1517 lines
143 KiB
TeX
1517 lines
143 KiB
TeX
\documentclass[journal=jctcce,manuscript=article,layout=traditional]{achemso}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amsmath,amssymb,amsfonts,physics,float,lscape,soul,rotating,longtable}
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\usepackage[version=4]{mhchem}
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\usepackage[normalem]{ulem}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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\newcommand{\titou}[1]{\textcolor{purple}{#1}}
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\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\trash}[1]{\textcolor{purple}{\sout{#1}}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\mr}{\multirow}
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\newcommand{\ra}{\rightarrow}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\ie}{\textit{i.e.}}
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% energies
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\newcommand{\EFCI}{E_\text{FCI}}
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\newcommand{\EexCI}{E_\text{exCI}}
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\newcommand{\EsCI}{E_\text{sCI}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\PsisCI}{\Psi_\text{sCI}}
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\newcommand{\Ndet}{N_\text{det}}
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\newcommand{\ex}[6]{$^{#1}#2_{#3}^{#4}(#5 \rightarrow #6)$}
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% methods
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\newcommand{\TDDFT}{TD-DFT}
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\newcommand{\CASSCF}{CASSCF}
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\newcommand{\CASPT}{CASPT2}
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\newcommand{\NEV}{NEVPT2}
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\newcommand{\PNEV}{PC-NEVPT2}
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\newcommand{\SNEV}{SC-NEVPT2}
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\newcommand{\AD}{ADC(2)}
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\newcommand{\AT}{ADC(3)}
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\newcommand{\CCD}{CC2}
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\newcommand{\CCSD}{CCSD}
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\newcommand{\STEOM}{STEOM-CCSD}
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\newcommand{\CCT}{CC3}
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\newcommand{\EOMCCSD}{EOM-CCSD}
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\newcommand{\CCSDT}{CCSDT}
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\newcommand{\CCSDTQ}{CCSDTQ}
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\newcommand{\CCSDTQP}{CCSDTQP}
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\newcommand{\CI}{CI}
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\newcommand{\sCI}{sCI}
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\newcommand{\exCI}{exCI}
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\newcommand{\FCI}{FCI}
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% basis
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\newcommand{\Pop}{6-31+G(d)}
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\newcommand{\AVDZ}{\emph{aug}-cc-pVDZ}
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\newcommand{\AVTZ}{\emph{aug}-cc-pVTZ}
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\newcommand{\DAVTZ}{d-\emph{aug}-cc-pVTZ}
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\newcommand{\AVQZ}{\emph{aug}-cc-pVQZ}
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\newcommand{\AVFZ}{\emph{aug}-cc-pV5Z}
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\newcommand{\DAVQZ}{d-\emph{aug}-cc-pVQZ}
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\newcommand{\TAVQZ}{t-\emph{aug}-cc-pVQZ}
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\newcommand{\AVPZ}{\emph{aug}-cc-pV5Z}
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\newcommand{\DAVPZ}{d-\emph{aug}-cc-pV5Z}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\MaxP}{Max($+$)}
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\newcommand{\MaxN}{Max($-$)}
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% greek shortcut
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\newcommand{\pis}{\pi^\star}
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\newcommand{\Ryd}{\mathrm{R}}
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\renewcommand\floatpagefraction{.99}
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\renewcommand\topfraction{.99}
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\renewcommand\bottomfraction{.99}
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\renewcommand\textfraction{.01}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
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\newcommand{\Pisa}{Dipartimento di Chimica e Chimica Industriale, University of Pisa, Via Moruzzi 3, 56124 Pisa, Italy}
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\title{Highly-Accurate Reference Excitation Energies and Benchmarks: Medium Size Molecules}
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\author{Pierre-Fran{\c c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation[LCPQ, Toulouse]{\LCPQ}
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\author{Filippo Lipparini}
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\affiliation[DC, Pisa]{\Pisa}
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\email{filippo.lipparini@unipi.it}
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\author{Martial Boggio-Pasqua}
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\affiliation[LCPQ, Toulouse]{\LCPQ}
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\author{Anthony Scemama}
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\affiliation[LCPQ, Toulouse]{\LCPQ}
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\author{Denis Jacquemin}
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\email{Denis.Jacquemin@univ-nantes.fr}
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\affiliation[UN, Nantes]{\CEISAM}
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\begin{document}
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\begin{abstract}
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Following our previous work focussing on compounds containing up to 3 non-hydrogen atoms [\emph{J. Chem. Theory Comput.} {\bfseries 14} (2018) 4360--4379], we present here highly-accurate vertical transition energies
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obtained for 27 molecules encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione,
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diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine.
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To obtain these energies, we use equation-of-motion coupled cluster theory up to the highest technically possible excitation order for these systems ({\CCT}, {\CCSDT}, and {\CCSDTQ}), selected configuration interaction ({\sCI}) calculations (with tens of millions of determinants in the reference space),
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as well as the multiconfigurational $n$-electron valence state perturbation theory (NEVPT2) method.
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All these approaches are applied in combination with diffuse-containing atomic basis sets. For all transitions, we report at least {\CCT}/{\AVQZ} vertical excitation
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energies as well as {\CCT}/{\AVTZ} oscillator strengths for each dipole-allowed transition. We show that {\CCT} almost systematically delivers transition energies in agreement with higher-level theoretical methods with a typically deviation of $\pm 0.04$ eV, except for transitions with a dominant double excitation character where the error is much larger.
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The present contribution gathers a large, diverse and accurate set of more than 200 highly-accurate transition energies for states of various natures
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(valence, Rydberg, singlet, triplet, $n \ra \pi^\star$, $\pi \ra \pi^\star$, \ldots).
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We use this series of theoretical best estimates to benchmark a series of popular methods for excited state calculations: CIS(D), {\AD},
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{\CCD}, {\STEOM}, {\CCSD}, CCSDR(3), CCSDT-3, and {\CCT}. The results of these benchmarks are compared to the available literature data.
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\end{abstract}
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\clearpage
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%
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% I. Introduction
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%
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\section{Introduction}
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Accurately describing transition energies between the electronic ground state (GS) and excited states (ES) remains an important challenge in quantum chemistry. When dealing with large compounds in complex environments,
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one is typically limited to the use of time-dependent density-functional theory (TD-DFT), \cite{Cas95,Ulr12b,Ada13a} a successful yet far from flawless approach. In particular, to perform TD-DFT calculations, one must
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choose an ``appropriate'' exchange-correlation functional, which is difficult yet primordial as the impact of the exchange-correlation functional is exacerbated in TD-DFT as compared to DFT. \cite{Lau13} Such selection can, of course, rely
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on the intrinsic features of each exchange-correlation functional, \eg, it is well-known that range-separated hybrids provide a more physically-sound description of long-range charge-transfer transitions than semi-local exchange-correlation functionals. \cite{Dre04,Pea08}
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However, to obtain a quantitative assessment of the accuracy that can be expected from TD-DFT calculations, benchmarks are needed. This is why many assessment of TD-DFT performances for various properties are
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available. \cite{Lau13} While several of these benchmarks use experimental data as reference, typically band shapes \cite{Die04,Die04b,Avi13,Cha13,Lat15b,Mun15,Vaz15,San16b} or 0-0 energies,
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\cite{Die04b,Goe10a,Jac12d,Chi13b,Win13,Fan14b,Jac14a,Jac15b,Loo19b} using theoretical best estimates (TBE) obtained with more refined levels of theory as references, \cite{Sch08,Sau09,Sil10b,Sil10c,Sch17,Loo18a}
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is advantageous as it allows comparisons on a perfectly equal footing (same geometry, vertical transitions, no environmental effects...). For such works, the challenge is in fact to obtain accurate TBE, as high-level theories
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generally come with a dreadful scaling with system size and, in addition, typically require large atomic basis sets to deliver transition energies close to the basis set limit.
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More than 20 years ago, Serrano-Andr\`es, Roos, and their coworkers proposed an impressive series of reference transition energies for several typical conjugated organic molecules (butadiene, furan, pyrrole, tetrazine...).
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\cite{Ful92,Ser93,Ser93b,Ser93c,Lor95b,Mer96,Mer96b,Roo96,Ser96b} To this end, they relied on the Complete Active Space Second-Order Perturbation Theory ({\CASPT}) approach with the largest active spaces and basis sets
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they could offer at that time, typically using experimental GS geometries. Beyond comparisons with experiments, which are always challenging when computing vertical transition energies, \cite{San16b} there was no approach
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available at that time to ascertain the accuracy of the obtained transition energies. These {\CASPT} values were latter used to assess the performances of TD-DFT combined with various exchange-correlation functionals, \cite{Toz99b,Bur02} and remained for a long
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time the best references available. A decade ago, Thiel and coworkers defined TBE for 104 singlet and 63 triplet valence ES in 28 small and medium conjugated CNOH organic molecules. \cite{Sch08,Sil10b,Sil10c} These TBE
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were computed on MP2/6-31G(d) structures with several levels of theories, notably {\CASPT} and various Coupled-Cluster levels ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default reference approach used by Thiel to define his
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first series of TBE was {\CASPT}, \cite{Sch08} the majority of the most recent TBE (so-called ``TBE-2'' in Ref. \citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level, often using a basis set extrapolation technique.
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In more details, CC3/TZVP values were typically corrected for basis set effects by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works used Thiel's TBE for
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assessing low-order methods, \cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18,Hel19} highlighting
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their value for the community. In contrast, the number of extensions of this original set remains quite limited, i.e., {\CCSDT}/TZVP reference energies computed for 17 singlet states of six molecules appeared in 2014, \cite{Kan14}
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and 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide) have been described in 2017. \cite{Kan17}
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This has motivated us to propose two years ago a set of 106 transition energies for which it was technically possible to reach the Full Configuration Interaction (FCI) limit by performing high-order Coupled-Cluster (up to {\CCSDTQP}) and selected
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CI (sCI) transition energy calculations on {\CCT} GS structures. \cite{Loo18a} We used these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} was on the {\FCI} spot, whereas we could not
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detect significant differences of accuracies between {\CCT} and {\CCSDT}, both being very accurate with mean absolute errors (MAE) as small as 0.03 eV compared to {\FCI} for ES with a single excitation character. These conclusions
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agree well with earlier studies. \cite{Wat13,Kan14,Kan17} We also recently proposed a set of 20 TBE for transitions presenting a large double excitation character. \cite{Loo19c} For such transitions, one can distinguish the ES
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as a function of $\%T_1$, the percentage of single excitation calculated at the {\CCT} level. For ES with a significant yet not dominant double excitation character, such as the famous $A_g$ ES of butadiene ($\%T_1$=75\%),
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CC methods including triples in the GS treatment deliver rather accurate estimates (MAE of 0.11 eV with {\CCT} and 0.06 eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or
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the generally robust $N$-Electron Valence State Perturbation Theory ({\NEV}). In contrast, for ES with a dominant double excitation character, e.g., the low-lying $n,n \rightarrow \pi^\star,\pi^\star$ excitation in nitrosomethane ($\%T_1$=2\%),
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single-reference methods are not suited (MAE of 0.86 eV with {\CCT} and 0.42 eV with {\CCSDT}) and multi-reference methods are, in practice, required to obtain accurate results. \cite{Loo19c} Obviously, a clear limit of our 2018 work\cite{Loo18a}
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is that we treated only compounds containing 1--3 non-hydrogen atoms, hence introducing a significant chemical bias. Therefore we have decided to go for larger molecules and we consider in the present contribution organic
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compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large bases remains a dream, and, the convergence of {\sCI} with the number of determinants is slower
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as well, so that extrapolating to the {\FCI} limit with a ca. 0.01 eV error bar is rarely doable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work,\cite{Loo18a} is beyond reach.
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Anticipating this problem, we have previously demonstrated that one can very accurately estimate such limit by correcting values obtained with a high-level of theory and a double-$\zeta$ basis set by {\CCT} results obtained with a
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larger basis, \cite{Loo18a} and we glibally follow such strategy here. In addition, we also performed {\NEV} calculations in an effort to check the consistency of our estimates, especially for ES with intermediate $\%T_1$ values.
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Using this protocol, we define a set of more than 200 {\AVTZ} reference transition energies, most being within $\pm$ 0.03 eV for the {\FCI} limit. These reference energies are obtained on {\CCT} geometries and further basis set
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corrections up to at least quadruple-$\zeta$, are also provided with {\CCT}. Together with the results obtained in our two earlier works, \cite{Loo18a,Loo19c} the present TBE will hopefully contribute to climb a further step
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on the ES accuracy staircase.
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%
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% II. Computational Details
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%
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\section{Computational Details}
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\label{sec-met}
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Unless otherwise stated, all transition energies are computed in the frozen-core approximation (with a large core for the sulfur atoms).
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Pople's {\Pop} and Dunning's \emph{aug}-cc-pVXZ (X $=$ D, T, Q, and 5) atomic basis sets are systematically employed in our excited-state calculations.
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Here, we globally follow the same procedure as in Ref.~\citenum{Loo18a}, so that we only briefly outline the various theoretical methods that we have employed in the subsections below.
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\subsection{Geometries}
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Consistently with our previous work, \cite{Loo18a} we systematically use {\CCT}/{\AVTZ} GS geometries obtained without applying the frozen-core approximation.
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The cartesian coordinates (in bohr) of each compound can be found in the Supporting Information (SI).
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Several structures have been extracted from previous contributions, \cite{Bud17,Jac18a,Bre18a} whereas the missing structures were optimized using DALTON \cite{dalton} and/or CFOUR, \cite{cfour} applying default parameters in both cases.
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\subsection{Selected Configuration Interaction methods}
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All the sCI calculations have been performed in the frozen-core approximation with the latest version of QUANTUM PACKAGE \cite{Gar19} using the Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI) algorithm to select the
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most important determinants in the FCI space. Instead of generating all possible excited determinants like a conventional CI calculation, the iterative CIPSI algorithm performs a sparse exploration of the FCI space via a selection of the most relevant
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determinants using a second-order perturbative criterion. At each iteration, the variational (or reference) space is enlarged with new determinants. CIPSI can be seen as a deterministic version of the FCIQMC algorithm developed by Alavi and
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coworkers. \cite{Boo09} We refer the interested reader to Ref.~\citenum{Gar19} where our implementation of the CIPSI algorithm is detailed.
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Excited-state calculations are performed within a state-averaged formalism which means that the CIPSI algorithm select determinants simultaneously for the GS and ES. Therefore, all electronic states share the same set of determinants with different CI coefficients.
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Our implementation of the CIPSI algorithm for ES is detailed in Ref.~\citenum{Sce19}. For each system, a preliminary sCI calculation is performed using Hartree-Fock orbitals in order to generate sCI wavefunctions with at least 5,000,000 determinants.
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State-averaged natural orbitals are then computed based on this wavefunction, and a new, larger sCI calculation is performed with this new set of orbitals. This has the advantage to produce a smoother and faster convergence of the sCI energy to the FCI limit.
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For the largest systems, an additional iteration is sometimes required in order to obtain better quality natural orbitals and hence well-converged calculations.
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The total sCI energy is defined as the sum of the (zeroth-order) variational energy (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction which takes into account the external determinants, \ie,
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the determinants which do not belong to the variational space but are linked to the reference space via a non-zero matrix element. The magnitude of this second-order correction, $E^{(2)}$, provides a qualitative idea of the ``distance" to the FCI limit.
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For maximum efficiency, the total sCI energy is linearly extrapolated to $E^{(2)} = 0$ (which effectively corresponds to the FCI limit) using the two largest sCI wavefunctions. These extrapolated total energies simply (labeled as FCI in the remaining of the paper)
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are then used to computed vertical excitation energies. Although it is not possible to provide a theoretically-sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest sCI wavefunction and its corresponding
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extrapolated value. We believe that it provides a very safe estimate of the extrapolation error. Additional information about the sCI wavefunctions and excitation energies as well as their extrapolated values can be found in the SI.
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\subsection{NEVPT2}
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The {\NEV} calculations have been performed with MOLPRO \cite{molpro} within the partially-contracted scheme ({\PNEV}), which is theoretically superior to its strongly-contracted version due to the larger number of perturbers and greater flexibility. \cite{Ang01,Ang01b,Ang02}
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These {\NEV} calculations are performed on top of a state-averaged complete-active-space self-consistent field (SA$n$-CASSCF) calculation (where $n$ is the number of states that has been averaged).
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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 the SI.
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\hl{Martial: you to play}
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\hl{xxxxx}
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\hl{xxxxx}
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\hl{xxxxx}
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\hl{xxxxx}
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\subsection{Other wavefunction calculations}
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For the other levels of theory, we apply a variety of programs, namely, CFOUR,\cite{cfour} DALTON,\cite{dalton} GAUSSIAN,\cite{Gaussian16} ORCA,\cite{Nee12} MRCC,\cite{Rol13,mrcc} and Q-CHEM. \cite{Sha15} CFOUR is used for
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{\CCT}, \cite{Chr95b,Koc97} CCSDT-3, \cite{Wat96,Pro10} {\CCSDT} \cite{Nog87} and {\CCSDTQ}\cite{Kuc91}; Dalton for {\CCD}, \cite{Chr95,Hat00} {\CCSD},\cite{Pur82} CCSDR(3), \cite{Chr96b} and {\CCT} \cite{Chr95b,Koc97}; Gaussian
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for CIS(D); \cite{Hea94,Hea95} ORCA for the similarity-transformed EOM-CCSD ({\STEOM})\cite{Noo97,Dut18}; MRCC for {\CCSDT} \cite{Nog87} and {\CCSDTQ}; \cite{Kuc91} and Q-Chem for {\AD}. \cite{Dre15}
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Default program settings were applied. We note that for {\STEOM} we report only states that are characterized by an active character percentage of 98\%\ or larger.
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%
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% III. Results & Discussion
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%
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\section{Main results}
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\label{sec-res}
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In the following, we present the results obtained for molecules containing four, five, and six (non-hydrogen) atoms. In all cases, we test several atomic basis sets and push the CC excitation order as high as technically possible.
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Given that the {\sCI} results converges rather slowly for these larger systems, we provide an estimated error bar for these extrapolated {\FCI} values (\emph{vide supra}). In most cases, these extrapolated FCI reference data are used as a safety net to demonstrate the
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consistency of the approaches rather than as definitive TBE (see next Section). We also show the results of {\NEV}/{\AVTZ} calculations for all relevant states to have a further consistency check. We underline that, except
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when specifically discussed, all ES present a dominant single-excitation character (see also next Section), so that we do not expect serious CC breakdowns. This is especially true for the triplet ES that are known to show
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very large \%$T_1$ for the vast majority of states, \cite{Sch08} and we consequently put our maximal computational effort on determining accurate transition energies for singlet states. To assign the different ES, we use literature data, as well as
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usual criteria, i.e., relative energies, symmetries and compositions of the underlying MOs, as well as oscillator strengths. This allows clear-cut assignment for the vast majority of the cases. There are however some
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state/method combination for which strong mixing between ES of the same symmetry makes unambiguous assignments beyond reach, which is a typical problem in such works. Such cases are however not statistically
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relevant and are therefore unlikely to change any of our main conclusions.
