FCI2
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Manuscript/FCI2.tex
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\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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% energies
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\newcommand{\EFCI}{E_\text{FCI}}
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\newcommand{\EexCI}{E_\text{exCI}}
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\newcommand{\EsCI}{E_\text{sCI}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\PsisCI}{\Psi_\text{sCI}}
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\newcommand{\Ndet}{N_\text{det}}
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\newcommand{\ex}[6]{$^{#1}#2_{#3}^{#4}(#5 \rightarrow #6)$}
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% methods
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\newcommand{\TDDFT}{TD-DFT}
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\newcommand{\CASSCF}{CASSCF}
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\newcommand{\CASPT}{CASPT2}
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\newcommand{\NEV}{NEVPT2}
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\newcommand{\PNEV}{PC-NEVPT2}
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\newcommand{\SNEV}{SC-NEVPT2}
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\newcommand{\AD}{ADC(2)}
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\newcommand{\AT}{ADC(3)}
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\newcommand{\CCD}{CC2}
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\newcommand{\CCSD}{CCSD}
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\newcommand{\CCT}{CC3}
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\newcommand{\EOMCCSD}{EOM-CCSD}
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\newcommand{\CCSDT}{CCSDT}
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\newcommand{\CCSDTQ}{CCSDTQ}
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\newcommand{\CCSDTQP}{CCSDTQP}
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\newcommand{\CI}{CI}
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\newcommand{\sCI}{sCI}
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\newcommand{\exCI}{exCI}
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\newcommand{\FCI}{FCI}
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% basis
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\newcommand{\Pop}{6-31+G(d)}
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\newcommand{\AVDZ}{\emph{aug}-cc-pVDZ}
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\newcommand{\AVTZ}{\emph{aug}-cc-pVTZ}
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\newcommand{\DAVTZ}{d-\emph{aug}-cc-pVTZ}
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\newcommand{\AVQZ}{\emph{aug}-cc-pVQZ}
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\newcommand{\AVFZ}{\emph{aug}-cc-pV5Z}
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\newcommand{\DAVQZ}{d-\emph{aug}-cc-pVQZ}
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\newcommand{\TAVQZ}{t-\emph{aug}-cc-pVQZ}
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\newcommand{\AVPZ}{\emph{aug}-cc-pV5Z}
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\newcommand{\DAVPZ}{d-\emph{aug}-cc-pV5Z}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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% greek shortcut
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\newcommand{\pis}{\pi^\star}
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\newcommand{\Ryd}{\mathrm{R}}
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\newcommand{\SI}{Supporting Information}
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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 focussed on compounds containing 1--3 non-hydrogen atoms [\emph{J. Chem. Theory Comput.} {\bfseries 14} (2018) 4360--4379], we present highly-accurate vertical transition energies
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obtained for molecules encompassing 4, 5, and 6 non-hydrogen atoms, namely: \hl{complete alphabetic list here}.
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To obtain these energies, we use both Coupled-Cluster approaches up to the highest possible order ({\CCT}, {\CCSDT}, and {\CCSDTQ}), the selected Configuration Interaction approach up to several millions determinants,
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and the $N$-Electron Valence State Perturbation Theory (NEVPT2), all approaches being combined with a diffuse-containing atomic basis sets. For all states, we report at least {\CCT}/{\AVQZ} transition
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energies and as well as {\CCT}/{\AVTZ} oscillator strengths for all dipole-allowed transitions. \hl{xxx nb and nature of states xxx}. We show that {\CCT} almost systematically delivers transition energies
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very close to the theoretical best estimates ($\pm$ 0.04 eV) but for transitions presenting a dominant double excitation character. We use a series of \hl{xxxx} theoretical best estimates to benchmark a series of popular
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methods for excited state calculations [CIS(D), {\CCD}, STEOM-CCSD, {\CCSD}, CCSDR(3), {\AD}, and {\AT})] \hl{xxxxx}. The results of these benchmarks are compared to 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} which is a successful, yet far from flawless, theory. In particular, to perform TD-DFT calculations, one needs
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to choose an ``appropriate'' exchange-correlation functional (XCF), which is difficultas the impact of the XCF on the results is exacerbated in TD-DFT as compared to DFT. \cite{Lau13} Such selection can of course rely
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on the intrinsic features of each XCF, e.g., it is well-known that range-separated hybrids provide a more physically-sound description of long-range charge-transfer (CT) transitions than semi-local XCF. \cite{Dre04,Pea08}
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However, to obtain a quantitative assessment of the accuracy that can be expected from TD-DFT calculations, benchmarks are needed. This is why many assessment of TD-DFT performances for various properties are
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available. \cite{Lau13} While several of these benchmarks use experimental data as reference, typically band shapes \cite{Die04,Die04b,Avi13,Cha13,Lat15b,Mun15,Vaz15,San16b} or 0-0 energies,
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\cite{Die04b,Goe10a,Jac12d,Chi13b,Win13,Fan14b,Jac14a,Jac15b,Loo19b} using theoretical best estimates (TBE) obtained with a more refined level of theory as references, \cite{Sch08,Sau09,Sil10b,Sil10c,Sch17,Loo18a}
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is advantageous as it allows comparisons on a perfectly equal footing (same geometry, vertical transitions, no environmental effects...). However, obtaining accurate TBE is not an easy task as high-level theories
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generally come with a dreadful scaling with system size and, additionally, typically require large atomic basis sets to deliver transition energies close to the basis set limit.
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More than 20 years ago, Serrano-Andr\`es, Roos, and their coworkers proposed an impressive series of reference transition energies for several typical conjugated organic molecules (butadiene, furan, pyrrole, tetrazine...).
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\cite{Ser93,Ser93b,Ser93c,Mer96,Mer96b,Roo96,Ser96b} To this end, they relied on the Complete Active Space Second-Order Perturbation Theory ({\CASPT}) approach with the largest active spaces and basis sets
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they could offer at that time, and they typically used experimental GS geometries. Beyond comparisons with experiments, which are always challenging when computing vertical transition energies, \cite{San16b} there was no approach
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available at that time to ascertain the accuracy of the obtained transition energies. These {\CASPT} values were latter used to assess the performances of TD-DFT combined with various XCF, \cite{Toz99b,Bur02} and remained for a long
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time the best references available. A decade ago, Thiel and coworkers defined TBE for 104 singlet and 63 triplet valence ES in 28 small and medium conjugated CNOH organic molecules. \cite{Sch08,Sil10b,Sil10c} These TBE
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were computed on MP2/6-31G(d) structures with various levels of theories, notably {\CASPT} and various Coupled-Cluster levels ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default reference approach used by Thiel for his
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first series of TBE was {\CASPT}, \cite{Sch08} the majority of the most recent TBE (so-called ``TBE-2'' in Ref. \citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level, often using a basis set extrapolation technique to obtain these
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estimates. In more details, CC3/TZVP values were typically corrected for basis set effects by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works used Thiel's TBE for
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assessing low-order methods, \cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18} highlighting
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their value for the community. In contrast, the number of extensions of this original set remains quite limited, e.g., {\CCSDT}/TZVP reference energies computed for 17 singlet states of six molecules appeared in 2014, \cite{Kan14}
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and 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide) have been described in 2017. \cite{Kan17}
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This has motivated us to propose in 2018 a set of 106 transition energies for which it was technically possible to reach the Full Configuration Interaction (FCI) limit by combining high-order Coupled-Cluster (up to {\CCSDTQP}) and selected
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CI (sCI) transition energy calculations on {\CCT} GS structures. \cite{Loo18a} We used these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} was on the {\FCI} spot, whereas we could not
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detect significant differences of accuracies between {\CCT} and {\CCSDT}, both being very accurate with mean absolute errors (MAE) as small as 0.03 eV compared to {\FCI} for ES with a single excitation character. These conclusions
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agree well with earlier studies. \cite{Wat13,Kan14,Kan17} We also recently proposed a set of 20 TBE for transitions presenting a large double excitation character. \cite{Loo19c} For such transitions, one can distinguish the ES
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as a function of $\%T_1$, the percentage of single excitation calculated at the {\CCT} level. For ES with a significant yet not dominant double excitation character, such as the famous $A_g$ ES of butadiene ($\%T_1$=75\%),
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CC methods including triples in the GS treatment deliver rather accurate estimates (MAE of 0.11 eV with {\CCT} and 0.06 eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or
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the generally robust $N$-Electron Valence State Perturbation Theory ({\NEV}). In contrast, for ES with a dominant double excitation character, e.g., the low-lying $n,n \rightarrow \pi^\star,\pi^\star$ excitation in nitrosomethane ($\%T_1$=2\%),
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single-reference methods are not suited (MAE of 0.86 eV with {\CCT} and 0.42 eV with {\CCSDT} and multi-reference methods are in practice required to obtain accurate results. \cite{Loo19c} In Ref. \citenum{Loo18a} we also reported a relatively
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limited accuracy of the third-order Algebraic Diagrammatic Construction scheme [{\AT}] and larger errors with {\CCD} than with {\CCSD} for the set of 106 single-excitation TBE, both results being in par with previously published
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conclusions. \cite{Sch08,Sau09,Har14} Obviously, a possible origin of these discrepancies is that we treated only compounds containing 1--3 non-hydrogen atoms, hence introducing a significant chemical bias. Therefore we have decided to
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go for larger molecules and we consider in the present contribution organic compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large bases remains a dream, and,
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the convergence of {\sCI} with the number of determinants is slower, so that extrapolating to the {\FCI} limit with a ca. 0.01 eV error bar is also rarely doable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates,
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as in our earlier work,\cite{Loo18a} is beyond reach. Anticipating this problem, we have previously demonstrated that one can very accurately estimate such limit by correcting values obtained with a high-level of theory and a double-$\zeta$
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basis set by {\CCT} results obtained with a larger basis, \cite{Loo18a} and we follow such strategy here. In addition, we also performed {\NEV} calculations in an effort to check the consistency of our estimates, especially for ES with
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intermediate $\%T_1$ values. Using this protocol, we define a set of \hl{xxxxxxxx} {\AVTZ} reference transition energies, most being within $\pm$ 0.03 eV for the {\FCI} limit. These reference energies are obtained on {\CCT} geometries and
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further basis set corrections at least up to quadruple-$\zeta$, are also provided with {\CCT}. Together with the results obtained in our two earlier works, \cite{Loo18a,Loo19c} the present TBE will hopefully contribute to reach a further step
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on the accuracy staircase.
