ADC(2), SOS
Note that these two codes have distinct SOS implementations, as explained in Ref.~\cite{Krauter_2013}.
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.
%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.
%When comparisons between various codes/implementations were possible, we could not detect variations in the transition energies larger than $0.01$ eV.
For radicals, 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.
Finally, the composite approach, ADC(2.5), which follows the spirit of Grimme's and Hobza's MP2.5 approach \cite{Pitonak_2009} by averaging the ADC(2) and ADC(3) excitation energies, is also tested in the following \cite{Loos_2020d}.
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 tested \cite{Angeli_2001a,Angeli_2001b,Angeli_2002}.
Concerning the NEVPT2 calculations (which are also performed with MOLPRO), the partially-contracted (PC) and strongly-contracted (SC) variants have been 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.
In the case of double excitations \cite{Loos_2019}, 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} when there is a strong mixing between states with same spin and spatial symmetries.
The 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, unless otherwise stated.
The 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.
@ -253,6 +261,8 @@ The definition of the active space considered for each system as well as the num
\subsubsection{Estimating the extrapolation error}
\label{sec:error}
%------------------------------------------------
In this section, we present our scheme to estimate the extrapolation error in SCI calculations.
This new protocol is then applied to five- and six-membered ring molecules where SCI calculations are particularly challenging even for small basis sets.
For the $m$th excited state (where $m = 0$ corresponds to the ground state), we usually estimate its FCI energy $E_{\text{FCI}}^{(m)}$ by performing a linear extrapolation of its variational energy $E_\text{var}^{(m)}$ as a function of its rPT2 correction $E_{\text{rPT2}}^{(m)}$ as follows
\begin{equation}
@ -307,6 +317,78 @@ Only the last $M>2$ computed energy differences are considered. $M$ is chosen su
If all the values of $P(\mathcal{G})$ are below $0.8$, $M$ is chosen such that $P(\mathcal{G})$ is maximal.
A Python code associated with this procedure is provided in the {\SupInf}.
The singlet and triplet excitation energies obtained at the FCI/6-31+G(d) level are reported in Table \ref{tab:cycles} alongside the computed error bars estimated with the method presented above based on Gaussian random variables.
For the sake of comparison, we also report the CC3 and CCSDT vertical energies from Ref.~\cite{Loos_2020b} computed in the same basis.
The estimated values of the excitation energies obtained via a three-point linear extrapolation considering the three largest CIPSI wave functions are also gathered in Table \ref{tab:cycles}.
In this case, the error bar is estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
This strategy has been considered in some of our previous works \cite{Loos_2020b,Loos_2020c,Loos_2020e}.
The deviation from the CCSDT excitation energies for the same set of excitations are depicted in Fig.~\ref{fig:errors}, where the red dots correspond to the excitation energies and error bars estimated via the present method, and the blue dots correspond to the excitation energies obtained via a three-point linear fit using the three largest CIPSI wave functions, and error bars estimated via the extrapolation distance.
These results are a good balance between well-behaved and ill-behaved cases.
For example, cyclopentadiene and furan correspond to well-behaved cases where the two flavor of excitation energy estimates are nearly identical and the error bars associated with these two methods overlap nicely.
In these cases, one can observe that our method based on Gaussian random variables provides almost systematically smaller error bars.
Even in less idealistic situations (like in imidazole, pyrrole, and thiophene), the results are very satisfactory.
The six-membered rings correspond to much more challenging cases for SCI methods, and even for these systems the newly-developed method provides realistic error bars.
The present scheme has also been tested on much smaller systems when one can easily tightly converged the CIPSI calculations.
In these cases, the agreement is nearly perfect in every cases.
Some of these results can be found in the {\SupInf}.
%%% TABLE I %%%
\begin{table}
\centering
\caption{Singlet and triplet excitation energies (in eV) obtained at the CC3, CCSDT, and FCI levels of theory with the 6-31+G* basis set for various five- and six-membered rings.