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\subsection{Four-atom molecules}
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\subsubsection{Cyanoacetylene, cyanogen, and diacetylene}
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\begin{sidewaystable}[htp]
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\caption{\small Vertical transition energies determined in cyanoacetylene, cyanogen, and diaectylene. All states have a valence $\pi \rightarrow \pi^\star$ character. All values are in eV.}
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\label{Table-1}
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\begin{footnotesize}
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\begin{tabular}{l|p{.5cm}p{1.0cm}p{1.2cm}p{1.4cm}|p{.5cm}p{1.0cm}p{1.2cm}p{1.4cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}|p{.5cm}|p{.6cm}p{.6cm}}
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\hline
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\multicolumn{14}{c}{Cyanoacetylene}\\
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& \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{\AVQZ} & \multicolumn{1}{c}{\AVFZ} & \multicolumn{2}{c}{Litt.}\\
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State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV} & {\CCT} & {\CCT}& Th.$^a$ & Exp.$^b$ \\
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\hline
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$^1\Sigma^-$ &6.02&6.04&6.02&6.02$\pm$0.01 &5.92&5.92&5.91&5.84$\pm$0.09 &5.80&5.81&5.78& 5.79 &5.79 &5.46&4.77\\
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$^1\Delta$ &6.29&6.31&6.29&6.28$\pm$0.01 &6.17&6.19&6.17&6.14$\pm$0.05 &6.08&6.09&6.10& 6.06 &6.06 &5.81&5.48\\
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$^3\Sigma^+$ &4.44&4.45& &4.45$\pm$0.03 &4.43&4.43& &4.41$\pm$0.06 &4.45&4.44&4.45& 4.46 &4.47 &&\\
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$^3\Delta$ &5.35&5.34& &5.32$\pm$0.03 &5.28&5.27& &5.20$\pm$0.08 &5.22&5.21&5.19& 5.22 &5.22 &&\\
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$^1A''$[F]$^c$ &3.70&3.72&3.70&3.67$\pm$0.03 &3.60&3.62&3.60&3.59$\pm$0.02 &3.54&3.56&3.50& 3.54 & &&\\
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\hline
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\multicolumn{14}{c}{Cyanogen}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{\AVQZ} & \multicolumn{1}{c}{\AVFZ} & \multicolumn{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV}& {\CCT} & {\CCT}& Exp.$^d$ \\
|
||
\hline
|
||
$^1\Sigma_u^-$ &6.62&6.63&6.62&6.58$\pm$0.03 &6.52&6.52&6.51&6.44$\pm$0.08 &6.39&6.40&6.32& 6.38 &6.38 &5.63\\
|
||
$^1\Delta_u$ &6.88&6.89&6.88&6.87$\pm$0.02 &6.77&6.78&6.77&6.74$\pm$0.04 &6.66&6.67&6.66& 6.64 &6.64 &5.99\\
|
||
$^3\Sigma_u^+$ &4.92&4.92&4.94&4.91$\pm$0.06 &4.89&4.89& &4.87$\pm$0.07 &4.90&4.89&4.88& 4.91 &4.91 &4.13\\
|
||
$^1\Sigma_u^-$[F]$^c$ &5.27&5.28&5.26&5.31$\pm$0.05 &5.19&5.20&5.18&5.26$\pm$0.09 &5.06&5.07&4.97& 5.05 &5.05 & \\
|
||
\hline
|
||
\multicolumn{14}{c}{Diacetylene}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{\AVQZ} & \multicolumn{1}{c}{\AVFZ} & \multicolumn{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV}& {\CCT} & {\CCT}& Exp.$^e$ \\
|
||
\hline
|
||
$^1\Sigma_u^-$ &5.57&5.58&5.56&5.52$\pm$0.06 &5.44&5.45&5.43&5.47$\pm$0.02 &5.34&5.35&5.33& 5.33 &5.33 &4.81\\
|
||
$^1\Delta_u$ &5.83&5.85& &5.84$\pm$0.01 &5.69&5.70&5.69&5.69$\pm$0.02 &5.61&5.62&5.61& 5.60 &5.60 &5.06\\
|
||
$^3\Sigma_u^+$ &4.07&4.08&4.09&4.04$\pm$0.07 &4.06&4.06& &4.07$\pm$0.04 &4.08& &4.08& 4.10 &4.11 &2.7 \\
|
||
$^3\Delta_u$ &4.93&4.93&4.92&4.94$\pm$0.01 &4.86&4.85& &4.85$\pm$0.02 &4.80&4.79&4.78& 4.80 &4.80 &3.21\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Luo08};}
|
||
$^b${Experimental 0-0 energies from Refs. \citenum{Job66a} and \citenum{Job66b} (vacuum UV experiments);}
|
||
$^c${Vertical fluorescence energy from the lowest excited-state;}
|
||
$^d${Experimental 0-0 energies from Refs. \citenum{Cal63} ($^3\Sigma_u^+$), \citenum{Bel69} ($^1\Sigma_u^-$), and \citenum{Fis72} ($^1\Delta_u$), all analyzing vacuum electronic spectra;}
|
||
$^e${Experimental 0-0 energies from Ref. \citenum{Hai79} (singlet ES, vacuum UV experiment) and Ref. \citenum{All84} (triplet ES, EELS). In the latter contribution, the 2.7 eV value for the $^3\Sigma_u^+$
|
||
state is the onset, whereas an estimate of the vertical energy of 4.2$\pm$0.2 eV is given for the $^3\Delta_u$ state.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{sidewaystable}
|
||
|
||
The ES of these three closely-related linear molecules containing two triple bonds have been quite rarely investigated with theory, \cite{Fis03,Pat06,Luo08,Loo18b,Loo19a} though (rather old) measurements of their
|
||
0-0 energies are available for several ES. \cite{Cal63,Job66a,Job66b,Bel69,Fis72,Har77,Hai79,All84} Our main results are collected in Tables \ref{Table-1} and S1. We consider only low-lying
|
||
valence $\pi \rightarrow \pi^\star$ transitions, and, all have a strongly dominant single excitation character (\%$T_1 > 90$, \emph{vide infra}). For cyanoacetylene, the {\FCI} estimates come with small error bars
|
||
with {\Pop} and one notes an excellent agreement between these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning's double-$\zeta$ basis for which the {\FCI} uncertainities are however
|
||
larger. Using the {\CCSDTQ} values as references, it appears that the previously obtained {\CASPT} estimates\cite{Luo08} are too low and that the {\CCT} transition energies are slightly more accurate than
|
||
their CCSDT counterparts, although all CC estimates of Table \ref{Table-1} come in a very tight energetic window for a given basis set. There is also a superb agreement between the CC and {\NEV} values with the
|
||
{\AVTZ} basis set. All these facts give high confidence that the CC estimates can be trusted. The basis set effects are quite significant for the valence excited-states of cyanoacetylene with successive drops of the transitions
|
||
energies by ca. -0.10 eV, when going from {\Pop} to {\AVDZ} and from {\AVDZ} to {\AVTZ}, but for the lowest triplet state that appears less sensitive to the selected basis. As expected, the changes when further
|
||
extending the basis set to quadruple and quintuple-$\zeta$ are trifling, and the same holds when adding a second set of diffuse functions in the basis, or when correlating the core electrons (see the SI). Obviously,
|
||
both cyanogen and diacetylene yield very similar trends, with limited methodological effects and quite large basis set effects, but for the transition to the $^3\Sigma_u^+$ ES. We note that all CC3 and CCSDT values are, at worst,
|
||
within $\pm$ 0.02 eV of the {\FCI} window, i.e., all methods presented in Table \ref{Table-1} provide very consistent estimates. Across all states reported in that Table with {\AVTZ}, the average absolute deviation between {\NEV} and {\CCT}
|
||
({\CCSDT}) is as small as 0.02 (0.03) eV, the lowest absorption and emission energies of cyanogen being the only two cases showing significant deviations. As a final note, all our vertical absorption (emission) energies are significantly bigger
|
||
(smaller) than the experimentally measured 0-0 energies, as it should. We refer the interested reader to previous works, \cite{Fis03,Loo19a} for comparisons of {\CASPT} and {\CCT} values with measured 0-0 energies
|
||
for these three compounds.
|
||
|
||
|
||
\subsubsection{Cyclopropenone, cyclopropenethione, and methylenecyclopropene}
|
||
|
||
These three related compounds present a three-member $sp^2$ carbon cycle conjugated to an external $\pi$ bond. While the ES of methylenecyclopropene has regularly been investigated with theoretical tools in the past,
|
||
\cite{Mer96,Roo96,Car10b,Lea12,Gua13,Dad14,Gua14,Sch17,Bud17} the only investigations of vertical transitions we could find for the two other derivatives are the detailed 2002 {\CASPT} study of Serrano-Andr\'es and
|
||
coworkers on both compounds, \cite{Ser02} and a more recent work reporting the three lowest-lying singlet states of cyclopropenone at the {\CASPT}/6-31G level.\cite{Liu14b}
|
||
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in cyclopropenone, cyclopropenethione, and methylenecyclopropene. All values are in eV.}
|
||
\label{Table-2}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{.9cm}p{1.1cm}p{1.4cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{12}{c}{Cyclopropenone}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Exp.$^b$ \\
|
||
\hline
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &4.32&4.34&4.36& 4.38$\pm$0.02 &4.22&4.23 &4.21&4.24&4.04 &4.25&4.13 \\%4.50 in Liu14b MS-CASPT2/6-31G
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &5.68&5.65&5.65& 5.64$\pm$0.06 &5.59&5.56 &5.57&5.55&5.85 &5.59&5.5 \\%5.18 in Liu14b
|
||
$^1B_2 (n \rightarrow 3s)$ &6.39&6.38&6.41& &6.21&6.19 &6.32&6.31&6.51 &6.90&6.22 \\%6.44 in Liu14b
|
||
$^1B_2 (\pi \rightarrow \pi^\star$) &6.70&6.67&6.68& &6.56&6.54 &6.54&6.53&6.82 &5.96&6.1 \\
|
||
$^1B_2 (n \rightarrow 3p)$ &6.92&6.91&6.94& &6.88&6.86 &6.96&6.95&7.07 &7.24&6.88 \\
|
||
$^1A_1 (n \rightarrow 3p)$ &7.00&7.00&7.03& &6.88&6.87 &7.00&6.99&7.28 &7.28& \\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &8.51&8.49&8.51& &8.32&8.29 &8.28&8.26&8.19 &7.80&$\sim$8.1 \\
|
||
$^3B_1 (n \rightarrow \pi^\star)$ &4.02&4.03& & 4.00$\pm$0.07 &3.90&3.92 &3.91&3.93&3.51 &4.05& \\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.92&4.92& & 4.95$\pm$0.00 &4.90&4.89 &4.89&4.88&5.10 &4.81& \\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &5.48&5.44& & &5.38&5.35 &5.37&5.35&5.60 &5.56& \\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &6.89&6.88& & &6.79&6.78 &6.83&6.79&7.16 &6.98& \\
|
||
\hline
|
||
\multicolumn{12}{c}{Cyclopropenethione}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ \\
|
||
\hline
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &3.46&3.44&3.44& 3.45$\pm$0.01 &3.47&3.45 &3.43&3.41&3.52 &3.23 & \\
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &3.45&3.44&3.45& 3.44$\pm$0.05 &3.42&3.42 &3.43&3.44&3.50 &3.47 & \\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &4.67&4.64&4.62& 4.59$\pm$0.09 &4.66&4.64 &4.64&4.62&4.77 &4.34 & \\
|
||
$^1B_2 (n \rightarrow 3s)$ &5.26&5.24&5.27& &5.23&5.21 &5.34&5.31&5.35 &4.98 & \\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &5.53&5.52&5.51& &5.52&5.50 &5.49&5.47&5.54 &5.52 & \\
|
||
$^1B_2 (n \rightarrow 3p)$ &5.83&5.81&5.83& &5.86&5.84 &5.93&5.90&5.99 &5.88 & \\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &3.33&3.31& & 3.29$\pm$0.03 &3.34&3.32 &3.30& &3.38 &3.20 & \\
|
||
$^3B_1 (n \rightarrow \pi^\star)$ &3.34&3.33& & &3.30&3.30 &3.31&3.32&3.40 &3.30 & \\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.01&4.00& & 4.03$\pm$0.03 &4.03&4.02 &4.02& &4.17 &3.86 & \\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.06&4.04& & &4.09&4.07 &4.03& &4.13 &3.99 & \\
|
||
\hline
|
||
\multicolumn{12}{c}{Methylenecyclopropene}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV}& Th.$^c$ & Th.$^d$ & Exp.$^e$\\
|
||
\hline
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &4.38&4.37&4.34& 4.32$\pm$0.03 &4.32&4.31 &4.31&4.31&4.37 &4.13&4.36 &4.01\\
|
||
$^1B_1 (\pi \rightarrow 3s)$ &5.65&5.66&5.66& &5.35&5.35 &5.44&5.44&5.49 &5.32&5.44 &5.12\\
|
||
$^1A_2 (\pi \rightarrow 3p)$ &5.97&5.98&5.98& 5.92$\pm$0.10 &5.86&5.88 &5.95&5.96&6.00 &5.83& &\\
|
||
$^1A_1(\pi \rightarrow \pi^\star)$$^f$ &6.17&6.18&6.17& 6.20$\pm$0.01 &6.15&6.15 &6.13&6.13&6.36 & &6.13 &6.02\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &3.50&3.50& & 3.44$\pm$0.06 &3.49&3.49$^g$&3.50&3.49&3.66 &3.24& &\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.74&4.74& & 4.67$\pm$0.10 &4.74&4.74$^g$&4.74& &4.87 &4.52& &\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} of Ref. \citenum{Ser02};}
|
||
$^b${Electron impact experiment of Ref. \citenum{Har74}, note that the 5.5 eV peak was assigned differently in the original paper, and we follow here the analysis of Serrano-Andr\'es\cite{Ser02}
|
||
whereas the 6.1 eV assignment was ``supposed'' in the original paper; experimental $\lambda_{\mathrm{max}}$ have been measured at at 3.62 eV and 6.52 eV for the $^1B_1 (n \rightarrow \pi^\star)$ and
|
||
$^1B_2 (\pi \rightarrow \pi^\star$) transitions, respectively; \cite{Bre72}}
|
||
$^c${{\CASPT} of Refs. \citenum{Mer96} and \citenum{Roo96};}
|
||
$^d${ {\CCT} of Ref. \citenum{Sch17};}
|
||
$^e${$\lambda_{\mathrm{max}}$ in pentane at -78$^o$C from Ref. \citenum{Sta84};}
|
||
$^f${Significant state mixing with the $^1A_1(\pi \rightarrow 3p)$ transition, rendering unambiguous attribution difficult;}
|
||
$^g${As can be seen in the SI, our {\FCI} estimates are 3.45$\pm$0.04 and 4.79$\pm$0.02 for the two lowest triplet states of methylenecyclopropene with {\AVDZ} hinting that the CC3 and CCSDT
|
||
results might be slightly too low for the second transition. }
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
|
||
Our results are listed in Tables \ref{Table-2} and S2. As above, considering the Pople's basis set, we note very small differences between {\CCT}, {\CCSDT}, and {\CCSDTQ} results, the latter method giving transition energies
|
||
systematically falling within the {\FCI} extrapolation incertitude, but in one case, the lowest totally symmetric state of methylenecyclopropene for which the {\CCSDTQ} value is ``off'' by 0.02 eV only. Depending on the state, it is
|
||
either {\CCT} or {\CCSDT} that is the closest to {\CCSDTQ}. In fact, considering all {\CCSDTQ}/{\Pop} data listed in Table \ref{Table-2} as benchmark, the mean absolute deviation of {\CCT} and {\CCSDT} are 0.019 and 0.016 eV, respectively,
|
||
hinting that the improvements brought by the latter more computationally intensive method are limited for these compounds. For the lowest $B_2$ state of methylenecyclopropene, one of the most challenging cases (\%$T_1 = 85\%$),
|
||
it is clear from the {\exCI} value that only {\CCSDTQ} is really close to the spot, the {\CCT} and {\CCSDT} results being slightly too large by ca. 0.05 eV. It seems likely that the same pattern appears for the corresponding state in
|
||
cyclopropenethione, although in that case the FCI error bar is large, preventing definitive conclusions. Interestingly, the quite small Pople basis set provides data within ca. 0.10 eV of basis set convergence at {\CCT} level for 80\%\ of
|
||
the transitions. There are of course exceptions to this rule, e.g., the strongly dipole-allowed $^1A_1 (\pi \rightarrow \pi^\star)$ ES of cyclopropenone and the $^1B_1(\pi \rightarrow 3s)$ ES of methylenecyclopropene are significantly
|
||
too blueshifted with the Pople's basis set (Table S2). For cyclopropenone, our {\CCSDT}/{\AVTZ} estimates do agree reasonably well with the {\CASPT} data of Serrano-Andr\'es, but for the $^1B_2 (\pi \rightarrow \pi^\star)$ state
|
||
that we locate significantly higher in energy and the three Rydberg states that CC foresees at significantly smaller energies. The current {\NEV} results are globally in better agreement with the CC values than the older {\CASPT} estimates,
|
||
though some non-negligible deviations pertain. Even if comparisons with experiments should be made very cautiously, we note that the CC data are clearly more coherent with the electron impact measurements\cite{Har74} for the Rydberg
|
||
states than the original {\CASPT} values. For cyclopropenethione, we obtain transition energies typically in agreement or larger than those obtained with {\CASPT}, \cite{Ser02} though there is no obvious relationship between the
|
||
valence/Rydberg nature of the considered ES and the relative {\CASPT} error. The average absolute deviation between our {\NEV} and {\CCT} results is 0.08 eV only. Eventually for methylenecyclopropene, our values logically agree
|
||
very well with the recent estimates of Schwabe and Goerigk, \cite{Sch17} obtained at the {\CCT}/{\AVTZ} level on a different geometry, whereas the available {\CASPT} values \cite{Mer96,Roo96} appear too low as compared to the
|
||
current {\NEV} and {\CCSDT} values. For this compound, the available experimental data being wavelength of maximal absorption determined in condensed phase, \cite{Sta84} only a qualitative match is logically reached between theory and experiment.
|
||
|
||
\subsubsection{Acrolein, butadiene, and glyoxal}
|
||
|
||
Let us now turn to three pseudo-linear $\pi$-conjugated systems that have been the subject to several ES investigations before, namely, acrolein, \cite{Aqu03,Sah06,Car10b,Lea12,Gua13,Mai14,Aza17b,Sch17,Bat17}
|
||
butadiene, \cite{Dal04,Sah06,Sch08,Sil10c,Li11,Wat12,Dad12,Lea12,Ise12,Ise13,Sch17,Shu17,Sok17,Chi18,Cop18,Tra19,Loo19c} and glyoxal, \cite{Sta97b,Koh03,Hat05c,Sah06,Lea12,Poo14,Sch17,Aza17b,Loo18b}
|
||
that we all consider in their most stable \emph{trans} conformation in the following. Amongst these works, it is worth highlighting the detailed theoretical investigation by Saha, Ehara, and Nakatsuji, who reported a huge
|
||
number of ES in these three systems using a coherent theoretical Symmetry-Adapted-Cluster Configuration-Interaction (SAC-CI) protocol. \cite{Sah06} Our results are listed in Tables \ref{Table-3} and S3.