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%
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% II. Computational Details
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%
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\section{Computational Details}
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\label{sec-met}
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All our transition energies are computed in the frozen-core approximation (large cores for S), except when noted, and we globally follow the same procedure as in Ref. \citenum{Loo18a}, so that we only briefly outline the models used 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). Cartesian coordinates for all compounds can be found in the Supporting
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Information (SI). Some structures have been taken from previous contributions, \cite{Bud17,Jac18a,Bre18a} whereas we performed additional optimizations using Dalton \cite{dalton} and/or CFOUR, \cite{cfour} applying default parameters
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in both cases, to obtain the missing structures.
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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 a new 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{Alavi} 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 wave functions with at least 5,000,000 determinants.
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State-averaged natural orbitals are then computed based on this wave function, 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 (5 and 6-membered rings), 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, i.e.,
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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 wave functions. These extrapolated total energies (labeled as exFCI 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 wave function and its corresponding
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extrapolated value. More theoretical and technical details about our extrapolation procedure can be found in Ref.~\citenum{Gar19}. Additional information about the sCI wave functions 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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\hl{Martial: you to play}
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In the following, {\NEV} stands systematically for {\PNEV}. + active space in the SI ?
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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-CCSD)\cite{Noo97,Dut18}; MRCC for {\CCSDT} \cite{Nog87} and {\CCSDTQ}; \cite{Kuc91} and Q-Chem for {\AD} and {\AT}. \cite{Dre15}
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Default program setting are applied. We underline that for the STEOM-CCSD we report only states that are characterized by an active character percentage of 98\%\ or larger. For all calculations, we use Pople's {\Pop} as well as
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the well-known Dunning's \emph{aug}-cc-pVXZ (X $=$ D, T, Q, and 5) atomic basis sets.
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%
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% III. Results & Discussion
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%
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\section{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 expansion as far as technically doable.
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Given that the {\sCI} results converges rather slowly for these rather large systems, we provide an error bar for these extrapolated {\FCI} values. In most cases, the latter 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 {\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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an ultra-large \%$T_1$, \cite{Sch08} and we consequently put our maximal computational effort on determining accurate transition energies for singlet states. To assign the different ES, we used literature data, as well as
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usual criteria, i.e., relative energies, symmetries and compositions of the underlying MOs, as well as oscillator strengths. This allowed 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 calculation. Such cases are however not statistically
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relevant and are therefore unlikely to change our conclusions.
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\subsection{Four-atom molecules}
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\subsubsection{Cyanoacetylene, cyanogen, and diacetylene}
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The ES of these three closely-related linear molecules containing two triple bonds have been quite rarely investigated with theory, \cite{Fis03,Pat06,Luo08,Loo18b,Loo19a} though (rather old) measurements of their
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0-0 energies are available for several excited states. \cite{Cal63,Job66a,Job66b,Bel69,Fis72,Har77,Hai79,All84} Our main results are collected in Tables \ref{Table-1} and S1 in the SI. We consider only low-lying
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||||||
|
valence $\pi \rightarrow \pi^\star$ transitions, and, all have a strongly dominant single excitation character (\%$T_1 > 90$, \emph{vide infra}). For cyanoacetylene, the {\FCI} estimates come with small error bars
|
||||||
|
with {\Pop} and one notes an excellent agreement between these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning's double-$\zeta$ basis for which the {\FCI} errors bars are however
|
||||||
|
larger. Using the {\CCSDTQ} values as references, it appears that the previously obtained {\CASPT} estimates\cite{Luo08} are too low and that the {\CCT} transition energies are slightly more accurate than
|
||||||
|
their CCSDT counterparts, although all CC estimates of Table \ref{Table-1} come in a very tight energetic window for a given basis set . There is also a superb agreement between the CC and {\NEV} values with the
|
||||||
|
{\AVTZ} basis set. All these facts give high confidence that the CC estimates can be trusted. The basis set effects are quite significant for the valence excited-states of cyanoacetylene with successive drops of the transitions
|
||||||
|
energies by ca. -0.10 eV, when going from {\Pop} to {\AVDZ} and from {\AVDZ} to {\AVTZ}, but for the lowest triplet state that appears less sensitive to the selected basis. As expected, the changes when further
|
||||||
|
extending the basis set to quadruple and quintuple-$\zeta$ are trifling, and the same holds when adding a second set of diffuse functions in the basis, or when correlating the core electrons (see the SI). Obviously,
|
||||||
|
both cyanogen and diacetylene yield very similar trends, with limited methodological effects and quite large basis set effects, but for the transitions to the $^3\Sigma_u^+$ ES. We note that all CC values are, at worst,
|
||||||
|
within $\pm$ 0.02 eV of the {\exCI} window, i.e., all methods presented in Table \ref{Table-1} provide very consistent estimates. Across all states reported in that Table with {\AVTZ}, the average deviations between {\NEV} and {\CCT}
|
||||||
|
(or {\CCSDT}) is as small as 0.03 eV, the lowest absorption and emission of cyanogen being the only two cases showing significant deviations. As a final note, all our vertical transition energies are significantly bigger
|
||||||
|
than the experimentally measured 0-0 energies, as it should. We refer the interested reader to previous works, \cite{Fis03,Loo19a} for comparisons of {\CASPT} and {\CCT} values with measured 0-0 energies
|
||||||
|
for these linear compounds.
|
||||||
|
|
||||||
|
\begin{sidewaystable}[htp]
|
||||||
|
\caption{\small Vertical transition energies determined in cyanoacetylene, cyanogen, and diaectylene. All states have a valence $\pi \rightarrow \pi^\star$ character. All values are in eV.}
|
||||||
|
\label{Table-1}
|
||||||
|
\begin{footnotesize}
|
||||||
|
\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}}
|
||||||
|
\hline
|
||||||
|
\multicolumn{14}{c}{Cyanoacetylene}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{\AVQZ} & \multicolumn{1}{c}{\AVFZ} & \multicolumn{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$0.03 &5.28&5.27& &5.29$\pm$0.08 &5.22&5.21&5.19& 5.22 &5.22 &&\\
|
||||||
|
\hline
|
||||||
|
\multicolumn{14}{c}{Cyanogen}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{\AVQZ} & \multicolumn{1}{c}{\AVFZ} & \multicolumn{1}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV}& {\CCT} & {\CCT}& Exp.$^c$ \\
|
||||||
|
\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]$^d$ &5.27&5.28&5.26&5.21$\pm$0.05 &5.19&5.20&5.18&5.26$\pm$0.09 &5.06&5.07&4.97& 5.05 &5.05 & \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{14}{c}{Diacetylene}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{\AVQZ} & \multicolumn{1}{c}{\AVFZ} & \multicolumn{1}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI}& {\CCT} & {\CCSDT} & {\NEV}& {\CCT} & {\CCT}& Exp.$^e$ \\
|
||||||
|
\hline
|
||||||
|
$^1\Sigma_u^-$ &5.57&5.58&5.56&5.52$\pm$0.06 &5.44&5.45&5.43&5.47$\pm$0.02 &5.34&5.35&5.33& 5.33 &5.33 &4.81\\
|
||||||
|
$^1\Delta_u$ &5.83&5.85& &5.84$\pm$0.01 &5.69&5.70&5.69&5.69$\pm$0.02 &5.61&5.62&5.61& 5.60 &5.60 &5.06\\
|
||||||
|
$^3\Sigma_u^+$ &4.07&4.08&4.09&4.04$\pm$0.07 &4.06&4.06& &4.07$\pm$0.04 &4.08& &4.08& 4.10 &4.11 &2.7 \\
|
||||||
|
$^3\Delta_u$ &4.93&4.93&4.92&4.94$\pm$0.01 &4.86&4.85& &4.85$\pm$0.02 &4.80&4.79&4.78& 4.80 &4.80 &3.21\\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\end{footnotesize}
|
||||||
|
\begin{flushleft}
|
||||||
|
\begin{footnotesize}
|
||||||
|
$^a${{\CASPT} values from Ref. \citenum{Luo08};}
|
||||||
|
$^b${Experimental 0-0 energies from Refs. \citenum{Job66a} and \citenum{Job66b} (vacuum UV experiments);}
|
||||||
|
$^c${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;}
|
||||||
|
$^d${Vertical fluorescence energy from the lowest excited-state;}
|
||||||
|
$^e${Experimental 0-0 energies from Ref. \citenum{Hai79} (singlet ES, vacuum UV experiment) and Ref. \citenum{All84} (triplet ES, EELS). In the latter paper, the 2.7 eV value for the $^3\Sigma_u^+$
|
||||||
|
state is the onset, whereas an estimate of the vertical energy of 4.2$\pm$0.2 eV is given for the $^3\Delta_u$ state.}
|
||||||
|
\end{footnotesize}
|
||||||
|
\end{flushleft}
|
||||||
|
\end{sidewaystable}
|
||||||
|
|
||||||
|
%
|
||||||
|
\clearpage
|
||||||
|
%
|
||||||
|
|
||||||
|
\subsubsection{Cyclopropenone, cyclopropenethione, and methylenecyclopropene}
|
||||||
|
|
||||||
|
These three related compounds present a three-member $sp^2$ carbon cycle conjugated to an external $\pi$ bond. While the ES of methylenecyclopropene has regularly been investigated with theoretical tools in the past,
|
||||||
|
\cite{Mer96,Roo96,Car10b,Lea12,Gua13,Dad14,Gua14,Sch17,Bud17} the only investigations of vertical transitions we could find for the two other derivatives are the 2002 detailed {\CASPT} study of Serrano-Andr\'es and
|
||||||
|
coworkers on both compounds, \cite{Ser02} and a more recent work reporting three low-lying singlet states of cyclopropenone at the {\CASPT}/6-31G level.\cite{Liu14b}
|
||||||
|
|
||||||
|
Our results are listed in Tables \ref{Table-2} and S2. As above, considering the Pople's basis set, we note very small differences between {\CCT}, {\CCSDT}, and {\CCSDTQ} results, the latter method giving transition energies
|
||||||
|
systematically falling within the {\FCI} extrapolation incertitude. Depending on the state, it is either {\CCT} or {\CCSDT} that is the closest to {\CCSDTQ}. In fact, considering all {\CCSDTQ}/{\Pop} data listed in Table \ref{Table-2},
|
||||||
|
the mean absolute deviation of {\CCT} and {\CCSDT} are 0.019 and 0.016 eV, respectively, hinting that the improvements brought by the latter more computationally intensive method are limited. For the lowest $B_2$ state of n
|
||||||
|
methylenecyclopropene, one of the most challenging cases (\%$T_1 = 85$), it is clear from the {\exCI} value that only {\CCSDTQ} is really close to the spot, the {\CCT} and {\CCSDT} results being slightly too large. It seems
|
||||||
|
likely that the same trends appears for the same state in cyclopropenethione. Interestingly, the quite small Pople basis set provides data within ca. 0.10 eV of basis set convergence at {\CCT} level for 80\%\ of the transitions.