The error bars reported in parenthesis correspond to one standard deviation.}
\label{tab:cycles}
\begin{threeparttable}
\begin{tabular}{lccccc}
\headrow
\thead{Molecule} & \thead{Transition} & \thead{CC3} & \thead{CCSDT} & \thead{FCI$^a$} & \thead{FCI$^b$}\\
\mc{6}{c}{Five-membered rings} \\
Cyclopentadiene & $^1 B_2 (\pi \ra \pis)$ & 5.79 & 5.80 & 5.80(2) & 5.79(2) \\
& $^3 B_2 (\pi \ra \pis)$ & 3.33 & 3.33 & 3.32(4) & 3.29(7) \\
Furan & $^1A_2(\pi \ra 3s)$ & 6.26 & 6.28 & 6.31(5) & 6.37(1) \\
& $^3B_2(\pi \ra \pis)$ & 4.28 & 4.28 & 4.26(4) & 4.22(7) \\
Imidazole & $^1A''(\pi \ra 3s)$ & 5.77 & 5.77 & 5.78(5) & 5.96(14) \\
& $^3A'(\pi \ra \pis)$ & 4.83 & 4.81 & 4.82(7) & 4.65(22) \\
Pyrrole & $^1A_2(\pi \ra 3s)$ & 5.25 & 5.25 & 5.23(7) & 5.31(1) \\
& $^3B_2(\pi \ra \pis)$ & 4.59 & 4.58 & 4.54(7) & 4.37(23) \\
Thiophene & $^1A_1(\pi \ra \pis)$ & 5.79 & 5.77 & 5.75(8) & 5.73(9) \\
& $^3B_2(\pi \ra \pis)$ & 3.95 & 3.94 & 3.98(1) & 3.99(2) \\
\mc{6}{c}{Six-membered rings} \\
Benzene & $^1B_{2u}(\pi \ra \pis)$ & 5.13 & 5.10 & 5.06(9) & 5.21(7) \\
& $^3B_{1u}(\pi \ra \pis)$ & 4.18 & 4.16 & 4.28(6) & 4.17(7) \\
Cyclopentadienone & $^1A_2(n \ra \pis)$ & 3.03 & 3.03 & 3.08(2) & 3.13(3) \\
& $^3B_2(\pi \ra \pis)$ & 2.30 & 2.32 & 2.37(5) & 2.10(25) \\
Pyrazine & $^1B_{3u}(n \ra \pis)$ & 4.28 & 4.28 & 4.26(9) & 4.10(25) \\
& $^3B_{3u}(n \ra \pis)$ & 3.68 & 3.68 & 3.70(3) & 3.70(1) \\
Tetrazine & $^1B_{3u}(n \ra \pis)$ & 2.53 & 2.54 & 2.56(5) & 5.07(16) \\
& $^3B_{3u}(n \ra \pis)$ & 1.87 & 1.88 & 1.91(3) & 4.04(49) \\
Pyridazine & $^1B_1(n \ra \pis)$ & 3.95 & 3.95 & 3.97(10)& 3.60(43) \\
& $^3B_1(n \ra \pis)$ & 3.27 & 3.26 & 3.27(15)& 3.46(14) \\
Pyridine & $^1B_1(n \ra \pis)$ & 5.12 & 5.10 & 5.15(12)& 4.90(24) \\
& $^3A_1(\pi \ra \pis)$ & 4.33 & 4.31 & 4.42(85)& 3.68(1.05) \\
Pyrimidine & $^1B_1(n \ra \pis)$ & 4.58 & 4.57 & 4.64(11)& 2.54(5) \\
& $^3B_1(n \ra \pis)$ & 4.20 & 4.20 & 4.55(37)& 2.18(27) \\
Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\
& $^3A_2''(n \ra \pis)$ & 4.40 & 4.40 & 4.45(39)& 4.73(6) \\
%\hiderowcolors
\hline % Please only put a hline at the end of the table
\end{tabular}
\begin{tablenotes}
\item $^a$ Excitation energies and error bars estimated via the present method (see Sec.~\ref{sec:error}).
\item $^b$ Excitation energies obtained via a three-point linear fit using the three largest CIPSI variational wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
\end{tablenotes}
\end{threeparttable}
\end{table}
%%% FIGURE 2 %%%
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{errors}
\caption{Deviation from the CCSDT excitation energies for singlet and triplet excitation energies (in eV) of five- and six-membered rings obtained at the FCI/6-31+G* level of theory. Red dots: excitation energies and error bars estimated via the present method (see Sec.~\ref{sec:error}). Blue dots: excitation energies obtained via a three-point linear fit using the three largest CIPSI wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
The error bars corresponds to one standard deviation.}
\label{fig:errors}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The QUEST database}
\label{sec:QUEST}
@ -320,7 +402,7 @@ Each of the five subsets making up the QUEST dataset is detailed below.
Throughout the present article, we report several statistical indicators: the mean signed error (MSE), mean absolute error (MAE), root-mean square error (RMSE), and standard deviation of the errors (SDE).
%%% FIGURE 2 %%%
\begin{figure}[ht]
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{fig2}
\caption{Molecules each of the five subsets making up the present QUEST dataset of highly-accurate vertical excitation energies:
@ -333,7 +415,7 @@ Throughout the present article, we report several statistical indicators: the me
%=======================
The QUEST\#1 benchmark set \cite{Loos_2018a} consists of 110 vertical excitation energies (as well as oscillator strengths) from 18 molecules with sizes ranging from one to three non-hydrogen atoms (water, hydrogen sulfide, ammonia, hydrogen chloride, dinitrogen, carbon monoxide, acetylene, ethylene, formaldehyde, methanimine, thioformaldehyde, acetaldehyde, cyclopropene, diazomethane, formamide, ketene, nitrosomethane, and the smallest
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 former set, we systematically employed 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-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.
@ -379,207 +461,195 @@ Likewise, the excitation energies obtained with CCSD are much less satisfying fo
%=======================
The QUEST\#5 subset is composed by additional accurate excitation energies that we have produced for the present article.
