1551 lines
137 KiB
TeX
1551 lines
137 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]{$^{#1}#2_{#3}^{#4}(#5 \ra #6)$}
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\newcommand{\Td}{\%T_1}
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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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\newcommand{\AVDZ}{aVDZ}
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\newcommand{\AVTZ}{aVTZ}
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\newcommand{\DAVTZ}{daVTZ}
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\newcommand{\AVQZ}{aVQZ}
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\newcommand{\DAVQZ}{daVQZ}
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\newcommand{\TAVQZ}{taVQZ}
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\newcommand{\AVPZ}{aV5Z}
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\newcommand{\DAVPZ}{dapV5Z}
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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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\newcommand{\Val}{\mathrm{V}}
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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 (CCT3, EOM-CCSDT, and EOM-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 CC3/aug-cc-pVQZ vertical excitation
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energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 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, $n \ra \pis$, $\pi \ra \pis$, \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), ADC(2),
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CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, and CC3. 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 within TD-DFT as compared to DFT. \cite{Lau13} Such selection can, of course, rely
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on the intrinsic features of the various exchange-correlation functional families, \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 cannot be avoided. This is why so many assessments of TD-DFT performances for various properties are
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available. \cite{Lau13}
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While several of these benchmarks rely on 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}), reference from theoretical best estimates (TBE) based on state-of-the-art computational methods \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, etc). In such a case, the challenge is in fact to obtain accurate TBE, as these top-notch theoretical models
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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 complete basis set (CBS) limit.
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More than 20 years ago, Serrano-Andr\`es, Roos, and collaborators compiled an impressive series of reference transition energies for several typical conjugated organic molecules (butadiene, furan, pyrrole, tetrazine, \ldots).
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\cite{Ful92,Ser93,Ser93b,Ser93c,Lor95b,Mer96,Mer96b,Roo96,Ser96b} To this end, they relied on experimental GS geometries and the complete-active-space second-order perturbation theory ({\CASPT}) approach with the largest active spaces and basis sets
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one could dream of at the time.
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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 time the best theoretical references available on the market.
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However, beyond comparisons with experiments, which are always challenging when computing vertical transition energies, \cite{San16b} there was no approach available at that time to ascertain the accuracy of these transition energies.
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Nowadays, it is common knowledge that CASPT2 has the tendency of underestimating vertical excitation energies in organic molecules.
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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 (CC) variants ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default theoretical protocol used by Thiel and coworkers to define their
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first series of TBE used {\CASPT}, \cite{Sch08} the vast majority of their most recent TBE (the so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level of theory, often using a basis set extrapolation technique.
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More specifically, CC3/TZVP values were corrected for basis set incompleteness errors by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works exploited 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 further
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their value for the electronic structure community. In contrast, the number of extensions/improvements of this original set remains quite limited.
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For example, K\'ann\'ar and Szalay computed, in 2014, {\CCSDT}/TZVP reference energies for 17 singlet states of six molecules. \cite{Kan14}
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Three years later, the same authors reported 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide). \cite{Kan17}
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Following the same philosophy, two years ago, we reported a set of 106 transition energies for which it was technically possible to reach the full configuration interaction (FCI) limit by performing high-order CC (up to {\CCSDTQP}) and selected
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CI (SCI) calculations on {\CCT}/{\AVTZ} GS structures. \cite{Loo18a} We exploited these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} yields near-{\FCI} quality excitation energies, whereas we could not
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detect any significant differences between {\CCT} and {\CCSDT} transition energies, both being very accurate with mean absolute errors (MAE) as small as $0.03$ eV compared to {\FCI}.
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Although these conclusions agree well with earlier studies, \cite{Wat13,Kan14,Kan17} they obviously only hold for single excitations, \ie, transitions with $\Td$ in the range $80$--$100\%$. Therefore, we also recently proposed a set of 20 TBE for transitions exhibiting a significant double-excitation character (\ie, with $\Td$ typically below $80\%$). \cite{Loo19c}
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Unsurprisingly, our results clearly evidenced that the error in CC methods is intimately related to the $\Td$ value.
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For example, ES with a significant yet \titou{not dominant} double excitation character, such as the infamous $A_g$ ES of butadiene ($\Td = 75\%$),
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CC methods including triples 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, \eg, the low-lying $(n,n) \ra (\pis,\pis)$ excitation in nitrosomethane ($\Td = 2\%$),
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single-reference methods have been found to be unsuitable with MAEs of $0.86$ and $0.42$ eV for {\CCT} and {\CCSDT}, respectively.
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In this case, multireference methods are required to obtain accurate results. \cite{Loo19c}
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A clear limit from our 2018 work \cite{Loo18a} was the sizes of the compounds put together in our set.
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These were limited to $1$--$3$ non-hydrogen atoms, hence introducing a potential ``chemical'' bias. Therefore, we have decided, in the present contribution, to consider larger molecules with organic
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compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large one-electron basis sets is elusive. Moreover, the convergence of the {\SCI} energy with respect to the number of determinants is obviously slower for these larger compounds, hence
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extrapolating to the {\FCI} limit with an error of $\sim 0.01$ eV is rarely achievable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work, \cite{Loo18a} is definitely out of reach here.
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Anticipating this problem, we have recently investigated bootstrap CBS extrapolation techniques. \cite{Loo18a,Loo19c}
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In particular, we have demonstrated that, following an ONIOM-like scheme, \cite{Chu15} one can very accurately estimate such limit by correcting high-level values obtained in a small basis by the difference between {\CCT} results obtained in a larger basis and in the same small basis.\cite{Loo18a}
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We globally follow such strategy here. In addition, we also perform {\NEV} calculations in an effort to check the consistency of our estimates. It is especially critical for ES with intermediate $\Td$ 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 of the {\FCI} limit. These reference energies are obtained on {\CCT}/{\AVTZ} geometries and further basis set
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corrections (up to quadruple-$\zeta$ at least) are also provided for {\CCT}. Together with the results obtained in our two earlier works, \cite{Loo18a,Loo19c} the present TBE will hopefully contribute to climb a rung higher on the ES accuracy ladder.
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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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In the following, we employ the aVXZ shorthand notations for these diffuse-containing Dunning basis sets.
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Various statistical quantities are reported in the remaining of this paper: the mean signed error (MSE), mean absolute error (MAE), root mean square error (RMSE), standard deviation of the errors (SDE), as well as the positive [\MaxP] and negative [\MaxN] maximum errors.
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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, \ie, 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 (in eV) of cyanoacetylene, cyanogen, and diaectylene. All states have a valence $\pi \ra \pis$ character.}
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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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\mc{14}{c}{Cyanoacetylene}\\
|
||
& \mc{4}{c}{\Pop} & \mc{4}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{\AVQZ} & \mc{1}{c}{\AVPZ} & \mc{2}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV} & {\CCT} & {\CCT}& Th.$^a$ & Exp.$^b$ \\
|
||
\hline
|
||
$^1\Sigma^-$ &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\\
|
||
$^1\Delta$ &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\\
|
||
$^3\Sigma^+$ &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 &&\\
|
||
$^3\Delta$ &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 &&\\
|
||
$^1A''$[F]$^c$ &3.70&3.72&3.70&3.67$\pm$0.03 &3.60&3.62&3.60&3.59$\pm$0.02 &3.54&3.56&3.50& 3.54 & &&\\
|
||
\hline
|
||
\mc{14}{c}{Cyanogen}\\
|
||
& \mc{4}{c}{\Pop} & \mc{4}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{\AVQZ} & \mc{1}{c}{\AVPZ} & \mc{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV}& {\CCT} & {\CCT}& Exp.$^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$0.09 &5.06&5.07&4.97& 5.05 &5.05 & \\
|
||
\hline
|
||
\mc{14}{c}{Diacetylene}\\
|
||
& \mc{4}{c}{\Pop} & \mc{4}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{\AVQZ} & \mc{1}{c}{\AVPZ} & \mc{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV}& {\CCT} & {\CCT}& Exp.$^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} results from Ref.~\citenum{Luo08};}
|
||
$^b${Experimental 0-0 energies from Refs.~\citenum{Job66a} and \citenum{Job66b} (vacuum UV experiments);}
|
||
$^c${Vertical fluorescence energy of 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 ($4.2 \pm 0.2$ 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 theoretically investigated, \cite{Fis03,Pat06,Luo08,Loo18b,Loo19a} though (rather old) experimental 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 $\pi \ra \pis$ transitions, which have all a strongly dominant single excitation character (\%$T_1 > 90\%$, \emph{vide infra}). For cyanoacetylene, the {\FCI}/{\Pop} estimates come with small error bars, and one notices an excellent agreement between these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning's double-$\zeta$ basis set results for which the {\FCI} uncertainties are however
|
||
larger. Using the {\CCSDTQ} values as references, it appears that the previously obtained {\CASPT} estimates\cite{Luo08} are, as expected, 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, for a given basis set, in a very tight energetic window. There is also a superb agreement between the CC and {\NEV} values with the
|
||
{\AVTZ} basis set. All these facts provide strong evidences that the CC estimates can be fully trusted for these three linear systems. The basis set effects are quite significant for the valence excited-states of cyanoacetylene with successive drops of the transition
|
||
energies by approximately $0.10$ eV, when going from {\Pop} to {\AVDZ} and from {\AVDZ} to {\AVTZ}. The lowest triplet state appears less basis set sensitive, though. As expected,
|
||
extending further the basis set size (to quadruple and quintuple-$\zeta$) leaves the results pretty much unchanged. The same observation holds when adding a second set of diffuse functions, 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, except for the $^1\Sigma_g^+ \ra {} ^3\Sigma_u^+$ transitions. We note that all CC3 and CCSDT values are, at worst,
|
||
within $\pm 0.02$ eV of the {\FCI} window, \ie, all methods presented in Table \ref{Table-1} provide very consistent estimates. For all the states reported in this Table, the average absolute deviation between {\NEV}/{\AVTZ} and {\CCT}/{\AVTZ}
|
||
({\CCSDT}/{\AVTZ}) 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 between theoretical ({\CASPT} and {\CCT}) and experimental 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 have 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 a detailed {\CASPT} study of Serrano-Andr\'es and
|
||