|
||||||
|
There are of course exceptions to this rule, e.g., the strongly dipole-allowed $^1A_1\ \pi \rightarrow \pi^\star$ ES of cyclopropenone and the $^1B_1\ \pi \rightarrow \sigma^\star$ ES of methylenecyclopropene are significantly too
|
||||||
|
blueshifted with the Pople's basis set (Table S2). For cyclopropenone, our {\CCSDT}/{\AVTZ} estimates do agree reasonably well with the data of Serrano-Andr\'es, but for the $^1B_2\ \pi \rightarrow \pi^\star$ state that we locate
|
||||||
|
significantly higher in energy and the three Rydberg states that CC foresees at significantly smaller energies. The current {\NEV} results are globally in better agreement with the CC values, though some non-negligible deviations
|
||||||
|
pertain. Although comparisons with experiments should be made very cautiously, we note that the CC data are clearly more coherent with the electron impact measurements\cite{Har74} for the Rydberg states. For cyclopropenethione,
|
||||||
|
we obtain transition energies typically in agreement or larger than those obtained with {\CASPT}, \cite{Ser02} though there is no obvious relationship between the valence/Rydberg nature of the considered ES and the relative
|
||||||
|
{\CASPT} error. \hl{commenter les NEV} Eventually methylenecyclopropene, our values logically agree very well with the recent estimates of Schwabe and Goerigk, \cite{Sch17} obtained at the {\CCT}/{\AVTZ} level on a different geometry,
|
||||||
|
whereas the available {\CASPT} values \cite{Mer96,Roo96} appear comparatively too low. For this compound, the available experimental data being wavelength of maximal absorption determined in condensed phase, \cite{Sta84}
|
||||||
|
only a qualitative match is logically reached between theory and experiment.
|
||||||
|
|
||||||
|
\begin{table}[htp]
|
||||||
|
\caption{\small Vertical transition energies determined in cyclopropenone, cyclopropenethione, and methylenecyclopropene. All values are in eV.}
|
||||||
|
\label{Table-2}
|
||||||
|
\begin{footnotesize}
|
||||||
|
\begin{tabular}{l|p{.5cm}p{.9cm}p{1.1cm}p{1.4cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Cyclopropenone}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Exp.$^b$ \\
|
||||||
|
\hline
|
||||||
|
$^1B_1 (n \rightarrow \pi^\star)$ &4.32&4.34&4.36& 4.38$\pm$0.02 &4.22&4.23 &4.21&4.24&4.04 &4.25&4.13 \\%4.50 in Liu14b MS-CASPT2/6-31G
|
||||||
|
$^1A_2 (n \rightarrow \pi^\star)$ &5.68&5.65&5.65& 5.64$\pm$0.06 &5.59&5.56 &5.57&5.55&5.85 &5.59&5.5 \\%5.18 in Liu14b
|
||||||
|
$^1B_2 (n \rightarrow 3s)$ &6.39&6.38&6.41& &6.21&6.19 &6.32&6.31&6.51 &6.90&6.22 \\%6.44 in Liu14b
|
||||||
|
$^1B_2 (\pi \rightarrow \pi^\star$) &6.70&6.67&6.68& &6.56&6.54 &6.54&6.53&6.82 &5.96&6.1 \\
|
||||||
|
$^1B_2 (n \rightarrow 3p)$ &6.92&6.91&6.94& &6.88&6.86 &6.96&6.95&7.04 &7.24&6.88 \\
|
||||||
|
$^1A_1 (n \rightarrow 3p)$ &7.00&7.00&7.03& &6.88&6.87 &7.00&6.99&7.28 &7.28& \\
|
||||||
|
$^1A_1 (\pi \rightarrow \pi^\star)$ &8.51&8.49&8.51& &8.32&8.29 &8.28&8.26&8.19 &7.80&$\sim$8.1 \\
|
||||||
|
$^3B_1 (n \rightarrow \pi^\star)$ &4.02&4.03& & 4.00$\pm$0.07 &3.90&3.92 &3.91&3.93& &4.05& \\
|
||||||
|
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.92&4.92& & 4.92$\pm$0.03 &4.90&4.89 &4.89&4.88& &4.81& \\
|
||||||
|
$^3A_2 (n \rightarrow \pi^\star)$ &5.48&5.44& & &5.38&5.35 &5.37&5.35& &5.56& \\
|
||||||
|
$^3A_1 (\pi \rightarrow \pi^\star)$ &6.89&6.88& & &6.79&6.78 &6.83&6.79& &6.98& \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Cyclopropenethione}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ \\
|
||||||
|
\hline
|
||||||
|
$^1A_2 (n \rightarrow \pi^\star)$ &3.46&3.44&3.44& 3.45$\pm$0.01 &3.47&3.45 &3.43&3.41& &3.23 & \\
|
||||||
|
$^1B_1 (n \rightarrow \pi^\star)$ &3.45&3.44&3.45& 3.44$\pm$0.05 &3.42&3.42 &3.43&3.44& &3.47 & \\
|
||||||
|
$^1B_2 (\pi \rightarrow \pi^\star)$ &4.67&4.64&4.62& 4.59$\pm$0.09 &4.66&4.64 &4.64&4.62& &4.34 & \\
|
||||||
|
$^1B_2 (n \rightarrow 3s)$ &5.26&5.24&5.27& &5.23&5.21 &5.34&5.31& &4.98 & \\
|
||||||
|
$^1A_1 (\pi \rightarrow \pi^\star)$ &5.53&5.52&5.51& &5.52&5.50 &5.49&5.47& &5.52 & \\
|
||||||
|
$^1B_2 (n \rightarrow 3p)$ &5.83&5.81&5.83& &5.86&5.84 &5.93&5.90& &5.88 & \\
|
||||||
|
$^3A_2 (n \rightarrow \pi^\star)$ &3.33&3.31& & 3.29$\pm$0.03 &3.34&3.32 &3.30&\hl{?} & &3.20 & \\
|
||||||
|
$^3B_1 (n \rightarrow \pi^\star)$ &3.34&3.33& & &3.30&3.30 &3.31&3.32& &3.30 & \\
|
||||||
|
$^3B_2 (\pi \rightarrow \pi^\star)$ &4.01&4.00& & 4.03$\pm$0.03 &4.03&4.02 &4.02&\hl{?} & &3.86 & \\
|
||||||
|
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.06&4.04& & &4.09&4.07 &4.03&\hl{?} & &3.99 & \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Methylenecyclopropene}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV}& Th.$^c$ & Th.$^d$ & Exp.$^e$\\
|
||||||
|
\hline
|
||||||
|
$^1B_2 (\pi \rightarrow \pi^\star)$ &4.38&4.37&4.34& 4.32$\pm$0.03 &4.32&4.31 &4.31&4.31& &4.13&4.36 &4.01\\
|
||||||
|
$^1B_1 (\pi \rightarrow \sigma^\star)$ &5.65&5.66&5.66& &5.35&5.35 &5.44&5.44& &5.32&5.44 &5.12\\
|
||||||
|
$^1A_2 (\pi \rightarrow 3p)$ &5.97&5.98&5.98& 5.92$\pm$0.10 &5.86&5.88 &5.95&5.96& &5.83& &\\
|
||||||
|
$^1A_1(\pi \rightarrow \pi^\star)$ &6.17&6.18&6.17& &6.15&6.15 &6.13&6.13& & &6.13 &6.02\\
|
||||||
|
$^3B_2 (\pi \rightarrow \pi^\star)$ &3.50&3.50& & 3.44$\pm$0.06 &3.49&3.49 &3.50&\hl{?} & &3.24& &\\
|
||||||
|
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.74&4.74& & 4.67$\pm$0.10 &4.74&4.74 &4.74&\hl{?} & &4.52& &\\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\end{footnotesize}
|
||||||
|
\begin{flushleft}
|
||||||
|
\begin{footnotesize}
|
||||||
|
$^a${{\CASPT} values of Ref. \citenum{Ser02};}
|
||||||
|
$^b${Electron impact experiment of Ref. \citenum{Har74}, note that the 5.5 eV peak was assigned differently in the original paper, and we follow here the analysis of Serrano-Andr\'es\cite{Ser02}
|
||||||
|
whereas the 6.1 eV assignment was "supposed" in the original paper; experimental $\lambda_{\mathrm{max}}$ have been measured at at 3.62 eV and 6.52 eV for the $^1B_1 (n \rightarrow \pi^\star)$ and
|
||||||
|
$^1B_2 (\pi \rightarrow \pi^\star$) transitions, respectively; \cite{Bre72}}
|
||||||
|
$^c${{\CASPT} values of Refs. \citenum{Mer96} and \citenum{Roo96};}
|
||||||
|
$^d${ {\CCT} values of Ref. \citenum{Sch17};}
|
||||||
|
$^e${$\lambda_{\mathrm{max}}$ in pentane at -78$^o$C from Ref. \citenum{Sta84}.}