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in acrolein, butadiene, and glyoxal. All values are in eV.}
|
||
\label{Table-3}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{p{2.9cm}|p{.5cm}p{.9cm}p{1.1cm}p{1.45cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{12}{c}{Acrolein}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Th.$^b$ & Exp.$^c$ \\
|
||
\hline
|
||
$^1A'' (n \rightarrow \pi^\star)$ &3.83&3.80& &3.85$\pm$0.01&3.77&3.74& 3.74&3.73&3.76 &3.63&3.83&3.71 \\
|
||
$^1A' (\pi \rightarrow \pi^\star)$ &6.83&6.86& &6.59$\pm$0.05$^f$&6.67&6.70& 6.65&6.69&6.67 &6.10&6.92&6.41 \\
|
||
$^1A'' (n \rightarrow \pi^\star)$ &6.94&6.89& & &6.75&6.72& 6.75& &7.16 &6.26&7.40& \\
|
||
$^1A' (n \rightarrow 3s)$ &7.22&7.23& & &6.99&7.00& 7.07& &7.05 &6.97&7.19&7.08 \\
|
||
$^3A'' (n \rightarrow \pi^\star)$ &3.55&3.53& &3.60$\pm$0.01&3.47&3.45& 3.46& &3.46 &3.39&3.61& \\
|
||
$^3A' (\pi \rightarrow \pi^\star)$ &3.94&3.95& &3.98$\pm$0.03&3.95&3.95& 3.94& &3.95 &3.81&3.87& \\
|
||
$^3A' (\pi \rightarrow \pi^\star)$ &6.25&6.23& & &6.22&6.21& 6.19& &6.23 & &6.21& \\
|
||
$^3A'' (n \rightarrow \pi^\star)$ &6.81&6.74& & &6.60& & 6.61& &6.83 & &7.36& \\
|
||
\hline
|
||
\multicolumn{12}{c}{Butadiene}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^b$ & Th.$^d$ & Exp$^e$ \\
|
||
\hline
|
||
$^1B_u (\pi \rightarrow \pi^\star)$ &6.41&6.43&6.41&6.41$\pm$0.02 &6.25&6.27& 6.22&6.24 &6.68 &6.33&6.36&5.92\\
|
||
$^1B_g (\pi \rightarrow 3s)$ &6.53&6.55&6.54& &6.26&6.27& 6.33&6.34 &6.44 &6.18&6.32&6.21\\
|
||
$^1A_g (\pi \rightarrow \pi^\star)$ &6.73&6.63&6.56&6.55$\pm$0.04$^f$ &6.68&6.59& 6.67&6.60 &6.70 &6.56&6.60& \\
|
||
$^1A_u (\pi \rightarrow 3p)$ &6.87&6.89&6.87& &6.57&6.59& 6.64&6.66 &6.84 &6.45&6.56&6.64\\
|
||
$^1A_u (\pi \rightarrow 3p)$ &6.93&6.95&6.94&6.95$\pm$0.01 &6.73&6.74& 6.80&6.81 &7.01 &6.65&6.74&6.80\\
|
||
$^1B_u (\pi \rightarrow 3p)$ &7.98&8.00&7.98& &7.86&7.87& 7.68& &7.45 &7.08&7.02&7.07\\
|
||
$^3B_u (\pi \rightarrow \pi^\star)$ &3.35&3.36& &3.37$\pm$0.03 &3.36&3.36& 3.36& &3.40 &3.20& &3.22\\
|
||
$^3A_g (\pi \rightarrow \pi^\star)$ &5.22&5.22& & &5.21&5.21& 5.20& &5.30 &5.08& &4.91\\
|
||
$^3B_g (\pi \rightarrow 3s)$ &6.46&6.47& &6.40$\pm$0.03 &6.20&6.21& 6.28& &6.38 &6.14& &\\
|
||
\hline
|
||
\multicolumn{12}{c}{Glyoxal}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^b$ & Th.$^g$ & Exp.$^h$\\
|
||
\hline
|
||
$^1A_u (n \rightarrow \pi^\star)$ &2.94&2.94&2.94& 2.93$\pm$0.03 &2.90&2.90& 2.88&2.88 &2.90 &3.10&2.93 &2.8 \\
|
||
$^1B_g (n \rightarrow \pi^\star)$ &4.34&4.32&4.31&4.28$\pm$0.06 &4.30&4.28& 4.27&4.25 &4.30 &4.68&4.39 &$\sim$4.4\\
|
||
$^1A_g (n,n \rightarrow \pi^\star,\pi^\star)$&6.74&6.24&5.67&5.60$\pm$0.09$^f$ &6.70&6.22& 6.76&6.35 &5.52 &5.66& &\\
|
||
$^1B_g (n \rightarrow \pi^\star)$ &6.81&6.83&6.79& &6.59&6.61& 6.58&6.61 &6.64 &7.54&6.63 &7.45\\
|
||
$^1B_u (n \rightarrow 3p)$ &7.72&7.74&7.76& &7.55&7.56& 7.67&7.69 &7.84 &7.83&7.61 &$\sim$7.7\\
|
||
$^3A_u (n \rightarrow \pi^\star)$ &2.55&2.55& &2.54$\pm$0.04 &2.49&2.49& 2.49&2.49 &2.49 &2.63& &2.5\\
|
||
$^3B_g (n \rightarrow \pi^\star)$ &3.97&3.95& & &3.91&3.90& 3.90&3.89 &3.99 &4.12& &$\sim$3.8\\
|
||
$^3B_u (\pi \rightarrow \pi^\star)$ &5.22&5.20& & &5.20&5.19& 5.17&5.15 &5.17 &5.35& &$\sim$5.2\\
|
||
$^3A_g (\pi \rightarrow \pi^\star)$ &6.35&6.35& & &6.34&6.34& 6.30&6.30 &6.33 & & &\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} of Ref. \citenum{Aqu03};}
|
||
$^b${SAC-CI of Ref. \citenum{Sah06};}
|
||
$^c${Vacuum UV spectra from Ref. \citenum{Wal45}; for the lowest state, the same 3.71 eV value is reported in Ref. \citenum{Bec70}.}
|
||
$^d${MR-AQCC of Ref. \citenum{Dal04}, theoretical best estimates listed for the lowest $B_u$ and $A_g$ states;}
|
||
$^e${Electron impact experiment from Refs. \citenum{Fli78} and \citenum{Doe81} for the singlet states and from Ref. \citenum{Mos73} for the two lowest triplet transitions;
|
||
note that for the lowest $B_u$ state, there is a vibrational structure with peaks at 5.76, 5.92, and 6.05 eV;}
|
||
$^f${From Ref. \citenum{Loo19c};}
|
||
$^g${{\CCT} of Ref. \citenum{Sch17};}
|
||
$^h${Electron impact experiment from Ref. \citenum{Ver80} but for the second $^1B_g$ ES for which the value is from another work; \cite{Rob85b} note that
|
||
for the lowest $^1B_g$ ($^1B_u$) ES, a range of 4.2--4.5 (7.4--7.9) eV is given Ref. \citenum{Ver80}. }
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
|
||
Acrolein, due to its lower symmetry and high density of ES with mixed characters, is challenging for theory and {\CCSDTQ} calculations were technically impossible despite our efforts. For the lowest $n \rightarrow \pi^\star$
|
||
transitions of both spin symmetries, the {\FCI} estimates come with a tiny error bar, and it is obvious that the CC estimates are slightly too low, especially with {\CCSDT}. Nevertheless, at the exception the second singlet
|
||
and triplet $A''$ ES, the {\CCT} and {\CCSDT} estimates are within $\pm$0.03 eV of each other. These $A''$ ES are also the only two transitions for which the discrepancies between {\CCT} and {\NEV} exceed 0.20 eV. This hints
|
||
at a good accuracy for all other transitions. This statement is supported by the fact that the current CC values are nearly systematically bracketed by previous {\CASPT} (lower bound)\cite{Aqu03} and SAC-CI (higher bound)\cite{Sah06} results,
|
||
consistently with the typical error signs of these two models. For the two lowest triplet states, the present {\CCT}/{\AVTZ} values are also within $\pm$0.05 eV of recent MR-CI estimates (3.50 and 3.89 eV). \cite{Mai14} As
|
||
can be seen in Table S3, {\AVTZ} allows being very close from basis set convergence, the largest variation when going to {\AVQZ} (+0.04 eV) is obtained for the second $^1A'$ ES of Rydberg nature. As the experimental data
|
||
are limited to measured UV spectra, \cite{Wal45,Bec70} one has therefore to be cautious in establishing TBE for acrolein (\emph{vide infra}).
|
||
|
||
The nature and relative energies of the lowest bright $B_u$ and dark $A_g$ ES of butadiene have been puzzling theoretical chemists for many years. It is beyond our scope to provide an exhaustive
|
||
list of previous calculations and experimental estimates for these two hallmark ES, and we refer the readers to Refs. \citenum{Wat12} and \citenum{Shu17} for overviews and references. For the $B_u$ transition
|
||
the best previous TBE we are aware of is the 6.21 eV value obtained by Watson and Chan using a computational strategy similar to ours. \cite{Wat12} Our {\CCSDT}/{\AVTZ} value of 6.24 eV is obviously compatible
|
||
with this reference value, and our TBE value is actually 6.21 eV as well (\emph{vide infra}). For the $A_g$ state, we believe that our previous basis-corrected exCI estimate of 6.50 eV \cite{Loo19c} remains
|
||
the most accurate available to date. These two values are slightly smaller than the heath-bath CI data obtained by Chien \emph{et al.} with a double-$\zeta$ basis and a slightly different geometry: 6.45 and 6.58 eV for $B_u$ and $A_g$,
|
||
respectively. \cite{Chi18} One can of course find many other estimates, e.g., at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} levels, \cite{Sok17} for these two ES.
|
||
More globally, in butadiene, we find an excellent coherence between the {\CCT}, {\CCSDT}, and {\CCSDTQ} estimates, that all fall in a $\pm$0.02 eV window. Unsurprisingly, this does not apply for the
|
||
already mentioned $^1A_g$ ES that is 0.2 and 0.1 eV too high with the two former CC methods, consistent with the large electronic reorganization taking place in that state. For all the other butadiene ES listed in
|
||
Table \ref{Table-3}, both {\CCT} and {\CCSDT} can be trusted. We also note that the {\NEV} estimates are within 0.1--0.2 eV of the CC values, but for the lowest $B_u$ ES, which is very dependent on the selected
|
||
active space (see the SI). Finally, as can be seen in Table S3, {\AVTZ} is sufficient for most ES, but a significant basis set effect exists for the Rydberg $^1B_u (\pi \rightarrow 3p)$ ES with an energy decrease as large as -0.12 eV
|
||
when going from {\AVTZ} to {\AVQZ}. For the records, we note that the available electron impact data \cite{Mos73,Fli78,Doe81} provide the very same ES ordering value as our calculations.
|
||
|
||
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, i.e., highly consistent CC estimates, reasonable agreement between {\NEV} and {\CCT} estimates, and limited basis set effects beyond {\AVTZ}
|
||
but for the considered Rydberg state (see Tables \ref{Table-3} and S3). This Rydberg $^1B_u (n \rightarrow 3p)$ state also hows a comparatively large deviation between {\CCT} and {\CCSDTQ}, that is 0.04 eV. More interestingly,
|
||
glyoxal presents a low-lying ``true'' double ES, $^1A_g (n,n \rightarrow \pi^\star,\pi^\star)$, a transition that is totally unseen by approaches that do not explicitly include double excitations during the calculation of transition energies, e.g., TD-DFT, {\CCSD},
|
||
or {\AD}. Compared to the {\exCI} values, the {\CCT} and {\CCSDT} estimates for this transition are too large by ca. 1.0 and 0.5 eV, respectively, whereas both the {\CCSDTQ} and {\NEV} approaches are much closer to the spot, as already
|
||
mentioned in our previous work. \cite{Loo19c} For the other transitions, the present {\CCT} estimates are logically coherent with the values of Ref. \citenum{Sch17} obtained with the same approach on a different
|
||
geometry, and remain slightly lower than the SAC-CI estimates of Ref. \citenum{Sah06}. Once more, the experimental data \cite{Ver80,Rob85b} make an unhelpful guide in view of the targeted accuracy.
|
||
|
||
\subsubsection{Acetone, cyanoformaldehyde, isobutene, propynal, thioacetone, and thiopropynal}
|
||
|
||
Let us now turn towards six other four-atom compounds. There are several earlier estimates of vertical transition energies for both acetone \cite{Gwa95,Mer96b,Roo96,Wib98,Toz99b,Wib02,Sch08,Sil10c,Car10,Pas12,Ise12,Gua13,Sch17}
|
||
and isobutene. \cite{Wib02,Car10,Ise12} To the best of our knowledge, the previous computational efforts were mainly focussed on 0-0 energies of the lowest-lying states for the four other compounds. \cite{Koh03,Hat05c,Sen11b,Loo18b,Loo19a}
|
||
There are also a few experimental values available for all six derivatives. \cite{Bir73,Jud83,Bra74,Sta75,Joh79,Jud83,Jud84c,Rob85,Pal87,Kar91b,Xin93} Our main data are reported in Tables \ref{Table-4}
|
||
and S4.
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in acetone, cyanonformaldehyde, isobutene, propynal, thioacetone, and thiopropynal. All values are in eV.}
|
||
\label{Table-4}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{.9cm}p{1.1cm}p{1.4cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{12}{c}{Acetone}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Th.$^b$ & Exp.$^c$ \\
|
||
\hline
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &4.55&4.52&4.53&4.60$\pm$0.05 &4.50&4.48& 4.48&4.46&4.48 &4.18&4.18&4.48\\
|
||
$^1B_2 (n \rightarrow 3s)$ &6.65&6.64&6.68& &6.31&6.30& 6.43&6.42&6.81 &6.58&6.58&6.36\\
|
||
$^1A_2 (n \rightarrow 3p)$ &7.83&7.83&7.87& &7.37&7.36& 7.45&7.43 &7.65 &7.34&7.34&7.36\\
|
||
$^1A_1 (n \rightarrow 3p)$ &7.81&7.81&7.84& &7.39&7.38& 7.48&7.48&7.75 &7.26&7.26&7.41\\
|
||
$^1B_2 (n \rightarrow 3p)$ &7.87&7.87&7.91& &7.56&7.55& 7.59&7.58 &7.91 &7.48&7.48&7.45\\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &4.21&4.19& &4.18$\pm$0.04 &4.16&4.14& 4.15& &4.20 &3.90&3.90&4.15\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &6.32&6.30& & &6.31&6.28& 6.28& &6.28 &5.98&5.98&\\
|
||
\hline
|
||
\multicolumn{12}{c}{Cyanoformaldehyde}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.}\\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^d$ \\
|
||
\hline
|
||
$^1A'' (n \rightarrow \pi^\star)$ &3.91&3.89& &3.92$\pm$0.02 &3.86&3.84&3.83 &3.81&3.98& 3.26\\
|
||
$^1A'' (\pi \rightarrow \pi^\star)$ &6.64&6.67& &6.60$\pm$0.07 &6.51&6.54&6.42 &6.46&6.44& \\
|
||
$^3A'' (n \rightarrow \pi^\star)$ &3.53&3.51& &3.48$\pm$0.06 &3.47&3.45&3.46 & &3.58& \\
|
||
$^3A' (\pi \rightarrow \pi^\star)$ &5.07&5.07& & &5.03&5.03&5.01 & &5.35& \\
|
||
\hline
|
||
\multicolumn{12}{c}{Isobutene}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^e$ & Exp.$^f$ & Exp.$^g$ \\
|
||
\hline
|
||
$^1B_1 (\pi \rightarrow 3s)$ &6.77&6.77& &6.78$\pm$0.08 &6.39&6.39& 6.45&6.46&6.63&6.40&6.15&6.17 \\
|
||
$^1A_1 (\pi \rightarrow 3p)$ &7.16&7.17& &7.16$\pm$0.02 &7.00&7.00& 7.00&7.01&7.20&6.96& &6.71 \\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.52&4.53& &4.56$\pm$0.02 &4.54&4.54& 4.53& &4.61& &4.21 &4.3 \\
|
||
\hline
|
||
\multicolumn{12}{c}{Propynal}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.} \\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^h$ \\
|
||
\hline
|
||
$^1A'' (n \rightarrow \pi^\star)$ &3.90&3.87& &3.84$\pm$0.06 &3.85&3.82&3.82 &3.80&3.95& 3.24\\
|
||
$^1A'' (\pi \rightarrow \pi^\star)$ &5.69&5.73& &5.64$\pm$0.08 &5.59&5.62&5.51 &5.54&5.50& \\
|
||
$^3A'' (n \rightarrow \pi^\star)$ &3.56&3.54& &3.54$\pm$0.04 &3.50&3.48&3.49 & &3.59& 2.99\\
|
||
$^3A' (\pi \rightarrow \pi^\star)$ &4.46&4.47& &4.44$\pm$0.08 &4.40&4.44&4.43 & &4.63& \\
|
||
\hline
|
||
\multicolumn{12}{c}{Thioacetone}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^i$ \\
|
||
\hline
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &2.58&2.56&2.56&2.61$\pm$0.05 &2.59&2.57&2.55 &2.53&2.55 &2.33\\
|
||
$^1B_2 (n \rightarrow 4s)$ &5.65&5.64&5.66& &5.44&5.43&5.55 &5.54&5.72 &5.49\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.09&6.10&6.07& &5.97&5.98&5.90 &5.91&6.24 &5.64\\
|
||
$^1B_2 (n \rightarrow 4p)$ &6.59&6.59&6.59& &6.45&6.44&6.51 & &6.62 &6.40\\
|
||
$^1A_1 (n \rightarrow 4p)$ &6.95&6.95&6.96& &6.54&6.53&6.61 &6.60&6.52 &6.52\\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &2.36&2.34& &2.36$\pm$0.00 &2.36&2.35&2.34 & &2.32 &2.14\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &3.45&3.45& & &3.51&3.50&3.46 & &3.46 &\\
|
||
\hline
|
||
\multicolumn{12}{c}{Thiopropynal}\\
|
||
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^j$ \\
|
||
\hline
|
||
$^1A'' (n \rightarrow \pi^\star)$ &2.09&2.06& &2.08$\pm$0.01 &2.09&2.06&2.05 &2.03&2.05 &1.82\\
|
||
$^3A'' (n \rightarrow \pi^\star)$ &1.84&1.82& & &1.83&1.82&1.81 & &1.81 &1.64\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} of Ref. \citenum{Mer96b};}
|
||
$^b${EOM-CCSD of Ref. \citenum{Gwa95};}
|
||
$^c${Two lowest singlet states: various experiments summarized in Ref. \citenum{Rob85}; three next singlet states: REMPI experiments from Ref. \citenum{Xin93}; lowest triplet: trapped electron measurements from Ref. \citenum{Sta75};}
|
||
$^d${0-0 energy reported in Ref. \citenum{Kar91b};}
|
||
$^e${EOM-CCSD from Ref. \citenum{Car10};}
|
||
$^f${Energy loss experiment from Ref. \citenum{Joh79};}
|
||
$^g${VUV experiment from Ref. \citenum{Pal87} (we report the lowest of the $\pi \rightarrow 3p$ state for the $^1A_1$ state)};
|
||
$^h${0-0 energies of Refs. \citenum{Bra74} (singlet) and \citenum{Bir73} (triplet);}
|
||
$^i${0-0 energies from Ref. \citenum{Jud83};}
|
||
$^i${0-0 energies from Ref. \citenum{Jud84c}.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
For acetone, one should clearly distinguish the valence ES, for which both methodological and basis set effects are small, and the Rydberg transitions that, not only are very
|
||
sensitive to the basis set, but are upshifted by ca. 0.04 eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} transition energies of Merch\'an and coworkers listed on the
|
||
r.h.s. of Table \ref{Table-4} are quite clearly too small, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the details of the calculations, as the Urban group obtained {\CASPT}
|
||
excitation energies of 4.40, 4.09 and 6.22 eV for the $^1A_2$, $^3A_2$, and $^3A_1$ ES, \cite{Pas12} in much better agreement with ours. Their estimates for the three $n \rightarrow 3p$ transitions of 7.52, 7.57, and 7.53 eV
|
||
for the $^1A_2$, $^1A_1$, and $^1B_2$ ES also systematically fall within 0.10 eV of our current CC values, whereas for these three ES, the current {\NEV} values are quite clearly too large.
|
||
|
||
In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to the CC expansion, as illustrated by the maximal discrepancies of $\pm$0.02 eV between the {\CCT} and {\CCSDTQ} results
|
||
with the {\Pop} basis set. While the lowest $n \rightarrow \pi^\star$ transition of both spin symmetries are rather insensitive to the selected basis set, all other states need quite large bases to be correctly described (Table S4).
|
||
As expected our theoretical vertical transition energies show the same ranking but are systematically larger than the available experimental 0-0 energies.
|
||
|
||
For the isoleectronic isobutene, we considered two singlet Rydberg and one triplet valence ES. For all three cases, we note very nice agreement between {\CCT} and {\CCSDT} results for all considered basis sets, the
|
||
CC results being also within or very close to the {\FCI} estimates with Pople's basis set. The match with the {\CCSD} results of Caricato and coworkers, \cite{Car10} is also very satisfying.