This new set gathers small molecules as well as larger molecules (aza-naphthalene, benzoquinone, cyclopentadienone, cyclopentadienethione, hexatriene, maleimide, naphthalene, nitroxyl, streptocyanine-C3, streptocyanine-C5, and thioacrolein).
Each of these molecules are discussed below and comparisons are made with literature data.
QUEST\#5 does also provide additional FCI/6-31+G* estimates of the lowest singlet and triplet transitions for the five- and six-membered rings considered in QUEST\#3.
The extrapolation errors for these quite challenging calculations are computed with the scheme presented in Sec.~\ref{sec:error}.
This new set gathers 13 new systems composed by small molecules as well as larger molecules (aza-naphthalene, benzoquinone, cyclopentadienone, cyclopentadienethione, diazirine, hexatriene, maleimide, naphthalene, nitroxyl, octatetraene, streptocyanine-C3, streptocyanine-C5, and thioacrolein).
For these new transitions, we report at least CCSDT/aug-cc-pVTZ vertical energies.
The interested reader will find in the {\SupInf} a detailed discussion for each of these molecules in which comparisons are made with literature data.
%--------------------------------------
\subsubsection{Toward larger molecules}
%--------------------------------------
\alert{Here comes Denis' discussion of each new molecule.}
\begin{table}[bt]
\centering
\caption{Singlet and triplet excitation energies of various molecules obtained at the CC3, CCSDT, NEVPT2, and FCI levels of theory.}
\begin{threeparttable}
\begin{tabular}{lccrrr}
\headrow
& & \mc{4}{c}{6-31+G*} \\
\thead{Molecule} & \thead{Transition} & \thead{CC3} & \thead{CCSDT} & \thead{NEVPT2} & \thead{FCI}\\
Aza-naphthalene
& $^1B_{3g}(n \ra \pis)$ \\
& $^1B_{2u}(\pi \ra \pis)$ \\
& $^1B_{1u}(n \ra \pis)$ \\
& $^1B_{2g}(n \ra \pis)$ \\
& $^1B_{2g}(n \ra \pis)$ \\
& $^1B_{1u}(n \ra \pis)$ \\
& $^1A_u(n \ra \pis)$ \\
& $^1B_{3u}(\pi \ra \pis)$ \\
& $^1A_g(\pi \ra \pis)$ \\
& $^1A_u(n \ra \pis)$ \\
& $^1A_g(n \ra 3s)$ \\
& $^3B_{3g}(n \ra \pis)$ \\
& $^3B_{2u}(\pi \ra \pis)$ \\
& $^3B_{3u}(\pi \ra \pis)$ \\
& $^3B_{1u}(n \ra \pis)$ \\
& $^3B_{2g}(n \ra \pis)$ \\
& $^3B_{2g}(n \ra \pis)$ \\
& $^3B_{3u}(\pi \ra \pis)$ \\
& $^3A_u(n \ra \pis)$ \\
Benzoquinone
& $^1 B_{1g}(n \ra \pis)$ & & & & \\
& $^1 A_{u}(n \ra \pis)$ & & & & \\
& $^1 A_{g}(\double)$ & & & & \\
& $^1 B_{3g}(\pi \ra \pis)$ & & & & \\
& $^1 B_{3u}(n \ra \pis)$ & & & & \\
& $^1 B_{2g}(n \ra \pis)$ & & & & \\
& $^1 A_{u}(n \ra \pis)$ & & & & \\
& $^1 B_{1g}(n \ra \pis)$ & & & & \\
& $^1 B_{2g}(n \ra \pis)$ & & & & \\
& $^3 B_{1g}(n \ra \pis)$ & & & & \\
& $^3 A_{u}(n \ra \pis)$ & & & & \\
& $^3 B_{1u}(\pi \ra \pis)$ & & & & \\
& $^3 B_{3g}(\pi \ra \pis)$ & & & & \\