|
||||||
|
\end{footnotesize}
|
||||||
|
\end{flushleft}
|
||||||
|
\end{table}
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
%
|
||||||
|
|
||||||
|
\subsubsection{Acrolein, butadiene, and glyoxal}
|
||||||
|
|
||||||
|
Let us now turn to three pseudo-linear $\pi$-conjugated systems that have been the subject to several ES investigations before, namely, acrolein, \cite{Aqu03,Sah06,Car10b,Lea12,Gua13,Mai14,Aza17b,Sch17,Bat17}
|
||||||
|
butadiene, \cite{Dal04,Sah06,Sch08,Sil10c,Li11,Wat12,Dad12,Lea12,Ise12,Ise13,Sch17,Shu17,Sok17,Chi18,Cop18,Tra19,Loo19c} and glyoxal, \cite{Sta97b,Koh03,Hat05c,Sah06,Lea12,Poo14,Sch17,Aza17b,Loo18b}
|
||||||
|
that we all consider in their most stable \emph{trans} conformation in the following. Amongst these works, it is worth highlighting the detailed theoretical investigation by Saha, Ehara, and Nakatsuji, who reported a huge
|
||||||
|
number of ES in these three systems using a coherent theoretical Symmetry-Adapted-Cluster Configuration-Interaction (SAC-CI) protocol. \cite{Sah06} Our results are listed in Tables \ref{Table-3} and S3.
|
||||||
|
|
||||||
|
\hl{All below to be completed with NEV}
|
||||||
|
|
||||||
|
Acrolein, due to its lower symmetry and high density of ES with mixed characters, is challenging for theory and {\CCSDTQ} calculations were technically impossible despite our efforts. For the lowest $n \rightarrow \pi^\star$
|
||||||
|
transitions of both spin symmetries, the {\FCI} estimates come with a tiny error bar, and it is obvious that the CC estimates are too slightly low, especially with {\CCSDT}. Nevertheless, at the exception the second singlet
|
||||||
|
and triplet $A''$ ES, the {\CCT} and {\CCSDT} estimates are within $\pm$ 0.03 eV of each other, hinting at a reasonable accuracy. Additionally, the CC values are nearly systematically bracketed by the {\CASPT}
|
||||||
|
(lower bound)\cite{Aqu03} and SAC-CI (higher bound)\cite{Sah06} results, consistently with the typical error signs of these two models. For the two lowest triplet states, the present {\CCT}/{\AVTZ} values are also
|
||||||
|
within $\pm$0.05 eV of recent MR-CI estimates (3.50 and 3.89 eV). \cite{Mai14} As can be seen in Table S3, {\AVTZ} is sufficient to be very close from basis set convergence, the largest variation when going to {\AVQZ}
|
||||||
|
being +0.04 eV for the second $^1A'$ ES of Rydberg nature. As the experimental data are limited to measured UV spectra, \cite{Wal45,Bec70} one has therefore to be cautious in establishing TBE for acrolein
|
||||||
|
(\emph{vide infra}).
|
||||||
|
|
||||||
|
The nature and relative energies of the lowest bright $B_u$ and dark $A_g$ ES of butadiene have been puzzling theoretical chemists for many years. It is beyond our scope to provide an exhaustive
|
||||||
|
list of previous calculations and experimental estimates for these two hallmark ES, and we refer the reader to Refs. \citenum{Wat12} and \citenum{Shu17} for overviews and references. For the $B_u$ transition
|
||||||
|
the best previous TBE we are aware of is the 6.21 eV value obtained by Watson and Chan using a computational strategy similar to ours. \cite{Wat12} Our {\CCSDT}/{\AVTZ} value of 6.24 eV is obviously compatible
|
||||||
|
with this reference value, and our TBE value is actually 6.21 eV as well (\emph{vide infra}). For the $A_g$ state, we believe that our previous basis-corrected exCI estimate of 6.50 eV \cite{Loo19c} remains
|
||||||
|
the most accurate available to date. These two values are slightly smaller than the heath-bath CI data obtained by Chien \emph{et al.} with a double-$\zeta$ basis and a slightly different geometry: 6.45 and 6.58 eV for $B_u$ and $A_g$,
|
||||||
|
respectively. \cite{Chi18} One can of course find many other estimates, e.g., at the SAC-CI, \cite{Sah06} {\CCT}, \cite{Sil10c,Sch17} {\CASPT}, \cite{Sil10c} and {\NEV} levels, \cite{Sok17} for these two states.
|
||||||
|
More globally, in butadiene, we find an excellent coherence between the {\CCT}, {\CCSDT}, and {\CCSDTQ} estimates, that all fall in a $\pm$0.02 eV window. Unsurprisingly, this rule does not apply for the
|
||||||
|
already mentioned $^A_g$ ES that is 0.2 and 0.1 eV too high with the two former methods, consistent with the large electronic reorganization taking place in that state. For all the other butadiene ES listed in
|
||||||
|
Table \ref{Table-3}, both {\CCT} and {\CCSDT} can be trusted. Finally, as can be seen in Table S3, {\AVTZ} is sufficient for most ES, but one notes a significant basis set effect for the Rydberg $^1B_u\ \pi \rightarrow 3p$ ES
|
||||||
|
with an energy decrease as large as -0.12 eV when going from {\AVTZ} to {\AVQZ}. For the records, we note that the available electron impact data \cite{Mos73,Fli78,Doe81} provide the same ES ordering value as
|
||||||
|
our calculations.
|
||||||
|
|
||||||
|
Globally, the conclusions obtained for acrolein and butadiene pertain for glyoxal, i.e., highly consistent CC estimates and limited basis set effects beyond {\AVTZ} but for the Rydberg state (see Tables
|
||||||
|
\ref{Table-3} and S3). This state shows also a comparatively large deviation between {\CCT} and {\CCSDTQ}, that is 0.04 eV. More interestingly, glyoxal presents a low-lying ``true'' double ES,
|
||||||
|
$^1A_g\ n,n \rightarrow \pi^\star,\pi^\star$, a state that is totally unseen by approaches that do not explicitly include double excitations during the calculation of transition energies, e.g., TD-DFT, {\CCSD},
|
||||||
|
and {\AD}. Compared to the {\exCI} values, the {\CCT} and {\CCSDT} estimates for this transition are too large by ca. 1.0 and 0.5 eV, respectively, whereas {\CCSDTQ} is close to the spot, as already
|
||||||
|
mentioned in our previous work. \cite{Loo19c} For the other transitions, the present {\CCT} estimates are logically coherent with the values of Ref. \citenum{Sch17} obtained with the same approach on a different
|
||||||
|
geometry and remain slightly lower than the SAC-CI estimates of Ref. \citenum{Sah06}. Again experimental data \cite{Ver80,Rob85b} make an unhelpful guide in view of the targeted accuracy.
|
||||||
|
|
||||||
|
Finally, considering the {\CCSDTQ}/{\Pop} values of Table \ref{Table-3} as reference, we obtain mean absolute errors of 0.012 and 0.017 eV with {\CCT} and {\CCSDT}, respectively for the ES with $\%T_1 > 80$\%.