|
||
|
||
For the three remaining compounds, namely, cyanoformaldehyde, propynal, and thiopropynal, we report low-lying valence transitions all showing a largely dominant single excitation character. The basis set
|
||
effects are clearly under control (they are only significant for the second $^1A''$ ES of cyanoformaldehyde) and we could not detect any variation larger than 0.03 eV between the {\CCT} and {\CCSDT} values for
|
||
a given basis, hinting that the CC values should be close to the spot, as confirmed by the {\FCI} data.
|
||
|
||
\subsubsection{Intermediate conclusions}
|
||
\label{sec-ic}
|
||
|
||
As we have seen for the 15 four-atom molecules considered here, we found extremely consistent transition energies between the tested CC approaches and the {\FCI} estimates in the vast majority of cases,
|
||
Importantly, we confirm the previous conclusions obtained on smaller compounds:\cite{Loo18a} i) {\CCSDTQ} values systematically fall within, or are extremely close from, the {\FCI} error bar,
|
||
ii) both {\CCT} and {\CCSDT} are also highly trustable when the considered ES does not show a very strong double excitation character. Indeed, considering all the 54 ``single transitions''
|
||
for which {\CCSDTQ} estimates could be obtained (only excluding the lowest $^1A_g$ ES of butadiene and glyoxal), we determined trifling mean signed errors (MSE of 0.00 eV), tiny
|
||
MAE (0.01 and 0.02 eV), and small maximal deviations (0.05 and 0.04 eV) for {\CCT} and {\CCSDT}, respectively. This clearly indicates that these two approaches provide chemically-accurate
|
||
estimates (errors $<$ 1 kcal.mol$^{-1}$ or 0.043 eV) for most electronic transitions. Interestingly, some of us have shown that {\CCT} also provides chemically-accurate 0-0 energies as compared
|
||
to experimental values for most valence transitions. \cite{Loo18b,Loo19a,Sue19} When comparing the {\NEV} and {\CCT} ({\CCSDT}) results obtained with {\AVTZ} for all transitions in four-atom molecules,
|
||
one obtains a mean signed deviation of +0.09 (+0.09) eV and a mean absolute deviation of 0.11 (0.12) eV, considering all 91 (65) ES for which comparisons are possible, again excluding only
|
||
the lowest $^1A_g$ states of butadiene and glyoxal. Although the error cannot be fully ascribed to the multi-reference method, that is additionally dependent of the selected active space, it seems to
|
||
indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with the {\CASPT} approach that, from the comparisons discussed above,
|
||
generally undershoots the transition energies.
|
||
|
||
\subsection{Five-atom molecules}
|
||
|
||
Let us now turn towards five-member rings. We treat here five classical derivatives that have been considered in several previous theoretical studies (\emph{vide infra}): cyclopentadiene, furan, imidazole, pyrrole, and thiophene. As the most
|
||
advanced levels of theory used in the previous Section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds, one has to rely on the nature of the ES and the consistency between the results of
|
||
different approaches to deduce TBE.
|
||
|
||
For furan, previous theoretical works have been performed with almost all possible wavefunction approaches, \cite{Ser93b,Nak96,Tro97b,Chr98b,Chr98c,Wan00,Gro03,Pas06b,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Hol15,Sch17} but the
|
||
present work is, to the best of our knowledge, the first to disclose {\CCSDT} values as well as {\CCT} energies obtained with a quadruple-$\zeta$ basis set. Our results for ten low-lying states are listed in Tables \ref{Table-5} and S5.
|
||
All computed singlet (triplet) transitions show $\%T_1$ in the 92--94 \%\ (97--99) \%\ range, and consistently the maximal discrepancies between the {\CCT} and {\CCSDT} transition energies are small (0.04 eV). In addition there is a
|
||
good consistency between the present data and both the {\NEV} results of Ref. \citenum{Pas06b} and the MR-CC values of Ref. \citenum{Li10c} for almost all transitions, but the $^1B_2 (\pi \rightarrow 3p)$ excitation that we predict to
|
||
be significantly higher than in most previous works, even after accounting for the quite large basis set effects (-0.10 eV between {\AVTZ} and {\AVQZ} estimates, see Table S5). We trust that our estimate is the most accurate to date for that ES.
|
||
Interestingly, the recent {\AT} values of Ref. \citenum{Hol15} are smaller by ca. -0.2 eV as compared to {\CCSDT} values for all transitions (see Table \ref{Table-6}), consistent with the error sign we found in smaller compounds with ADC(3). \cite{Loo18a}
|
||
Eventually, we note that the experimental data, \cite{Vee76b,Fli76,Rob85b} provide the same state ordering as our calculations.
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in furan and pyrrole. All values are in eV.}
|
||
\label{Table-5}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{15}{c}{Furan}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{7}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$ &Th.$^d$ &Th.$^e$ &Exp.$^f$&Exp.$^g$\\
|
||
\hline
|
||
$^1A_2 (\pi \rightarrow 3s)$ &6.26&6.28 &6.00&6.00 &6.08&6.09& &5.92&6.13&5.94&5.91&6.10&5.91 & \\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &6.50&6.52 &6.37&6.39 &6.34&6.37& &6.04&6.42&6.51&6.10&6.42&6.04 & 6.06\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.71&6.67 &6.62&6.58 &6.58&6.56& &6.16&6.71&6.89&6.44& & & 6.44 \\
|
||
$^1B_1 (\pi \rightarrow 3p)$ &6.76&6.77 &6.55&6.56 &6.63&6.64& &6.46&6.68&6.46&6.45&6.66&6.47 & \\
|
||
$^1A_2 (\pi \rightarrow 3p)$ &6.97&6.99 &6.73&6.74 &6.80&6.81& &6.59&6.79&6.61&6.60&6.83&6.61 & \\
|
||
$^1B_2 (\pi \rightarrow 3p)$ &7.53&7.54 &7.39&7.40 &7.23& & &6.48&6.91&6.87&6.72&7.36&6.75 & \\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.28&4.28 &4.25&4.23 &4.22& & &3.99& &4.26& & & &4.0\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &5.56&5.54 &5.51&5.49 &5.48& & &5.15& &5.53& & & &5.2\\
|
||
$^3A_2 (\pi \rightarrow 3s)$ &6.18&6.19 &5.94&5.94 &6.02& & &5.86& &5.89& & & &\\
|
||
$^3B_1 (\pi \rightarrow 3p)$ &6.69&6.71 &6.51&6.51 &6.59& & &6.42& &6.41& & & &\\
|
||
\hline
|
||
\multicolumn{15}{c}{Pyrrole}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{7}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^a$ &Th.$^h$ &Th.$^c$ &Th.$^i$ &Th.$^j$ &Exp.$^l$&Exp.$^l$\\
|
||
\hline
|
||
$^1A_2 (\pi \rightarrow 3s)$ &5.25&5.25 &5.15&5.14 &5.24&5.24&& 5.08&5.45&5.10&5.20&5.27&5.22&\\
|
||
$^1B_1 (\pi \rightarrow 3p)$ &5.99&5.98 &5.89&5.87 &5.98&6.00&& 5.85&6.21&5.79&5.95&6.00& &\\
|
||
$^1A_2 (\pi \rightarrow 3p)$ &6.27&6.27 &5.94&5.93 &6.01& && 5.83&6.14&5.81&5.94&7.03& &5.87\\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &6.33&6.33 &6.28&6.28 &6.25&6.26&& 5.92&6.95&5.96&6.04&6.08& &5.98\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.43&6.40 &6.35&6.32 &6.32&6.30&& 5.92&6.59&6.53&6.37&6.15& &\\
|
||
$^1B_2 (\pi \rightarrow 3p)$ &7.20&7.20 &7.00&7.00 &6.83& && 5.78&6.26&6.61&6.57& &\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.59&4.58 &4.56&4.54 &4.53& && 4.27& &4.53& & &4.21\\
|
||
$^3A_2 (\pi \rightarrow 3s)$ &5.22&5.22 &5.12&5.12 &5.21& && 5.04& &5.07& & &5.1\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &5.54&5.54 &5.49&5.48 &5.46& && 5.16& &5.53& & &\\
|
||
$^3B_1 (\pi \rightarrow 3p)$ &5.91&5.90 &5.82&5.81 &5.92& && 5.82& &5.74& & &\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Ser93b};}
|
||
$^b${{\NEV} from Ref. \citenum{Pas06b};}
|
||
$^c${MR-CC from Ref. \citenum{Li10c};}
|
||
$^d${{\AT} from Ref. \citenum{Hol15};}
|
||
$^e${{\CCT} from Ref. \citenum{Sch17};}
|
||
$^f${Various experiments summarized in Ref. \citenum{Wan00};}
|
||
$^g${Electron impact from Ref. \citenum{Vee76b}: for the $^1A_1$ state two values (6.44 and 6.61 eV) are reported, whereas for the two lowest triplet states, 3.99 eV and 5.22 eV values can be found in Ref. \citenum{Fli76};}
|
||
$^h${{\NEV} from Ref. \citenum{Pas06c};}
|
||
$^i${Best estimate from Ref. \citenum{Chr99}, based on CC calculations;}
|
||
$^j${XMS-{\CASPT} from Ref. \citenum{Hei19};}
|
||
$^k${Electron impact from Refs. \citenum{Vee76b} and \citenum{Fli76b};}
|
||
$^l${Vapour UV spectra from Refs. \citenum{Pal03b}, \citenum{Hor67}, and \citenum{Bav76}.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
Like furan, pyrrole has been extensively investigated previously using a large palette of approaches. \cite{Ser93b,Nak96,Tro97,Pal98,Chr99,Wan00,Roo02,Pal03b,Pas06c,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Nev14,Sch17,Hei19}
|
||
We report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All considered transitions have very large $\%T_1$ but for the totally symmetric $\pi \rightarrow \pi^\star$ excitation ($\%T_1 = 86\%$). For all states, we found
|
||
highly consistent {\CCT} and {\CCSDT} results, often significantly larger than older multi-reference estimates, \cite{Ser93b,Roo02,Li10c} but in nice agreement with the very recent XMS-{\CASPT} results of the Gonzalez'
|
||
group, \cite{Hei19} at the exception of the $^1A_2 (\pi \rightarrow 3p)$ transition. The match obtained with the 20-years old extrapolated CC values of Christiansen and coworkers \cite{Chr99} is also remarkable but for the two
|
||
$B_2$ transitions that were reported as significantly mixed in that venerable work. For the lowest singlet ES, the {\FCI}/{\Pop} value is 5.24$\pm$0.02 eV confirming the performances of both {\CCT} and {\CCSDT} for that transition.
|
||
As can be seen in Table S5, {\AVTZ} yields basis-set converged transition energies, but, like in furan, for the Rydberg $^1B_2 (\pi \rightarrow 3p)$ transition that is significantly redshifted (-0.09 eV) when going to the
|
||
quadruple-$\zeta$ basis set. As mentioned in Thiel's work, \cite{Sch08} the experimental spectra of pyrrole are quite broad and the few available experiments \cite{Hor67,Bav76,Fli76b,Vee76b,Pal98,Pal03b} can only be used as general guidelines.
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in cyclopentadiene, imidazole, and thiophene. All values are in eV.}
|
||
\label{Table-6}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{15}{c}{Cyclopentadiene}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{7}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$ &Th.$^d$ &Exp.$^e$ &Exp.$^f$&Exp.$^g$\\
|
||
\hline
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &5.79&5.80 &5.59&5.60 &5.54&5.56&5.65 &5.27 &5.54&5.19 &5.58 &5.26 & &5.20\\
|
||
$^1A_2 (\pi \rightarrow 3s)$ &6.08&6.08 &5.70&5.70 &5.77&5.78&5.92 &5.65 &5.58&5.62 &5.79 &5.68 &5.63 &\\
|
||
$^1B_1 (\pi \rightarrow 3p)$ &6.57&6.58 &6.34&6.34 &6.40&6.41&6.42 &6.24 &6.17&6.24 &6.43 & &6.35 &\\
|
||
$^1A_2 (\pi \rightarrow 3p)$ &6.67&6.67 &6.39&6.39 &6.45&6.46&6.59 &6.30 &6.21&6.25 &6.47 & & &6.26\\
|
||
$^1B_2 (\pi \rightarrow 3p)$ &7.06&7.07 &6.55&6.55 &6.56&6.56&6.58 &6.25 &6.22&6.27 &6.58 & &6.31 &\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.67&6.60 &6.59&6.53 &6.57&\hl{XXX}&6.75 &6.31 &6.76&6.42 &6.65 & & &$\sim$6.2\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &3.33&3.33 &3.32&3.31 &3.32& &3.41 &3.15 &3.40& & &3.10 &\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &5.16&5.15 &5.14&5.13 &5.12& &5.36 &4.90 &5.18& & &$>$4.7 & &\\
|
||
$^3A_2 (\pi \rightarrow 3s)$ &6.01&6.02 &5.65&5.65 &5.73& &5.73 &5.63 &5.56& & &\\
|
||
$^3B_1 (\pi \rightarrow 3p)$ &6.51&6.52 &6.30&6.30 &6.36& &6.40 &6.25 &6.19& & &\\
|
||
\hline
|
||
\multicolumn{15}{c}{Imidazole}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{7}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & & {\NEV} &Th.$^h$ &Th.$^i$ &&&Exp.$^j$\\
|
||
\hline
|
||
$^1A'' (\pi \rightarrow 3s)$ &5.77&5.77 &5.60&5.60 &5.71&& &5.71& &&&$\sim$5.2\\
|
||
$^1A' (\pi \rightarrow \pi^\star)$ &6.51&6.51 &6.43&6.43 &6.41&& &6.72&6.25&&&$\sim$6.4\\
|
||
$^1A'' (n \rightarrow \pi^\star)$ &6.66&6.66 &6.42&6.42 &6.50&& &6.52&6.65&&& \\
|
||
$^1A' (\pi \rightarrow 3p)$ &7.04&7.02 &6.93&6.89 &6.87&& &6.49& &&&\\
|
||
$^3A' (\pi \rightarrow \pi^\star)$ &4.83&4.81 &4.78& &4.75&& &4.49&4.65&&&\\
|
||
$^3A'' (\pi \rightarrow 3s)$ &5.72&5.72 &5.57&5.56 &5.67&& &5.68& &&&\\
|
||
$^3A' (\pi \rightarrow \pi^\star)$ &5.88&5.88 &5.78& &5.74&& &5.47&5.64&&&\\
|
||
$^3A'' (n \rightarrow \pi^\star)$ &6.48&6.46 &6.37&6.35 &6.33&& &6.07&6.25&&&\\
|
||
\hline
|
||
\multicolumn{15}{c}{Thiophene}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{7}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^k$ &Th.$^l$ &Th.$^m$ &Th.$^n$ &Exp.$^o$ &Exp.$^p$&Exp.$^q$\\
|
||
\hline
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &5.79&5.77 &5.70&5.68 &5.65&5.64& &5.33&5.41&5.70&5.64&5.16&5.13&5.16\\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &6.23&6.24 &6.05&6.06 &5.96&5.98& &5.72&5.72&6.10&5.97&5.99&5.83&\\
|
||
$^1A_2 (\pi \rightarrow 3s)$ &6.26&6.26 &6.07&6.06 &6.14&6.14& &5.93&5.70&6.05&6.23& & &\\
|
||
$^1B_1 (\pi \rightarrow 3p)$ &6.18&6.17 &6.19&6.17 &6.14&6.14& &6.30&5.87&6.30&6.17& & &6.71\\
|
||
$^1A_2 (\pi \rightarrow 3p)$ &6.32& &6.33&6.31 &6.25&6.21& &6.35&6.03&6.28&6.33& &\\
|
||
$^1B_1 (\pi \rightarrow 3s)$ &6.62&6.62 &6.42&6.41 &6.50&6.49& &6.23&6.12&6.36&6.68& & &6.47\\
|
||
$^1B_2 (\pi \rightarrow 3p)$ &7.45&7.44 &7.45&7.44 &7.29&7.29& &6.56&6.41&6.81&6.97& & &6.60\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &7.50&7.46 &7.41& &7.35& & &6.69&7.32&7.71&7.74&6.61&\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &3.95&3.94 &3.96&3.94 &3.94& & &3.75&3.94& &3.96&&3.74&\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.90&4.90 &4.82&4.81 &4.77& & &4.50&4.86& &4.87&&4.62&\\
|
||
$^3B_1 (\pi \rightarrow 3p)$ &6.00&5.98 &6.01&5.99 &5.95& & &5.90&5.94& &6.01&\\
|
||
$^3A_2 (\pi \rightarrow 3s)$ &6.20&6.20 &6.01&6.00 &6.09& & &5.88&5.75& &5.83&\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Ser93b};}
|
||
$^b${SAC-CI from Ref. \citenum{Wan00b};}
|
||
$^c${MR-MP from Ref. \citenum{Nak96};}
|
||
$^d${{\CCT} from Ref. \citenum{Sch17};}
|
||
$^e${Electron impact from Ref. \citenum{Fru79};}
|
||
$^f${Gas phase absorption from Ref. \citenum{McD91b};}
|
||
$^g${Energy loss from Ref. \citenum{McD85} for the two valence state; two-photon resonant experiment from Ref. \citenum{Sab92} for the $^1A_2$ Rydberg ES;}
|
||
$^h${{\CASPT} from Ref. \citenum{Ser96b};}
|
||
$^i${{\CCT} from Ref. \citenum{Sil10c};}
|
||
$^j${Gas-phase experimental estimates from Ref. \citenum{Dev06};}
|
||
$^k${{\CASPT} from Ref. \citenum{Ser93c};}
|
||
$^l${SAC-CI from Ref. \citenum{Wan01};}
|
||
$^m${CCSDR(3) from Ref. \citenum{Pas07};}%, this work also contains {\NEV} estimates;}
|
||
$^n${TBE from Ref. \citenum{Hol14}, based on EOM-CCSD for singlet and ADC(2) for triplets;}
|
||
$^o${0-0 energies from Ref. \citenum{Dil72};}
|
||
$^p${0-0 energies from Ref. \citenum{Var82} for the singlets and energy loss experiment from Ref. \citenum{Hab03} for the triplet ES;}
|
||
$^q${0-0 energies from Ref. \citenum{Hol14}.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
Although a quite significant array of previous wavefunction studies has been performed for cyclopentadiene not only at the {\CASPT}, \cite{Ser93b,Sch08,Sil10c} and CC \cite{Sch08,Sil10b,Sch17} levels but also with
|
||
SAC-CI \cite{Wan00b} and various multi-reference approaches, \cite{Nak96,She09b} this compound has been less intensively studied than furan and pyrrole (\emph{vide infra}), probably due to the presence of the
|
||
methylene group that renders the computations significantly more expensive. All transitions listed in Tables \ref{Table-6} and S6 are characterized by $\%T_1$ exceeding 93\%\ but for the $^1A_1 (\pi \rightarrow \pi^\star)$
|
||
excitation that has a nature similar to the lowest $A_g$ state of butadiene ($\%T_1 = 79\%$). Consistently, the {\CCT} and {\CCSDT} results are nearly identical for all ES but for that transition. By comparing the results
|
||
obtained for this $A_1 (\pi \rightarrow \pi^\star)$ transition to its butadiene counterpart, one can infer that the {\CCSDT} estimate is probably too large by ca. 0.04--0.08 eV, and that the {\NEV} value is likely not more accurate
|
||
than the {\CCSDT} one. This statement is also in line with the results of Ref. \citenum{Loo19c}. For the two $B_2 (\pi \rightarrow \pi^\star)$ transitions, we could obtain a {\FCI}/{\Pop} estimates of 5.78$\pm$0.02 eV and
|
||
(singlet) 3.33$\pm$0.05 eV (triplet); the {\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As can be seen in Tables \ref{Table-6} and S6, the basis set effects are rather moderate
|
||
for all transitions, with no variations larger than 0.10 eV (0.02 eV) between {\AVDZ} and {\AVTZ} ({\AVTZ} and {\AVQZ}). When comparing to literature data, our values are unsurprisingly consistent with the {\CCT} values of
|
||
Schwabe and Goerigk, \cite{Sch17} and tend to be significantly larger than earlier {\CASPT} \cite{Ser93b,Sil10c} and MR-MP \cite{Nak96} estimates. As expected, a few gas-phase experiments are available as well for this
|
||
derivative, \cite{Fru79,McD85,McD91b,Sab92} but hardly allow to make the final call.