coworkers in 2002, \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 (in eV) for cyclopropenone, cyclopropenethione, and methylenecyclopropene.}
|
||
\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
|
||
\mc{12}{c}{Cyclopropenone}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{2}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Exp.$^b$ \\
|
||
\hline
|
||
$^1B_1 (n \ra \pis)$ &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
|
||
$^1A_2 (n \ra \pis)$ &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
|
||
$^1B_2 (n \ra 3s)$ &6.39&6.38&6.41& &6.21&6.19 &6.32&6.31&6.51 &6.90&6.22 \\%6.44 in Liu14b
|
||
$^1B_2 (\pi \ra \pis$) &6.70&6.67&6.68& &6.56&6.54 &6.54&6.53&6.82 &5.96&6.1 \\
|
||
$^1B_2 (n \ra 3p)$ &6.92&6.91&6.94& &6.88&6.86 &6.96&6.95&7.07 &7.24&6.88 \\
|
||
$^1A_1 (n \ra 3p)$ &7.00&7.00&7.03& &6.88&6.87 &7.00&6.99&7.28 &7.28& \\
|
||
$^1A_1 (\pi \ra \pis)$ &8.51&8.49&8.51& &8.32&8.29 &8.28&8.26&8.19 &7.80&$\sim$8.1 \\
|
||
$^3B_1 (n \ra \pis)$ &4.02&4.03& & 4.00$\pm$0.07 &3.90&3.92 &3.91&3.93&3.51 &4.05& \\
|
||
$^3B_2 (\pi \ra \pis)$ &4.92&4.92& & 4.95$\pm$0.00 &4.90&4.89 &4.89&4.88&5.10 &4.81& \\
|
||
$^3A_2 (n \ra \pis)$ &5.48&5.44& & &5.38&5.35 &5.37&5.35&5.60 &5.56& \\
|
||
$^3A_1 (\pi \ra \pis)$ &6.89&6.88& & &6.79&6.78 &6.83&6.79&7.16 &6.98& \\
|
||
\hline
|
||
\mc{12}{c}{Cyclopropenethione}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{2}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ \\
|
||
\hline
|
||
$^1A_2 (n \ra \pis)$ &3.46&3.44&3.44& 3.45$\pm$0.01 &3.47&3.45 &3.43&3.41&3.52 &3.23 & \\
|
||
$^1B_1 (n \ra \pis)$ &3.45&3.44&3.45& 3.44$\pm$0.05 &3.42&3.42 &3.43&3.44&3.50 &3.47 & \\
|
||
$^1B_2 (\pi \ra \pis)$ &4.67&4.64&4.62& 4.59$\pm$0.09 &4.66&4.64 &4.64&4.62&4.77 &4.34 & \\
|
||
$^1B_2 (n \ra 3s)$ &5.26&5.24&5.27& &5.23&5.21 &5.34&5.31&5.35 &4.98 & \\
|
||
$^1A_1 (\pi \ra \pis)$ &5.53&5.52&5.51& &5.52&5.50 &5.49&5.47&5.54 &5.52 & \\
|
||
$^1B_2 (n \ra 3p)$ &5.83&5.81&5.83& &5.86&5.84 &5.93&5.90&5.99 &5.88 & \\
|
||
$^3A_2 (n \ra \pis)$ &3.33&3.31& & 3.29$\pm$0.03 &3.34&3.32 &3.30& &3.38 &3.20 & \\
|
||
$^3B_1 (n \ra \pis)$ &3.34&3.33& & &3.30&3.30 &3.31&3.32&3.40 &3.30 & \\
|
||
$^3B_2 (\pi \ra \pis)$ &4.01&4.00& & 4.03$\pm$0.03 &4.03&4.02 &4.02& &4.17 &3.86 & \\
|
||
$^3A_1 (\pi \ra \pis)$ &4.06&4.04& & &4.09&4.07 &4.03& &4.13 &3.99 & \\
|
||
\hline
|
||
\mc{12}{c}{Methylenecyclopropene}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV}& Th.$^c$ & Th.$^d$ & Exp.$^e$\\
|
||
\hline
|
||
$^1B_2 (\pi \ra \pis)$ &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\\
|
||
$^1B_1 (\pi \ra 3s)$ &5.65&5.66&5.66& &5.35&5.35 &5.44&5.44&5.49 &5.32&5.44 &5.12\\
|
||
$^1A_2 (\pi \ra 3p)$ &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 \ra \pis)$$^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 \ra \pis)$ &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 \ra \pis)$ &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} results from Ref.~\citenum{Ser02};}
|
||
$^b${Electron impact experiment from 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 $3.62$ eV and $6.52$ eV for the $^1B_1$ ($n \ra \pis$) and
|
||
$^1B_2$ ($\pi \ra \pis$) transitions, respectively; \cite{Bre72}}
|
||
$^c${{\CASPT} results from Refs.~\citenum{Mer96} and \citenum{Roo96};}
|
||
$^d${{\CCT} results from 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 \ra 3p$) transition, yielding unambiguous attribution difficult;}
|
||
$^g${As can be seen in the SI, our {\FCI}/{\AVDZ} estimates are $3.45 \pm 0.04$ and $4.79 \pm 0.02$ eV for the two lowest triplet states of methylenecyclopropene 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 {\Pop} basis set, we notice very small differences between {\CCT}, {\CCSDT}, and {\CCSDTQ}, the latter method giving transition energies
|
||
systematically falling within the {\FCI} extrapolation incertitude, except 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 closest to {\CCSDTQ}. In fact, considering the {\CCSDTQ}/{\Pop} data listed in Table \ref{Table-2} as reference, the MAE of {\CCT} and {\CCSDT} is $0.019$ and $0.016$ eV, respectively,
|
||
hinting that the improvement brought by the latter, more expensive method is limited for this set of compounds. For the lowest $B_2$ state of methylenecyclopropene, one of the most challenging cases (\%$T_1 = 85\%$),
|
||
it is clear from the {\FCI} value that only {\CCSDTQ} is close, the {\CCT} and {\CCSDT} results being slightly too large by $\sim 0.05$ eV. It seems reasonable to believe that the same observation can be made for the corresponding state of
|
||
cyclopropenethione, although in that case the FCI error bar is too large to prevent any definitive conclusion. Interestingly, at the {\CCT} level of theory, the rather small {\Pop} basis set provides data within $0.10$ eV of the CBS limit for 80\%\ of
|
||
the transitions. There are, of course, exceptions to this rule, \eg, the strongly dipole-allowed $^1A_1 (\pi \ra \pis)$ ES of cyclopropenone and the $^1B_1(\pi \ra 3s)$ ES of methylenecyclopropene which are significantly
|
||
over blueshifted with the Pople basis set (Table S2). For cyclopropenone, our {\CCSDT}/{\AVTZ} estimates do agree reasonably well with the {\CASPT} data of Serrano-Andr\'es, except for the $^1B_2 (\pi \ra \pis)$ state
|
||
that we locate significantly higher in energy and the three Rydberg states that our CC calculations predict at significantly lower energies. The present {\NEV} results are globally in better agreement with the CC values,
|
||
though non-negligible deviations pertain. Even if comparisons with experiment should be made very cautiously, we note that, for the Rydberg states, the present CC data are clearly more coherent with the electron impact measurements\cite{Har74} than the original {\CASPT} values. For cyclopropenethione, we typically obtain transition energies in agreement or larger than those obtained with {\CASPT}, \cite{Ser02} though there is no obvious relationship between the
|
||
valence/Rydberg nature of the ES and the relative {\CASPT} error. The average absolute deviation between our {\NEV} and {\CCT} results is $0.08$ eV only. Finally, in the case of methylenecyclopropene, our values logically agree
|
||
very well with the recent estimates of Schwabe and Goerigk, \cite{Sch17} obtained at the {\CCT}/{\AVTZ} level of theory on a different geometry.
|
||
As anticipated, the available {\CASPT} values \cite{Mer96,Roo96} appear too low as compared to the
|
||
present {\NEV} and {\CCSDT} values. For this compound, the available experimental data are based on the wavelength of maximal absorption determined in condensed phase. \cite{Sta84} Hence, only a qualitative match is reached between theory and experiment.
|
||
|
||
\subsubsection{Acrolein, butadiene, and glyoxal}
|
||
|
||
Let us now turn our attention to the excited states of three pseudo-linear $\pi$-conjugated systems that have been the subject to several investigations in the past, 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}
|
||
Amongst these works, it is worth highlighting the detailed theoretical investigation of Saha, Ehara, and Nakatsuji, who reported a huge
|
||
number of ES for these three systems using a coherent theoretical protocol based on the symmetry-adapted-cluster configuration interaction (SAC-CI) method. \cite{Sah06}
|
||
In the following, these three molecules are considered in their most stable \emph{trans} conformation.
|
||
Our results are listed in Tables \ref{Table-3} and S3.
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies (in eV) of acrolein, butadiene, and glyoxal.}
|
||
\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
|
||
\mc{12}{c}{Acrolein}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Th.$^b$ & Exp.$^c$ \\
|
||
\hline
|
||
$^1A'' (n \ra \pis)$ &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 \ra \pis)$ &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 \ra \pis)$ &6.94&6.89& & &6.75&6.72& 6.75& &7.16 &6.26&7.40& \\
|
||
$^1A' (n \ra 3s)$ &7.22&7.23& & &6.99&7.00& 7.07& &7.05 &6.97&7.19&7.08 \\
|
||
$^3A'' (n \ra \pis)$ &3.55&3.53& &3.60$\pm$0.01&3.47&3.45& 3.46& &3.46 &3.39&3.61& \\
|
||
$^3A' (\pi \ra \pis)$ &3.94&3.95& &3.98$\pm$0.03&3.95&3.95& 3.94& &3.95 &3.81&3.87& \\
|
||
$^3A' (\pi \ra \pis)$ &6.25&6.23& & &6.22&6.21& 6.19& &6.23 & &6.21& \\
|
||
$^3A'' (n \ra \pis)$ &6.81&6.74& & &6.60& & 6.61& &6.83 & &7.36& \\
|
||
\hline
|
||
\mc{12}{c}{Butadiene}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^b$ & Th.$^d$ & Exp$^e$ \\
|
||
\hline
|
||
$^1B_u (\pi \ra \pis)$ &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 \ra 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 \ra \pis)$ &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 \ra 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 \ra 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 \ra 3p)$ &7.98&8.00&7.98& &7.86&7.87& 7.68& &7.45 &7.08&7.02&7.07\\
|
||
$^3B_u (\pi \ra \pis)$ &3.35&3.36& &3.37$\pm$0.03 &3.36&3.36& 3.36& &3.40 &3.20& &3.22\\
|
||
$^3A_g (\pi \ra \pis)$ &5.22&5.22& & &5.21&5.21& 5.20& &5.30 &5.08& &4.91\\
|
||
$^3B_g (\pi \ra 3s)$ &6.46&6.47& &6.40$\pm$0.03 &6.20&6.21& 6.28& &6.38 &6.14& &\\
|
||
\hline
|
||
\mc{12}{c}{Glyoxal}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^b$ & Th.$^g$ & Exp.$^h$\\
|
||
\hline
|
||
$^1A_u (n \ra \pis)$ &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 \ra \pis)$ &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 \ra \pis,\pis)$&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 \ra \pis)$ &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 \ra 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 \ra \pis)$ &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 \ra \pis)$ &3.97&3.95& & &3.91&3.90& 3.90&3.89 &3.99 &4.12& &$\sim$3.8\\
|
||
$^3B_u (\pi \ra \pis)$ &5.22&5.20& & &5.20&5.19& 5.17&5.15 &5.17 &5.35& &$\sim$5.2\\
|
||
$^3A_g (\pi \ra \pis)$ &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} results from Ref.~\citenum{Aqu03};}
|
||
$^b${SAC-CI results from 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 results from 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} results from Ref.~\citenum{Sch17};}
|
||
$^h${Electron impact experiment from Ref.~\citenum{Ver80} except 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 in 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 from a theoretical point of view, and {\CCSDTQ} calculations were technically impossible despite our efforts. For the lowest $n \ra \pis$
|
||
transitions of both spin manifolds, the {\FCI} estimates come with a tiny error bar, and it is obvious that the CC excitation energies are slightly too low, especially with {\CCSDT}. Nevertheless, at the exception of the second singlet
|
||
and triplet $A''$ ES, the {\CCT} and {\CCSDT} transition energies 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.
|
||
\titou{This hints at a good accuracy for all other transitions.}
|
||
This statement is supported by the fact that the present CC values are nearly systematically bracketed by previous {\CASPT} (lower bound)\cite{Aqu03} and SAC-CI (upper bound)\cite{Sah06} results,
|
||
consistently with the typical error sign of these two models. For the two lowest triplet states, the present {\CCT}/{\AVTZ} values are also within $\pm 0.05$ eV of recent MRCI estimates ($3.50$ and $3.89$ eV). \cite{Mai14} As
|
||
can be seen in Table S3, the {\AVTZ} basis set delivers excitation energies very close to the CBS limit: the largest variation \titou{when going to {\AVQZ}} ($+0.04$ eV) is obtained for the second $^1A'$ Rydberg ES. As experimental data
|
||
are limited to measured UV spectra, \cite{Wal45,Bec70} one has 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 bamboozled theoretical chemists for many years. It is beyond the scope of the present study to provide an exhaustive
|
||
list of previous calculations and experimental measurements for these two hallmark ES, and we refer the readers to Refs.~\citenum{Wat12} and \citenum{Shu17} for a general and broader overview and the corresponding references. For the $B_u$ transition,
|
||
we believe that the most solid TBE 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 their 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 set corrected {\FCI} estimate of $6.50$ eV \cite{Loo19c} remains
|
||
the most accurate available to date. These two values are slightly smaller than the semi-stochastic 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} For these two thoroughly studied ES, one can, of course, find many other estimates, \eg, at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} \cite{Sok17} levels.
|
||
Globally, for 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 to the
|
||
already mentioned $^1A_g$ ES that is $0.2$ and $0.1$ eV too high with the two former CC methods, \titou{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, except for the lowest $B_u$ ES for which the associated excitation energy is highly dependent on the selected
|
||
active space (see the SI). Finally, as can be seen in Table S3, {\AVTZ} produces near-CBS excitation energies for most ES.
|
||
However, a significant basis set effect exists for the Rydberg $^1B_u (\pi \ra 3p)$ ES with an energy lowering 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 as our calculations.
|
||
|
||
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, \ie, highly consistent CC estimates, reasonable agreement between {\NEV} and {\CCT}, and limited basis set effects beyond {\AVTZ}, except for the $^1B_u (n \ra 3p)$ Rydberg state (see Tables \ref{Table-3} and S3). This Rydberg state also indicates a unexpectedly large deviation of $0.04$ eV between {\CCT} and {\CCSDTQ}. More interestingly,
|
||
glyoxal presents a genuine low-lying double ES of $^1A_g$ symmetry. The corresponding $(n,n \ra \pis,\pis)$ transition is totally unseen by approaches that cannot model double excitations, \eg, TD-DFT, {\CCSD},
|
||
or {\AD}. Compared to the {\FCI} values, the {\CCT} and {\CCSDT} estimates associated with this transition are too large by $\sim 1.0$ and $\sim 0.5$ eV, respectively, whereas both the {\CCSDTQ} and {\NEV} approaches are much closer, 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} are unhelpful 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 studies reporting estimates of the 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, for the four other compounds, the previous computational efforts were mainly focussed on the 0-0 energies of the lowest-lying states. \cite{Koh03,Hat05c,Sen11b,Loo18b,Loo19a}
|
||
There are also very few experimental data available for these six derivatives. \cite{Bir73,Jud83,Bra74,Sta75,Joh79,Jud83,Jud84c,Rob85,Pal87,Kar91b,Xin93}
|
||
Our main results are reported in Tables \ref{Table-4} and S4.