|
||||||
|
|
||||||
|
\begin{table}[htp]
|
||||||
|
\caption{\small Vertical transition energies determined in acrolein, butadiene, and glyoxal. All values are in eV.}
|
||||||
|
\label{Table-3}
|
||||||
|
\begin{footnotesize}
|
||||||
|
\begin{tabular}{l|p{.5cm}p{.9cm}p{1.1cm}p{1.5cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Acrolein}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Th.$^b$ & Exp.$^c$ \\
|
||||||
|
\hline
|
||||||
|
$^1A'' (n \rightarrow \pi^\star)$ &3.83&3.80& &3.85$\pm$0.01&3.77&3.74& 3.74&3.73& &3.63&3.83&3.71 \\
|
||||||
|
$^1A' (\pi \rightarrow \pi^\star)$ &6.83&6.86& & &6.67&6.70& 6.65&6.69& &6.10&6.92&6.41 \\
|
||||||
|
$^1A'' (n \rightarrow \pi^\star)$ &6.94&6.89& & &6.75&6.72& 6.75& & &6.26&7.40& \\
|
||||||
|
$^1A' (n \rightarrow 3s)$ &7.22&7.23& & &6.99&7.00& 7.07& & &6.97&7.19&7.08 \\
|
||||||
|
$^3A'' (n \rightarrow \pi^\star)$ &3.55&3.53& &3.60$\pm$0.01&3.47&3.45& 3.46& & &3.39&3.61& \\
|
||||||
|
$^3A' (\pi \rightarrow \pi^\star)$ &3.94&3.95& &3.97$\pm$0.03&3.95&3.95& 3.94& & &3.81&3.87& \\
|
||||||
|
$^3A' (\pi \rightarrow \pi^\star)$ &6.25&6.23& & &6.22&\hl{?}& 6.19& & & &6.21& \\
|
||||||
|
$^3A'' (\pi \rightarrow 3s)$ &6.81&6.74& & &6.60&\hl{?}& 6.61& & & &7.36& \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Butadiene}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^b$ & Th.$^d$ & Exp$^e$ \\
|
||||||
|
\hline
|
||||||
|
$^1B_u (\pi \rightarrow \pi^\star)$ &6.41&6.43&6.41&6.41$\pm$0.02 &6.25&6.27& 6.22&6.24 & &6.33&6.36&5.92\\
|
||||||
|
$^1B_g (\pi \rightarrow 3s)$ &6.53&6.55&6.54& &6.26&6.27& 6.33&6.34 & &6.18&6.32&6.21\\
|
||||||
|
$^1A_g (\pi \rightarrow \pi^\star)$ &6.73&6.63&6.56&6.55$\pm$0.03$^f$ &6.68&6.59& 6.67&6.60 & &6.56&6.60& \\
|
||||||
|
$^1A_u (\pi \rightarrow 3p)$ &6.87&6.89&6.87& &6.57&6.59& 6.64&6.66 & &6.45&6.56&6.64\\
|
||||||
|
$^1A_u (\pi \rightarrow 3p)$ &6.93&6.95&6.94&6.95$\pm$0.01 &6.73&6.74& 6.80&6.81 & &6.65&6.74&6.80\\
|
||||||
|
$^1B_u (\pi \rightarrow 3p)$ &7.98&8.00&7.98& &7.86&7.87& 7.68& & &7.08&7.02&7.07\\
|
||||||
|
$^3B_u (\pi \rightarrow \pi^\star)$ &3.35&3.36& &3.37$\pm$0.03 &3.36&3.36& 3.36& & &3.20& &3.22\\
|
||||||
|
$^3A_g (\pi \rightarrow \pi^\star)$ &5.22&5.22& & &5.21&5.21& 5.20& & &5.08& &4.91\\
|
||||||
|
$^3B_g (\pi \rightarrow 3s)$ &6.46&6.47& & &6.20&6.21& 6.28& & &6.14& &\\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Glyoxal}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^b$ & Th.$^g$ & Exp.$^h$\\
|
||||||
|
\hline
|
||||||
|
$^1A_u (n \rightarrow \pi^\star)$ &2.94&2.94&2.94& 2.93$\pm$0.03 &2.90&2.90& 2.88&2.88 & &3.10&2.93 &2.8 \\
|
||||||
|
$^1B_g (n \rightarrow \pi^\star)$ &4.34&4.32&4.31&4.28$\pm$0.06 &4.30&4.28& 4.27&4.25 & &4.68&4.39 &$\sim$4.4\\
|
||||||
|
$^1A_g (n,n \rightarrow \pi^\star,\pi^\star)$&6.74&6.24&5.67&5.60$\pm$0.01$^f$ &6.70&6.22& 6.76&6.35 & &5.66& &\\
|
||||||
|
$^1B_g (n \rightarrow \pi^\star)$ &6.81&6.83&6.79& &6.59&6.61& 6.58&6.61 & &7.54&6.63 &7.45\\
|
||||||
|
$^1B_u (n \rightarrow 3p)$ &7.72&7.74&7.76& &7.55&7.56& 7.67&7.69 & &7.83&7.61 &$\sim$7.7\\
|
||||||
|
$^3A_u (n \rightarrow \pi^\star)$ &2.55&2.55& &2.54$\pm$0.04 &2.49&2.49& 2.49&2.49 & &2.63& &2.5\\
|
||||||
|
$^3B_g (n \rightarrow \pi^\star)$ &3.97&3.95& & &3.91&3.90& 3.90&3.89 & &4.12& &$\sim$3.8\\
|
||||||
|
$^3B_u (\pi \rightarrow \pi^\star)$ &5.22&5.20& & &5.20&5.19& 5.17&5.15 & &5.35& &$\sim$5.2\\
|
||||||
|
$^3A_g (\pi \rightarrow \pi^\star)$ &6.35&6.35& & &6.34&6.34& 6.30&6.30 & & & &\\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\end{footnotesize}
|
||||||
|
\begin{flushleft}
|
||||||
|
\begin{footnotesize}
|
||||||
|
$^a${{\CASPT} values of Ref. \citenum{Aqu03};}
|
||||||
|
$^b${SAC-CI values of Ref. \citenum{Sah06};}
|
||||||
|
$^c${Vacuum UV spectra from Ref. \citenum{Wal45}. For the lowest state, the same 3.71 eV value is reported in Ref. \citenum{Bec70}.}
|
||||||
|
$^d${MR-AQCC values of Ref. \citenum{Dal04}, theoretical best estimates listed for the lowest $B_u$ and $A_g$ states;}
|
||||||
|
$^e${Electron impact experiment from Refs. \citenum{Fli78} and \citenum{Doe81} for the singlet states and from Ref. \citenum{Mos73} for the two lowest triplet transitions;
|
||||||
|
Note that for the lowest $B_u$ state, there is a vibrational structure with peaks at 5.76, 5.92, and 6.05 eV;}
|
||||||
|
$^f${From Ref. \citenum{Loo19c};}
|
||||||
|
$^g${{\CCT} values of Ref. \citenum{Sch17};}
|
||||||
|
$^h${Electron impact experiment from Ref. \citenum{Ver80} but for the second $^1B_g$ ES for which the value is from another work; \cite{Rob85b} note that
|
||||||
|
for the lowest $^1B_g$ ($^1B_u$) ES, a range of 4.2--4.5 (7.4--7.9) eV is given Ref. \citenum{Ver80}. }
|
||||||
|
\end{footnotesize}
|
||||||
|
\end{flushleft}
|
||||||
|
\end{table}
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
\subsubsection{Acetone, cyanoformaldehyde, isobutene, propynal, thioacetone, and thiopropynal}
|
||||||
|
|
||||||
|
There are earlier estimates of vertical transition energies for both acetone \cite{Gwa95,Mer96b,Roo96,Wib98,Toz99b,Wib02,Sch08,Sil10c,Car10,Pas12,Ise12,Gua13,Sch17} and isobutene. \cite{Wib02,Car10,Ise12}
|
||||||
|
To the best of our knowledge, the previous computational efforts were mainly focussed on 0-0 energies of the lowest-lying states for the four other compounds. \cite{Koh03,Hat05c,Sen11b,Loo18b,Loo19a}
|
||||||
|
There are also a few experimental values available for these six derivatives. \cite{Bir73,Jud83,Bra74,Sta75,Joh79,Jud83,Jud84c,Rob85,Pal87,Kar91b,Xin93} Our main data are reported in Tables \ref{Table-4}
|
||||||
|
and S4.