Cyclopentadienone
& $^1A_2(n \ra \pis)$ \\
& $^1B_2(\pi \ra \pis)$ \\
& $^1B_1(\double)$ \\
& $^1A_1(\double)$ \\
& $^1A_1(\pi \ra \pis)$ \\
& $^3B_2(\pi \ra \pis)$ \\
& $^3A_2( \ra \pis)$ \\
& $^3A_1(\pi \ra \pis)$ \\
& $^3B_1(\double)$ \\
Cyclopentadienethione
& $^1A_2(n \ra \pis)$ \\
& $^1B_2(\pi \ra \pis)$ \\
& $^1B_1(\double)$ \\
& $^1A_1(\pi \ra \pis)$ \\
& $^1A_1(\double)$ \\
& $^3A_2(n \ra \pis)$ \\
& $^3B_2(\pi \ra \pis)$ \\
& $^3A_1(\pi \ra \pis)$ \\
& $^3B_1(\double)$ \\
Hexatriene
& $^1B_u(\pi \ra \pis)$ \\
& $^1A_g(\pi \ra \pis)$ \\
& $^1A_u(\pi \ra 3s)$ \\
& $^1B_g(\pi \ra 3p)$ \\
& $^3B_u(\pi \ra \pis)$ \\
& $^3A_g(\pi \ra \pis)$ \\
Maleimide
& $^1B_1(n \ra \pis)$ \\
& $^1A_2(n \ra \pis)$ \\
& $^1B_2 (\pi \ra \pis)$ \\
& $^1B_2(\pi \ra \pis)$ \\
& $^1B_2(n \ra 3s)$ \\
& $^3B_1(n \ra \pis)$ \\
& $^3B_2(\pi \ra \pis)$ \\
& $^3B_2(\pi \ra \pis)$ \\
& $^3A_2(n \ra \pis)$ \\
Naphthalene
& $^1B_{3u}(\pi \ra \pis)$ \\
& $^1B_{2u}(\pi \ra \pis)$ \\
& $^1A_u(\pi \ra 3s)$ \\
& $^1B_{1g}(\pi \ra \pis)$ \\
& $^1A_g(\pi \ra \pis)$ \\
& $^1B_{3g}(\pi \ra 3p)$ \\
& $^1B_{2g}(\pi \ra 3p)$ \\
& $^1B_{3u}(\pi \ra \pis)$ \\
& $^1B_{1u}(\pi \ra 3s)$ \\
& $^1B_{2u}(\pi \ra \pis)$ \\
& $^1B_{1g}(\pi \ra \pis)$ \\
& $^1A_g(\pi \ra \pis)$ \\
& $^3B_{2u}(\pi \ra \pis)$ \\
& $^3B_{3u}(\pi \ra \pis)$ \\
& $^3B_{1g}(\pi \ra \pis)$ \\
& $^3B_{2u}(\pi \ra \pis)$ \\
& $^3B_{3u}(\pi \ra \pis)$ \\
& $^3A_g(\pi \ra \pis)$ \\
& $^3B_{1g}(\pi \ra \pis)$ \\
& $^3A_g(\pi \ra \pis)$ \\
Nitroxyl
& $^1A''(n \ra \pis)$ \\
& $^1A'(\double)$ \\
& $^1A'$ \\
& $^3A''(n \ra \pis)$ \\
& $^3A'(\pi \ra \pis)$ \\
Streptocyanine-C3
& $^1B_2(\pi \ra \pis)$ \\
& $^3B_2(\pi \ra \pis)$ \\
Streptocyanine-C5
& $^1B_2(\pi \ra \pis)$ \\
& $^3B_2(\pi \ra \pis)$ \\
Thioacrolein
& $^1A''(n \ra \pis)$ \\
& $^3A''(n \ra \pis)$ \\
%\begin{table}[bt]
%\centering
%\caption{Singlet and triplet excitation energies of various molecules obtained at the CC3, CCSDT, NEVPT2, and FCI levels of theory.}
%\begin{threeparttable}
%\begin{tabular}{lccrrr}
%\headrow
% & & \mc{4}{c}{6-31+G*} \\
%\thead{Molecule} & \thead{Transition} & \thead{CC3} & \thead{CCSDT} & \thead{NEVPT2} & \thead{FCI}\\
%Aza-naphthalene
% & $^1B_{3g}(n \ra \pis)$ \\
% & $^1B_{2u}(\pi \ra \pis)$ \\
% & $^1B_{1u}(n \ra \pis)$ \\
% & $^1B_{2g}(n \ra \pis)$ \\
% & $^1B_{2g}(n \ra \pis)$ \\
% & $^1B_{1u}(n \ra \pis)$ \\
% & $^1A_u(n \ra \pis)$ \\
% & $^1B_{3u}(\pi \ra \pis)$ \\
% & $^1A_g(\pi \ra \pis)$ \\
% & $^1A_u(n \ra \pis)$ \\
% & $^1A_g(n \ra 3s)$ \\
% & $^3B_{3g}(n \ra \pis)$ \\
% & $^3B_{2u}(\pi \ra \pis)$ \\
% & $^3B_{3u}(\pi \ra \pis)$ \\
% & $^3B_{1u}(n \ra \pis)$ \\
% & $^3B_{2g}(n \ra \pis)$ \\
% & $^3B_{2g}(n \ra \pis)$ \\
% & $^3B_{3u}(\pi \ra \pis)$ \\