|
||
|
||
\begin{table}[htp]
|
||
\caption{\small Vertical transition energies (in eV) of acetone, cyanonformaldehyde, isobutene, propynal, thioacetone, and thiopropynal.}
|
||
\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
|
||
\mc{12}{c}{Acetone}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{3}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Th.$^b$ & Exp.$^c$ \\
|
||
\hline
|
||
$^1A_2 (n \ra \pis)$ &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 \ra 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 \ra 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 \ra 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 \ra 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 \ra \pis)$ &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 \ra \pis)$ &6.32&6.30& & &6.31&6.28& 6.28& &6.28 &5.98&5.98&\\
|
||
\hline
|
||
\mc{12}{c}{Cyanoformaldehyde}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{Litt.}\\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^d$ \\
|
||
\hline
|
||
$^1A'' (n \ra \pis)$ &3.91&3.89& &3.92$\pm$0.02 &3.86&3.84&3.83 &3.81&3.98& 3.26\\
|
||
$^1A'' (\pi \ra \pis)$ &6.64&6.67& &6.60$\pm$0.07 &6.51&6.54&6.42 &6.46&6.44& \\
|
||
$^3A'' (n \ra \pis)$ &3.53&3.51& &3.48$\pm$0.06 &3.47&3.45&3.46 & &3.58& \\
|
||
$^3A' (\pi \ra \pis)$ &5.07&5.07& & &5.03&5.03&5.01 & &5.35& \\
|
||
\hline
|
||
\mc{12}{c}{Isobutene}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{3}{c}{Litt.}\\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^e$ & Exp.$^f$ & Exp.$^g$ \\
|
||
\hline
|
||
$^1B_1 (\pi \ra 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 \ra 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 \ra \pis)$ &4.52&4.53& &4.56$\pm$0.02 &4.54&4.54& 4.53& &4.61& &4.21 &4.3 \\
|
||
\hline
|
||
\mc{12}{c}{Propynal}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{Litt.} \\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^h$ \\
|
||
\hline
|
||
$^1A'' (n \ra \pis)$ &3.90&3.87& &3.84$\pm$0.06 &3.85&3.82&3.82 &3.80&3.95& 3.24\\
|
||
$^1A'' (\pi \ra \pis)$ &5.69&5.73& &5.64$\pm$0.08 &5.59&5.62&5.51 &5.54&5.50& \\
|
||
$^3A'' (n \ra \pis)$ &3.56&3.54& &3.54$\pm$0.04 &3.50&3.48&3.49 & &3.59& 2.99\\
|
||
$^3A' (\pi \ra \pis)$ &4.46&4.47& &4.44$\pm$0.08 &4.40&4.44&4.43 & &4.63& \\
|
||
\hline
|
||
\mc{12}{c}{Thioacetone}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^i$ \\
|
||
\hline
|
||
$^1A_2 (n \ra \pis)$ &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 \ra 4s)$ &5.65&5.64&5.66& &5.44&5.43&5.55 &5.54&5.72 &5.49\\
|
||
$^1A_1 (\pi \ra \pis)$ &6.09&6.10&6.07& &5.97&5.98&5.90 &5.91&6.24 &5.64\\
|
||
$^1B_2 (n \ra 4p)$ &6.59&6.59&6.59& &6.45&6.44&6.51 & &6.62 &6.40\\
|
||
$^1A_1 (n \ra 4p)$ &6.95&6.95&6.96& &6.54&6.53&6.61 &6.60&6.52 &6.52\\
|
||
$^3A_2 (n \ra \pis)$ &2.36&2.34& &2.36$\pm$0.00 &2.36&2.35&2.34 & &2.32 &2.14\\
|
||
$^3A_1 (\pi \ra \pis)$ &3.45&3.45& & &3.51&3.50&3.46 & &3.46 &\\
|
||
\hline
|
||
\mc{12}{c}{Thiopropynal}\\
|
||
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^j$ \\
|
||
\hline
|
||
$^1A'' (n \ra \pis)$ &2.09&2.06& &2.08$\pm$0.01 &2.09&2.06&2.05 &2.03&2.05 &1.82\\
|
||
$^3A'' (n \ra \pis)$ &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} results from Ref.~\citenum{Mer96b};}
|
||
$^b${EOM-CCSD results from 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 results 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 \ra 3p$ state for the $^1A_1$ state)};
|
||
$^h${0-0 energies from 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 both very
|
||
sensitive to the basis set, and 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
|
||
right panel of Table \ref{Table-4} are clearly too low, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the \titou{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 of the three $n \ra 3p$ transitions, $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 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}/{\Pop} and {\CCSDTQ}/{\Pop} results.
|
||
While the lowest $n \ra \pis$ transition of both spin symmetries are rather basis set insensitive, all the other states need quite large one-electron 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 molecule, we have considered two singlet Rydberg state and one triplet valence ES. For these three cases, we note, for each basis, a very nice agreement between {\CCT} and {\CCSDT}, the
|
||
CC results being also very close to the {\FCI} estimates obtained with the Pople basis set. The similarity 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 with a definite 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, alluding that the CC values are very accurate.
|
||
This is further 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 CC and {\FCI} estimates in the vast majority of the cases.
|
||
Importantly, we confirm our previous conclusions obtained on smaller compounds: \cite{Loo18a} i) {\CCSDTQ} values systematically fall within (or are extremely close to) the {\FCI} error bar,
|
||
ii) both {\CCT} and {\CCSDT} are also highly trustable when the considered ES does not exhibit a strong double excitation character. Indeed, considering the 54 ``single'' excitations
|
||
for which {\CCSDTQ} estimates could be obtained (only excluding the lowest $^1A_g$ ES of butadiene and glyoxal), we determined negligible 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 below $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 the 91 (65) ES for which comparisons are possible (again excluding only the lowest $^1A_g$ states of butadiene and glyoxal),
|
||
one obtains a MSE of $+0.09$ ($+0.09$) eV and a MAE of $0.11$ ($0.12$) eV.
|
||
This seems to indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with {\CASPT} that is known to generally underestimate transition energies, as further illustrated and discussed above.
|
||
|
||
\subsection{Five-atom molecules}
|
||
|
||
We now consider five-member rings, and, in particular, five common 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 $\Td$ 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 \ra 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 (in eV) of furan and pyrrole.}
|
||
\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
|
||
\mc{15}{c}{Furan}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra 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 \ra \pis)$ &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 \ra \pis)$ &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 \ra 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 \ra 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 \ra 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 \ra \pis)$ &4.28&4.28 &4.25&4.23 &4.22& & &3.99& &4.26& & & &4.0\\
|
||
$^3A_1 (\pi \ra \pis)$ &5.56&5.54 &5.51&5.49 &5.48& & &5.15& &5.53& & & &5.2\\
|
||
$^3A_2 (\pi \ra 3s)$ &6.18&6.19 &5.94&5.94 &6.02& & &5.86& &5.89& & & &\\
|
||
$^3B_1 (\pi \ra 3p)$ &6.69&6.71 &6.51&6.51 &6.59& & &6.42& &6.41& & & &\\
|
||
\hline
|
||
\mc{15}{c}{Pyrrole}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra 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 \ra 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 \ra 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 \ra \pis)$ &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 \ra \pis)$ &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 \ra 3p)$ &7.20&7.20 &7.00&7.00 &6.83& && 5.78&6.26&6.61&6.57& &\\
|
||
$^3B_2 (\pi \ra \pis)$ &4.59&4.58 &4.56&4.54 &4.53& && 4.27& &4.53& & &4.21\\
|
||
$^3A_2 (\pi \ra 3s)$ &5.22&5.22 &5.12&5.12 &5.21& && 5.04& &5.07& & &5.1\\
|
||
$^3A_1 (\pi \ra \pis)$ &5.54&5.54 &5.49&5.48 &5.46& && 5.16& &5.53& & &\\
|
||
$^3B_1 (\pi \ra 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} results from Ref.~\citenum{Ser93b};}
|
||
$^b${{\NEV} results from Ref.~\citenum{Pas06b};}
|
||
$^c${MR-CC results from Ref.~\citenum{Li10c};}
|
||
$^d${{\AT} results from Ref.~\citenum{Hol15};}
|
||
$^e${{\CCT} results 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, Two values ($3.99$ eV and $5.22$ eV) can be found in Ref.~\citenum{Fli76};}
|
||
$^h${{\NEV} results from Ref.~\citenum{Pas06c};}
|
||
$^i${Best estimate from Ref.~\citenum{Chr99}, based on CC calculations;}
|
||
$^j${XMS-{\CASPT} results 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 $\Td$ but for the totally symmetric $\pi \ra \pis$ excitation ($\Td = 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 \ra 3p)$ transition. The match obtained with the twenty 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 \ra 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 (in eV) of cyclopentadiene, imidazole, and thiophene.}
|
||
\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
|
||
\mc{15}{c}{Cyclopentadiene}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra \pis)$ &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 \ra 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 \ra 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 \ra 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 \ra 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 \ra \pis)$ &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 \ra \pis)$ &3.33&3.33 &3.32&3.31 &3.32& &3.41 &3.15 &3.40& & &3.10 &\\