|
||||||
|
|
||||||
|
For acetone, one should clearly distinguish the valence ES, for which both methodological and basis set effects are small \hl{T2: please check the lowest S transition}, and the Rydberg transitions that, not only are very
|
||||||
|
sensitive to the basis set, but are upshifted by ca. 0.04 eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} estimates of Merch\'an and coworkers listed on the
|
||||||
|
r.h.s. of Table \ref{Table-4} are quite clearly too low, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the details of the calculations, as the Urban group obtained {\CASPT}
|
||||||
|
excitation energies of 4.40, 4.09 and 6.22 eV for the $^1A_2$, $^3A_2$, and $^3A_1$ ES, \cite{Pas12} in much better agreement with ours. Their estimates for the three $n \rightarrow 3p$ transitions of 7.52, 7.57, and 7.53 eV
|
||||||
|
for the $^1A_2$, $^1A_1$, and $^1B_2$ ES also systematically fall within 0.10 eV of our current CC values. \hl{NEVPT ?}
|
||||||
|
|
||||||
|
\hl{Thioacetone: en attente des CCSDTQ}
|
||||||
|
|
||||||
|
For the isoleectronic isobutene, we considered two singlet Rydberg and one triplet valence ES. For all three cases, we note very nice agreement between {\CCT} and {\CCSDT} results for all considered basis sets, these
|
||||||
|
CC results being also within or very close to the {\FCI} estimates with Pople's basis set. The match with the {\CCSD} results of Caricato and coworkers, \cite{Car10} is also very satisfying. \hl{CCSDTQ doable ?}
|
||||||
|
|
||||||
|
For the three remaining compounds, namely, cyanoformaldehyde, propynal, and thiopropynal, we have considered low-lying valence transitions all showing a largely dominant single excitation character. The basis set
|
||||||
|
effects are clearly under control (they are only significant for the second $^1A''$ ES of cyanoformaldehyde) and we could not detect any variation larger than 0.03 eV between the {\CCT} and {\CCSDT} values for
|
||||||
|
a given basis, hinting that the CC values should be close to the spot, as confirmed by the extrapolated {\FCI} data listed in Table \ref{Table-4}.
|
||||||
|
|
||||||
|
|
||||||
|
\begin{table}[htp]
|
||||||
|
\caption{\small Vertical transition energies determined in acetone, cyanonformaldehyde, isobutene, propynal, thioacetone, and thiopropynal. All values are in eV.}
|
||||||
|
\label{Table-4}
|
||||||
|
\begin{footnotesize}
|
||||||
|
\begin{tabular}{l|p{.5cm}p{.9cm}p{1.1cm}p{1.4cm}|p{.5cm}p{.9cm}|p{.5cm}p{.9cm}p{1.2cm}|p{.6cm}p{.6cm}p{.6cm}}
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Acetone}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^a$ & Th.$^b$ & Exp.$^c$ \\
|
||||||
|
\hline
|
||||||
|
$^1A_2 (n \rightarrow \pi^\star)$ &4.55&4.52&4.53&4.60$\pm$0.05 &4.50&4.48& 4.48&4.46& &4.18&4.18&4.48\\
|
||||||
|
$^1B_2 (n \rightarrow 3s)$ &6.65&6.64&6.68& &6.31&6.30& 6.43&6.42& &6.58&6.58&6.36\\
|
||||||
|
$^1A_2 (n \rightarrow 3p)$ &7.83&7.83&7.87& &7.37&7.36& 7.45& & &7.34&7.34&7.36\\
|
||||||
|
$^1A_1 (n \rightarrow 3p)$ &7.81&7.81&7.84& &7.39&7.38& 7.48&7.48& &7.26&7.26&7.41\\
|
||||||
|
$^1B_2 (n \rightarrow 3p)$ &7.87&7.87&7.91& &7.56&7.55& 7.59& & &7.48&7.48&7.45\\
|
||||||
|
$^3A_2 (n \rightarrow \pi^\star)$ &4.21&4.19& &4.18$\pm$0.04 &4.16&4.14& 4.15& & &3.90&3.90&4.15\\
|
||||||
|
$^3A_1 (\pi \rightarrow \pi^\star)$ &6.32&6.30& & &6.31&6.28& 6.28& & &5.98&5.98&\\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Cyanoformaldehyde}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.}\\\
|
||||||
|
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^d$ \\
|
||||||
|
\hline
|
||||||
|
$^1A'' (n \rightarrow \pi^\star)$ &3.91&3.89& &3.92$\pm$0.02 &3.86&3.84& 3.83&3.81 && 3.26\\
|
||||||
|
$^1A'' (\pi \rightarrow \pi^\star)$ &6.64&6.67& &6.60$\pm$0.07 &6.51&6.54& 6.42&6.46&& \\
|
||||||
|
$^3A'' (n \rightarrow \pi^\star)$ &3.53&3.51& &3.48$\pm$0.06 &3.47&3.45& 3.46& && \\
|
||||||
|
$^3A' (\pi \rightarrow \pi^\star)$ &5.07&5.07& & &5.03&5.03& 5.01& && \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Isobutene}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\\
|
||||||
|
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Th.$^e$ & Exp.$^f$ & Exp.$^g$ \\
|
||||||
|
\hline
|
||||||
|
$^1B_1 (\pi \rightarrow 3s)$ &6.77&6.77& &6.78$\pm$0.08 &6.39&6.39& 6.45&6.46&&6.40&6.15&6.17 \\
|
||||||
|
$^1A_1 (\pi \rightarrow 3p)$ &7.16&7.17& &7.16$\pm$0.02 &7.00&7.00& 7.00&7.01&&6.96& &6.71 \\
|
||||||
|
$^3A_1 (\pi \rightarrow \pi^\star)$ &4.52&4.53& &4.56$\pm$0.02 &4.54&4.54& 4.53& && &4.21 &4.3 \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Propynal}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.} \\
|
||||||
|
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^h$ \\
|
||||||
|
\hline
|
||||||
|
$^1A'' (n \rightarrow \pi^\star)$ &3.90&3.87& &3.84$\pm$0.06 &3.85&3.82& 3.82&3.80&& 3.24\\
|
||||||
|
$^1A'' (\pi \rightarrow \pi^\star)$ &5.69&5.73& &5.63$\pm$0.09 &5.59&5.62& 5.51&5.54&& \\
|
||||||
|
$^3A'' (n \rightarrow \pi^\star)$ &3.56&3.54& &3.54$\pm$0.04 &3.50&3.48& 3.49& && 2.99\\
|
||||||
|
$^3A' (\pi \rightarrow \pi^\star)$ &4.46&4.47& &4.44$\pm$0.08 &4.40&4.44& 4.43& && \\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Thioacetone}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & {\CCSDTQ} & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^i$ \\
|
||||||
|
\hline
|
||||||
|
$^1A_2 (n \rightarrow \pi^\star)$ &2.58&2.56& &2.61$\pm$0.05 &2.59&2.57& 2.55&2.53 & &2.33\\
|
||||||
|
$^1B_2 (n \rightarrow 4s)$ &5.65&5.64& &5.60$\pm$0.04 &5.44&5.43& 5.55&5.54 & &5.49\\
|
||||||
|
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.09&6.10& & &5.97&5.98& 5.90&5.91 & &5.64\\
|
||||||
|
$^1A_1 (n \rightarrow 4p)$ &6.95&6.95& & &6.54&6.53& 6.61& & &6.52\\
|
||||||
|
$^1B_2 (n \rightarrow 4p)$ &6.59&6.59& & &6.45&6.44& 6.51& & &6.40\\
|
||||||
|
$^3A_2 (n \rightarrow \pi^\star)$ &2.36&2.34& &2.36$\pm$0.00 &2.36&2.35& 2.34& & &2.14\\
|
||||||
|
$^3A_1 (\pi \rightarrow \pi^\star)$ &3.45&3.45& & &3.51&3.50& 3.46& & &\\
|
||||||
|
\hline
|
||||||
|
\multicolumn{12}{c}{Thiopropynal}\\
|
||||||
|
& \multicolumn{4}{c}{\Pop} & \multicolumn{2}{c}{\AVDZ}& \multicolumn{3}{c}{\AVTZ} & \multicolumn{1}{c}{Litt.}\\
|
||||||
|
State & {\CCT} & {\CCSDT} & & {\FCI} & {\CCT} & {\CCSDT}& {\CCT} & {\CCSDT} & {\NEV} & Exp$^j$ \\
|
||||||
|
\hline
|
||||||
|
$^1A'' (n \rightarrow \pi^\star)$ &2.09&2.06& &2.07$\pm$0.02 &2.09&2.06& 2.05&2.03 & &1.82\\
|
||||||
|
$^3A'' (n \rightarrow \pi^\star)$ &1.84&1.82& & &1.83&1.82& 1.81& & &1.64\\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\end{footnotesize}
|
||||||
|
\begin{flushleft}
|
||||||
|
\begin{footnotesize}
|
||||||
|
$^a${{\CASPT} values of Ref. \citenum{Mer96b};}
|
||||||
|
$^b${EOM-CCSD values of Ref. \citenum{Gwa95};}
|
||||||
|
$^c${Two lowest singlet states: various experiments summarized in Ref. \citenum{Rob85}; three next singlet states: REMPI experiments from Ref. \citenum{Xin93}; lowest triplet: trapped electron measurements from Ref. \citenum{Sta75};}
|
||||||
|
$^d${0-0 energy reported in Ref. \citenum{Kar91b};}
|
||||||
|
$^e${EOM-CCSD values from Ref. \citenum{Car10};}
|
||||||
|
$^f${Energy loss experiment from Ref. \citenum{Joh79};}
|
||||||
|
$^g${VUV experiment from Ref. \citenum{Pal87} (we report the lowest of the $\pi \rightarrow 3p$ state for the $^1A_1$ state)};
|
||||||
|
$^h${0-0 energies reported in Ref. \citenum{Bra74} (singlet) and \citenum{Bir73} (triplet);}
|
||||||
|
$^i${0-0 energies reported in Ref. \citenum{Jud83};}
|
||||||
|
$^i${0-0 energies reported in Ref. \citenum{Jud84c}.}
|
||||||
|
\end{footnotesize}
|
||||||
|
\end{flushleft}
|
||||||
|
\end{table}
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
\subsection{Five-atom molecules}
|
||||||
|
|
||||||
|
|
||||||
|
Thiophene:
|
||||||
|
For instance, this problem occurs for the second $B_2$ state of thiophene for which mixing with the third
|
||||||
|
$B_2$ state renders the assignment of the valence ($\pi \rightarrow \pi^\star$) or Rydberg ($\pi \rightarrow 3p$) nature uneasy at the CC3 level. The same problems occurs with other levels of theory. \cite{Ser93c,Wan01}
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
\section{Theoretical Best Estimates}
|
||||||
|
|
||||||
|
In Table \ref{Table-tbe}, we present our theoretical best estimates obtained with the {\AVTZ} basis set and further corrected for basis set effects. The details of the approaches used are given in Table \ref{Table-tbe}.