% & $^3A_u(n \ra \pis)$ \\
%Benzoquinone
% & $^1 B_{1g}(n \ra \pis)$ & & & & \\
% & $^1 A_{u}(n \ra \pis)$ & & & & \\
% & $^1 A_{g}(\double)$ & & & & \\
% & $^1 B_{3g}(\pi \ra \pis)$ & & & & \\
% & $^1 B_{3u}(n \ra \pis)$ & & & & \\
% & $^1 B_{2g}(n \ra \pis)$ & & & & \\
% & $^1 A_{u}(n \ra \pis)$ & & & & \\
% & $^1 B_{1g}(n \ra \pis)$ & & & & \\
% & $^1 B_{2g}(n \ra \pis)$ & & & & \\
% & $^3 B_{1g}(n \ra \pis)$ & & & & \\
% & $^3 A_{u}(n \ra \pis)$ & & & & \\
% & $^3 B_{1u}(\pi \ra \pis)$ & & & & \\
% & $^3 B_{3g}(\pi \ra \pis)$ & & & & \\
%Cyclopentadienone
% & $^1A_2(n \ra \pis)$ \\
% & $^1B_2(\pi \ra \pis)$ \\
% & $^1B_1(\double)$ \\
% & $^1A_1(\double)$ \\
% & $^1A_1(\pi \ra \pis)$ \\
% & $^3B_2(\pi \ra \pis)$ \\
% & $^3A_2( \ra \pis)$ \\
% & $^3A_1(\pi \ra \pis)$ \\
% & $^3B_1(\double)$ \\
%Cyclopentadienethione
% & $^1A_2(n \ra \pis)$ \\
% & $^1B_2(\pi \ra \pis)$ \\
% & $^1B_1(\double)$ \\
% & $^1A_1(\pi \ra \pis)$ \\
% & $^1A_1(\double)$ \\
% & $^3A_2(n \ra \pis)$ \\
% & $^3B_2(\pi \ra \pis)$ \\
% & $^3A_1(\pi \ra \pis)$ \\
% & $^3B_1(\double)$ \\
%Hexatriene
% & $^1B_u(\pi \ra \pis)$ \\
% & $^1A_g(\pi \ra \pis)$ \\
% & $^1A_u(\pi \ra 3s)$ \\
% & $^1B_g(\pi \ra 3p)$ \\
% & $^3B_u(\pi \ra \pis)$ \\
% & $^3A_g(\pi \ra \pis)$ \\
%Maleimide
% & $^1B_1(n \ra \pis)$ \\
% & $^1A_2(n \ra \pis)$ \\
% & $^1B_2 (\pi \ra \pis)$ \\
% & $^1B_2(\pi \ra \pis)$ \\
% & $^1B_2(n \ra 3s)$ \\
% & $^3B_1(n \ra \pis)$ \\
% & $^3B_2(\pi \ra \pis)$ \\
% & $^3B_2(\pi \ra \pis)$ \\
% & $^3A_2(n \ra \pis)$ \\
%Naphthalene
% & $^1B_{3u}(\pi \ra \pis)$ \\
% & $^1B_{2u}(\pi \ra \pis)$ \\
% & $^1A_u(\pi \ra 3s)$ \\
% & $^1B_{1g}(\pi \ra \pis)$ \\
% & $^1A_g(\pi \ra \pis)$ \\
% & $^1B_{3g}(\pi \ra 3p)$ \\
% & $^1B_{2g}(\pi \ra 3p)$ \\
% & $^1B_{3u}(\pi \ra \pis)$ \\
% & $^1B_{1u}(\pi \ra 3s)$ \\
% & $^1B_{2u}(\pi \ra \pis)$ \\
% & $^1B_{1g}(\pi \ra \pis)$ \\
% & $^1A_g(\pi \ra \pis)$ \\
% & $^3B_{2u}(\pi \ra \pis)$ \\
% & $^3B_{3u}(\pi \ra \pis)$ \\
% & $^3B_{1g}(\pi \ra \pis)$ \\
% & $^3B_{2u}(\pi \ra \pis)$ \\
% & $^3B_{3u}(\pi \ra \pis)$ \\
% & $^3A_g(\pi \ra \pis)$ \\
% & $^3B_{1g}(\pi \ra \pis)$ \\
% & $^3A_g(\pi \ra \pis)$ \\
%Nitroxyl
% & $^1A''(n \ra \pis)$ \\
% & $^1A'(\double)$ \\
% & $^1A'$ \\
% & $^3A''(n \ra \pis)$ \\
% & $^3A'(\pi \ra \pis)$ \\
%Streptocyanine-C3
% & $^1B_2(\pi \ra \pis)$ \\
% & $^3B_2(\pi \ra \pis)$ \\
%Streptocyanine-C5
% & $^1B_2(\pi \ra \pis)$ \\
% & $^3B_2(\pi \ra \pis)$ \\
%Thioacrolein
% & $^1A''(n \ra \pis)$ \\
% & $^3A''(n \ra \pis)$ \\
%\hiderowcolors
\hline % Please only put a hline at the end of the table
\end{tabular}
%\hline % Please only put a hline at the end of the table
%\end{tabular}
%\begin{tablenotes}
%\item JKL, just keep laughing; MN, merry noise.