|
||
$^3A_1 (\pi \ra \pis)$ &5.16&5.15 &5.14&5.13 &5.12& &5.36 &4.90 &5.18& & &$>$4.7 & &\\
|
||
$^3A_2 (\pi \ra 3s)$ &6.01&6.02 &5.65&5.65 &5.73& &5.73 &5.63 &5.56& & &\\
|
||
$^3B_1 (\pi \ra 3p)$ &6.51&6.52 &6.30&6.30 &6.36& &6.40 &6.25 &6.19& & &\\
|
||
\hline
|
||
\mc{15}{c}{Imidazole}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{7}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & & {\NEV} &Th.$^h$ &Th.$^i$ &&&Exp.$^j$\\
|
||
\hline
|
||
$^1A'' (\pi \ra 3s)$ &5.77&5.77 &5.60&5.60 &5.71&& &5.71& &&&$\sim$5.2\\
|
||
$^1A' (\pi \ra \pis)$ &6.51&6.51 &6.43&6.43 &6.41&& &6.72&6.25&&&$\sim$6.4\\
|
||
$^1A'' (n \ra \pis)$ &6.66&6.66 &6.42&6.42 &6.50&& &6.52&6.65&&& \\
|
||
$^1A' (\pi \ra 3p)$ &7.04&7.02 &6.93&6.89 &6.87&& &6.49& &&&\\
|
||
$^3A' (\pi \ra \pis)$ &4.83&4.81 &4.78& &4.75&& &4.49&4.65&&&\\
|
||
$^3A'' (\pi \ra 3s)$ &5.72&5.72 &5.57&5.56 &5.67&& &5.68& &&&\\
|
||
$^3A' (\pi \ra \pis)$ &5.88&5.88 &5.78& &5.74&& &5.47&5.64&&&\\
|
||
$^3A'' (n \ra \pis)$ &6.48&6.46 &6.37&6.35 &6.33&& &6.07&6.25&&&\\
|
||
\hline
|
||
\mc{15}{c}{Thiophene}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra \pis)$ &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 \ra \pis)$ &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 \ra 3s)$ &6.26&6.26 &6.07&6.06 &6.14&6.14& &5.93&5.70&6.05&6.23& & &\\
|
||
$^1B_1 (\pi \ra 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 \ra 3p)$ &6.32& &6.33&6.31 &6.25&6.21& &6.35&6.03&6.28&6.33& &\\
|
||
$^1B_1 (\pi \ra 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 \ra 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 \ra \pis)$ &7.50&7.46 &7.41& &7.35& & &6.69&7.32&7.71&7.74&6.61&\\
|
||
$^3B_2 (\pi \ra \pis)$ &3.95&3.94 &3.96&3.94 &3.94& & &3.75&3.94& &3.96&&3.74&\\
|
||
$^3A_1 (\pi \ra \pis)$ &4.90&4.90 &4.82&4.81 &4.77& & &4.50&4.86& &4.87&&4.62&\\
|
||
$^3B_1 (\pi \ra 3p)$ &6.00&5.98 &6.01&5.99 &5.95& & &5.90&5.94& &6.01&\\
|
||
$^3A_2 (\pi \ra 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} results from Ref.~\citenum{Ser93b};}
|
||
$^b${SAC-CI results from Ref.~\citenum{Wan00b};}
|
||
$^c${MR-MP results from Ref.~\citenum{Nak96};}
|
||
$^d${{\CCT} results 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 states; two-photon resonant experiment from Ref.~\citenum{Sab92} for the $^1A_2$ Rydberg ES;}
|
||
$^h${{\CASPT} results from Ref.~\citenum{Ser96b};}
|
||
$^i${{\CCT} results from Ref.~\citenum{Sil10c};}
|
||
$^j${Gas-phase experimental estimates from Ref.~\citenum{Dev06};}
|
||
$^k${{\CASPT} results from Ref.~\citenum{Ser93c};}
|
||
$^l${SAC-CI results from Ref.~\citenum{Wan01};}
|
||
$^m${CCSDR(3) results 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 triplets;}
|
||
$^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 $\Td$ exceeding 93\%\ but for the $^1A_1 (\pi \ra \pis)$
|
||
excitation that has a nature similar to the lowest $A_g$ state of butadiene ($\Td = 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 \ra \pis)$ 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 \ra \pis)$ 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 $\Td$ 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 $\Td$ above
|
||
90\%\ but for the two $^1A_1$ transitions that show $\Td$ 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 \eg, a strong mixing between the second and third singlet $B_2$ ES rendering the assignment of the valence
|
||
($\pi \ra \pis$) or Rydberg ($\pi \ra 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 \ra 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 $\Td$ 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 (in eV) of benzene.}
|
||
\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
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra \pis)$ &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 \ra \pis)$ &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 \ra 3s)$ &6.75&6.76 &6.46&6.46 &6.52&6.52& &6.38&6.51&6.42& & &6.33\\
|
||
$^1A_{2u} (\pi \ra 3p)$ &7.24&7.25 &7.02&7.02 &7.08&7.08& &6.86&6.97&7.06& & &6.93\\
|
||
$^1E_{2u} (\pi \ra 3p)$ &7.34&7.35 &7.09&7.09 &7.15&7.15& &6.91&7.03&7.12& & &6.95\\
|
||
$^3B_{1u} (\pi \ra \pis)$ &4.18&4.16 &4.19&4.17 &4.18& & &3.89&4.15&3.88&4.11&3.95&\\
|
||
$^3E_{1u}(\pi \ra \pis)$ &4.95&4.94 &4.89&4.88 &4.86& & &4.49&4.86&4.72&4.75&4.75&\\
|
||
$^3B_{2u} (\pi \ra \pis)$ &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} results from Ref.~\citenum{Lor95b};}
|
||
$^b${{\CCT} results from Ref.~\citenum{Chr96c};}
|
||
$^c${SAC-CI results from Ref.~\citenum{Li07b};}
|
||
$^d${RASPT2(18,18) results 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 (in eV) of pyrazine and tetrazine.}
|
||
\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
|
||
\mc{14}{c}{Pyrazine}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra \pis)$ &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 \ra \pis)$ &5.08&5.08 &4.98&4.98 &4.97&4.98& &4.36&4.93&5.24&4.93&\\
|
||
$^1B_{2u} (\pi \ra \pis)$ &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 \ra \pis)$ &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 \ra 3s)$ &6.74&6.73 &6.54&6.53 &6.66&6.65& & &6.83&7.07&6.46&\\
|
||
$^1B_{1g} (n \ra \pis)$ &6.87&6.87 &6.75&6.75 &6.73&6.74& &6.26&6.73& &6.73&&6.10\\
|
||
$^1B_{1u} (\pi \ra \pis)$ &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 \ra 3p)$ &7.36&7.37 &7.13&7.14 &7.20&7.21& & &7.31&7.08& &\\
|
||
$^1B_{2u} (n \ra 3p)$ &7.39&7.39 &7.14&7.13 &7.25& & & &7.45&7.67&7.06&\\
|
||
$^1B_{1u} (\pi \ra 3s)$ &7.56&7.55 &7.38&7.37 &7.45& & &7.28&7.50&7.73&7.31&&7.67\\
|
||
$^1B_{1u} (\pi \ra \pis)$ &8.19&8.23 &7.99&8.03 &7.94& & &7.43&7.96&8.24&8.08&\\
|
||
$^3B_{3u} (n \ra \pis)$ &3.68&3.68 &3.60&3.60 &3.59& & &3.16& & & &3.33\\
|
||
$^3B_{1u} (\pi \ra \pis)$ &4.39&4.36 &4.40&4.36 &4.39& & &4.15& & & &4.04\\
|
||
$^3B_{2u} (\pi \ra \pis)$ &4.56&4.55 &4.46&4.45 &4.40& & &4.28& & & &$\sim$4.4\\
|
||
$^3A_{u} (n \ra \pis)$ &5.05&5.05 &4.93&4.93 &4.93& & &4.19& & & &4.2\\
|
||
$^3B_{2g} (n \ra \pis)$ &5.18&5.17 &5.11&5.11 &5.08& & &4.81& & & &4.49\\
|
||
$^3B_{1u} (\pi \ra \pis)$ &5.38&5.37 &5.32&5.31 &5.29& & &4.98& & & &\\
|
||
\hline
|
||
\mc{14}{c}{Tetrazine}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{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 \ra \pis)$ &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 \ra \pis)$ &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& &\\
|
||
$^1B_{1g} (n \ra \pis)$ &5.01&5.02 &4.97&4.98 &4.91&4.93& &4.51&4.73&4.47&4.53&4.87&\\
|
||
$^1B_{2u} (\pi \ra \pis)$ &5.29&5.26 &5.27&5.25 &5.23&5.21& &4.89&4.90& &5.59&5.08&4.97\\
|
||
$^1B_{2g} (n \ra \pis)$ &5.56&5.52 &5.53&5.50 &5.46&5.45& &5.05&5.09&4.92&5.59&5.28&\\
|
||
$^1A_{u} (n \ra \pis)$ &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})$$^m$ &7.64& &7.62& &7.62& &6.15 &5.16&6.30&5.26&6.01& &5.92\\
|
||
$^1B_{2g} (n \ra \pis)$ &6.24&6.22 &6.17&6.16 &6.13& & &5.48&6.16&5.78&6.05&6.16&\\
|
||
$^1B_{1g} (n \ra \pis)$ &7.04&7.04 &6.98&6.98 &6.92& & &5.99&6.73&6.20&6.92&6.80&\\
|
||
$^3B_{3u} (n \ra \pis)$ &1.87&1.88 &1.86&1.86 &1.85& & &1.45&1.71& & &1.87&1.7\\
|
||
$^3A_{u} (n \ra \pis)$ &3.48&3.49 &3.43&3.44 &3.44& & &2.81&3.47& & &3.49&2.90\\
|
||
$^3B_{1g} (n \ra \pis)$ &4.25&4.25 &4.23&4.23 &4.20& & &3.76&3.97& & &4.18&\\
|
||
$^3B_{1u} (\pi \ra \pis)$ &4.54&4.49 &4.54&4.49 &4.54& & &4.25&3.67& & &4.36&\\
|
||
$^3B_{2u} (\pi \ra \pis)$ &4.65&4.64 &4.58&4.58 &4.52& & &4.29&4.35& & &4.39&\\
|
||
$^3B_{2g} (n \ra \pis)$ &5.11&5.11 &5.09&5.08 &5.05& & &4.67&4.78& & &4.89&\\
|
||
$^3A_{u} (n \ra \pis)$ &5.17&5.17 &5.15&5.15 &5.11& & &4.85&4.89& & &4.96&\\
|
||
$^3B_{3g} (\mathrm{double})$$^m$ &7.35& &7.33& &7.35& &5.51 &5.08& & & & &\\
|
||
$^3B_{1u} (\pi \ra \pis)$ &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} results from Ref.~\citenum{Web99};}
|
||
$^b${{\STEOM} results from Ref.~\citenum{Noo99};}
|
||
$^c${SAC-CI results from Ref.~\citenum{Li07b};}
|
||
$^d${{\CCT} results from Ref.~\citenum{Sch17};}
|
||
$^e${\titou{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} results from Ref.~\citenum{Rub99};}
|
||
$^h${Ext-{\STEOM} results from Ref.~\citenum{Noo00};}
|
||
$^i${GVVPT2 results from Ref.~\citenum{Dev08};}
|
||
$^j${{\NEV} results from Ref.~\citenum{Ang09};}
|
||
$^k${{\CCT} results from Ref.~\citenum{Sil10c};}
|
||
$^l${From Ref.~\citenum{Pal97}, the singlets are from EEL, except 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 $(n,n \ra \pis, \pis)$ 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 $\Td > 85\%$, but for $^1B_{1g} (n \ra \pis)$, 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 ($\Td < 10\%$), 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 $\Td$ 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 $^3B_{1u} (\pi \ra \pis)$ 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 $\Td$ 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 $^3B_2 (\pi \ra \pis)$ 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 $\Td$ 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 $^1B_2 (\pi \ra \pis)$ 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 (in eV) of pyridazine and pyridine.}