|
||||||
|
For all states with a dominant single-excitation character, that is when $\%T_1 > 85$\%, we relied on CC results, using the incremental strategy to obtain the our TBE. For the other transitions, we relied either
|
||||||
|
on the current or previous FCI \cite{Loo19c} or the {\NEV} values shown here as references \hl{correct ?}. In a few cases, e.g., the second singlet and triplet $A''$ ES of acrolein, the affordable CC expansion
|
||||||
|
yields quite large changes from one level of theory to the other, and we believe such estimates are not robust enough. \hl{que faire: italique ? enlever ?}
|
||||||
|
To determine the basis set corrections above triple-$\zeta$, we mainly used the {\CCT}/{\AVQZ} data.
|
||||||
|
For several of the smallest compounds, we also provide in the {\SI}, {\CCT}/{\DAVQZ} transition energies. However, we did not used these values as reference. This is because, the use of second set of
|
||||||
|
diffuse orbitals tends to modify the computed transition energies significantly only when it induces a more complex orbital mixture as well. This indicates that state mixing takes place, even if we use the
|
||||||
|
{\CCT} approach. We also stick to the frozen-core approximation for two reasons: i) the corrections brought by ``full-correlation'' are generally trifling (typically $\pm$ 0.02 eV) for the compounds under study
|
||||||
|
(see the {\SI}); and ii) it would probably be in principle necessary to add core polarization functions for such calculations.
|
||||||
|
|
||||||
|
\hl{On ferait une erreur totale CC3 versus CCSDTQ et CCSDT versus CCSDTQ pour choisir la meilleure approche ? Pas clair si CC3 ou CCSDTQ avec les donnees qu'on a jusqu'ici...Au moins discuter
|
||||||
|
de cela}
|
||||||
|
|
||||||
|
\hl{Decrire le set}
|
||||||
|
|
||||||
|
It should also be recalled that all our data are obtained on {\CCT){\AVTZ} geometries, consistently with our previous work, \cite{Loo18a} and therefore offer a consistent set of transition energies.
|
||||||
|
\hl{in total}
|
||||||
|
|
||||||
|
\renewcommand*{\arraystretch}{.55}
|
||||||
|
\LTcapwidth=\textwidth
|
||||||
|
|
||||||
|
\begin{footnotesize}
|
||||||
|
\begin{longtable}{llcccccc}
|
||||||
|
\caption{\small TBE values determined for all considered states. For each state, we provide the oscillator strength and percentage of single excitations obtained at the \CCT/{\AVTZ} level,
|
||||||
|
as well TBE obtained with the same basis set together with the method used to obtain that TBE. In the right-most columns, we list the values obtained by including further basis set corrections,
|
||||||
|
that have always been obtained at the \CCT level. All values are in the FC approximation.} \label{Table-tbe}\\
|
||||||
|
\hline
|
||||||
|
& & & & \multicolumn{2}{l}{TBE/{\AVTZ}} & \multicolumn{2}{l}{TBE/CBS} \\
|
||||||
|
& State & $f$ & $\%T_1$ & Value & Method$^a$ & Value & Corr. \\
|
||||||
|
\hline
|
||||||
|
\endfirsthead
|
||||||
|
\hline
|
||||||
|
& & & & \multicolumn{2}{l}{TBE/{\AVTZ}} & \multicolumn{2}{l}{TBE/CBS} \\
|
||||||
|
& State & $f$ & $\%T_1$ & Value & Method $^a$ & Value & Corr. \\
|
||||||
|
\hline
|
||||||
|
\endhead
|
||||||
|
\hline \multicolumn{7}{r}{{Continued on next page}} \\
|
||||||
|
\endfoot
|
||||||
|
\hline
|
||||||
|
\endlastfoot
|
||||||
|
Acetone &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 91.1 & 4.47 & B & 4.48 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ &0.000 & 90.5 & 6.46 & B & 6.51 & {\AVQZ} \\
|
||||||
|
&$^1A_2 (\mathrm{R}; n \rightarrow 3p)$ & & 90.9 & 7.45 & C & 7.48 & {\AVQZ} \\
|
||||||
|
&$^1A_1 (\mathrm{R}; n \rightarrow 3p)$ &0.004 & 90.6 & 7.51 & B & 7.44 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 3p)$ &0.029 & 91.2 & 7.63 & C & 7.64 & {\AVQZ} \\
|
||||||
|
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.8 & 4.13 & D & 4.15 & {\AVQZ} \\
|
||||||
|
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.7 & 6.25 & D & 6.27 & {\AVQZ} \\
|
||||||
|
Acrolein &$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 87.6 & 3.76 & G & 3.77 & {\AVQZ} \\
|
||||||
|
&$^1A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.344 & 91.2 & 6.69 & {\CCSDT}/{\AVTZ} & 6.69 & {\AVQZ} \\
|
||||||
|
&$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 79.4 & \hl{6.72} & \hl{xxxx} &\hl{ 6.74} & {\AVQZ} \\
|
||||||
|
&$^1A' (\mathrm{R}; n \rightarrow 3s)$ &0.109 & 89.4 & 7.08 & D & 7.12 & {\AVQZ} \\
|
||||||
|
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.0 & 3.51 & G & 3.52 & {\AVQZ} \\
|
||||||
|
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.6 & 3.94 & D & 3.95 & {\AVQZ} \\
|
||||||
|
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 6.17 & E & 6.18 & {\AVQZ} \\
|
||||||
|
&$^3A'' (\mathrm{R}; \pi \rightarrow 3s)$ & & 92.7 & \hl{6.61} & \hl{xxxxx} & \hl{6.62} & {\AVQZ} \\
|
||||||
|
Butadiene &$^1B_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.664 & 93.3 & 6.22 & B & 6.21 & {\AVQZ} \\
|
||||||
|
&$^1B_g (\mathrm{R}; \pi \rightarrow 3s)$ & & 94.1 & 6.33 & B & 6.35 & {\AVQZ} \\
|
||||||
|
&$^1A_g (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 75.1 & 6.50 & F & 6.50 & {\AVQZ} \\
|
||||||
|
&$^1A_u (\mathrm{R}; \pi \rightarrow 3p)$ &0.001 & 94.1 & 6.64 & B & 6.66 & {\AVQZ} \\
|
||||||
|
&$^1A_u (\mathrm{R}; \pi \rightarrow 3p)$ &0.049 & 94.1 & 6.80 & B & 6.82 & {\AVQZ} \\
|
||||||
|
&$^1B_u (\mathrm{R}; \pi \rightarrow 3p)$ &0.055 & 93.8 & 7.68 & C & 7.54 & {\AVQZ} \\
|
||||||
|
&$^3B_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 3.36 & D & 3.37 & {\AVQZ} \\
|
||||||
|
&$^3A_g (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.7 & 5.20 & D & 5.21 & {\AVQZ} \\
|
||||||
|
&$^3B_g (\mathrm{V}; \pi \rightarrow 3s)$ & & 97.9 & 6.29 & D & 6.31 & {\AVQZ} \\
|
||||||
|
Cyanoacetylene &$^1\Sigma^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.3 & 5.80 & A & 5.79 & {\AVFZ} \\
|
||||||
|
&$^1\Delta (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.0 & 6.07 & A & 6.05 & {\AVFZ} \\
|
||||||
|
&$^3\Sigma^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.44 & {\CCSDT}/{\AVTZ} & 4.46 & {\AVFZ} \\
|
||||||
|
&$^3\Delta (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 5.21 & {\CCSDT}/{\AVTZ} & 5.21 & {\AVFZ} \\
|
||||||
|
Cyanoformaldehyde &$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.001 & 89.8 & 3.81 & {\CCSDT}/{\AVTZ} & 3.82 & {\AVQZ} \\
|
||||||
|
&$^1A'' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.000 & 91.9 & 6.46 & {\CCSDT}/{\AVTZ} & 6.45 & {\AVQZ} \\
|
||||||
|
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.6 & 3.44 & D & 3.45 & {\AVQZ} \\
|
||||||
|
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.4 & 5.01 & D & 5.00 & {\AVQZ} \\
|
||||||
|
Cyanogen & $^1\Sigma_u^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.1 & 6.39 & A & 6.38 & {\AVFZ} \\
|
||||||
|
& $^1\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 93.4 & 6.66 & A & 6.64 & {\AVFZ} \\
|
||||||
|
& $^3\Sigma_u^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.91 & B & 4.89 & {\AVFZ} \\
|
||||||
|
& $^1\Sigma_u^- [\mathrm{F}] (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 93.4 & 5.05 & A & 5.03 & {\AVFZ} \\
|
||||||
|
Cyclopropenone &$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 87.7 & 4.26 & B & 4.28 & {\AVFZ} \\
|
||||||
|
&$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 91.0 & 5.55 & B & 5.56 & {\AVFZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ &0.003 & 90.8 & 6.34 & B & 6.39 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star$) &0.047 & 86.5 & 6.54 & B & 6.56 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 3p)$ &0.018 & 91.1 & 6.98 & B & 7.01 & {\AVQZ} \\
|
||||||
|
&$^1A_1 (\mathrm{R}; n \rightarrow 3p)$ &0.003 & 91.2 & 7.02 & B & 7.08 & {\AVFZ} \\
|
||||||
|
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.320 & 90.8 & 8.28 & B & 8.26 & {\AVFZ} \\
|
||||||
|
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 96.0 & 3.93 & {\CCSDT}/{\AVTZ} & 3.95 & {\AVQZ} \\
|
||||||
|
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.9 & 4.88 & {\CCSDT}/{\AVTZ} & 4.90 & {\AVQZ} \\
|
||||||
|
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.5 & 5.35 & {\CCSDT}/{\AVTZ} & 5.37 & {\AVQZ} \\
|
||||||
|
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.1 & 6.79 & {\CCSDT}/{\AVTZ} & 6.81 & {\AVFZ} \\
|
||||||
|