|
||
|
||
|
||
Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remaining the 1996 {\CASPT} work of Serrano-Andres \emph{et al}, \cite{Ser96b} and the
|
||
basis-set extrapolated {\CCT} investigation of Silva-Junior \emph{et al} for the valence transitions. \cite{Sil10c} The experimental data in gas-phase are also limited.\cite{Dev06}
|
||
Our results are displayed in Tables \ref{Table-6} and S6. The {\CCT} and {\CCSDT} values are quite consistent despite the fact that the $\%T_1$ of the two singlet $A'$ states
|
||
are slightly smaller 90\%. For all eight considered transitions, the basis set effects are moderate and {\AVTZ} yield results within 0.03 eV of {\AVQZ} (Table S6 in the SI). \hl{NEV ?}
|
||
|
||
Finally, the ES thiophene, which is one of the most important building block in organic electronic devices, were the subject of a few previous theoretical investigations, \cite{Ser93c,Pal99,Wan01,Kle02,Pas07,Hol14} that unveiled a series of
|
||
transitions that were not yet characterized in the available measurements. \cite{Dil72,Fli76,Fli76b,Var82,Hab03,Pal99,Hol14} To our knowledge, the present work is the first to report CC calculations obtained with (iterative)
|
||
triples and therefore constitutes the most accurate estimates to date. Indeed, all the transitions listed in Tables \ref{Table-6} and S6 are characterized by a largely dominant single excitation character, with $\%T_1$ above
|
||
90\%\ but for the two $^1A_1$ transitions that show $\%T_1$ of 88\%\ and 87\%. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for the lowest totally symmetric transition. Thiophene is
|
||
also a typical compound in which unambiguous characterization of the nature of the ES is difficult, with e.g., a strong mixing between the second and third singlet $B_2$ ES rendering the assignment of the valence
|
||
($\pi \rightarrow \pi^\star$) or Rydberg ($\pi \rightarrow 3p$) character of that transittion uneasy at the {\CCT} level. We note that contradictory assignments can be found in the literature. \cite{Ser93c,Wan01,Pas07} As for the
|
||
previously discussed isostructural systems, the only ES that undergoes significant basis set effects beyond {\AVTZ} is the Rydberg $^1B_2 (\pi \rightarrow 3p)$ (-0.09 eV when upgrading to {\AVQZ}, see Table S6).
|
||
The data of Table \ref{Table-6} are globally in good agreement with the previously reported values with discrepancies that are however significant for the three highest-lying singlet states.
|
||
|
||
\subsection{Six-atom molecules}
|
||
|
||
Let us now turn towards six-member cycles playing a key role in chemistry: benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine. To the best of our knowledge, the present work
|
||
is the first to propose {\CCSDT} reference energies as well as {\CCT} values obtained with a {\AVQZ} for all these compounds. Of course, these systems have been investigated before, and beyond Thiel's
|
||
benchmarks, \cite{Sch08,Sil10b,Sil10c} it is worth to point out the investigations of Del Bene and coworkers, \cite{Del97b} performed with a CC approach including perturbative corrections for the triples, and of Nooijen, \cite{Noo99}
|
||
who used {\STEOM}, to study the ES of all these derivatives with a theoretically consistent protocol. However, these two works considered the singlet ES only.
|
||
|
||
\subsubsection{Benzene, pyrazine, and tetrazine}
|
||
|
||
\hl{Martial: please do check Rydberg assignments throughout}
|
||
|
||
These three highly-symmetric systems allow to directly perform {\CCSDT}/{\AVTZ} calculations for singlet states without the need for basis set extrapolation. Benzene was studied many times
|
||
before, \cite{Sob93,Lor95b,Chr96c,Pac96,Del97b,Noo99,Hal02,Li07b,Sch08,Dev08,Sil10b,Sil10c,Li11,Lea12,Kan14,Sch17,Dut18,Sha19,Loo19c} and we report in Tables \ref{Table-7} and S7 estimates obtained for
|
||
five singlet and three triplet ES, all characterized by $\%T_1$ exceeding 90\%\ but the lowest singlet (86\%). As can be seen, the two CC approaches are again yielding very consistent transitions energies
|
||
and {\AVTZ} is essentially providing basis set converged transition energies. The present estimates are also very consistent with early {\CCT}\cite{Chr96c} and very recent RASPT2 values. \cite{Sha19}
|
||
For both the singlet and triplet transitions, our values are slightly larger than available electron impact/multi-photon measurements. \cite{Doe69,Nak80,Joh76,Joh83,Hir91}
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in benzene All values are in eV.}
|
||
\label{Table-7}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$ &Th.$^d$ &Exp.$^e$ &Exp.$^f$\\
|
||
\hline
|
||
$^1B_{2u} (\pi \rightarrow \pi^\star)$ &5.13&5.10 &5.11&5.08 &5.09&5.06& &4.84&5.08&5.06&5.03& &4.90\\
|
||
$^1B_{1u} (\pi \rightarrow \pi^\star)$ &6.68&6.69 &6.50&6.50 &6.44&6.45& &6.30&6.54&6.22&6.23& &6.20\\
|
||
$^1E_{1g} (\pi \rightarrow 3s)$ &6.75&6.76 &6.46&6.46 &6.52&6.52& &6.38&6.51&6.42& & &6.33\\
|
||
$^1A_{2u} (\pi \rightarrow 3p)$ &7.24&7.25 &7.02&7.02 &7.08&7.08& &6.86&6.97&7.06& & &6.93\\
|
||
$^1E_{2u} (\pi \rightarrow 3p)$ &7.34&7.35 &7.09&7.09 &7.15&7.15& &6.91&7.03&7.12& & &6.95\\
|
||
$^3B_{1u} (\pi \rightarrow \pi^\star)$ &4.18&4.16 &4.19&4.17 &4.18& & &3.89&4.15&3.88&4.11&3.95&\\
|
||
$^3E_{1u}(\pi \rightarrow \pi^\star)$ &4.95&4.94 &4.89&4.88 &4.86& & &4.49&4.86&4.72&4.75&4.75&\\
|
||
$^3B_{2u} (\pi \rightarrow \pi^\star)$ &6.06&6.06 &5.86&5.86 &5.81& & &5.49&5.88&5.54&5.67&5.60&\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Lor95b};}
|
||
$^b${{\CCT} from Ref. \citenum{Chr96c};}
|
||
$^c${SAC-CI from Ref. \citenum{Li07b};}
|
||
$^d${RASPT2(18,18) from Ref. \citenum{Sha19};}
|
||
$^e${Electron impact from Ref. \citenum{Doe69};}
|
||
$^f${Jet-cooled experiment from Ref. \citenum{Hir91} for the two lowest states, multi-photon experiments from Refs. \citenum{Joh76} and \citenum{Joh83} for the Rydberg states.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in pyrazine and tetrazine. All values are in eV.}
|
||
\label{Table-8}
|
||
\begin{footnotesize}
|
||
\vspace{-0.3 cm}
|
||
\begin{tabular}{p{2.18cm}|p{.5cm}p{1.cm}|p{.5cm}p{.9cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{14}{c}{Pyrazine}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^e$ &Exp.$^f$\\
|
||
\hline
|
||
$^1B_{3u} (n \rightarrow \pi^\star)$ &4.28&4.28 &4.19&4.19 &4.14&4.15& &3.83&4.12&4.25&4.19&3.93\\
|
||
$^1A_{u} (n \rightarrow \pi^\star)$ &5.08&5.08 &4.98&4.98 &4.97&4.98& &4.36&4.93&5.24&4.93&\\
|
||
$^1B_{2u} (\pi \rightarrow \pi^\star)$ &5.10&5.08 &5.07&5.05 &5.03&5.02& &4.79&4.75&4.84&5.19&4.8&4.81\\
|
||
$^1B_{2g} (n \rightarrow \pi^\star)$ &5.86&5.85 &5.78&5.77 &5.71&5.71& &5.50&5.85&6.04&5.81&5.19\\
|
||
$^1A_{g} (n \rightarrow 3s)$ &6.74&6.73 &6.54&6.53 &6.66&6.65& & &6.83&7.07&6.46&\\
|
||
$^1B_{1g} (n \rightarrow \pi^\star)$ &6.87&6.87 &6.75&6.75 &6.73&6.74& &6.26&6.73& &6.73&&6.10\\
|
||
$^1B_{1u} (\pi \rightarrow \pi^\star)$ &7.10&7.11 &6.92&6.93 &6.86&6.88& &6.60&6.89&6.68&6.99&6.5&6.51\\
|
||
$^1B_{1g} (n \rightarrow 3p)$ &7.36&7.37 &7.13&7.14 &7.20&7.21& & &7.31&7.08& &\\
|
||
$^1B_{2u} (n \rightarrow 3p)$ &7.39&7.39 &7.14&7.13 &7.25& & & &7.45&7.67&7.06&\\
|
||
$^1B_{1u} (\pi \rightarrow 3s)$ &7.56&7.55 &7.38&7.37 &7.45& & &7.28&7.50&7.73&7.31&&7.67\\
|
||
$^1B_{1u} (\pi \rightarrow \pi^\star)$ &8.19&8.23 &7.99&8.03 &7.94& & &7.43&7.96&8.24&8.08&\\
|
||
$^3B_{3u} (n \rightarrow \pi^\star)$ &3.68&3.68 &3.60&3.60 &3.59& & &3.16& & & &3.33\\
|
||
$^3B_{1u} (\pi \rightarrow \pi^\star)$ &4.39&4.36 &4.40&4.36 &4.39& & &4.15& & & &4.04\\
|
||
$^3B_{2u} (\pi \rightarrow \pi^\star)$ &4.56&4.55 &4.46&4.45 &4.40& & &4.28& & & &$\sim$4.4\\
|
||
$^3A_{u} (n \rightarrow \pi^\star)$ &5.05&5.05 &4.93&4.93 &4.93& & &4.19& & & &4.2\\
|
||
$^3B_{2g} (n \rightarrow \pi^\star)$ &5.18&5.17 &5.11&5.11 &5.08& & &4.81& & & &4.49\\
|
||
$^3B_{1u} (\pi \rightarrow \pi^\star)$ &5.38&5.37 &5.32&5.31 &5.29& & &4.98& & & &\\
|
||
\hline
|
||
\multicolumn{14}{c}{Tetrazine}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^g$ &Th.$^h$ &Th.$^i$&Th.$^j$ &Th.$^k$ &Exp.$^l$\\
|
||
\hline
|
||
$^1B_{3u} (n \rightarrow \pi^\star)$ &2.53&2.54 &2.49&2.50 &2.46&2.47& &1.96&2.22&2.01&2.29&2.46&2.35\\
|
||
$^1A_{u} (n \rightarrow \pi^\star)$ &3.75&3.75 &3.69&3.70 &3.67&3.69& &3.06&3.62&3.09&3.41&3.78&3.6\\
|
||
$^1A_{g} (\mathrm{double})$$^m$ &6.22&5.86 &6.22&5.86 &6.21&5.96&4.61 &4.37&5.06&4.34&4.66& &\\
|
||
$^1B_{1g} (n \rightarrow \pi^\star)$ &5.01&5.02 &4.97&4.98 &4.91&4.93& &4.51&4.73&4.47&4.53&4.87&\\
|
||
$^1B_{2u} (\pi \rightarrow \pi^\star)$ &5.29&5.26 &5.27&5.25 &5.23&5.21& &4.89&4.90& &5.59&5.08&4.97\\
|
||
$^1B_{2g} (n \rightarrow \pi^\star)$ &5.56&5.52 &5.53&5.50 &5.46&5.45& &5.05&5.09&4.92&5.59&5.28&\\
|
||
$^1A_{u} (n \rightarrow \pi^\star)$ &5.61&5.61 &5.59&5.59 &5.52&5.53& &5.28&5.23&5.32&5.95&5.39&5.5\\
|
||
$^1B_{3g} (\mathrm{double})$$^m$ &7.64& &7.62& &7.62& &6.15 &5.16&6.30&5.26&6.01& &5.92\\
|
||
$^1B_{2g} (n \rightarrow \pi^\star)$ &6.24&6.22 &6.17&6.16 &6.13& & &5.48&6.16&5.78&6.05&6.16&\\
|
||
$^1B_{1g} (n \rightarrow \pi^\star)$ &7.04&7.04 &6.98&6.98 &6.92& & &5.99&6.73&6.20&6.92&6.80&\\
|
||
$^3B_{3u} (n \rightarrow \pi^\star)$ &1.87&1.88 &1.86&1.86 &1.85& & &1.45&1.71& & &1.87&1.7\\
|
||
$^3A_{u} (n \rightarrow \pi^\star)$ &3.48&3.49 &3.43&3.44 &3.44& & &2.81&3.47& & &3.49&2.90\\
|
||
$^3B_{1g} (n \rightarrow \pi^\star)$ &4.25&4.25 &4.23&4.23 &4.20& & &3.76&3.97& & &4.18&\\
|
||
$^3B_{1u} (\pi \rightarrow \pi^\star)$ &4.54&4.49 &4.54&4.49 &4.54& & &4.25&3.67& & &4.36&\\
|
||
$^3B_{2u} (\pi \rightarrow \pi^\star)$ &4.65&4.64 &4.58&4.58 &4.52& & &4.29&4.35& & &4.39&\\
|
||
$^3B_{2g} (n \rightarrow \pi^\star)$ &5.11&5.11 &5.09&5.08 &5.05& & &4.67&4.78& & &4.89&\\
|
||
$^3A_{u} (n \rightarrow \pi^\star)$ &5.17&5.17 &5.15&5.15 &5.11& & &4.85&4.89& & &4.96&\\
|
||
$^3B_{3g} (\mathrm{double})$$^m$ &7.35& &7.33& &7.35& &5.51 &5.08& & & & &\\
|
||
$^3B_{1u} (\pi \rightarrow \pi^\star)$ &5.51&5.50 &5.46&5.46 &5.42& & &5.09&5.31& & &5.32&\\
|
||
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.5 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Web99};}
|
||
$^b${{\STEOM} from Ref. \citenum{Noo99};}
|
||
$^c${SAC-CI from Ref. \citenum{Li07b};}
|
||
$^d${{\CCT} from Ref. \citenum{Sch17};}
|
||
$^e${Dip spectroscopy from Ref. \citenum{Oku90} ($B_{3u}$ and $B_{2g}$ states) and EEL from Ref. \citenum{Wal91} (other states);}
|
||
$^f${UV max from Ref. \citenum{Bol84};}
|
||
$^g${{\CASPT} from Ref. \citenum{Rub99};}
|
||
$^h${Ext-{\STEOM} from Ref. \citenum{Noo00};}
|
||
$^i${GVVPT2 from Ref. \citenum{Dev08};}
|
||
$^j${{\NEV} from Ref. \citenum{Ang09};}
|
||
$^k${{\CCT} from Ref. \citenum{Sil10c};}
|
||
$^l${From Ref. \citenum{Pal97}, the singlets are from EEL, but for the 4.97 and 5.92 eV values that are from VUV; the triplets are from EEL, and other triplet peaks are mentioned at 4.21, 4.6, and 5.2 eV but not identified;}
|
||
$^m${all these doubly ES have a $(n,n \rightarrow \pi^\star, \pi^\star)$ character.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
|
||
Numerous previous theoretical estimates are available for both pyrazine, \cite{Ful92,Del97b,Web99,Noo99,Li07b,Sch08,Sau09,Sil10c,Woy10,Car10,Lea12,Kan14,Sch17,Dut18} and tetrazine, \cite{Sta96,Del97b,Rub99,Noo99,Ada00,Noo00,Ang09,Sch08,Sau09,Sil10b,Sil10c,Car10,Lea12,Kan14,Sch17,Dut18,Pas18b}
|
||
for which the $D_{2h}$ symmetry helps distinguishing the different ES. Our results are collected in Tables \ref{Table-8} and S8. In pyrazine, all transitions are characterized by $\%T_1 > 85\%$, but for $^1B_{1g} (n \rightarrow \pi^\star)$, and the
|
||
changes in going from {\CCT} to {\CCSDT} are always trifling but for the highest-lying singlet state considered here. When going from triple-$\zeta$ to quadruple-$\zeta$, the variations do not exceed 0.04 eV, even for the four considered Rydberg
|
||
ES. This indicates that one can probably be highly confident in the present estimates. Again, the previous {\CASPT} estimates, \cite{Ful92,Web99,Sch08} appear to be globally too low, whereas the unconventional CASPT3 results that are
|
||
available, \cite{Woy10} are too large. The same holds for the SAC-CI results. \cite{Li07b} In fact we obtain globally our best match with the {\STEOM} values of Nooijen (but for the highest ES), \cite{Noo99} and recent {\CCT} estimates.
|
||
\cite{Sch17}. The experimental data we are aware of, \cite{Bol84,Oku90,Wal91,Ste11c} do not report all transitions, but provide globally a similar ranking for the triplet transitions.
|
||
|
||
|
||
For tetrazine, we consider valence ES only, but three transitions present a true double excitation nature ($\%T_1 < 10\%$), for which {\CCT} nor {\CCSDT} can not be viewed as reliable, and the best approach is likely {\NEV}. \cite{Loo19c}
|
||
For all other transitions, the $\%T_1$ are in the 80-90\%\ range for singlets and larger than 95\%\ for triplets, and the results of the two CC approaches are very consistent, but for the lowest $^3B_{1u} (\pi \rightarrow \pi^\star)$ excitation.
|
||
In all other cases, there is a good consistency between the values we obtained with the two CC models, and the basis set effects are very small beyond {\AVTZ} with maximal variations of 0.02 eV only (Table S8). The present values are
|
||
almost systematically larger than previous {\CASPT},\cite{Rub99} {\STEOM}, \cite{Noo00} and GVVPT2 \cite{Dev08} estimates, and are globally in agreement with Thiel's {\CCT}/{\AVTZ} values, \cite{Sil10c} although we note variations
|
||
of ca. 0.20 eV for some specific transitions like the $B_{2g}$ transitions, likely due to the use of different geometries in that work. The experimental EEL values from Palmer's work, \cite{Pal97} show a reasonable agreement with our estimates.
|
||
|
||
|
||
\subsubsection{Pyridazine, pyridine, pyrimidine, and triazine}
|
||
|
||
Those four azabenzenes, of $C_{2v}$ and $D_{3h}$ symmetry, are also popular molecules for ES calculations. \cite{Pal91,Ful92,Wal92,Lor95,Del97b,Noo97,Noo99,Fis00,Cai00b,Wan01b,Sch08,Sil10b,Sil10c,Car10,Lea12,Kan14,Sch17,Dut18}
|
||
Our results for pyridazine and pyridine are collected in Tables \ref{Table-9} and S9. For the former compounds, the available wavefunction results \cite{Pal91,Ful92,Del97b,Noo99,Fis00,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18}
|
||
have all considered the singlet transitions only, at the exception of rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} The $\%T_1$ are larger than 85\%\ (95\%) for the singlet (triplet) transitions,
|
||
and the only state for which there is a variation larger than 0.03 eV between the {\AVDZ} {\CCT} and {\CCSDT} energies, but for the $^3B_2 (\pi \rightarrow \pi^\star)$ transition. \hl{BASIS SETS} For the valence singlet
|
||
ES, we find again a quite good agreement with previous {\STEOM} \cite{Noo99} and CC \cite{Del97b,Sil10c} estimates, but are again significantly higher than {\CASPT} estimates. \cite{Ful92,Sil10c} For the triplets, the
|
||
present data represents the best published to date. Interestingly, beyond the usually cited experiments, \cite{Inn88,Pal91} there is a very recent experimental EEL analysis for pyridazine, \cite{Lin19} that localized
|
||
almost all ES. The transition energies reported in this very recent effort are systematically smaller than our CC estimates, by ca. -0.20 eV, but remarkably show exactly the exact same ranking.