|
||
\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
|
||
\mc{13}{c}{Pyridazine}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{2}{c}{\AVTZ} & \mc{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^e$ &Exp.$^f$\\
|
||
\hline
|
||
$^1B_1 (n \ra \pis)$ &3.95&3.95 &3.86&3.86 &3.83& &3.48&3.76&3.65&3.85&&3.36\\
|
||
$^1A_2 (n \ra \pis)$ &4.49&4.48 &4.39&4.39 &4.37& &3.66&4.46&4.28&4.44&&4.02\\
|
||
$^1A_1 (\pi \ra \pis)$ &5.36&5.32 &5.33&5.30 &5.29& &4.86&4.92&4.86&5.20&5.0&5.01\\
|
||
$^1A_2 (n \ra \pis)$ &5.88&5.86 &5.80&5.78 &5.74& &5.09&5.66&5.52&5.66&&5.61\\
|
||
$^1B_2 (n \ra 3s)$ &6.26&6.27 &6.06&6.06 &6.17& & &6.45& & && \\
|
||
$^1B_1 (n \ra \pis)$ &6.51&6.51 &6.41&6.41 &6.37& &5.80&6.41&6.20&6.33&&6.00\\
|
||
$^1B_2 (\pi \ra \pis)$ &6.96&6.97 &6.79&6.80 &6.74& &6.61&6.77&6.44&6.68&&6.50\\
|
||
$^3B_1 (n \ra \pis)$ &3.27&3.26 &3.20&3.20 &3.19& & & & & &&3.06\\
|
||
$^3A_2 (n \ra \pis)$ &4.19&4.19 &4.11&4.11 &4.11& & & & & &&3.55\\
|
||
$^3B_2 (\pi \ra \pis)$ &4.39&4.36 &4.39&4.35 &4.38& & & & & &4.0&4.33\\
|
||
$^3A_1 (\pi \ra \pis)$ &4.93&4.94 &4.87&4.86 &4.83& & & & & &4.4&4.68\\
|
||
\hline
|
||
\mc{13}{c}{Pyridine}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{2}{c}{\AVTZ} & \mc{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\NEV} &Th.$^g$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^h$ &Exp.$^i$ \\
|
||
\hline
|
||
$^1B_1 (n \ra \pis)$ &5.12&5.10 &5.01&5.00 &4.96& &4.91&4.90&4.80&4.95&5.24&4.78\\
|
||
$^1B_2 (\pi \ra \pis)$ &5.23&5.20 &5.21&5.18 &5.17& &4.84&4.82&4.81&5.12&4.99&4.99\\
|
||
$^1A_2 (n \ra \pis)$ &5.55&5.54 &5.41&5.41 &5.40& &5.17&5.31&5.24&5.41&5.43&5.40\\
|
||
$^1A_1 (\pi \ra \pis)$ &6.84&6.84 &6.64&6.63 &6.63& &6.42&6.62&6.36&6.60&6.38& \\
|
||
$^1A_1 (n \ra 3s)$ &6.92&6.92 &6.71&6.71 &6.76& &6.70&6.96&6.64& &6.28&6.25\\
|
||
$^1A_2 (\pi \ra 3s)$ &6.98&6.99 &6.74&6.75 &6.81& &6.75&6.90&6.53& & & \\
|
||
$^1B_2 (\pi \ra \pis)$$^j$ &7.50&7.52 &7.40&\hl{xxx} &\hl{xxx} & &7.48&7.29&7.14&7.33&7.22&7.20\\
|
||
$^1B_1 (\pi \ra 3p)$ &7.54&7.55 &7.32& \hl{xxx} &7.38& &7.25&7.42&7.10& & & \\
|
||
$^1A_1 (\pi \ra \pis)$ &7.56& &7.34&\hl{xxx} &7.39& &7.23&7.37&7.26&7.39&7.22&6.39\\
|
||
$^3A_1 (\pi \ra \pis)$ &4.33&4.31 &4.34&4.31 &4.33& &4.05& & &4.28& &3.86\\
|
||
$^3B_1 (n \ra \pis)$ &4.57&4.56 &4.47&4.47 &4.46& &4.41& & &4.42& &4.12\\
|
||
$^3B_2 (\pi \ra \pis)$ &4.92&4.91 &4.83&4.83 &4.79& &4.56& & &4.72& &4.47\\
|
||
$^3A_1 (\pi \ra \pis)$ &5.14&5.13 &5.08&\hl{xxx} &5.05& &4.73& & &4.96& & \\
|
||
$^3A_2 (n \ra \pis)$ &5.51&5.49 &5.37&5.36 &5.35& &5.10& & &5.53& &5.40\\
|
||
$^3B_2 (\pi \ra \pis)$ &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} results from Ref.~\citenum{Ful92};}
|
||
$^b${{\STEOM} results 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 $\Td$ larger than $85\%$ for singlet and $95\%$ for triplet, so that {\CCT} and {\CCSDT} are highly coherent, but possibly in one case, the
|
||
$^3A_1 (\pi \ra \pis)$ 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 (in eV) of pyrimidine and triazine.}
|
||
\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
|
||
\mc{14}{c}{Pyrimidine}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & & {\NEV} &Th.$^a$ &Th.$^b$ &Th.$^c$&Th.$^d$ &Exp.$^e$ &Exp.$^f$\\
|
||
\hline
|
||
$^1B_1 (n \ra \pis)$ &4.58&4.57 &4.48&4.48 &4.44&& &4.26&4.40&4.32&4.24& 4.2 &4.18 \\
|
||
$^1A_2 (n \ra \pis)$ &4.99&4.99 &4.89&4.88 &4.86&& &4.49&4.72&4.74&4.74& &4.69 \\
|
||
$^1B_2 (\pi \ra \pis)$ &5.47&5.44 &5.44&5.41 &5.41&& &5.47&5.04&5.29&5.01& 5.12 &5.18 \\
|
||
$^1A_2 (n \ra \pis)$ &6.07&6.06 &5.98&5.97 &5.93&& & &5.94&5.98&5.84& 6.05 &5.67 \\
|
||
$^1B_1 (n \ra \pis)$ &6.39& &6.29&6.29 &6.26&& &6.03&6.18&6.35&6.11& &6.02 \\
|
||
$^1B_2 (n \ra 3s)$ &6.81&6.80 &6.61&6.59 &6.72&& & &6.85&6.84&6.57& & \\
|
||
$^1A_1 (\pi \ra \pis)$ &7.08&7.09 &6.93&6.94 &6.87&& &7.10&6.87&6.86&6.57& 6.7 &6.69 \\
|
||
$^3B_1 (n \ra \pis)$ &4.20&4.20 &4.12&4.11 &4.10&& &3.81& &4.11& & &3.85 \\
|
||
$^3A_1 (\pi \ra \pis)$ &4.55&4.52 &4.56&4.52 &4.55&& &4.35& &4.39& & &4.42 \\
|
||
$^3A_2 (n \ra \pis)$ &4.77&4.76 &4.67&4.67 &4.66&& &4.24& &4.71& & &4.18 \\
|
||
$^3B_2 (\pi \ra \pis)$ &5.08&5.08 &5.00&5.00 &4.96&& &4.83& &4.81& & &4.93 \\
|
||
\hline
|
||
\mc{14}{c}{Triazine}\\
|
||
& \mc{2}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{6}{c}{Litt.}\\
|
||
State & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\CCT} & {\CCSDT} & {\NEV} &Th.$^g$ &Th.$^b$ &Th.$^d$&Th.$^h$ &Exp.$^h$ \\
|
||
\hline
|
||
$^1A_1'' (n \ra \pis)$ &4.85&4.84 &4.76&4.74 &4.73&4.72& &4.11&4.58&4.49&4.70 &\\
|
||
$^1A_2'' (n \ra \pis)$ &4.84&4.84 &4.78&4.78 &4.74&4.75& &4.30&4.74&4.54&4.71 &4.59\\
|
||
$^1E'' (n \ra \pis)$ &4.89&4.89 &4.82&4.81 &4.78&4.78& &4.32&4.69&4.56&4.75 &3.97\\
|
||
$^1A_2' (\pi \ra \pis)$ &5.84&5.80 &5.81&5.78 &5.78&5.75& &5.59&5.35&5.36&5.71 &5.70\\
|
||
$^1A_1' (\pi \ra \pis)$ &7.45&7.45 &7.31&7.31 &7.24&7.24& & &7.21&6.90&7.18 &6.86\\
|
||
$^1E' (n \ra 3s)$ &7.44&7.41 &7.24&7.21 &7.35&7.32& & &7.38&7.16& & \\
|
||
$^1E'' (n \ra \pis)$ &7.89&7.86 &7.82&7.80 &7.79&7.78& & & &7.78&7.78 &\\
|
||
$^1E' (\pi \ra \pis)$ &8.12&8.13 &7.97& &7.92&7.94& & &7.82&7.72&7.84 &7.76\\
|
||
$^3A_2'' (n \ra \pis)$ &4.40&4.40 &4.35&4.35 &4.33& & &3.87&\\
|
||
$^3E'' (n \ra \pis)$ &4.59&4.59 &4.52&4.52 &4.51& & &4.04&\\
|
||
$^3A_1'' (n \ra \pis)$ &4.87& &4.78&4.76 &4.75& & &4.15&\\
|
||
$^3A_1' (\pi \ra \pis)$ &4.88&4.85 &4.88&4.85 &4.88& & & &\\
|
||
$^3E' (\pi \ra \pis)$ &5.70&5.68 &5.64& &5.61& & & &\\
|
||
$^3A_2' (\pi \ra \pis)$ &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}
|
||
|
||
Table \ref{Table-tbe} reports our two sets of TBE: a set obtained with the {\AVTZ} basis set and one set with an additional correction for the one-electron basis set incompleteness error. The details of our protocol employed to generate these TBE are also provided in Table \ref{Table-tbe}.
|
||
For all states with a dominant single-excitation character (that is when $\Td > 80\%$), we rely on CC results using an incremental strategy to generate these TBE. For ES with $\Td$ between $70\%$ and $80\%$,
|
||
our previous works indicated that {\CCSDT} tends to overshoot the transition energies by roughly $0.05$--$0.10$ eV, but that {\NEV} errors tend to be, on average, slightly larger. \cite{Loo19c} Therefore, if {\CCSDTQ} or {\FCI}
|
||
results are not available, it is extremely difficult to make the final call. For the other transitions, we relied either on the current or previous FCI data or the {\NEV} values as reference. The italicized transition energies in Table \ref{Table-tbe} are believed to be (relatively) less accurate. This is the case when: i) {\NEV} results have to be selected; ii) the CC calculations yield quite large changes in excitation energies while incrementing the excitation order by one unit despite large $\Td$;
|
||
and iii) there is a very strong 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}/{\AVPZ} results. For several compounds, we also provide in the SI, {\CCT}/d-{\AVQZ} transition energies (\ie, with an additional set of diffuse functions). However, we do not consider such values as reference because
|
||
the addition of a second set of diffuse functions only significantly modifies the transition energies for strongly-mixing ES.
|
||
We also stick to the frozen-core approximation for two reasons: i) the effect of correlating the core electrons is generally negligible (typically $\pm 0.02$ eV) for the compounds under study (see the SI for examples); and ii) it would be,
|
||
in principle, necessary to add core polarization functions in such cases.
|
||
|
||
Table \ref{Table-tbe} encompasses 238 ES, each of them obtained, at least, at the {\CCSDT} level. This set can be decomposed as follows: 144 singlet and 94 triplet transitions, or 174 valence (99 $\pi \ra \pis$, 71
|
||
$n \ra \pis$ and 4 double excitations) and 64 Rydberg transitions. Amongst these transition energies, fifteen can be considered as ``unsafe'' and are reported in italics accordingly. This definitely corresponds to a very significant extension of our previous
|
||
ES data sets (see Introduction).
|
||
Taken all together, they offer a consistent, diverse and accurate ensemble of transition energies for approximately 350 electronic transitions of various natures in small and medium-sized organic molecules.
|
||
Table \ref{Table-tbe} also reports 90 oscillator strengths, $f$, which makes it, to the best of our knowledge, the largest set of {\CCT}/{\AVTZ} oscillator strengths reported to date, the previous effort being mostly performed
|
||
at the {\CCT}/TZVP level for Thiel's set. \cite{Kan14} It should also be pointed out that all these data are obtained on {\CCT}/{\AVTZ} geometries, consistently with our previous works. \cite{Loo18a,Loo19c}
|
||
|
||
\renewcommand*{\arraystretch}{.55}
|
||
\LTcapwidth=\textwidth
|
||
|
||
\begin{footnotesize}
|
||
\begin{longtable}{p{3.3cm}lcccccc}
|
||