%\end{tablenotes}
\end{threeparttable}
\end{table}
%-----------------------------------------------------------------------
\subsubsection{FCI excitation energies for five- and six-membered rings}
%-----------------------------------------------------------------------
\alert{Here comes Anthony's new CIPSI numbers for the five- and six-membered rings.}
The singlet and triplet excitation energies obtained at the FCI/6-31+G(d) level are reported in Table \ref{tab:cycles} alongside the CC3 and CCSDT values in the same basis from Ref.~\cite{Loos_2020b}.
\begin{table}[bt]
\centering
\caption{Singlet and triplet excitation energies obtained at the CC3, CCSDT, and FCI levels of theory with the 6-31+G* basis set for various five- and six-membered rings.}
\label{tab:cycles}
\begin{threeparttable}
\begin{tabular}{lccrr}
\headrow
\thead{Molecule} & \thead{Transition} & \thead{CC3} & \thead{CCSDT} & \thead{FCI}\\
\mc{5}{c}{Five-membered rings} \\
Cyclopentadiene & $^1 B_2 (\pi \ra \pis)$ & 5.79 & 5.80 & 5.797(15) \\
& $^3 B_2 (\pi \ra \pis)$ & 3.33 & 3.33 & 3.321(35) \\
Furan & $^1A_2(\pi \ra 3s)$ & 6.26 & 6.28 & 6.310(46) \\
& $^3B_2(\pi \ra \pis)$ & 4.28 & 4.28 & 4.262(39) \\
Imidazole & $^1A''(\pi \ra 3s)$ & 5.77 & 5.77 & 5.777(48) \\
& $^3A'(\pi \ra \pis)$ & 4.83 & 4.81 & 4.823(74) \\
Pyrrole & $^1A_2(\pi \ra 3s)$ & 5.25 & 5.25 & 5.225(68) \\
& $^3B_2(\pi \ra \pis)$ & 4.59 & 4.58 & 4.540(68) \\
Thiophene & $^1A_1(\pi \ra \pis)$ & 5.79 & 5.77 & 5.748(79) \\
& $^3B_2(\pi \ra \pis)$ & 3.95 & 3.94 & 3.980(13) \\
\mc{5}{c}{Six-membered rings} \\
Benzene & $^1B_{2u}(\pi \ra \pis)$ & 5.13 & 5.10 & 5.063(86) \\
& $^3B_{1u}(\pi \ra \pis)$ & 4.18 & 4.16 & 4.276(57) \\
Cyclopentadienone & $^1A_2(n \ra \pis)$ & 3.03 & 3.03 & 3.084(17) \\
& $^3B_2(\pi \ra \pis)$ & 2.30 & 2.32 & 2.369(47) \\
Pyrazine & $^1B_{3u}(n \ra \pis)$ & 4.28 & 4.28 & 4.259(91) \\
& $^3B_{3u}(n \ra \pis)$ & 3.68 & 3.68 & 3.697(30) \\
Tetrazine & $^1B_{3u}(n \ra \pis)$ & 2.53 & 2.54 & 2.563(50) \\
& $^3B_{3u}(n \ra \pis)$ & 1.87 & 1.88 & 1.914(32) \\
Pyridazine & $^1B_1(n \ra \pis)$ & 3.95 & 3.95 & 3.969(97) \\
& $^3B_1(n \ra \pis)$ & 3.27 & 3.26 & 3.265(146) \\
Pyridine & $^1B_1(n \ra \pis)$ & 5.12 & 5.10 & 5.153(118) \\
& $^3A_1(\pi \ra \pis)$ & 4.33 & 4.31 & 4.419(85) \\
Pyrimidine & $^1B_1(n \ra \pis)$ & 4.58 & 4.57 & 4.639(106) \\
& $^3B_1(n \ra \pis)$ & 4.20 & 4.20 & 4.553(374) \\
Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.769(132) \\
& $^3A_2''(n \ra \pis)$ & 4.40 & 4.40 & 4.448(389) \\
%\hiderowcolors
\hline % Please only put a hline at the end of the table
\end{tabular}
%\begin{tablenotes}
%\item JKL, just keep laughing; MN, merry noise.
%\end{tablenotes}
\end{threeparttable}
\end{table}
%\end{threeparttable}
%\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical best estimates}
\label{sec:TBE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We discuss in this section the generation of the TBEs obtained with the aug-cc-pVTZ basis as well as oscillator strengths for most transitions.
An exhaustive list of TBEs can be found in the {\SupInf}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Benchmarks}
\label{sec:bench}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we report a comprehensive benchmark of various lower-order methods on the entire QUEST set which is composed by more than \alert{470} excitations.
Statistical quantities are reported in Table \ref{tab:stat}.
Additionally, we also provide a specific analysis for each type of excited states.
Hence, the statistical values are reported for various types of excited states and molecular sizes for the MSE and MAE.
\begin{sidewaystable}
\scriptsize
\centering
\caption{Mean signed error (MSE), mean absolute error (MAE), root-mean-square error (RMSE), standard deviation of the errors (SDE), as well as the maximum positive [Max(+)] and negative [Max($-$)] errors with respect to the TBE/aug-cc-pVTZ.
For the MSE and MAE, the statistical values are reported for various types of excited states and molecular sizes.