|
||
|
||
For pyridine, that has been more thoroughly investigated with wavefunction approaches, \cite{Ful92,Lor95,Del97b,Noo97,Noo99,Cai00b,Wan01b,Sch08,Sil10b,Sil10c,Car10,Kan14,Sch17,Dut18} and for which we could found two
|
||
detailed EEL experiments, \cite{Wal90,Lin16} the general trends described for pyridazine pertain: i) large $\%T_1$ and good CC consistency for all transitions listed in Table \ref{Table-9}; ii) \hl{basis}; iii) good agreement with previous
|
||
CC estimates; and iv) same ranking of the ES as in the most recent measurements. \cite{Lin16} Beyond those aspects, it is worth to underline that the second $^1B_2 (\pi \rightarrow \pi^\star)$ ES is strongly mixed with a nearby
|
||
Rydberg transition that is separated by only 0.03 eV at the {\CCT}/{AVTZ} level, making the analysis particularly challenging for that specific transition. \hl{Keep or not A1 transitiion}
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in pyridazine and pyridine. All values are in eV.}
|
||
\label{Table-9}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{13}{c}{Pyridazine}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{2}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^e$ &Exp.$^f$\\
|
||
\hline
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &3.95&3.95 &3.86&3.86 &3.83& &3.48&3.76&3.65&3.85&&3.36\\
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &4.49&4.48 &4.39&4.39 &4.37& &3.66&4.46&4.28&4.44&&4.02\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &5.36&5.32 &5.33&5.30 &5.29& &4.86&4.92&4.86&5.20&5.0&5.01\\
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &5.88&5.86 &5.80&5.78 &5.74& &5.09&5.66&5.52&5.66&&5.61\\
|
||
$^1B_2 (n \rightarrow 3s)$ &6.26&6.27 &6.06&6.06 &6.17& & &6.45& & && \\
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &6.51&6.51 &6.41&6.41 &6.37& &5.80&6.41&6.20&6.33&&6.00\\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &6.96&6.97 &6.79&6.80 &6.74& &6.61&6.77&6.44&6.68&&6.50\\
|
||
$^3B_1 (n \rightarrow \pi^\star)$ &3.27&3.26 &3.20&3.20 &3.19& & & & & &&3.06\\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &4.19&4.19 &4.11&4.11 &4.11& & & & & &&3.55\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.39&4.36 &4.39&4.35 &4.38& & & & & &4.0&4.33\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.93&4.94 &4.87&4.86 &4.83& & & & & &4.4&4.68\\
|
||
\hline
|
||
\multicolumn{13}{c}{Pyridine}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{2}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\NEV} &Th.$^g$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^h$ &Exp.$^i$ \\
|
||
\hline
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &5.12&5.10 &5.01&5.00 &4.96& &4.91&4.90&4.80&4.95&5.24&4.78\\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &5.23&5.20 &5.21&5.18 &5.17& &4.84&4.82&4.81&5.12&4.99&4.99\\
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &5.55&5.54 &5.41&5.41 &5.40& &5.17&5.31&5.24&5.41&5.43&5.40\\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.84&6.84 &6.64&6.63 &6.63& &6.42&6.62&6.36&6.60&6.38& \\
|
||
$^1A_1 (n \rightarrow 3s)$ &6.92&6.92 &6.71&6.71 &6.76& &6.70&6.96&6.64& &6.28&6.25\\
|
||
$^1A_2 (\pi \rightarrow 3s)$ &6.98&6.99 &6.74&6.75 &6.81& &6.75&6.90&6.53& & & \\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$$^j$ &7.50&7.52 &7.40&\hl{xxx} &\hl{xxx} & &7.48&7.29&7.14&7.33&7.22&7.20\\
|
||
$^1B_1 (\pi \rightarrow 3p)$ &7.54&7.55 &7.32& \hl{xxx} &7.38& &7.25&7.42&7.10& & & \\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &7.56& &7.34&\hl{xxx} &7.39& &7.23&7.37&7.26&7.39&7.22&6.39\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.33&4.31 &4.34&4.31 &4.33& &4.05& & &4.28& &3.86\\
|
||
$^3B_1 (n \rightarrow \pi^\star)$ &4.57&4.56 &4.47&4.47 &4.46& &4.41& & &4.42& &4.12\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.92&4.91 &4.83&4.83 &4.79& &4.56& & &4.72& &4.47\\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &5.14&5.13 &5.08&\hl{xxx} &5.05& &4.73& & &4.96& & \\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &5.51&5.49 &5.37&5.36 &5.35& &5.10& & &5.53& &5.40\\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &6.46&6.45 &6.30&6.29 &6.25& &6.02& & &6.22& &6.09\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.4 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Ful92};}
|
||
$^b${{\STEOM} from Ref. \citenum{Noo99};}
|
||
$^c${EOM-CCSD({$\tilde{{T}}$}) from Ref. \citenum{Del97b};}
|
||
$^d${CC3-ext. from Ref. \citenum{Sil10c};}
|
||
$^e${EEL from Ref. \citenum{Pal91};}
|
||
$^f${EEL from Ref. \citenum{Lin19};}
|
||
$^g${{\CASPT} from Ref. \citenum{Lor95};}
|
||
$^h${EEL from Ref. \citenum{Wal90};}
|
||
$^i${EEL from Ref. \citenum{Lin16};}
|
||
$^j${Significant state mixing with a close-lying Rydberg transition, rendering unambiguous attribution difficult.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
The results obtained for both pyrimidine and triazine are listed in Tables \ref{Table-10} and S10. For the former derivative previous theoretical \cite{Ful92,Del97b,Ser97b,Noo99,Ohr01,Fis03b,Li07b,Sch08,Sil10b,Sil10c,Car10,Kan14,Sch17,Dut18}
|
||
and experimental \cite{Bol84,Pal90,Lin15} studies are rather extensive, as it can be viewed as the smallest model of DNA bases. For triazine, which does not have an abelian point group, one finds less theoretical studies,
|
||
\cite{Ful92,Wal92,Pal95,Del97b,Noo99,Oli05,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18}, especially for the triplet transitions. \cite{Wal92,Pal95,Oli05} The experimental data are also less numerous. \cite{Bol84,Wal92}
|
||
As in pyridazine and pyridine, all the ES listed in Table \ref{Table-10} show $\%T_1$ larger than 85\%\ for singlet and 95\%\ for triplet, so that {\CCT} and {\CCSDT} are highly coherent, but possibly in one case, the
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ transitions in pyrimidine. The basis set effects are also small, with no variations larger than 0.10 (0.03) eV between double-$\zeta$ and triple-$\zeta$ (triple-$\zeta$ and quadruple-$\zeta$) and only
|
||
slightly larger variations for the two Rydberg transitions. For both compounds, the current values are almost systematically larger than most previously published data. For the triplets of triazine, the three lowest ES estimated
|
||
by {\CASPT} previously are too low by ca. -0.5 eV.
|
||
|
||
s
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies determined in pyrimidine and triazine. All values are in eV.}
|
||
\label{Table-10}
|
||
\begin{footnotesize}
|
||
\begin{tabular}{l|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}p{.5cm}p{.5cm}p{.5cm}p{.6cm}p{.6cm}}
|
||
\hline
|
||
\multicolumn{14}{c}{Pyrimidine}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^e$ &Exp.$^f$\\
|
||
\hline
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &4.58&4.57 &4.48&4.48 &4.44&& &4.26&4.40&4.32&4.24& 4.2 &4.18 \\
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &4.99&4.99 &4.89&4.88 &4.86&& &4.49&4.72&4.74&4.74& &4.69 \\
|
||
$^1B_2 (\pi \rightarrow \pi^\star)$ &5.47&5.44 &5.44&5.41 &5.41&& &5.47&5.04&5.29&5.01& 5.12 &5.18 \\
|
||
$^1A_2 (n \rightarrow \pi^\star)$ &6.07&6.06 &5.98&5.97 &5.93&& & &5.94&5.98&5.84& 6.05 &5.67 \\
|
||
$^1B_1 (n \rightarrow \pi^\star)$ &6.39& &6.29&6.29 &6.26&& &6.03&6.18&6.35&6.11& &6.02 \\
|
||
$^1B_2 (n \rightarrow 3s)$ &6.81&6.80 &6.61&6.59 &6.72&& & &6.85&6.84&6.57& & \\
|
||
$^1A_1 (\pi \rightarrow \pi^\star)$ &7.08&7.09 &6.93&6.94 &6.87&& &7.10&6.87&6.86&6.57& 6.7 &6.69 \\
|
||
$^3B_1 (n \rightarrow \pi^\star)$ &4.20&4.20 &4.12&4.11 &4.10&& &3.81& &4.11& & &3.85 \\
|
||
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.55&4.52 &4.56&4.52 &4.55&& &4.35& &4.39& & &4.42 \\
|
||
$^3A_2 (n \rightarrow \pi^\star)$ &4.77&4.76 &4.67&4.67 &4.66&& &4.24& &4.71& & &4.18 \\
|
||
$^3B_2 (\pi \rightarrow \pi^\star)$ &5.08&5.08 &5.00&5.00 &4.96&& &4.83& &4.81& & &4.93 \\
|
||
\hline
|
||
\multicolumn{14}{c}{Triazine}\\
|
||
& \multicolumn{2}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^g$ &Th.$^b$ &Th.$^d$&Th.$^h$ &Exp.$^h$ \\
|
||
\hline
|
||
$^1A_1'' (n \rightarrow \pi^\star)$ &4.85&4.84 &4.76&4.74 &4.73&4.72& &4.11&4.58&4.49&4.70 &\\
|
||
$^1A_2'' (n \rightarrow \pi^\star)$ &4.84&4.84 &4.78&4.78 &4.74&4.75& &4.30&4.74&4.54&4.71 &4.59\\
|
||
$^1E'' (n \rightarrow \pi^\star)$ &4.89&4.89 &4.82&4.81 &4.78&4.78& &4.32&4.69&4.56&4.75 &3.97\\
|
||
$^1A_2' (\pi \rightarrow \pi^\star)$ &5.84&5.80 &5.81&5.78 &5.78&5.75& &5.59&5.35&5.36&5.71 &5.70\\
|
||
$^1A_1' (\pi \rightarrow \pi^\star)$ &7.45&7.45 &7.31&7.31 &7.24&7.24& & &7.21&6.90&7.18 &6.86\\
|
||
$^1E' (n \rightarrow 3s)$ &7.44&7.41 &7.24&7.21 &7.35&7.32& & &7.38&7.16& & \\
|
||
$^1E'' (n \rightarrow \pi^\star)$ &7.89&7.86 &7.82&7.80 &7.79&7.78& & & &7.78&7.78 &\\
|
||
$^1E' (\pi \rightarrow \pi^\star)$ &8.12&8.13 &7.97& &7.92&7.94& & &7.82&7.72&7.84 &7.76\\
|
||
$^3A_2'' (n \rightarrow \pi^\star)$ &4.40&4.40 &4.35&4.35 &4.33& & &3.87&\\
|
||
$^3E'' (n \rightarrow \pi^\star)$ &4.59&4.59 &4.52&4.52 &4.51& & &4.04&\\
|
||
$^3A_1'' (n \rightarrow \pi^\star)$ &4.87& &4.78&4.76 &4.75& & &4.15&\\
|
||
$^3A_1' (\pi \rightarrow \pi^\star)$ &4.88&4.85 &4.88&4.85 &4.88& & & &\\
|
||
$^3E' (\pi \rightarrow \pi^\star)$ &5.70&5.68 &5.64& &5.61& & & &\\
|
||
$^3A_2' (\pi \rightarrow \pi^\star)$ &6.85&6.84 &6.69&6.68 &6.63& & &4.76&\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{footnotesize}
|
||
\vspace{-0.4 cm}
|
||
\begin{flushleft}
|
||
\begin{footnotesize}
|
||
$^a${{\CASPT} from Ref. \citenum{Fis03b};}
|
||
$^b${{\STEOM} from Ref. \citenum{Noo99};}
|
||
$^c${SAC-CI from Ref. \citenum{Li07b};}
|
||
$^d${EOM-CCSD({$\tilde{{T}}$}) from Ref. \citenum{Del97b};}
|
||
$^e${UV max from Ref. \citenum{Bol84};}
|
||
$^f${EEL from Ref. \citenum{Lin15};}
|
||
$^g${{\CASPT} from Ref. \citenum{Oli05};}
|
||
$^h${CC3-ext. from Ref. \citenum{Sil10c}.}
|
||
\end{footnotesize}
|
||
\end{flushleft}
|
||
\end{table}
|
||
|
||
\clearpage
|
||
|
||
|
||
\section{Theoretical Best Estimates}
|
||
|
||
In Table \ref{Table-tbe}, we present our theoretical best estimates as obtained with the {\AVTZ} basis set or further corrected for basis set effects. The details of the approach used to get all the TBE are given as well.
|
||
For all states with a dominant single-excitation character, that is when $\%T_1 > 80$\%, we rely on CC results, using the incremental strategy to determine the TBE. For ES with $\%T_1$ between 70\%\ and 80\%,
|
||
our previous works indicated that {\CCSDT} tends to overshoot the transition energies by ca. 0.05--0.10 eV, but that the {\NEV} error tends to be slightly larger (on average). \cite{Loo19c} Therefore, if {\CCSDTQ} or {\FCI}
|
||
results are unavailable, it is hard to make the final call. For the other transitions, we relied either on the current or previous FCI or the {\NEV} values as reference. We indicate some transition energies
|
||
in italics in Table \ref{Table-tbe} to underline that they are (relatively) less accurate. This is the case when: i) {\NEV} results have to be selected; ii) the affordable CC calculations yield quite large changes from one expansion order to another
|
||
despite large $\%T_1$; and iii) there is a very large ES mixing making hard to follow a specific transition from one method (or one basis) to another. To determine the basis set corrections beyond augmented triple-$\zeta$,
|
||
we use the {\CCT}/{\AVQZ} or {\CCT}/{\AVFZ} results. For several compounds, we also provide in the SI, {\CCT}/{\DAVQZ} transition energies. However, we are not using these values as reference. This is because,
|
||
the addition of a second set of diffuse orbitals tends to modify the computed transition energies significantly only when it induces a more complex state mixing. We also stick to the
|
||
frozen-core approximation for two reasons: i) the corrections brought by ``full-correlation'' are generally trifling (typically $\pm$ 0.02 eV) for the compounds under study (see the SI for many examples); and ii) it would be,
|
||
in principle, necessary to add core polarization functions in such ``full'' calculations.
|
||
|
||
Table \ref{Table-tbe} encompasses 238 ES, all at least obtained at the {\CCSDT} level. This set that can be decomposed as follows: 144 singlet and 94 triplet transitions, or 174 valence (99 $\pi \rightarrow \pi^\star$, 71
|
||
$n \rightarrow \pi^\star$ and 4 double excitation) and 64 Rydberg transitions. Amongst the reported transition energies, fifteen can be considered as ``unsafe''. This is a significant extension of all previously
|
||
proposed ES datasets (see Introduction). The Table also reports 90 oscillator strengths, $f$ which is, to our knowledge, the largest set of {\CCT}/{\AVTZ} $f$ reported to date, the previous effort being mostly performed
|
||
at the {\CCT}/TZVP level for Thiel's set. \cite{Kan14} It should also be recalled that all these data are obtained on {\CCT}/{\AVTZ} geometries, consistently with our previous works. \cite{Loo18a,Loo19c} Taken
|
||
together they offer a consistent ensemble of transition energies of ca. 350 electronic transitions of various natures in small and medium-sized organic compounds.