\caption{\small TBE values (in eV) for all considered states alongside their corresponding oscillator strength, $f$, and percentage of single excitations, $\Td$, obtained at the \CCT/{\AVTZ} level.
|
||
The composite protocol to generate these TBE is also reported (see footnotes). In the right-most column, we list the TBE values obtained by including an additional correction (obtained at the {\CCT} level) for basis set incompleteness error.
|
||
Values displayed in italics are likely to be relatively less accurate. All values are obtained in the FC approximation.} \label{Table-tbe}\\
|
||
\hline
|
||
& & & & \mc{2}{l}{TBE/{\AVTZ}} & \mc{2}{l}{TBE/CBS} \\
|
||
& State & $f$ & $\Td$ & Value & Protocol$^a$ & Value & Corr. \\
|
||
\hline
|
||
\endfirsthead
|
||
\hline
|
||
& & & & \mc{2}{l}{TBE/{\AVTZ}} & \mc{2}{l}{TBE/CBS} \\
|
||
& State & $f$ & $\Td$ & Value & Protocol $^a$ & Value & Corr. \\
|
||
\hline
|
||
\endhead
|
||
\hline \mc{7}{r}{{Continued on next page}} \\
|
||
\endfoot
|
||
\hline
|
||
\endlastfoot
|
||
Acetone &$^1A_2 (\Val; n \ra \pis)$ & & 91.1 & 4.47 & B & 4.48 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3s)$ &0.000 & 90.5 & 6.46 & B & 6.51 & \AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; n \ra 3p)$ & & 90.9 & 7.47 & B & 7.44 & \AVQZ \\
|
||
&$^1A_1 (\mathrm{R}; n \ra 3p)$ &0.004 & 90.6 & 7.51 & B & 7.55 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3p)$ &0.029 & 91.2 & 7.62 & B & 7.63 & \AVQZ \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 97.8 & 4.13 & D & 4.15 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.7 & 6.25 & D & 6.27 & \AVQZ \\
|
||
Acrolein &$^1A'' (\Val; n \ra \pis)$ &0.000 & 87.6 & 3.73 & {\CCSDT}/\AVTZ & 3.74 & \AVQZ \\
|
||
&$^1A' (\Val; \pi \ra \pis)$ &0.344 & 91.2 & 6.69 & {\CCSDT}/\AVTZ & 6.69 & \AVQZ \\
|
||
&$^1A'' (\Val; n \ra \pis)$ &0.000 & 79.4 & \emph{6.72} & D &\emph{6.74} & \AVQZ \\
|
||
&$^1A' (\mathrm{R}; n \ra 3s)$ &0.109 & 89.4 & 7.08 & D & 7.12 & \AVQZ \\
|
||
&$^3A'' (\Val; n \ra \pis)$ & & 97.0 & 3.44 & D & 3.43 & \AVQZ \\
|
||
&$^3A' (\Val; \pi \ra \pis)$ & & 98.6 & 3.94 & D & 3.95 & \AVQZ \\
|
||
&$^3A' (\Val; \pi \ra \pis)$ & & 98.4 & 6.18 & D & 6.19 & \AVQZ \\
|
||
&$^3A'' (\Val; n \ra \pis)$ & & 92.7 & \emph{6.54} & E & \emph{6.55}& \AVQZ \\%remove
|
||
Benzene &$^1B_{2u} (\Val; \pi \ra \pis)$ & & 86.3 & 5.06 & {\CCSDT}/\AVTZ & 5.06 &\AVQZ \\
|
||
&$^1B_{1u} (\Val; \pi \ra \pis)$ & & 92.9 & 6.45 & {\CCSDT}/\AVTZ &6.44 &\AVQZ \\
|
||
&$^1E_{1g} (\mathrm{R}; \pi \ra 3s)$ & & 92.8 & 6.52 & {\CCSDT}/\AVTZ &6.54 &\AVQZ \\
|
||
&$^1A_{2u} (\mathrm{R}; \pi \ra 3p)$ &0.066 & 93.4 & 7.08 & {\CCSDT}/\AVTZ &7.10 &\AVQZ \\
|
||
&$^1E_{2u} (\mathrm{R}; \pi \ra 3p)$ & & 92.8 & 7.15 & {\CCSDT}/\AVTZ &7.16 &\AVQZ \\
|
||
&$^3B_{1u} (\Val; \pi \ra \pis)$ & & 98.6 & 4.16 & D &4.17 &\AVQZ \\
|
||
&$^3E_{1u}(\Val; \pi \ra \pis)$ & & 97.1 & 4.85 & D &4.86 &\AVQZ \\
|
||
&$^3B_{2u} (\Val; \pi \ra \pis)$ & & 98.1 & 5.81 & D & 5.81 &\AVQZ \\
|
||
Butadiene &$^1B_u (\Val; \pi \ra \pis)$ &0.664 & 93.3 & 6.22 & B & 6.21 & \AVQZ \\
|
||
&$^1B_g (\mathrm{R}; \pi \ra 3s)$ & & 94.1 & 6.33 & B & 6.35 & \AVQZ \\
|
||
&$^1A_g (\Val; \pi \ra \pis)$ & & 75.1 & 6.50 & F & 6.50 & \AVQZ \\
|
||
&$^1A_u (\mathrm{R}; \pi \ra 3p)$ &0.001 & 94.1 & 6.64 & B & 6.66 & \AVQZ \\
|
||
&$^1A_u (\mathrm{R}; \pi \ra 3p)$ &0.049 & 94.1 & 6.80 & B & 6.82 & \AVQZ \\
|
||
&$^1B_u (\mathrm{R}; \pi \ra 3p)$ &0.055 & 93.8 & 7.68 & C & 7.54 & \AVQZ \\
|
||
&$^3B_u (\Val; \pi \ra \pis)$ & & 98.4 & 3.36 & D & 3.37 & \AVQZ \\
|
||
&$^3A_g (\Val; \pi \ra \pis)$ & & 98.7 & 5.20 & D & 5.21 & \AVQZ \\
|
||
&$^3B_g (\mathrm{R}; \pi \ra 3s)$ & & 97.9 & 6.29 & D & 6.31 & \AVQZ \\
|
||
Cyanoacetylene &$^1\Sigma^- (\Val; \pi \ra \pis)$ & & 94.3 & 5.80 & A & 5.79 & \AVPZ\\
|
||
&$^1\Delta (\Val; \pi \ra \pis)$ & & 94.0 & 6.07 & A & 6.05 &\AVPZ\\
|
||
&$^3\Sigma^+ (\Val; \pi \ra \pis)$ & & 98.5 & 4.44 & {\CCSDT}/\AVTZ & 4.46 &\AVPZ \\
|
||
&$^3\Delta (\Val; \pi \ra \pis)$ & & 98.2 & 5.21 & {\CCSDT}/\AVTZ & 5.21 & \AVPZ\\
|
||
&$^1A'' [\mathrm{F}] (\Val; \pi \ra \pis)$ & 0.004 & 93.6 & 3.54 & A & 3.54 & \AVQZ \\
|
||
Cyanoformaldehyde &$^1A'' (\Val; n \ra \pis)$ & 0.001 & 89.8 & 3.81 & {\CCSDT}/\AVTZ & 3.82 & \AVQZ \\
|
||
&$^1A'' (\Val; \pi \ra \pis)$ & 0.000 & 91.9 & 6.46 & {\CCSDT}/\AVTZ & 6.45 & \AVQZ \\
|
||
&$^3A'' (\Val; n \ra \pis)$ & & 97.6 & 3.44 & D & 3.45 & \AVQZ \\
|
||
&$^3A' (\Val; \pi \ra \pis)$ & & 98.4 & 5.01 & D & 5.02 & \AVQZ \\
|
||
Cyanogen & $^1\Sigma_u^- (\Val; \pi \ra \pis)$ & & 94.1 & 6.39 & A & 6.38 & \AVPZ\\
|
||
& $^1\Delta_u (\Val; \pi \ra \pis)$ & & 93.4 & 6.66 & A & 6.64 & \AVPZ\\
|
||
& $^3\Sigma_u^+ (\Val; \pi \ra \pis)$ & & 98.5 & 4.91 & B & 4.93 & \AVPZ \\
|
||
& $^1\Sigma_u^- [\mathrm{F}] (\Val; \pi \ra \pis)$ & & 93.4 & 5.05 & A & 5.03 & \AVPZ \\
|
||
Cyclopentadiene &$^1B_2 (\Val; \pi \ra \pis)$ &0.084 & 93.8 & 5.56 & {\CCSDT}/\AVTZ & 5.55 & \AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3s)$ & & 94.0 & 5.78 & {\CCSDT}/\AVTZ & 5.80 & \AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \ra 3p)$ &0.037 & 94.2 & 6.41 & {\CCSDT}/\AVTZ & 6.42 & \AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3p)$ & & 93.8 & 6.46 & {\CCSDT}/\AVTZ & 6.47 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; \pi \ra 3p)$ &0.046 & 94.2 & 6.56 &{\CCSDT}/\AVTZ & 6.55 & \AVQZ\\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.001 & 78.9 & \emph{6.51} & D & \emph{6.51} & \AVQZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 98.4 & 3.31 & D & 3.31 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.6 & 5.11 & D & 5.12 & \AVQZ \\
|
||
&$^3A_2 (\mathrm{R}; \pi \ra 3s)$ & & 97.9 & 5.73 & D & 5.75 & \AVQZ \\
|
||
&$^3B_1 (\mathrm{R}; \pi \ra 3p)$ & & 97.9 & 6.36 & D & 6.38 & \AVQZ \\
|
||
Cyclopropenone &$^1B_1 (\Val; n \ra \pis)$ &0.000 & 87.7 & 4.26 & B & 4.28 & \AVPZ \\
|
||
&$^1A_2 (\Val; n \ra \pis)$ & & 91.0 & 5.55 & B & 5.56 &\AVPZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3s)$ &0.003 & 90.8 & 6.34 & B & 6.40 & \AVPZ \\
|
||
&$^1B_2 (\Val; \pi \ra \pis$) &0.047 & 86.5 & 6.54 & B & 6.56 & \AVPZ\\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3p)$ &0.018 & 91.1 & 6.98 & B & 7.01 & \AVPZ \\
|
||
&$^1A_1 (\mathrm{R}; n \ra 3p)$ &0.003 & 91.2 & 7.02 & B & 7.08 &\AVPZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.320 & 90.8 & 8.28 & B & 8.26 &\AVPZ \\
|
||
&$^3B_1 (\Val; n \ra \pis)$ & & 96.0 & 3.93 & {\CCSDT}/\AVTZ & 3.96 & \AVPZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 97.9 & 4.88 & {\CCSDT}/\AVTZ & 4.91 & \AVPZ \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 97.5 & 5.35 & {\CCSDT}/\AVTZ & 5.37 & \AVPZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.1 & 6.79 & {\CCSDT}/\AVTZ & 6.81 & \AVPZ\\
|
||
Cyclopropenethione &$^1A_2 (\Val; n \ra \pis)$ & & 89.6 & 3.41 & B & 3.41 & \AVPZ \\
|
||
&$^1B_1 (\Val; n \ra \pis)$ &0.000 & 84.8 & 3.45 & B & 3.48 & \AVPZ \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ &0.007 & 83.0 & 4.60 & B & 4.62 & \AVPZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3s)$ &0.048 & 91.8 & 5.34 & B & 5.40 & \AVPZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.228 & 89.0 & 5.46 & B & 5.46 & \AVPZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3p)$ &0.084 & 91.3 & 5.92 & B & 5.94 & \AVPZ \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 97.2 & 3.28 & D & 3.28 & \AVPZ \\
|
||
&$^3B_1 (\Val; n \ra \pis)$ & & 94.5 & 3.32 & {\CCSDT}/\AVTZ & 3.36 & \AVPZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 96.5 & 4.01 & D & 4.04 & \AVPZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.2 & 4.01 & D & 4.01 & \AVPZ \\
|
||
Diacetylene &$^1\Sigma_u^- (\Val; \pi \ra \pis)$ & & 94.4 & 5.33 & A & 5.32 & \AVPZ \\
|
||
&$^1\Delta_u (\Val; \pi \ra \pis)$ & & 94.1 & 5.61 & A & 5.60 & \AVPZ \\
|
||
&$^3\Sigma_u^+ (\Val; \pi \ra \pis)$ & & 98.5 & 4.10 & C & 4.13 & \AVPZ \\
|
||
&$^3\Delta_u (\Val; \pi \ra \pis)$ & & 98.2 & 4.78 & B & 4.78 &\AVPZ \\
|
||
Furan &$^1A_2 (\mathrm{R}; \pi \ra 3s)$ & & 93.8 & 6.09 &{\CCSDT}/\AVTZ & 6.11 &\AVQZ \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ &0.163 & 93.0 & 6.37 &{\CCSDT}/\AVTZ & 6.37 &\AVQZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.000 & 92.4 & 6.56 &{\CCSDT}/\AVTZ & 6.56 &\AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \ra 3p)$ &0.038 & 93.9 & 6.64 &{\CCSDT}/\AVTZ & 6.66 &\AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3p)$ & & 93.6 & 6.81 &{\CCSDT}/\AVTZ & 6.83 &\AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; \pi \ra 3p)$ &0.008 & 93.5 & 7.24 & D & 7.14 &\AVQZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 98.4 & 4.20 & D & 4.20 &\AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.1 & 5.46 & D & 5.47 &\AVQZ \\
|
||
&$^3A_2 (\mathrm{R}; \pi \ra 3s)$ & & 97.9 & 6.02 & D & 6.05 &\AVQZ \\
|
||
&$^3B_1 (\mathrm{R}; \pi \ra 3p)$ & & 97.9 & 6.59 & D & 6.61 &\AVQZ \\
|
||
Glyoxal &$^1A_u (\Val; n \ra \pis)$ & 0.000 & 91.0 & 2.88 & B & 2.88 & \AVPZ \\
|
||
&$^1B_g (\Val; n \ra \pis)$ & & 88.3 & 4.24 & B & 4.24 & \AVQZ \\
|
||
&$^1A_g (\Val; n,n \ra \pis,\pis)$ & & 0.5 & 5.61 & F & 5.60 & \AVPZ \\%to be remade with CCSDT correction ??