All quantities are given in eV. ``Count'' refers to the number of transitions considered for each method.}
\label{tab:stat}
\begin{threeparttable}
\begin{tabular}{llccccccccccccccc}
\headrow
& & \thead{CIS(D)} & \thead{CC2} & \thead{CCSD(2)} & \thead{STEOM-CCSD} & \thead{CCSD} & \thead{CCSDR(3)} & \thead{CCCSDT-3} & \thead{CC3}
& \thead{SOS-ADC(2)[TM]} & \thead{SOS-CC2[TM]} & \thead{SCS-CC2[TM]} & \thead{SOS-ADC(2) [QC]} & \thead{ADC(2)} & \thead{ADC(3)} & \thead{ADC(2.5)} \\
Count & & 429 & 431 & 427 & 360 & 431 & 259 & 251 & 431 & 430 & 430 & 430 & 430 & 426 & 423 & 423 \\
Max(+) & & 1.06 & 0.63 & 0.80 & 0.59 & 0.80 & 0.43 & 0.26 & 0.19 & 0.87 & 0.84 & 0.76 & 0.73 & 0.64 & 0.60 & 0.24 \\
Max($-$) & & -0.69 & -0.71 & -0.38 & -0.56 & -0.25 & -0.07 & -0.07 & -0.09 & -0.29 & -0.24 & -0.92 & -0.46 & -0.76 & -0.79 & -0.34 \\
MSE & & 0.13 & 0.02 & 0.18 & -0.01 & 0.10 & 0.04 & 0.04 & 0.00 & 0.18 & 0.21 & 0.15 & 0.02 & -0.01 & -0.12 & -0.06 \\
& singlet & 0.10 & -0.02 & 0.22 & 0.03 & 0.14 & 0.04 & 0.04 & 0.00 & 0.18 & 0.20 & 0.13 & 0.00 & -0.04 & -0.08 & -0.06 \\
& triplet & 0.19 & 0.08 & 0.14 & -0.07 & 0.03 & & & 0.00 & 0.19 & 0.22 & 0.17 & 0.04 & 0.04 & -0.18 & -0.07 \\
& valence & 0.20 & 0.10 & 0.20 & -0.06 & 0.10 & 0.06 & 0.05 & 0.00 & 0.19 & 0.24 & 0.20 & 0.02 & 0.04 & -0.16 & -0.06 \\
& Rydberg & -0.04 & -0.17 & 0.15 & 0.09 & 0.08 & 0.01 & 0.03 & -0.01 & 0.16 & 0.12 & 0.01 & 0.02 & -0.13 & -0.02 & -0.07 \\
& $n \ra \pis$ & 0.16 & 0.02 & 0.24 & -0.03 & 0.17 & 0.07 & 0.07 & 0.00 & 0.26 & 0.32 & 0.22 & 0.05 & -0.05 & -0.01 & -0.03 \\
& $\pi \ra \pis$& 0.25 & 0.17 & 0.20 & -0.07 & 0.06 & 0.05 & 0.04 & 0.00 & 0.15 & 0.19 & 0.19 & 0.00 & 0.12 & -0.27 & -0.07 \\
& 1--3 non-H & 0.10 & 0.03 & 0.03 & -0.02 & 0.04 & 0.01 & 0.01 & 0.00 & 0.13 & 0.16 & 0.11 & -0.01 & -0.01 & -0.17 & -0.09 \\
& 4 non-H & 0.13 & 0.04 & 0.12 & 0.00 & 0.09 & 0.03 & 0.04 & 0.00 & 0.19 & 0.26 & 0.19 & 0.03 & -0.04 & -0.10 & -0.07 \\
& 5--6 non-H & 0.17 & 0.02 & 0.30 & -0.01 & 0.11 & 0.05 & 0.05 & 0.00 & 0.21 & 0.20 & 0.14 & 0.03 & 0.03 & -0.10 & -0.04 \\
& 7--10 non-H & 0.15 & -0.03 & 0.42 & -0.05 & 0.22 & 0.10 & 0.08 & -0.01 & 0.26 & 0.29 & 0.19 & 0.05 & -0.06 & -0.02 & -0.04 \\
MSE & & 0.13 & 0.02 & 0.18 & -0.01 & 0.10 & 0.04 & 0.04 & 0.00 & 0.18 & 0.21 & 0.15 & 0.02 & -0.01 & -0.12 & -0.06 \\
SDE & & 0.24 & 0.20 & 0.21 & 0.13 & 0.12 & 0.05 & 0.04 & 0.02 & 0.17 & 0.16 & 0.16 & 0.15 & 0.20 & 0.22 & 0.08 \\
RMSE & & 0.29 & 0.22 & 0.28 & 0.15 & 0.16 & 0.07 & 0.06 & 0.03 & 0.25 & 0.26 & 0.22 & 0.17 & 0.21 & 0.26 & 0.10 \\
MAE & & 0.22 & 0.16 & 0.22 & 0.11 & 0.12 & 0.05 & 0.04 & 0.02 & 0.20 & 0.22 & 0.18 & 0.13 & 0.15 & 0.21 & 0.08 \\
& singlet & 0.22 & 0.16 & 0.25 & 0.10 & 0.14 & 0.05 & 0.04 & 0.02 & 0.21 & 0.22 & 0.17 & 0.14 & 0.16 & 0.20 & 0.09 \\
& triplet & 0.23 & 0.15 & 0.18 & 0.12 & 0.08 & & & 0.01 & 0.20 & 0.23 & 0.19 & 0.11 & 0.15 & 0.22 & 0.08 \\
& valence & 0.22 & 0.14 & 0.24 & 0.12 & 0.13 & 0.06 & 0.05 & 0.02 & 0.21 & 0.25 & 0.20 & 0.12 & 0.13 & 0.22 & 0.08 \\
& Rydberg & 0.22 & 0.21 & 0.19 & 0.10 & 0.08 & 0.03 & 0.03 & 0.02 & 0.20 & 0.15 & 0.13 & 0.14 & 0.21 & 0.18 & 0.09 \\
& $n \ra \pis$ & 0.18 & 0.08 & 0.28 & 0.08 & 0.17 & 0.07 & 0.07 & 0.01 & 0.26 & 0.32 & 0.22 & 0.11 & 0.10 & 0.14 & 0.07 \\
& $\pi \ra \pis$& 0.27 & 0.19 & 0.21 & 0.14 & 0.11 & 0.06 & 0.04 & 0.02 & 0.18 & 0.21 & 0.20 & 0.12 & 0.16 & 0.28 & 0.09 \\
& 1--3 non-H & 0.23 & 0.19 & 0.13 & 0.10 & 0.07 & 0.03 & 0.03 & 0.02 & 0.18 & 0.20 & 0.19 & 0.14 & 0.19 & 0.24 & 0.10 \\