|
||
|
||
\renewcommand*{\arraystretch}{.55}
|
||
\LTcapwidth=\textwidth
|
||
|
||
\begin{footnotesize}
|
||
\begin{longtable}{p{3.3cm}lcccccc}
|
||
\caption{\small TBE values determined for all considered states. For each state, we provide the oscillator strength and percentage of single excitations obtained at the \CCT/{\AVTZ} level,
|
||
as well TBE obtained with the same basis set together with the method used to obtain that TBE. In the right-most columns, we list the values obtained by including further basis set corrections,
|
||
that have always been obtained at the {\CCT} level. Values displayed in italics are likelt relatively less accurate. All values are in the FC approximation.} \label{Table-tbe}\\
|
||
\hline
|
||
& & & & \multicolumn{2}{l}{TBE/{\AVTZ}} & \multicolumn{2}{l}{TBE/CBS} \\
|
||
& State & $f$ & $\%T_1$ & Value & Method$^a$ & Value & Corr. \\
|
||
\hline
|
||
\endfirsthead
|
||
\hline
|
||
& & & & \multicolumn{2}{l}{TBE/{\AVTZ}} & \multicolumn{2}{l}{TBE/CBS} \\
|
||
& State & $f$ & $\%T_1$ & Value & Method $^a$ & Value & Corr. \\
|
||
\hline
|
||
\endhead
|
||
\hline \multicolumn{7}{r}{{Continued on next page}} \\
|
||
\endfoot
|
||
\hline
|
||
\endlastfoot
|
||
Acetone &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 91.1 & 4.47 & B & 4.48 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ &0.000 & 90.5 & 6.46 & B & 6.51 & AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; n \rightarrow 3p)$ & & 90.9 & 7.47 & B & 7.44 & AVQZ \\
|
||
&$^1A_1 (\mathrm{R}; n \rightarrow 3p)$ &0.004 & 90.6 & 7.51 & B & 7.55 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3p)$ &0.029 & 91.2 & 7.62 & B & 7.63 & AVQZ \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.8 & 4.13 & D & 4.15 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.7 & 6.25 & D & 6.27 & AVQZ \\
|
||
Acrolein &$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 87.6 & 3.73 & {\CCSDT}/AVTZ & 3.74 & AVQZ \\
|
||
&$^1A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.344 & 91.2 & 6.69 & {\CCSDT}/AVTZ & 6.69 & AVQZ \\
|
||
&$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 79.4 & \emph{6.72} & D &\emph{6.74} & AVQZ \\
|
||
&$^1A' (\mathrm{R}; n \rightarrow 3s)$ &0.109 & 89.4 & 7.08 & D & 7.12 & AVQZ \\
|
||
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.0 & 3.44 & D & 3.43 & AVQZ \\
|
||
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.6 & 3.94 & D & 3.95 & AVQZ \\
|
||
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 6.18 & D & 6.19 & AVQZ \\
|
||
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 92.7 & \emph{6.54} & E & \emph{6.55}& AVQZ \\%remove
|
||
Benzene &$^1B_{2u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 86.3 & 5.06 & {\CCSDT}/AVTZ & 5.06 &AVQZ \\
|
||
&$^1B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 92.9 & 6.45 & {\CCSDT}/AVTZ &6.44 &AVQZ \\
|
||
&$^1E_{1g} (\mathrm{R}; \pi \rightarrow 3s)$ & & 92.8 & 6.52 & {\CCSDT}/AVTZ &6.54 &AVQZ \\
|
||
&$^1A_{2u} (\mathrm{R}; \pi \rightarrow 3p)$ &0.066 & 93.4 & 7.08 & {\CCSDT}/AVTZ &7.10 &AVQZ \\
|
||
&$^1E_{2u} (\mathrm{R}; \pi \rightarrow 3p)$ & & 92.8 & 7.15 & {\CCSDT}/AVTZ &7.16 &AVQZ \\
|
||
&$^3B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.6 & 4.16 & D &4.17 &AVQZ \\
|
||
&$^3E_{1u}(\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.1 & 4.85 & D &4.86 &AVQZ \\
|
||
&$^3B_{2u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.1 & 5.81 & D & 5.81 &AVQZ \\
|
||
Butadiene &$^1B_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.664 & 93.3 & 6.22 & B & 6.21 & AVQZ \\
|
||
&$^1B_g (\mathrm{R}; \pi \rightarrow 3s)$ & & 94.1 & 6.33 & B & 6.35 & AVQZ \\
|
||
&$^1A_g (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 75.1 & 6.50 & F & 6.50 & AVQZ \\
|
||
&$^1A_u (\mathrm{R}; \pi \rightarrow 3p)$ &0.001 & 94.1 & 6.64 & B & 6.66 & AVQZ \\
|
||
&$^1A_u (\mathrm{R}; \pi \rightarrow 3p)$ &0.049 & 94.1 & 6.80 & B & 6.82 & AVQZ \\
|
||
&$^1B_u (\mathrm{R}; \pi \rightarrow 3p)$ &0.055 & 93.8 & 7.68 & C & 7.54 & AVQZ \\
|
||
&$^3B_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 3.36 & D & 3.37 & AVQZ \\
|
||
&$^3A_g (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.7 & 5.20 & D & 5.21 & AVQZ \\
|
||
&$^3B_g (\mathrm{R}; \pi \rightarrow 3s)$ & & 97.9 & 6.29 & D & 6.31 & AVQZ \\
|
||
Cyanoacetylene &$^1\Sigma^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.3 & 5.80 & A & 5.79 & AV5Z\\
|
||
&$^1\Delta (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.0 & 6.07 & A & 6.05 &AV5Z\\
|
||
&$^3\Sigma^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.44 & {\CCSDT}/AVTZ & 4.46 &AV5Z \\
|
||
&$^3\Delta (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 5.21 & {\CCSDT}/AVTZ & 5.21 & AV5Z\\
|
||
&$^1A'' [\mathrm{F}] (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.004 & 93.6 & 3.54 & A & 3.54 & AVQZ \\
|
||
Cyanoformaldehyde &$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.001 & 89.8 & 3.81 & {\CCSDT}/AVTZ & 3.82 & AVQZ \\
|
||
&$^1A'' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.000 & 91.9 & 6.46 & {\CCSDT}/AVTZ & 6.45 & AVQZ \\
|
||
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.6 & 3.44 & D & 3.45 & AVQZ \\
|
||
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 5.01 & D & 5.02 & AVQZ \\
|
||
Cyanogen & $^1\Sigma_u^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.1 & 6.39 & A & 6.38 & AV5Z\\
|
||
& $^1\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 93.4 & 6.66 & A & 6.64 & AV5Z\\
|
||
& $^3\Sigma_u^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.91 & B & 4.93 & AV5Z \\
|
||
& $^1\Sigma_u^- [\mathrm{F}] (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 93.4 & 5.05 & A & 5.03 & AV5Z \\
|
||
Cyclopentadiene &$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.084 & 93.8 & 5.56 & {\CCSDT}/AVTZ & 5.55 & AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 94.0 & 5.78 & {\CCSDT}/AVTZ & 5.80 & AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \rightarrow 3p)$ &0.037 & 94.2 & 6.41 & {\CCSDT}/AVTZ & 6.42 & AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3p)$ & & 93.8 & 6.46 & {\CCSDT}/AVTZ & 6.47 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; \pi \rightarrow 3p)$ &0.046 & 94.2 & 6.56 &{\CCSDT}/AVTZ & 6.55 & AVQZ\\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.001 & 78.9 & \emph{6.51} & D & \emph{6.51} & AVQZ \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 3.31 & D & 3.31 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.6 & 5.11 & D & 5.12 & AVQZ \\
|
||
&$^3A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 97.9 & 5.73 & D & 5.75 & AVQZ \\
|
||
&$^3B_1 (\mathrm{R}; \pi \rightarrow 3p)$ & & 97.9 & 6.36 & D & 6.38 & AVQZ \\
|
||
Cyclopropenone &$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 87.7 & 4.26 & B & 4.28 & AV5Z \\
|
||
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 91.0 & 5.55 & B & 5.56 &AV5Z \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ &0.003 & 90.8 & 6.34 & B & 6.40 & AV5Z \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star$) &0.047 & 86.5 & 6.54 & B & 6.56 & AV5Z\\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3p)$ &0.018 & 91.1 & 6.98 & B & 7.01 & AV5Z \\
|
||
&$^1A_1 (\mathrm{R}; n \rightarrow 3p)$ &0.003 & 91.2 & 7.02 & B & 7.08 &AV5Z \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.320 & 90.8 & 8.28 & B & 8.26 &AV5Z \\
|
||
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.0 & 3.93 & {\CCSDT}/AVTZ & 3.96 & AV5Z \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.9 & 4.88 & {\CCSDT}/AVTZ & 4.91 & AV5Z \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.5 & 5.35 & {\CCSDT}/AVTZ & 5.37 & AV5Z \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.1 & 6.79 & {\CCSDT}/AVTZ & 6.81 & AV5Z\\
|
||
Cyclopropenethione &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 89.6 & 3.41 & B & 3.41 & AV5Z \\
|
||
&$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 84.8 & 3.45 & B & 3.48 & AV5Z \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.007 & 83.0 & 4.60 & B & 4.62 & AV5Z \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ &0.048 & 91.8 & 5.34 & B & 5.40 & AV5Z \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.228 & 89.0 & 5.46 & B & 5.46 & AV5Z \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3p)$ &0.084 & 91.3 & 5.92 & B & 5.94 & AV5Z \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.2 & 3.28 & D & 3.28 & AV5Z \\
|
||
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 94.5 & 3.32 & {\CCSDT}/AVTZ & 3.36 & AV5Z \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 96.5 & 4.01 & D & 4.04 & AV5Z \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 4.01 & D & 4.01 & AV5Z \\
|
||
Diacetylene &$^1\Sigma_u^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.4 & 5.33 & A & 5.32 & AV5Z \\
|
||
&$^1\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.1 & 5.61 & A & 5.60 & AV5Z \\
|
||
&$^3\Sigma_u^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.10 & C & 4.13 & AV5Z \\
|
||
&$^3\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 4.78 & B & 4.78 &AV5Z \\
|
||
Furan &$^1A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 93.8 & 6.09 &{\CCSDT}/AVTZ & 6.11 &AVQZ \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.163 & 93.0 & 6.37 &{\CCSDT}/AVTZ & 6.37 &AVQZ \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.000 & 92.4 & 6.56 &{\CCSDT}/AVTZ & 6.56 &AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \rightarrow 3p)$ &0.038 & 93.9 & 6.64 &{\CCSDT}/AVTZ & 6.66 &AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3p)$ & & 93.6 & 6.81 &{\CCSDT}/AVTZ & 6.83 &AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; \pi \rightarrow 3p)$ &0.008 & 93.5 & 7.24 & D & 7.14 &AVQZ \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 4.20 & D & 4.20 &AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.1 & 5.46 & D & 5.47 &AVQZ \\
|
||
&$^3A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 97.9 & 6.02 & D & 6.05 &AVQZ \\
|
||
&$^3B_1 (\mathrm{R}; \pi \rightarrow 3p)$ & & 97.9 & 6.59 & D & 6.61 &AVQZ \\
|
||
Glyoxal &$^1A_u (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.000 & 91.0 & 2.88 & B & 2.88 & AV5Z \\
|
||
&$^1B_g (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.3 & 4.24 & B & 4.24 & AVQZ \\
|
||
&$^1A_g (\mathrm{V}; n,n \rightarrow \pi^\star,\pi^\star)$ & & 0.5 & 5.61 & F & 5.60 & AV5Z \\%to be remade with CCSDT correction ??
|
||
&$^1B_g (\mathrm{V}; n \rightarrow \pi^\star)$ & & 83.9 & 6.57 & B & 6.58 & AVQZ \\
|
||
&$^1B_u (\mathrm{R}; n \rightarrow 3p)$ & 0.095 & 91.7 & 7.71 & B & 7.78 & AV5Z \\
|
||
&$^3A_u (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.6 & 2.49 & {\CCSDT}/AVTZ & 2.50 & AV5Z \\
|
||
&$^3B_g (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.4 & 3.89 & {\CCSDT}/AVTZ & 3.90 & AVQZ \\
|
||
&$^3B_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 5.15 & {\CCSDT}/AVTZ & 5.17 & AV5Z \\
|
||
&$^3A_g (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.8 & 6.30 & {\CCSDT}/AVTZ & 6.31 & AV5Z \\
|
||
Imidazole &$^1A'' (\mathrm{R}; \pi \rightarrow 3s)$ & 0.001 & 93.0 & 5.71 & D & 5.73 & AVQZ \\
|
||
&$^1A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.124 & 89.6 & 6.41 & D & 6.41s & AVQZ \\
|
||
&$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.028 & 93.6 & 6.50 & D & 6.53 & AVQZ \\
|
||
&$^1A' (\mathrm{R};\pi \rightarrow 3p)$ & 0.035 & 88.9 & \emph{6.83} & D &\emph{6.82} & AVQZ \\
|
||
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.3 & 4.73 & E & 4.74 & AVQZ \\
|
||
&$^3A'' (\mathrm{R};(\pi \rightarrow 3s)$ & & 97.6 & 5.66 & D & 5.69 & AVQZ \\
|
||
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.9 & 5.74 & E & 5.75 & AVQZ \\
|
||
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.3 & 6.31 & D & 6.31 & AVQZ \\
|
||
Isobutene &$^1B_1 (\mathrm{R}; \pi \rightarrow 3s)$ & 0.006 & 94.1 & 6.46 & {\CCSDT}/AVTZ & 6.48 & AVQZ \\
|
||
&$^1A_1 (\mathrm{R}; \pi \rightarrow 3p)$ & 0.228 & 94.2 & 7.01 & {\CCSDT}/AVTZ & 7.00 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; (\pi \rightarrow \pi^\star)$ & & 98.9 & 4.53 & D & 4.54 & AVQZ \\
|
||
Methylenecyclopropene& $^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.011 & 85.4 & 4.28 & B & 4.29 & AV5Z \\
|
||
&$^1B_1 (\mathrm{R}; \pi \rightarrow 3s)$ & 0.005 & 93.6 & 5.44 & B & 5.47 & AV5Z \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3p)$ & & 93.3 & 5.96 & B & 5.99 & AVQZ \\
|
||
&$^1A_1(\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.224 & 92.8 & \emph{6.12} & B & \emph{6.03} & AV5Z \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.2 & 3.49 & {\CCSDT}/AVTZ & 3.49 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.6 & 4.74 & D & 4.75 & AV5Z \\
|
||
Propynal & $^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.000 & 89.0 & 3.80 & {\CCSDT}/AVTZ & 3.81 & AVQZ \\
|
||
&$^1A'' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.000 & 92.9 & 5.54 & {\CCSDT}/AVTZ & 5.53 & AVQZ \\
|
||
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.4 & 3.47 & D & 3.48 & AVQZ \\
|
||
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.3 & 4.47 & D & 4.48 & AVQZ \\
|
||
Pyrazine &$^1B_{3u} (\mathrm{V}; n \rightarrow \pi^\star)$ &0.006 & 90.1 & 4.15 & {\CCSDT}/AVTZ & 4.15 & AVQZ \\
|
||
&$^1A_{u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.6 & 4.98 & {\CCSDT}/AVTZ & 4.99 & AVQZ \\
|
||
&$^1B_{2u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.078 & 86.9 & 5.02 & {\CCSDT}/AVTZ & 5.01 & AVQZ \\
|
||
&$^1B_{2g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 85.6 & 5.71 & {\CCSDT}/AVTZ & 5.71 & AVQZ \\
|
||
&$^1A_{g} (\mathrm{R};n \rightarrow 3s)$ & & 91.1 & 6.65 & {\CCSDT}/AVTZ & 6.69 & AVQZ \\
|
||
&$^1B_{1g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 84.2 & 6.74 & {\CCSDT}/AVTZ & 6.74 & AVQZ \\
|
||
&$^1B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.063 & 92.8 & 6.88 & {\CCSDT}/AVTZ & 6.87 & AVQZ \\
|
||
&$^1B_{1g} (\mathrm{R};n \rightarrow 3p)$ & & 93.8 & 7.21 & {\CCSDT}/AVTZ & 7.24 & AVQZ \\
|
||
&$^1B_{2u} (\mathrm{R};n \rightarrow 3p)$ &0.037 & 90.8 & 7.24 & D & 7.28 &AVQZ \\
|
||
&$^1B_{1u} (\mathrm{R};\pi \rightarrow 3s)$ &0.128 & 91.4 & 7.44 & D & 7.47 &AVQZ \\
|
||
&$^1B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.285 & 90.5 & \emph{7.98}& D & \emph{7.97} &AVQZ \\
|
||
&$^3B_{3u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.3 & 3.59 & D & 3.59 & AVQZ \\
|
||
&$^3B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.35 & D & 4.36 & AVQZ \\
|
||
&$^3B_{2u} (\mathrm{V}; (\pi \rightarrow \pi^\star)$ & & 97.6 & 4.39 & D & 4.39 & AVQZ \\
|
||
&$^3A_{u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.1 & 4.93 & D & 4.94 & AVQZ \\
|
||
&$^3B_{2g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.0 & 5.08 & D & 5.09 & AVQZ \\
|
||
&$^3B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.0 & 5.28 & D & 5.28 & AVQZ \\
|
||
Pyridazine &$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 89.0 & 3.83 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 86.9 & 4.37 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 85.8 & 5.26 & D & 5.26 & AVQZ \\
|
||
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 86.2 & 5.72 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ & & 88.5 & 6.17 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 87.0 & 6.37 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 90.6 & 6.75 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.1 & 3.19 & D &3.20 & AVQZ \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.2 & 4.11 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & \emph{4.34} & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.3 & 4.82 & D & 4.81 & AVQZ \\
|
||
Pyridine &$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.4 & 4.95 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 86.5 & 5.14 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 87.9 & 5.40 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 92.1 & 6.62 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\mathrm{R}; n \rightarrow 3s)$ & & 89.7 & 6.76 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 93.2 & 6.82 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & \hl{xxx} & \emph{xxx} & &\emph{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_1 (\mathrm{R}; \pi \rightarrow 3p)$ & & 93.6 & 7.39 & E &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 90.5 & \hl{xxx} & &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.30 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.0 & 4.46 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.3 & 4.79 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.1 & 5.04 & E &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 95.8 & 5.36 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.7 & 6.24 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
Pyrimidine &$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.005 & 88.6 & 4.44 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.5 & 4.85 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.028 & 86.3 & 5.38 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 86.7 & 5.92 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ &0.005 & 86.7 & 6.26 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{R} ;n \rightarrow 3s)$ &0.005 & 90.3 & 6.70 & D &6.74 & AVQZ \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.036 & 91.5 & 6.88 & D &6.87 & AVQZ \\
|
||
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.8 & 4.09 & D &4.10 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.3 & \emph{4.51} & D &\emph{4.52} & AVQZ \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.5 & 4.66 & D &4.67 & AVQZ \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.4 & 4.96 & D &4.96 & AVQZ \\
|
||
Pyrrole &$^1A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 92.9 & 5.24 & {\CCSDT}/AVTZ & 5.27 & AVQZ \\
|
||
&$^1B_1 (\mathrm{R};\pi \rightarrow 3p)$ &0.015 & 92.4 & 6.00 & {\CCSDT}/AVTZ & 6.03 & AVQZ \\
|
||
&$^1A_2 (\mathrm{R};\pi \rightarrow 3p)$ & & 93.0 & 6.00 & D & 6.02 & AVQZ \\
|
||
&$^1B_2 (\mathrm{V}; (\pi \rightarrow \pi^\star)$ &0.164 & 92.5 & 6.26 & {\CCSDT}/AVTZ & 6.23 & AVQZ \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.001 & 86.3 & 6.30 & {\CCSDT}/AVTZ & 6.29 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R};\pi \rightarrow 3p)$ &0.003 & 92.6 & 6.83 & D & 6.74 & AVQZ \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.3 & 4.51 & D & 4.51 & AVQZ \\
|
||
&$^3A_2 (\mathrm{R};\pi \rightarrow 3s)$ & & 97.6 & 5.21 & D & 5.24 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.8 & 5.45 & D & 5.46 & AVQZ \\
|
||
&$^3B_1 (\mathrm{R};\pi \rightarrow 3p)$ & & 97.4 & 5.91 & D & 5.94 & AVQZ \\
|
||
Tetrazine &$^1B_{3u} (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.006 & 89.8 & 2.47 & {\CCSDT}/AVTZ & 2.46 & AVQZ \\
|
||
&$^1A_{u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 87.9 & 3.69 & {\CCSDT}/AVTZ & 3.70 & AVQZ \\
|
||
&$^1A_{g} (\mathrm{V}; n,n \rightarrow \pi^\star, \pi^\star)$ & & 0.7 & \emph{4.61} & {\NEV}/AVTZ & \emph{4.59} & AVQZ\\%to be remade with CCSDT correction ???