|
||
&$^1B_g (\Val; n \ra \pis)$ & & 83.9 & 6.57 & B & 6.58 & \AVQZ \\
|
||
&$^1B_u (\mathrm{R}; n \ra 3p)$ & 0.095 & 91.7 & 7.71 & B & 7.78 & \AVPZ \\
|
||
&$^3A_u (\Val; n \ra \pis)$ & & 97.6 & 2.49 & {\CCSDT}/\AVTZ & 2.50 & \AVPZ \\
|
||
&$^3B_g (\Val; n \ra \pis)$ & & 97.4 & 3.89 & {\CCSDT}/\AVTZ & 3.90 & \AVQZ \\
|
||
&$^3B_u (\Val; \pi \ra \pis)$ & & 98.5 & 5.15 & {\CCSDT}/\AVTZ & 5.17 & \AVPZ \\
|
||
&$^3A_g (\Val; \pi \ra \pis)$ & & 98.8 & 6.30 & {\CCSDT}/\AVTZ & 6.31 & \AVPZ \\
|
||
Imidazole &$^1A'' (\mathrm{R}; \pi \ra 3s)$ & 0.001 & 93.0 & 5.71 & D & 5.73 & \AVQZ \\
|
||
&$^1A' (\Val; \pi \ra \pis)$ & 0.124 & 89.6 & 6.41 & D & 6.41s & \AVQZ \\
|
||
&$^1A'' (\Val; n \ra \pis)$ & 0.028 & 93.6 & 6.50 & D & 6.53 & \AVQZ \\
|
||
&$^1A' (\mathrm{R};\pi \ra 3p)$ & 0.035 & 88.9 & \emph{6.83} & D &\emph{6.82} & \AVQZ \\
|
||
&$^3A' (\Val; \pi \ra \pis)$ & & 98.3 & 4.73 & E & 4.74 & \AVQZ \\
|
||
&$^3A'' (\mathrm{R};(\pi \ra 3s)$ & & 97.6 & 5.66 & D & 5.69 & \AVQZ \\
|
||
&$^3A' (\Val; \pi \ra \pis)$ & & 97.9 & 5.74 & E & 5.75 & \AVQZ \\
|
||
&$^3A'' (\Val; n \ra \pis)$ & & 97.3 & 6.31 & D & 6.31 & \AVQZ \\
|
||
Isobutene &$^1B_1 (\mathrm{R}; \pi \ra 3s)$ & 0.006 & 94.1 & 6.46 & {\CCSDT}/\AVTZ & 6.48 & \AVQZ \\
|
||
&$^1A_1 (\mathrm{R}; \pi \ra 3p)$ & 0.228 & 94.2 & 7.01 & {\CCSDT}/\AVTZ & 7.00 & \AVQZ \\
|
||
&$^3A_1 (\Val; (\pi \ra \pis)$ & & 98.9 & 4.53 & D & 4.54 & \AVQZ \\
|
||
Methylenecyclopropene& $^1B_2 (\Val; \pi \ra \pis)$ & 0.011 & 85.4 & 4.28 & B & 4.29 & \AVPZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \ra 3s)$ & 0.005 & 93.6 & 5.44 & B & 5.47 & \AVPZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3p)$ & & 93.3 & 5.96 & B & 5.99 & \AVQZ \\
|
||
&$^1A_1(\Val; \pi \ra \pis)$ & 0.224 & 92.8 & \emph{6.12} & B & \emph{6.03} & \AVPZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 97.2 & 3.49 & {\CCSDT}/\AVTZ & 3.49 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.6 & 4.74 & D & 4.75 & \AVPZ \\
|
||
Propynal & $^1A'' (\Val; n \ra \pis)$ & 0.000 & 89.0 & 3.80 & {\CCSDT}/\AVTZ & 3.81 & \AVQZ \\
|
||
&$^1A'' (\Val; \pi \ra \pis)$ & 0.000 & 92.9 & 5.54 & {\CCSDT}/\AVTZ & 5.53 & \AVQZ \\
|
||
&$^3A'' (\Val; n \ra \pis)$ & & 97.4 & 3.47 & D & 3.48 & \AVQZ \\
|
||
&$^3A' (\Val; \pi \ra \pis)$ & & 98.3 & 4.47 & D & 4.48 & \AVQZ \\
|
||
Pyrazine &$^1B_{3u} (\Val; n \ra \pis)$ &0.006 & 90.1 & 4.15 & {\CCSDT}/\AVTZ & 4.15 & \AVQZ \\
|
||
&$^1A_{u} (\Val; n \ra \pis)$ & & 88.6 & 4.98 & {\CCSDT}/\AVTZ & 4.99 & \AVQZ \\
|
||
&$^1B_{2u} (\Val; \pi \ra \pis)$ &0.078 & 86.9 & 5.02 & {\CCSDT}/\AVTZ & 5.01 & \AVQZ \\
|
||
&$^1B_{2g} (\Val; n \ra \pis)$ & & 85.6 & 5.71 & {\CCSDT}/\AVTZ & 5.71 & \AVQZ \\
|
||
&$^1A_{g} (\mathrm{R};n \ra 3s)$ & & 91.1 & 6.65 & {\CCSDT}/\AVTZ & 6.69 & \AVQZ \\
|
||
&$^1B_{1g} (\Val; n \ra \pis)$ & & 84.2 & 6.74 & {\CCSDT}/\AVTZ & 6.74 & \AVQZ \\
|
||
&$^1B_{1u} (\Val; \pi \ra \pis)$ &0.063 & 92.8 & 6.88 & {\CCSDT}/\AVTZ & 6.87 & \AVQZ \\
|
||
&$^1B_{1g} (\mathrm{R};n \ra 3p)$ & & 93.8 & 7.21 & {\CCSDT}/\AVTZ & 7.24 & \AVQZ \\
|
||
&$^1B_{2u} (\mathrm{R};n \ra 3p)$ &0.037 & 90.8 & 7.24 & D & 7.28 &\AVQZ \\
|
||
&$^1B_{1u} (\mathrm{R};\pi \ra 3s)$ &0.128 & 91.4 & 7.44 & D & 7.47 &\AVQZ \\
|
||
&$^1B_{1u} (\Val; \pi \ra \pis)$ &0.285 & 90.5 & \emph{7.98}& D & \emph{7.97} &\AVQZ \\
|
||
&$^3B_{3u} (\Val; n \ra \pis)$ & & 97.3 & 3.59 & D & 3.59 & \AVQZ \\
|
||
&$^3B_{1u} (\Val; \pi \ra \pis)$ & & 98.5 & 4.35 & D & 4.36 & \AVQZ \\
|
||
&$^3B_{2u} (\Val; (\pi \ra \pis)$ & & 97.6 & 4.39 & D & 4.39 & \AVQZ \\
|
||
&$^3A_{u} (\Val; n \ra \pis)$ & & 96.1 & 4.93 & D & 4.94 & \AVQZ \\
|
||
&$^3B_{2g} (\Val; n \ra \pis)$ & & 97.0 & 5.08 & D & 5.09 & \AVQZ \\
|
||
&$^3B_{1u} (\Val; \pi \ra \pis)$ & & 97.0 & 5.28 & D & 5.28 & \AVQZ \\
|
||
Pyridazine &$^1B_1 (\Val; n \ra \pis)$ & & 89.0 & 3.83 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\Val; n \ra \pis)$ & & 86.9 & 4.37 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ & & 85.8 & 5.26 & D & 5.26 & \AVQZ \\
|
||
&$^1A_2 (\Val; n \ra \pis)$ & & 86.2 & 5.72 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 3s)$ & & 88.5 & 6.17 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_1 (\Val; n \ra \pis)$ & & 87.0 & 6.37 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ & & 90.6 & 6.75 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_1 (\Val; n \ra \pis)$ & & 97.1 & 3.19 & D &3.20 & \AVQZ \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 96.2 & 4.11 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 98.5 & \emph{4.34} & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 97.3 & 4.82 & D & 4.81 & \AVQZ \\
|
||
Pyridine &$^1B_1 (\Val; n \ra \pis)$ & & 88.4 & 4.95 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ & & 86.5 & 5.14 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\Val; n \ra \pis)$ & & 87.9 & 5.40 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ & & 92.1 & 6.62 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\mathrm{R}; n \ra 3s)$ & & 89.7 & 6.76 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3s)$ & & 93.2 & 6.82 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ & & \hl{xxx} & \emph{xxx} & &\emph{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_1 (\mathrm{R}; \pi \ra 3p)$ & & 93.6 & 7.39 & E &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ & & 90.5 & \hl{xxx} & &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.5 & 4.30 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_1 (\Val; n \ra \pis)$ & & 97.0 & 4.46 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 97.3 & 4.79 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 97.1 & 5.04 & E &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 95.8 & 5.36 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 97.7 & 6.24 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
Pyrimidine &$^1B_1 (\Val; n \ra \pis)$ & 0.005 & 88.6 & 4.44 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\Val; n \ra \pis)$ & & 88.5 & 4.85 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ &0.028 & 86.3 & 5.38 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1A_2 (\Val; n \ra \pis)$ & & 86.7 & 5.92 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_1 (\Val; n \ra \pis)$ &0.005 & 86.7 & 6.26 & D &\hl{XXXX} & \hl{XXXXX} \\
|
||
&$^1B_2 (\mathrm{R} ;n \ra 3s)$ &0.005 & 90.3 & 6.70 & D &6.74 & \AVQZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.036 & 91.5 & 6.88 & D &6.87 & \AVQZ \\
|
||
&$^3B_1 (\Val; n \ra \pis)$ & & 96.8 & 4.09 & D &4.10 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.3 & \emph{4.51} & D &\emph{4.52} & \AVQZ \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 96.5 & 4.66 & D &4.67 & \AVQZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 97.4 & 4.96 & D &4.96 & \AVQZ \\
|
||
Pyrrole &$^1A_2 (\mathrm{R}; \pi \ra 3s)$ & & 92.9 & 5.24 & {\CCSDT}/\AVTZ & 5.27 & \AVQZ \\
|
||
&$^1B_1 (\mathrm{R};\pi \ra 3p)$ &0.015 & 92.4 & 6.00 & {\CCSDT}/\AVTZ & 6.03 & \AVQZ \\
|
||
&$^1A_2 (\mathrm{R};\pi \ra 3p)$ & & 93.0 & 6.00 & D & 6.02 & \AVQZ \\
|
||
&$^1B_2 (\Val; (\pi \ra \pis)$ &0.164 & 92.5 & 6.26 & {\CCSDT}/\AVTZ & 6.23 & \AVQZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.001 & 86.3 & 6.30 & {\CCSDT}/\AVTZ & 6.29 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R};\pi \ra 3p)$ &0.003 & 92.6 & 6.83 & D & 6.74 & \AVQZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 98.3 & 4.51 & D & 4.51 & \AVQZ \\
|
||
&$^3A_2 (\mathrm{R};\pi \ra 3s)$ & & 97.6 & 5.21 & D & 5.24 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 97.8 & 5.45 & D & 5.46 & \AVQZ \\
|
||
&$^3B_1 (\mathrm{R};\pi \ra 3p)$ & & 97.4 & 5.91 & D & 5.94 & \AVQZ \\
|
||
Tetrazine &$^1B_{3u} (\Val; n \ra \pis)$ & 0.006 & 89.8 & 2.47 & {\CCSDT}/\AVTZ & 2.46 & \AVQZ \\
|
||
&$^1A_{u} (\Val; n \ra \pis)$ & & 87.9 & 3.69 & {\CCSDT}/\AVTZ & 3.70 & \AVQZ \\
|
||
&$^1A_{g} (\Val; n,n \ra \pis, \pis)$ & & 0.7 & \emph{4.61} & {\NEV}/\AVTZ & \emph{4.59} & \AVQZ\\%to be remade with CCSDT correction ???
|
||
&$^1B_{1g} (\Val; n \ra \pis)$ & & 83.1 & 4.93 & {\CCSDT}/\AVTZ & 4.92 & \AVQZ \\
|
||
&$^1B_{2u} (\Val; \pi \ra \pis)$ & 0.055 & 85.4 & 5.21 & {\CCSDT}/\AVTZ & 5.20 & \AVQZ \\
|
||
&$^1B_{2g} (\Val; n \ra \pis)$ & & 81.7 & 5.45 & {\CCSDT}/\AVTZ & 5.45 & \AVQZ \\
|
||
&$^1A_{u} (\Val; n \ra \pis)$ & & 87.7 & 5.53 & {\CCSDT}/\AVTZ & 5.53 & \AVQZ \\
|
||
&$^1B_{3g} (\Val; n,n \ra \pis, \pis)$ & & 0.7 & \emph{6.15} & {\NEV}/\AVTZ & \emph{6.13} & \AVQZ\\
|
||
&$^1B_{2g} (\Val; n \ra \pis)$ & & 80.2 & 6.12 & D & 6.12 & \AVQZ \\
|
||
&$^1B_{1g} (\Val; n \ra \pis)$ & & 85.1 & 6.91 & D & 6.91 & \AVQZ \\
|
||
&$^3B_{3u} (\Val; n \ra \pis)$ & & 97.1 & 1.85 & D & 1.86 & \AVQZ \\
|
||
&$^3A_{u} (\Val; n \ra \pis)$ & & 96.3 & 3.45 & D & 3.46 & \AVQZ \\
|
||
&$^3B_{1g} (\Val; n \ra \pis)$ & & 97.0 & 4.20 & D & 4.21 & \AVQZ \\
|
||
&$^3B_{1u} (\Val; \pi \ra \pis)$ & & 98.5 & \emph{4.49} & D & \emph{4.49} & \AVQZ \\
|
||
&$^3B_{2u} (\Val; \pi \ra \pis)$ & & 97.5 & 4.52 & D & 4.52 & \AVQZ \\
|
||
&$^3B_{2g} (\Val; n \ra \pis)$ & & 96.4 & 5.04 & D & 5.04 & \AVQZ \\
|
||
&$^3A_{u} (\Val; n \ra \pis)$ & & 96.6 & 5.11 & D & 5.11 & \AVQZ \\
|
||
&$^3B_{3g} (\Val; n,n \ra \pis, \pis)$ & & 5.7 & \emph{5.51} &{\NEV}/\AVTZ & \emph{5.50} & \AVQZ\\
|
||
&$^3B_{1u} (\Val; \pi \ra \pis)$ & & 96.6 & 5.42 & D & 5.43 & \AVQZ \\
|
||
Thioacetone &$^1A_2 (\Val; n \ra \pis)$ & & 88.9 & 2.53 & B & 2.54 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 4s)$ & 0.052 & 91.3 & 5.56 & B & 5.61 & \AVQZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ & 0.242 & 90.6 & 5.88 & B & 5.86 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; n \ra 4p)$ & 0.028 & 92.4 & 6.51 & C & 6.52 & \AVQZ \\
|
||
&$^1A_1 (\mathrm{R}; n \ra 4p)$ & 0.023 & 91.6 & 6.61 &B & 6.64 & \AVQZ \\
|
||
&$^3A_2 (\Val; n \ra \pis)$ & & 97.4 & 2.33 & D & 2.34 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 98.7 & 3.45 & D & 3.46 & \AVQZ \\
|
||
Thiophene &$^1A_1 (\Val; \pi \ra \pis)$ &0.070 & 87.6 & 5.64 & {\CCSDT}/\AVTZ & 5.63 & \AVQZ \\
|
||
&$^1B_2 (\Val; \pi \ra \pis)$ &0.079 & 91.5 & 5.98 & {\CCSDT}/\AVTZ & 5.96 & \AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3s)$ & & 92.6 & 6.14 & {\CCSDT}/\AVTZ & 6.16 & \AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \ra 3p)$ &0.010 & 90.1 & 6.14 & {\CCSDT}/\AVTZ & 6.11 & \AVQZ \\
|
||
&$^1A_2 (\mathrm{R}; \pi \ra 3p)$ & & 91.8 & 6.21 & {\CCSDT}/\AVTZ & 6.18 & \AVQZ \\
|
||
&$^1B_1 (\mathrm{R}; \pi \ra 3s)$ &0.000 & 92.8 & 6.49 & {\CCSDT}/\AVTZ & 6.52 & \AVQZ \\
|
||
&$^1B_2 (\mathrm{R}; \pi \ra 3p)$ &0.082 & 92.4 & 7.29 & {\CCSDT}/\AVTZ & 7.18 & \AVQZ \\
|
||
&$^1A_1 (\Val; \pi \ra \pis)$ &0.314 & 86.5 & \emph{7.31}& E & \emph{7.29} & \AVQZ \\
|
||
&$^3B_2 (\Val; \pi \ra \pis)$ & & 98.2 & 3.92 & D & 3.91 & \AVQZ \\
|
||
&$^3A_1 (\Val; \pi \ra \pis)$ & & 97.7 & 4.76 & D & 4.76 & \AVQZ \\
|
||
&$^3B_1 (\mathrm{R}; \pi \ra 3p)$ & & 96.6 & 5.93 & D & 5.90 & \AVQZ \\
|
||
&$^3A_2 (\mathrm{R}; \pi \ra 3s)$ & & 97.5 & 6.08 & D & 5.98 & \AVQZ \\
|
||
Thiopropynal &$^1A'' (\Val; n \ra \pis)$ & 0.000 & 87.5 & 2.03 & {\CCSDT}/\AVTZ & 2.04 & \AVQZ \\
|
||
&$^3A'' (\Val; n \ra \pis)$ & & 97.2 & 1.80 & D & 1.81 & \AVQZ \\
|
||
Triazine &$^1A_1'' (\Val; n \ra \pis)$ & & 88.3 & 4.72 & {\CCSDT}/\AVTZ & 4.72 & \AVQZ \\
|
||
&$^1A_2'' (\Val; n \ra \pis)$ &0.014 & 88.3 & 4.75 & {\CCSDT}/\AVTZ & 4.74 & \AVQZ \\
|
||
&$^1E'' (\Val; n \ra \pis)$ & & 88.3 & 4.78 & {\CCSDT}/\AVTZ & 4.78 & \AVQZ \\
|
||