& 4 non-H & 0.22 & 0.19 & 0.15 & 0.11 & 0.11 & 0.03 & 0.04 & 0.02 & 0.19 & 0.26 & 0.22 & 0.13 & 0.18 & 0.23 & 0.08 \\
& 5--6 non-H & 0.21 & 0.12 & 0.30 & 0.12 & 0.13 & 0.06 & 0.05 & 0.01 & 0.22 & 0.21 & 0.15 & 0.11 & 0.11 & 0.19 & 0.07 \\
& 7--10 non-H & 0.24 & 0.11 & 0.42 & 0.12 & 0.23 & 0.10 & 0.08 & 0.02 & 0.27 & 0.29 & 0.19 & 0.12 & 0.14 & 0.16 & 0.07 \\
\hline
\end{tabular}
\end{threeparttable}
\end{sidewaystable}
@ -719,26 +789,22 @@ In 2006, he obtained a Research Engineer position from the \textit{``Centre Nati
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{biography}[example-image-1x1]{M.~Caffarel}
Please check with the journal's author guidelines whether author biographies are required. They are usually only included for review-type articles, and typically require photos and brief biographies (up to 75 words) for each author.
\bigskip
\bigskip
\begin{biography}[MCaffarel]{M.~Caffarel}
received his Ph.D. in Theoretical Physics and Chemistry from the Universit\'e Pierre et Marie Curie (Paris, France) in 1987, before moving to the University of Illinois at Urbana-Champaign for a two-year postdoctoral stay in the group of Prof.~David Ceperley.
He is currently working as a senior scientist at the ``Centre National de la Recherche Scientifique (CNRS)'' at the Laboratoire de Chimie et Physique Quantiques in Toulouse (France).
His research is mainly focused on the development and application of quantum Monte Carlo methods for theoretical chemistry and condensed-mater physics.
\end{biography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{biography}[example-image-1x1]{F.~Filippo}
Please check with the journal's author guidelines whether author biographies are required. They are usually only included for review-type articles, and typically require photos and brief biographies (up to 75 words) for each author.
\bigskip
\bigskip
\begin{biography}[FLipparini]{F.~Lipparini}
got his Ph.D. in Chemistry from Scuola Normale Superiore, Pisa in 2013. He worked as a Postdoc at the Universit\'e Pierre et Marie Curie in Paris and moved to Mainz, Germany, with a fellowship from the Alexander von Humboldt foundation and then as a regular postdoc. Since June 2017 he is assistant professor of Physical Chemistry at the department of Chemistry of the University of Pisa, in Italy. In 2014, he was awarded the ``Eolo Scrocco'' prize for young researcher in theoretical and computational chemistry by the Italian Chemical Society. His research focuses on mathematical methods and algorithms for computational chemistry, with a particular interest to their application to multiscale methods and electronic structure theory.
\end{biography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{biography}[example-image-1x1]{M.~Boggio-Pasqua}
Please check with the journal's author guidelines whether author biographies are required. They are usually only included for review-type articles, and typically require photos and brief biographies (up to 75 words) for each author.
\bigskip
\bigskip
\begin{biography}[MBoggioPasqua]{M.~Boggio-Pasqua}
is a CNRS researcher at the Laboratoire de Chimie et Physique Quantiques at the University of Toulouse III - Paul Sabatier. His main research interests are focused on the theoretical studies of photochemical processes in complex molecular systems including the description of excited-state reaction mechanisms based on static exploration of potential energy surfaces and simulations of the nonadiabatic dynamics.
\end{biography}
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