Cyclopropenethione &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 89.6 & 3.41 & B & 3.42 & {\AVQZ} \\
|
||||||
|
&$^1B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ &0.000 & 84.8 & 3.45 & B & 3.47 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.007 & 83.0 & 4.60 & B & 4.62 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 3s)$ &0.048 & 91.8 & 5.34 & B & 5.39 & {\AVQZ} \\
|
||||||
|
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ &0.228 & 89.0 & 5.46 & B & 5.46 & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 3p)$ &0.084 & 91.3 & 5.92 & B & 5.94 & {\AVQZ} \\
|
||||||
|
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.2 & 3.28 & D & 3.29 & {\AVQZ} \\
|
||||||
|
&$^3B_1 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 94.5 & 3.32 & {\CCSDT}/{\AVTZ} & 3.35 & {\AVQZ} \\
|
||||||
|
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 96.5 & 4.01 & D & 4.03 & {\AVQZ} \\
|
||||||
|
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 4.01 & D & 4.02 & {\AVQZ} \\
|
||||||
|
Diacetylene &$^1\Sigma_u^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.4 & 5.33 & A & 5.32 & {\AVFZ} \\
|
||||||
|
&$^1\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 94.1 & 5.61 & A & 5.60 & {\AVFZ} \\
|
||||||
|
&$^3\Sigma_u^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 4.10 & C & 4.07 & {\AVFZ} \\
|
||||||
|
&$^3\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.2 & 4.78 & B & 4.78 & {\AVFZ} \\
|
||||||
|
Glyoxal &$^1A_u (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.000 & 91.0 & 2.88 & B & 2.88 & {\AVQZ} \\
|
||||||
|
&$^1B_g (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.3 & 4.24 & B & 4.24 & {\AVQZ} \\
|
||||||
|
&$^1A_g (\mathrm{V}; n,n \rightarrow \pi^\star,\pi^\star)$ & & 0.5 & 5.61 & F & 5.61 & {\AVQZ} \\
|
||||||
|
&$^1B_g (\mathrm{V}; n \rightarrow \pi^\star)$ & & 83.9 & 6.57 & B & 6.58 & {\AVQZ} \\
|
||||||
|
&$^1B_u (\mathrm{R}; n \rightarrow 3p)$ & 0.095 & 91.7 & 7.71 & B & 7.75 & {\AVQZ} \\
|
||||||
|
&$^3A_u (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.6 & 2.49 & {\CCSDT}/{\AVTZ} & 2.49 & {\AVQZ} \\
|
||||||
|
&$^3B_g (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.4 & 3.89 & {\CCSDT}/{\AVTZ} & 3.90 & {\AVQZ} \\
|
||||||
|
&$^3B_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.5 & 5.15 & {\CCSDT}/{\AVTZ} & 5.16 & {\AVQZ} \\
|
||||||
|
&$^3A_g (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.8 & 6.30 & {\CCSDT}/{\AVTZ} & 6.31 & {\AVQZ} \\
|
||||||
|
Isobutene &$^1B_1 (\mathrm{R}; \pi \rightarrow 3s)$ & 0.006 & 94.1 & 6.46 & {\CCSDT}/{\AVTZ} & 6.48 & {\AVQZ} \\
|
||||||
|
&$^1A_1 (\mathrm{R}; \pi \rightarrow 3p)$ & 0.228 & 94.2 & 7.01 & {\CCSDT}/{\AVTZ} & 7.00 & {\AVQZ} \\
|
||||||
|
&$^3A_1 \mathrm{V}; (\pi \rightarrow \pi^\star)$ & & 98.9 & 4.53 & D & 4.54 & {\AVQZ} \\
|
||||||
|
Methylenecyclopropene& $^1B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.011 & 85.4 & 4.28 & B & 4.28 & {\AVQZ} \\
|
||||||
|
&$^1B_1 (\mathrm{R}; \pi \rightarrow \sigma^\star)$ & 0.005 & 93.6 & 5.44 & B & 5.47 & {\AVQZ} \\
|
||||||
|
&$^1A_2 (\mathrm{R}; \pi \rightarrow 3p)$ & & 93.3 & 5.96 & B & 5.99 & {\AVQZ} \\
|
||||||
|
&$^1A_1(\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.224 & 92.8 & 6.12 & B & 6.08 & {\AVQZ} \\
|
||||||
|
&$^3B_2 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 97.2 & 3.50 & D & 3.50 & {\AVQZ} \\
|
||||||
|
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.6 & 4.74 & D & 4.75 & {\AVQZ} \\
|
||||||
|
Propynal & $^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.000 & 89.0 & 3.80 & {\CCSDT}/{\AVTZ} & 3.81 & {\AVQZ} \\
|
||||||
|
&$^1A'' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.000 & 92.9 & \hl{5.54} & {\CCSDT}/{\AVTZ} & \hl{5.53} & {\AVQZ} \\
|
||||||
|
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.4 & 3.47 & D & 3.48 & {\AVQZ} \\
|
||||||
|
&$^3A' (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.3 & 4.47 & D & 4.48 & {\AVQZ} \\
|
||||||
|
Thioacetone &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 88.9 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 4s)$ & 0.052 & 91.3 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
|
||||||
|
&$^1A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & 0.242 & 90.6 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
|
||||||
|
&$^1A_1 (\mathrm{R}; n \rightarrow 4p)$ & 0.023 & 91.6 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
|
||||||
|
&$^1B_2 (\mathrm{R}; n \rightarrow 4p)$ & 0.028 & 92.4 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
|
||||||
|
&$^3A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.4 & 2.33 & D & 2.34 & {\AVQZ} \\
|
||||||
|
&$^3A_1 (\mathrm{V}; \pi \rightarrow \pi^\star)$ & & 98.7 & 3.45 & D & 3.46 & {\AVQZ} \\
|
||||||
|
Thiopropynal &$^1A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & 0.000 & 87.5 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
|
||||||
|
&$^3A'' (\mathrm{V}; n \rightarrow \pi^\star)$ & & 97.2 & \hl{xxx} & \hl{xxx} & \hl{xxx} & {\AVQZ} \\
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\end{longtable}
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\end{footnotesize}
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\begin{flushleft}\begin{footnotesize}\begin{singlespace}
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|
$^a${
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|
Method A: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\AVDZ} and {\CCSDT}/{\AVDZ} results;
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|
Method B: {\CCSDT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCSDT}/{\Pop} results;
|
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|
Method C: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDTQ}/{\Pop} and {\CCT}/{\Pop} results;
|
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|
Method D: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\AVDZ} and {\CCT}/{\AVDZ} results;
|
||||||
|
Method E: {\CCT}/{\AVTZ} value corrected by the difference between {\CCSDT}/{\Pop} and {\CCT}/{\Pop} results;
|
||||||
|
Method F: {exCI}/{\AVDZ} value (from Ref. \citenum{Loo19c}) corrected by the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ} results;
|
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|
Method G: {exCI}/{\Pop} value (this work) corrected by the difference between {\CCT}/{\AVTZ} and {\CCT}/{\Pop} results;
|
||||||
|
}
|
||||||
|
\end{singlespace}\end{footnotesize}\end{flushleft}
|
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|
|
||||||
|
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|
\section{Benchmarks}
|
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|
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\section{Conclusions and outlook}
|
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|
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||||||
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|
||||||
|
\begin{suppinfo}
|
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\end{suppinfo}
|
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|
|
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|
\begin{acknowledgement}
|
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|
D.J.~acknowledges the \emph{R\'egion des Pays de la Loire} for financial support. This research used resources of i) the GENCI-CINES/IDRIS (Grant 2016-08s015); ii) CCIPL (\emph{Centre de Calcul Intensif des Pays de Loire});
|
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|
iii) the Troy cluster installed in Nantes; and iv) CALMIP under allocations 2018-0510 and 2018-18005 (Toulouse).
|
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\end{acknowledgement}
|
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\bibliography{biblio-new}
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\clearpage
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\begin{tocentry}
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\begin{center}
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%\includegraphics[scale=.17]{TOC.png}
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\end{center}
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\end{tocentry}
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\end{document}
|
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3799
Mountain2.nb
3799
Mountain2.nb
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