|
||
&$^1B_{1g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 83.1 & 4.93 & {\CCSDT}/AVTZ & 4.92 & AVQZ \\
|
||
&$^1B_{2u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.055 & 85.4 & 5.21 & {\CCSDT}/AVTZ & 5.20 & AVQZ \\
|
||
&$^1B_{2g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 81.7 & 5.45 & {\CCSDT}/AVTZ & 5.45 & AVQZ \\
|
||
&$^1A_{u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 87.7 & 5.53 & {\CCSDT}/AVTZ & 5.53 & AVQZ \\
|
||
&$^1B_{3g} (\mathrm{V}; n,n \rightarrow \pi^\star, \pi^\star)$ & & 0.7 & \emph{6.15} & {\NEV}/AVTZ & \emph{6.13} & AVQZ\\
|
||
&$^1B_{2g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 80.2 & 6.12 & D & 6.12 & AVQZ \\
|
||
&$^1B_{1g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 85.1 & 6.91 & D & 6.91 & AVQZ \\
|
||
&$^3B_{3u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.1 & 1.85 & D & 1.86 & AVQZ \\
|
||
&$^3A_{u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.3 & 3.45 & D & 3.46 & AVQZ \\
|
||
&$^3B_{1g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.0 & 4.20 & D & 4.21 & AVQZ \\
|
||
&$^3B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & \emph{4.49} & D & \emph{4.49} & AVQZ \\
|
||
&$^3B_{2u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.5 & 4.52 & D & 4.52 & AVQZ \\
|
||
&$^3B_{2g} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.4 & 5.04 & D & 5.04 & AVQZ \\
|
||
&$^3A_{u} (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.6 & 5.11 & D & 5.11 & AVQZ \\
|
||
&$^3B_{3g} (\mathrm{V}; n,n \rightarrow \pi^\star, \pi^\star)$ & & 5.7 & \emph{5.51} &{\NEV}/AVTZ & \emph{5.50} & AVQZ\\
|
||
&$^3B_{1u} (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 96.6 & 5.42 & D & 5.43 & AVQZ \\
|
||
Thioacetone &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.9 & 2.53 & B & 2.54 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 4s)$ & 0.052 & 91.3 & 5.56 & B & 5.61 & AVQZ \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.242 & 90.6 & 5.88 & B & 5.86 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \rightarrow 4p)$ & 0.028 & 92.4 & 6.51 & C & 6.52 & AVQZ \\
|
||
&$^1A_1 (\mathrm{R}; n \rightarrow 4p)$ & 0.023 & 91.6 & 6.61 &B & 6.64 & AVQZ \\
|
||
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.4 & 2.33 & D & 2.34 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.7 & 3.45 & D & 3.46 & AVQZ \\
|
||
Thiophene &$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.070 & 87.6 & 5.64 & {\CCSDT}/AVTZ & 5.63 & AVQZ \\
|
||
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.079 & 91.5 & 5.98 & {\CCSDT}/AVTZ & 5.96 & AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 92.6 & 6.14 & {\CCSDT}/AVTZ & 6.16 & AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \rightarrow 3p)$ &0.010 & 90.1 & 6.14 & {\CCSDT}/AVTZ & 6.11 & AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3p)$ & & 91.8 & 6.21 & {\CCSDT}/AVTZ & 6.18 & AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \rightarrow 3s)$ &0.000 & 92.8 & 6.49 & {\CCSDT}/AVTZ & 6.52 & AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; \pi \rightarrow 3p)$ &0.082 & 92.4 & 7.29 & {\CCSDT}/AVTZ & 7.18 & AVQZ \\
|
||
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.314 & 86.5 & \emph{7.31}& E & \emph{7.29} & AVQZ \\
|
||
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 3.92 & D & 3.91 & AVQZ \\
|
||
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.7 & 4.76 & D & 4.76 & AVQZ \\
|
||
&$^3B_1 (\mathrm{R}; \pi \rightarrow 3p)$ & & 96.6 & 5.93 & D & 5.90 & AVQZ \\
|
||
&$^3A_2 (\mathrm{R}; \pi \rightarrow 3s)$ & & 97.5 & 6.08 & D & 5.98 & AVQZ \\
|
||
Thiopropynal &$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.000 & 87.5 & 2.03 & {\CCSDT}/AVTZ & 2.04 & AVQZ \\
|
||
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.2 & 1.80 & D & 1.81 & AVQZ \\
|
||
Triazine &$^1A_1'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.3 & 4.72 & {\CCSDT}/AVTZ & 4.72 & AVQZ \\
|
||
&$^1A_2'' (\mathrm{V}; n \rightarrow \pi^\star)$ &0.014 & 88.3 & 4.75 & {\CCSDT}/AVTZ & 4.74 & AVQZ \\
|
||
&$^1E'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.3 & 4.78 & {\CCSDT}/AVTZ & 4.78 & AVQZ \\
|
||
&$^1A_2' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 85.7 & 5.75 & {\CCSDT}/AVTZ &5.75 &AVQZ\\
|
||
&$^1A_1' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 90.4 & 7.24 & {\CCSDT}/AVTZ & 7.23 & AVQZ \\
|
||
&$^1E' (\mathrm{R}; n \rightarrow 3s)$ &0.016 & 90.9 & 7.32 & {\CCSDT}/AVTZ & 7.36 & AVQZ \\
|
||
&$^1E'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 82.6 & 7.78 & {\CCSDT}/AVTZ & 7.76 & AVQZ \\
|
||
&$^1E' (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.451 & 90.0 & 7.94 & {\CCSDT}/AVTZ & 7.93 & AVQZ \\
|
||
&$^3A_2'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.7 & 4.33 & D & 4.34 & AVQZ \\
|
||
&$^3E'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.6 & 4.51 & D & 4.51 & AVQZ \\
|
||
&$^3A_1'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.2 & 4.73 & D & 4.74 & AVQZ \\
|
||
&$^3A_1' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 4.85 & D & 4.86 & AVQZ \\
|
||
&$^3E' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 96.9 & 5.59 & E & 5.59 & AVQZ \\
|
||
&$^3A_2' (\mathrm{V}; (\pi \rightarrow \pi^\star)$ & & 97.6 & 6.62 & D & 6.61 & AVQZ \\
|
||
\end{longtable}
|
||
\end{footnotesize}
|
||
\begin{flushleft}\begin{footnotesize}\begin{singlespace}
|
||
\vspace{-0.6 cm}
|
||
$^a${
|
||
Method A: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\AVDZ} and {\CCSDT}/{\AVDZ} results;
|
||
Method B: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCSDT}/{\Pop} results;
|
||
Method C: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCT}/{\Pop} results;
|
||
Method D: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\AVDZ} and {\CCT}/{\AVDZ} results;
|
||
Method E: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\Pop} and {\CCT}/{\Pop} results;
|
||
Method F: {exCI}/{\AVDZ} value (from Ref. \citenum{Loo19c}) corrected by the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ} results.
|
||
}
|
||
\end{singlespace}\end{footnotesize}\end{flushleft}
|
||
|
||
\clearpage
|
||
|
||
\section{Benchmarks}
|
||
|
||
Having at hand such a large set of accurate transition energies, it seems natural to pursue previous benchmarking efforts. More specifically, we assess here the performances of eight popular wavefunction approaches, namely, CIS(D), {\AD},
|
||
{\CCD}, {\STEOM}, {\CCSD}, CCSDR(3), CCSDT-3 and {\CCT}. The complete list of results can be found in Table \hl{SXXX} in the SI. As all these approaches are single-reference methods, we have removed from the
|
||
benchmark not only the unsafe transition energies (in italics in Table \ref{Table-tbe}), but also the four transitions with a dominant double excitation character ($\%T_1 < 50\%$ listed in Table \ref{Table-tbe}).
|
||
Our global results are collected in Table \ref{Table-bench} that presents the MSE, MAE, root mean square deviation (RMS), standard deviation (SD), as well as the positive [\MaxP] and negative [\MaxN] maximum deviations.
|
||
Figure \ref{Fig-1} shows histograms of the error distributions for all eight methods. Before discussing the obtained results, let us underline two obvious bias of this benchmark: i) it encompasses only conjugated organic molecules
|
||
containing 4 to 6 non-hydrogen atoms; and ii) we mainly used {\CCSDTQ} (4 atoms) or {\CCSDT} (5--6 atoms) reference values. As discussed in Section \ref{sec-ic} and in our previous work, \cite{Loo18a} the MAE obtained
|
||
with these two methods are of the order of 0.01 and 0.03 eV, respectively. This means that any deviation (or difference of deviations) smaller than ca. 0.02--0.03 eV is likely irrelevant.
|
||
|
||
\renewcommand*{\arraystretch}{1.0}
|
||
\begin{table}[htp]
|
||
\caption{Mean signed error (MSE), mean absolute error (MAE), root-mean square deviation (RMS), standard deviation (SD), positive [\MaxP] and negative [\MaxN] maximal deviations with respect to the TBE.
|
||
All values are in eV and have been obtained with the {\AVTZ} basis set.}
|
||
\label{Table-bench}
|
||
\begin{tabular}{lccccccc}
|
||
\hline
|
||
Method & Nb. States & MSE &MAE &RMS &SD &\MaxP &\MaxN \\
|
||
\hline
|
||
CIS(D) &220 &0.16 &0.23 &0.29 &0.24 &0.96 &-0.69\\
|
||
{\AD} &217 &0.01 &0.14 &0.20 &0.19 &0.64 &-0.73\\
|
||
{\CCD} &222 &0.02 &0.15 &0.21 &0.20 &0.59 &-0.68\\
|
||
{\STEOM} &190 &0.00 &0.12 &0.15 &0.14 &0.59 &-0.42\\
|
||
{\CCSD} &222 &0.11 &0.13 &0.16 &0.12 &0.62 &-0.16\\
|
||
CCSDR(3) &133 &0.05 &0.05 &0.07 &0.05 &0.36 &-0.03\\
|
||
CCSDT-3 &126 &0.05 &0.05 &0.07 &0.04 &0.26 &0.00\\
|
||
{\CCT} &222 &0.00 &0.01 &0.02 &0.02 &0.17 &-0.05\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{table}
|
||
|
||
\begin{figure}[htp]
|
||
\includegraphics[scale=0.98,viewport=2cm 14.5cm 19cm 27.5cm,clip]{Figure-1.pdf}
|
||
\caption{Histograms of the error patterns obtained with various leveles of theory, taking the TBE/{\AVTZ} of Table \ref{Table-bench} as references. Note the different $Y$ scales.}
|
||
\label{Fig-1}
|
||
\end{figure}
|
||
|
||
Let us analyse the global performances of all methods, starting with the most refined models. The relative accuracies of {\CCT} and {\CCSDT}-3 as compared to {\CCSDT} remains an open question in the literature. \cite{Wat13,Dem13}
|
||
Indeed, to our knowledge, the only two previous reports discussing this specific aspect are limited to tiny compounds. \cite{Kan17,Loo18a} According to the results of Table \ref{Table-bench}, it appears that {\CCT} has the edge, although
|
||
{\CCSDT}-3 is closer to {\CCSDT} in formal terms. Indeed, {\CCSDT}-3 seems to provide slightly too large transition energies (MSE of +0.05 eV). These conclusions are qualitatively consistent with the analyses performed for smaller derivatives,
|
||
\cite{Kan17,Loo18a} but the amplitude of {\CCSDT}-3's errors are larger with the present set. Although the performances of {\CCT} might be unduly inflated by the use of {\CCSDT} and {\CCSDTQ} reference values, it is also clear that this
|
||
method very rarely fails (Figure \ref{Fig-1}). Consequently, {\CCT} transition energies can be viewed as very solid references for all transitions with a dominant single-excitation character. This conclusion is again consistent with previous
|
||
analyses performed for smaller compounds, \cite{Kan17,Loo18a} as well as with recent comparisons between theoretical and experimental 0-0 energies that have been performed by some of us on medium-sized molecules.\cite{Loo18b,Loo19a,Sue19}
|
||
To state it more bluntly: it appears likely that {\CCT} is even more accurate than previously thought. In addition, from all the comparisons made in this work, one can conclude that {\CCT} regularly outperforms {\CASPT} and {\NEV}, even when
|
||
these methods are combined with active space chosen by specialists, a statement that seems true as long as the considered ES does not show a strong multiple excitation character, that is, when $\%T_1 < 70\%$. The perturbative
|
||
inclusion of triples as made in CCSDR(3) offers a very small MAE (0.05 eV) for a much reduced computational cost as compared to {\CCSDT}. Nevertheless, as with {\CCSDT}-3, the CCSDR(3) transition energies have a clear tendency
|
||
of being too large, an error sign likely inherited from the parent {\CCSD} model. This 0.05 eV MAE for CCSDR(3) is rather similar to the one obtained for small compounds when comparing to {\FCI} (0.04 eV), \cite{Loo18a} and is also inline with the
|
||
2009 benchmark of Sauer et al. \cite{Sau09} {\CCSD} provides an interesting case. The calculated MSE (+0.11 eV), indicating an overestimation of the transition energies, fits well many previous reports,
|
||
\cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a} but is larger than the one determined for smaller molecules (+0.05 eV), \cite{Loo18a} hinting that the performances of {\CCSD} deteriorates when larger compounds
|
||
are considered. The {\CCSD} MAE (0.13 eV) is much smaller than the one reported by Thiel in its original work (0.49 eV) \cite{Sch08} but of the same order of magnitude as in the more recent study of Kannar and Szalay performed
|
||
on Thiel's set (0.18 eV for transitions with $\%T_1 > 90\%$ ). \cite{Kan14} In retrospect, the much larger value obtained by Thiel is likely related to the use of {\CASPT} reference values in the 2008 work. Indeed, as we have shown
|
||
in many of the proposed examples, {\CASPT} transitions energies tend to be significantly too low, therefore exacerbating {\CCSD}'s overestimation. The {\STEOM} approach, which received relatively less attention to date -- we are
|
||
aware of one detailed benchmark \cite{Dut18} only -- provides a smaller MSE than {\CCSD} and comparable MAE and RMS. The spread of the error is however slightly larger as can be seen in Figure \ref{Fig-1} and from the SD values
|
||
in Table \ref{Table-bench}. These trends are the same as for smaller compounds. \cite{Loo18a} For Thiel's set using {\CCT}/TZVP results as references, Dutta and coworkers also reported a rather good performance
|
||
of {\STEOM}, though in that case a slightly negative MSE is obtained, \cite{Dut18} which could possibly be due to the different basis sets used. It should be nevertheless stressed that we consider here only ``clean'' {\STEOM} results
|
||
(see Computational details), therefore removing several difficult cases that are included in the {\CCSD} statistics, e.g., the $A_g$ excitation in butadiene, which can slightly bias the relative accuracies when comparing the two models. Finally, for the three
|
||
second-order methods, namely CIS(D), {\AD}, and {\CCD}, that are often used as reference to benchmark TD-DFT for ``real-life'' applications, we obtain clearly worse performances for the former approach than for the two latter, that show very
|
||
similar statistical behaviors. These trends were also reported in several previous works. \cite{Hat05c,Jac18a,Sch08,Sil10c,Win13,Har14,Jac15b,Kan17,Loo18a} Interestingly, the {\CCD} MAE obtained here, 0.15 eV, is significantly
|
||
smaller than the one we found for the smaller compounds (0.22 eV): \cite{Loo18a} in contrast to {\CCSD}, {\CCD} seems to improve with molecular size. As above, Thiel's original MAE for {\CCD} (0.29 eV) was likely too large due
|
||
to the selection of {\CASPT} reference values. \cite{Sch08} As already noticed by Szalay's group, \cite{Kan14,Kan17} although the MSE of {\CCD} is smaller than the one of {\CCSD}, the standard deviation is significantly larger
|
||
with the former model, i.e., {\CCD} is less consistent in terms of trends than {\CCSD}.
|
||
|
||
In Table \ref{Table-bench2}, we report a decomposition of the MAE for different subsets of ES. Only singlet ES could be determined with both CCSDR(3) and CCSDT-3, which is why no value appears in the triplet
|
||
column for these two methods. A few interesting conclusions emerge from the displayed data. First, the errors for the singlet and triplet transitions are rather similar with all models, but with {\CCSD} that
|
||
is significantly more effective for the triplets. Dutta and coworkers obtained the same conclusions for Thiel's set with MAE of 0.20 eV and 0.11 eV for the singlet and triplet ES, respectively. \cite{Dut18}
|
||
When turning to the comparison between valence and Rydberg states, it is found that {\CCD} actually performs more effectively for the former, whereas {\CCSD} (and higher order methods) yields the opposite trend.
|
||
In fact {\CCD} has a tendency to overestimate the energies of the valence ES (MSE of +0.10 eV), but to underestimate their Rydberg counterparts (MSE of -0.17 eV), whereas {\CCSD} is much more consistent
|
||
with MSE of 0.12 and 0.09 eV, respectively (see the SI). This relatively poorer performance of {\CCD} as compared to {\CCSD} for Rydberg ES is again consistent with other benchmarks, \cite{Kan17,Dut18} although the MAE
|
||
for {\CCD} (0.18 eV) reported in Table \ref{Table-bench2} remains relatively small as compared to the one given in Ref. \citenum{Kan17}. This is likely a side effect of the consideration of (relatively) low-lying
|
||
Rydberg transitions in medium-sized molecules in the present work, whereas Kannar and Szalay (mostly) investigated higher-lying Rydberg in smaller compounds. Eventually, CIS(D), {\AD}, {\CCD}, and {\STEOM}
|
||
better describe $n\rightarrow\pi^\star$ transitions, whereas {\CCSD} seems more suited for $\pi\rightarrow\pi^\star$ transitions; the variations between the two subsets being probably not very significant for the
|
||
higher-order approaches. The former finding agrees with the results obtained for smaller compounds, \cite{Loo18a} as well for Thiel's set, \cite{Sch08,Kan14} whereas the latter, less expected conclusions, seems to be significantly
|
||
dependent on the selected subset of ES. \cite{Sch08,Kan17}
|
||
|
||
\renewcommand*{\arraystretch}{1.0}
|
||
\begin{table}[htp]
|
||
\caption{MAE (in eV) obtained with different methods for various classes of excited states.}
|
||
\label{Table-bench2}
|
||
\begin{tabular}{lcccccc}
|
||
\hline
|
||
Method & Singlet & Triplet & Valence & Rydberg & $n \ra \pis$ & $\pi \ra \pis$ \\
|
||
\hline
|
||
CIS(D) &0.21 &0.25 &0.26 &0.15 &0.22 &0.28\\
|
||
{\AD} &0.15 &0.13 &0.13 &0.17 &0.08 &0.17\\
|
||
{\CCD} &0.16 &0.15 &0.15 &0.18 &0.08 &0.20\\%moins grosse diff<66>rence entre V et Ry que dans Kan17
|
||
{\STEOM} &0.11 &0.13 &0.12 &0.12 &0.08 &0.15\\
|
||
{\CCSD} &0.16 &0.09 &0.15 &0.09 &0.20 &0.11\\%CCSD error smaller for pi-pi* than n-pi* different from Kan14
|
||
CCSDR(3) &0.05 & &0.07 &0.02 &0.08 &0.06\\
|
||
CCSDT-3 &0.05 & &0.06 &0.03 &0.08 &0.04\\
|
||
{\CCT} &0.01 &0.01 &0.01 &0.01 &0.01 &0.02\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{table}
|
||
|
||
\section{Conclusions and outlook}
|
||
|
||
We have determined highly-accurate vertical transition energies for a set of 27 medium-sized organic molecules containing from 4 to 6 (non-hydrogen) atoms. To this end, we used several theoretical levels
|
||
and basis sets, but our theoretical best estimates are mainly based on {\CCSDTQ} (4 atoms) or {\CCSDT} (5 and 6 atoms) values determined with diffuse containing basis sets. For the vast majority of the
|
||
listed excited-states, this contribution is the first to disclose (sometimes basis-set extrapolated) {\CCSDT}/{\AVTZ} and (true) {\CCT}/{\AVQZ} transition energies as well as {\CCT}/{\AVTZ} oscillator strengths
|
||
for all dipole-allowed transitions. The set contains a total of 238 transition energies and 90 oscillator strengths, including a reasonably good balance between singlet and triplet transition energies and valence
|
||
and Rydberg states. Amongst these 238 transitions, we consider that 224 are ``safe'' TBE, that is, that they are chemically accurate (mean error below $0.043$ eV or $1$ kcal.mol$^{-1}$ for the considered geometry),
|
||
allowing to establish a reasonable error bar for lower-cost ES models daily used by computational chemists. In this framework, we benchmarked eight popular methods, CIS(D), {\AD}, {\CCD}, {\STEOM}, {\CCSD},
|
||
CCSDR(3), CCSDT-3, and {\CCT}. It turned out that the latter approach is extremely accurate, and, very likely should be more trusted than {\CASPT} or {\NEV} but for ES dominated by a double
|
||
excitation character. Other methods including corrections for the triples yield a mean absolute deviation of ca. 0.05 eV, whereas none of the second-order approach is chemically accurate, with MAE
|
||
in the 0.12--0.23 eV range.
|
||
|
||
Paraphrasing Thiel and coworkers, \cite{Sch08} we hope that this new set of vertical transition energies, combined or not with the ones described in our previous works, \cite{Loo18a,Loo19c} will be useful for the community,
|
||
will stimulate further developments and analyses in the field, and will provide new grounds for appraising the \emph{pros} and \emph{cons} of ES approaches already available or under development. We can
|
||
crystal-ball that the emergence of new {\sCI} algorithms optimized for modern computer architectures will likely lead to the revision of some the present TBE, further allowing to climb one step on the accuracy
|
||
ladder.
|
||
|
||
\begin{suppinfo}
|
||
Geometries.
|
||
Basis set and frozen core effects.
|
||
Definition of the active spaces for the multi-reference calculations.
|
||
Additional details about the {\sCI} calculations and their extrapolation.
|
||
Benchmark data and further statistical analysis.
|
||
\end{suppinfo}
|
||
|
||
\begin{acknowledgement}
|
||
P.F.L.~would like to thank \textit{Centre National de la Recherche Scientifique} for funding.
|
||
D.J.~acknowledges the \emph{R\'egion des Pays de la Loire} for financial support. This research used resources of i) the GENCI-TGCC (Grant No.~2018-A0040801738); ii) CCIPL (\emph{Centre de Calcul Intensif des Pays de Loire});
|
||
iii) the Troy cluster installed in Nantes; and iv) CALMIP under allocations 2019-18005 (Toulouse).
|
||
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
|
||
\end{acknowledgement}
|
||
|
||
\bibliography{biblio-new}
|
||
\clearpage
|
||
|
||
\begin{tocentry}
|
||
\begin{center}
|
||
%\includegraphics[scale=.17]{TOC.png}
|
||
\end{center}
|
||
\end{tocentry}
|
||
|
||
|
||
\end{document}
|