&$^1A_2' (\Val; \pi \ra \pis)$ & & 85.7 & 5.75 & {\CCSDT}/\AVTZ &5.75 &\AVQZ\\
|
||
&$^1A_1' (\Val; \pi \ra \pis)$ & & 90.4 & 7.24 & {\CCSDT}/\AVTZ & 7.23 & \AVQZ \\
|
||
&$^1E' (\mathrm{R}; n \ra 3s)$ &0.016 & 90.9 & 7.32 & {\CCSDT}/\AVTZ & 7.36 & \AVQZ \\
|
||
&$^1E'' (\Val; n \ra \pis)$ & & 82.6 & 7.78 & {\CCSDT}/\AVTZ & 7.76 & \AVQZ \\
|
||
&$^1E' (\Val; \pi \ra \pis)$ &0.451 & 90.0 & 7.94 & {\CCSDT}/\AVTZ & 7.93 & \AVQZ \\
|
||
&$^3A_2'' (\Val; n \ra \pis)$ & & 96.7 & 4.33 & D & 4.34 & \AVQZ \\
|
||
&$^3E'' (\Val; n \ra \pis)$ & & 96.6 & 4.51 & D & 4.51 & \AVQZ \\
|
||
&$^3A_1'' (\Val; n \ra \pis)$ & & 96.2 & 4.73 & D & 4.74 & \AVQZ \\
|
||
&$^3A_1' (\Val; \pi \ra \pis)$ & & 98.2 & 4.85 & D & 4.86 & \AVQZ \\
|
||
&$^3E' (\Val; \pi \ra \pis)$ & & 96.9 & 5.59 & E & 5.59 & \AVQZ \\
|
||
&$^3A_2' (\Val; (\pi \ra \pis)$ & & 97.6 & 6.62 & D & 6.61 & \AVQZ \\
|
||
\end{longtable}
|
||
\end{footnotesize}
|
||
\begin{flushleft}\begin{footnotesize}\begin{singlespace}
|
||
\vspace{-0.6 cm}
|
||
$^a${
|
||
Protocol A: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\AVDZ} and {\CCSDT}/{\AVDZ};
|
||
Protocol B: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCSDT}/{\Pop};
|
||
Protocol C: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCT}/{\Pop};
|
||
Protocol D: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\AVDZ} and {\CCT}/{\AVDZ};
|
||
Protocol E: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\Pop} and {\CCT}/{\Pop};
|
||
Protocol F: {\FCI}/{\AVDZ} value (from Ref.~\citenum{Loo19c}) corrected by the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ}.
|
||
}
|
||
\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} of the SI. Because all these approaches are single-reference methods, we have removed from the
|
||
reference set the ``unsafe'' transition energies (in italics in Table \ref{Table-tbe}), as well as the four transitions with a dominant double excitation character (with $\Td < 50\%$ as listed in Table \ref{Table-tbe}).
|
||
A comprehensive list of results are collected in Table \ref{Table-bench} which, more specifically, gathers the MSE, MAE, RMSE, SDE, \MaxP, and \MaxN.
|
||
Figure \ref{Fig-1} shows histograms of the error distributions for these eight methods. Before discussing these, let us stress two obvious biases of this benchmark set: 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 statistical quantity smaller than $\sim 0.02$--$0.03$ eV is very likely to be irrelevant.
|
||
|
||
\renewcommand*{\arraystretch}{1.0}
|
||
\begin{table}[htp]
|
||
\caption{Mean signed error (MSE), mean absolute error (MAE), root-mean square error (RMSE), standard deviation of the errors (SDE), as well as the positive [\MaxP] and negative [\MaxN] maximal errors with respect to the TBE.
|
||
All these statistical quantities are reported in eV and have been obtained with the {\AVTZ} basis set.
|
||
``Count'' refers to the number of states.}
|
||
\label{Table-bench}
|
||
\begin{tabular}{lccccccc}
|
||
\hline
|
||
Method & Count & MSE &MAE &RMSE &SDE &\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 levels of theory, taking the TBE/{\AVTZ} of Table \ref{Table-bench} as references.
|
||
Note the difference of scaling in the vertical axes.}
|
||
\label{Fig-1}
|
||
\end{figure}
|
||
|
||
Let us analyse the global performances of all methods, starting with the most accurate and computationally demanding 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 the best of our knowledge, the only two previous studies discussing this specific aspect are limited to very small compounds. \cite{Kan17,Loo18a} According to the results gathered in Table \ref{Table-bench}, it appears that {\CCT} has a slight edge over {\CCSDT}-3, although
|
||
{\CCSDT}-3 is closer to {\CCSDT} in \titou{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 on smaller derivatives,
|
||
\cite{Kan17,Loo18a} but the amplitude of the {\CCSDT}-3 errors is larger in 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 extremely solid references for any transition with a dominant single-excitation character. This conclusion is again consistent with previous
|
||
analyses performed on smaller compounds, \cite{Kan17,Loo18a} as well as with recent comparisons between theoretical and experimental 0-0 energies performed by some of us on medium-sized molecules. \cite{Loo18b,Loo19a,Sue19}
|
||
To state it more boldly: it appears likely that {\CCT} is even more accurate than previously thought. In addition, thanks to the exhaustive and detailed comparisons made in the present work, we can safely conclude that {\CCT} regularly outperforms {\CASPT} and {\NEV}, even when
|
||
these methods are combined with large active spaces. This statement seems to hold as long as the considered ES does not show a strong multiple excitation character, that is, when $\Td < 70\%$.
|
||
|
||
The perturbative inclusion of triples as in CCSDR(3) yields a very small MAE ($0.05$ eV) for a much lighter 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 smaller compounds when comparing to {\FCI} ($0.04$ eV), \cite{Loo18a} and is also inline with the
|
||
2009 benchmark study 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 with several previous recent reports.
|
||
\cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a}
|
||
It is, nonetheless, larger than the one determined for smaller molecules ($+0.05$ eV), \cite{Loo18a} hinting that the performances of {\CCSD} deteriorates for larger compounds.
|
||
Moreover, {\CCSD} MAE of $0.13$ 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 K\'ann\'ar and Szalay performed
|
||
on Thiel's set ($0.18$ eV for transitions with $\Td > 90\%$). \cite{Kan14} Retrospectively, it is pretty obvious that Thiel's much larger value is very likely due to the {\CASPT} reference values. \cite{Sch08}
|
||
Indeed, as we have shown several times in the present study, {\CASPT} transitions energies tend to be significantly too low, therefore exacerbating the usual {\CCSD} overestimation.
|
||
|
||
With a single detailed benchmark study to date, \cite{Dut18} the {\STEOM} approach has received relatively little attention and its overall accuracy still needs to be corroborated.
|
||
It is noteworthy that {\STEOM} provides a smaller MSE than {\CCSD} and comparable MAE and RMSE. The spread of the error is however slightly larger as evidenced by Figure \ref{Fig-1} and the SDE values
|
||
reported in Table \ref{Table-bench}. These trends are the same as for smaller compounds. \cite{Loo18a} For Thiel's set, Dutta and coworkers also reported rather good performance
|
||
for {\STEOM} with respect to the {\CCT}/TZVP reference data, though a slightly negative MSE is obtained in their case. \cite{Dut18} This could well be due to the different basis set considered by these two studies. It should be nevertheless stressed that we, here, only consider only ``clean'' {\STEOM} results
|
||
(see Computational Details), therefore removing several difficult cases that are included in the {\CCSD} statistics, \eg, the $A_g$ excitation in butadiene, which can slightly bias the relative performance of {\STEOM} and {\CCSD}.
|
||
|
||
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, the performance of the former method clearly deteriorates compared to the two latter which exhibit very
|
||
similar statistical behaviors. These trends were also reported in 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 smaller compounds ($0.22$ eV). \cite{Loo18a} Therefore, in contrast to {\CCSD}, {\CCD} performance 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, \ie, {\CCD} is less consistent in terms of trends than {\CCSD}.
|
||
|
||
In Table \ref{Table-bench2}, we report a MAE decomposition for different subsets of ES. Note that, due to implementational limitations, only singlet ES could be computed with CCSDR(3) and CCSDT-3 which explains the lack of data for triplet ES.
|
||
A few interesting conclusions emerge from these results. First, the errors for singlet and triplet transitions are rather similar with all models, except for {\CCSD} that
|
||
is significantly more effective for triplets. Dutta and coworkers observed the same trend for Thiel's set with MAE of $0.20$ eV and $0.11$ eV for the singlet and triplet ES, respectively. \cite{Dut18}
|
||
Turning to the comparison between valence and Rydberg states, we find that {\CCD} provide a better description of the former, whereas {\CCSD} (and higher-order methods) yields the opposite trend.
|
||
In fact, {\CCD} has the clear tendency to overestimate valence ES energies (MSE of $+0.10$ eV), and to underestimate Rydberg ES energies (MSE of $-0.17$ eV).
|
||
{\CCSD} is found to be 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 perfectly coherent 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}. We believe that it is likely due to the distinct types of Rydberg states considered in these two studies. Indeed, we consider here (relatively) low-lying
|
||
Rydberg transitions in medium-sized molecules, while K\'ann\'ar and Szalay (mostly) investigated higher-lying Rydberg states in smaller compounds. Finally, CIS(D), {\AD}, {\CCD}, and {\STEOM}
|
||
better describe $n\ra\pis$ transitions, whereas {\CCSD} seems more suited for $\pi\ra\pis$ transitions.
|
||
The variations between the two subsets are probably not significant for the
|
||
higher-order approaches. The former observation agrees well with previous results on smaller compounds \cite{Loo18a} and the Thiel set, \cite{Sch08,Kan14} whereas \titou{the latter, less expected observation is likely dependent on the selected ES subset.} \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 computed 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 employed several state-of-the-art theoretical models with increasingly large diffuse basis sets.
|
||
However, most of our theoretical best estimates are based on {\CCSDTQ} (4 atoms) or {\CCSDT} (5 and 6 atoms) excitation energies. For the vast majority of the
|
||
listed excited states, the present contribution is the very first to disclose (sometimes basis-set extrapolated) {\CCSDT}/{\AVTZ} and (true) {\CCT}/{\AVQZ} transition energies as well as {\CCT}/{\AVTZ} oscillator strengths
|
||
for each dipole-allowed transition. Our set contains a total of 238 transition energies and 90 oscillator strengths, with a reasonably good balance between singlet, triplet, valence,
|
||
and Rydberg states. Amongst these 238 transitions, we believe that 224 are ``solid'' TBE, \ie, they are chemically accurate (MAE below $0.043$ eV or $1$ kcal.mol$^{-1}$) for the considered geometry.
|
||
It allows us to establish a reasonable error bar for several popular ES models with lower computational cost: CIS(D), {\AD}, {\CCD}, {\STEOM}, {\CCSD},
|
||
CCSDR(3), CCSDT-3, and {\CCT}. It turns out that the latter approach is extremely accurate, and, very likely should be considered as more robust and trustworthy than {\CASPT} or {\NEV}, except for ES with a predominant double
|
||
excitation character. Other methods including corrections for the triples yield a mean absolute deviation around $0.05$ eV, whereas none of the second-order approach has been found to be 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 models already available or currently 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, allowing to climb even higher 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}
|