QUEST articles and SI
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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{\ADC}[1]{ADC(#1)}
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\newcommand{\CC}[1]{CC#1}
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\newcommand{\CCSD}{CCSD}
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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{\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{\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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\title{A Mountaineering Strategy to Excited States: Highly-Accurate Reference Energies and Benchmarks}
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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{Anthony Scemama}
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\affiliation[LCPQ, Toulouse]{\LCPQ}
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\author{Aymeric Blondel}
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\affiliation[UN, Nantes]{\CEISAM}
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\author{Yann Garniron}
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\affiliation[LCPQ, Toulouse]{\LCPQ}
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\author{Michel Caffarel}
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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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Striving to define very accurate vertical transition energies, we perform both high-level coupled cluster (CC) calculations (up to {\CCSDTQP}) and selected configuration interaction ({\sCI}) calculations (up to several millions of
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determinants) for 17 small compounds (water, ammonia, hydrogen chloride, dinitrogen, carbon monoxide, acetylene, ethylene, formaldehyde, methanimine, thioformaldehyde, acetaldehyde, cyclopropene, diazomethane,
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formamide, ketene, nitrosomethane and the smallest streptocyanine). By systematically increasing the order of the CC expansion, the number of determinants in the CI expansion as well as the size of the one-electron basis set,
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we have been able to reach near full CI (FCI) quality transition energies, with an estimated deviation as small as $\pm 0.03$ eV. These calculations are carried out on {\CC{3}}/{\AVTZ} geometries, using a series of increasingly
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large atomic basis sets systematically including diffuse functions. In this way, we define a list of 106 transition energies for states of various characters (valence, Rydberg, $n \rightarrow \pis$, $\pi \rightarrow \pis$, singlet, triplet, etc.)
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to be used as references for further calculations. Benchmark transition energies are provided at the {\AVTZ} level as well as with additional basis set corrections, in order to obtain results close to the complete basis set limit.
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These reference data are used to benchmark a series of eleven excited-state wave function methods accounting for double and triple contributions, namely {\ADC{2}}, {\ADC{3}}, CIS(D), {\CC{2}}, STEOM-CCSD, {\CCSD},
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CCSDR(3), CCSDT-3, {\CC{3}}, {\CCSDT} and {\CCSDTQ}. It turns out that {\CCSDTQ} yields a negligible difference with the extrapolated {\CI} values with a mean absolute error as small as \IneV{0.01}, whereas the
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coupled cluster approaches including iterative triples are also very accurate (mean absolute error: \IneV{0.03}). Consequently, CCSDT-3 and {\CC{3}} can be used to define reliable benchmarks, whereas this does not
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hold for {\ADC{3}} that delivers quite large errors for this set of small compounds, with a clear trend to overcorrect the {\ADC{2}} values.
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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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Defining an effective method reliably providing accurate excited-state energies and properties remains a major challenge in theoretical chemistry. For practical applications, the most popular approaches are the complete active
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space self-consistent field ({\CASSCF}) and the time-dependent density functional theory ({\TDDFT}) methods for systems dominated by static and dynamic electron correlation effects, respectively. When these schemes are
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not sufficiently accurate, one often uses merthods including second-order perturbative corrections. For {\CASSCF}, a natural choice is {\CASPT}, \cite{And90} but this method rapidly becomes impractical for large compounds.
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If a single-reference method is sufficient, the most popular second-order approaches are probably the second-order algebraic diagrammatic construction, {\ADC{2}}, \cite{Dre15} and the second-order coupled cluster, {\CC{2}},
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methods, \cite{Chr95,Hat00} that both offer an attractive $\order*{N^5}$ scaling (where $N$ is the number of basis functions) allowing applications up to systems comprising ca.~100 atoms. Compared to {\TDDFT}, these
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approaches have the indisputable advantage of being free of the choice of a specific exchange-correlation functional. Using {\ADC{2}} or {\CC{2}} generally provides more systematic errors with respect to reference values than
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TD-DFT, although the improvements in terms of error magnitude are often rather moderate (at least for valence singlet states). \cite{Win13,Jac15b,Oru16} Importantly, both {\ADC{$n$}} and {\CC{$n$}} offer a systematic pathway
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for improvement via an increase of the expansion order $n$. For example, using {\CCSD}, {\CCSDT}, {\CCSDTQ}, etc., allows to check the quality of the obtained estimates. However, in practice, one can only contemplate such
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systematic approach and the ultimate choice of a method for excited-state calculations is often guided by previous benchmarks. These benchmark studies are either performed using experimental or theoretical reference values.
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While the former approach allows in principle to rely on an almost infinite pool of reference data, most measurements are performed in solution and provide absorption bands that can be compared to theory only with the use of
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extra approximations for modeling environmental and vibronic effects. Consequently, it is easier to use first-principle reference values as benchmarks, as they allow to assess theoretical methods more consistently (vertical values,
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same geometries, no environmental effects, etc). This is well illustrated by the recent contribution of Schwabe and Goerigk, \cite{Sch17} who decided to compute third-order response CC ({\CC{3}})\cite{Chr95b,Koc97} reference
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values instead of using the previously collected experimental values for the test set originally proposed by Gordon's group. \cite{Lea12}
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Whilst many benchmark sets have been proposed for excited states, \cite{Par02,Die04b,Gri04b,Rhe07,Pea08,Jac08b,Jac09c,Goe09,Car10,Lea12,Jac12d,Win13,Jac15b,Hoy16} the most praised database of theoretical excited
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state energies is undoubtedly the one set up by Thiel and his co-workers. In 2008, they proposed a large set of theoretical best estimates (TBE) for 28 small and medium CNOH organic compounds. \cite{Sch08}
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More precisely, using some literature values but mainly their own {\CC{3}}/TZVP and {\CASPT}/TZVP results computed on MP2/6-31G(d) geometries, these authors determined 104 singlet and 63 triplet reference
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excitation energies. The same group soon proposed {\AVTZ} TBE for the same set of compounds, \cite{Sil10b,Sil10c} though some {\CC{3}}/{\AVTZ} reference values were estimated by a basis set extrapolation technique.
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In their conclusion, they stated that they ``\emph{expect this benchmark set to be useful for validation and development purposes, and anticipate future improvements and extensions of this set through further
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high-level calculations}''.\cite{Sch08} The first prediction was soon realized. Indeed, both the TZVP and {\AVTZ} TBE were applied to benchmark various computationally-effective methods, including semi-empirical approaches,
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\cite{Sil10,Dom13,Voi14} {\TDDFT}, \cite{Sil08,Goe09,Jac09c,Roh09,Jac10c,Jac10g,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Mai16} the second-order polarization propagator approximation (SOPPA), \cite{Sau15} {\ADC{2}},
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\cite{Har14} the random phase approximation (RPA), \cite{Yan14b} as well as several {\CC{}} variants. \cite{Sau09,Pie15,Taj16,Ris17,Dut18} In contrast, even a decade after the original work appeared, the progresses aiming at
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improving and/or extending Thiel's set have been much less numerous. To the best of our knowledge, these extensions are limited to the more compact TZVP basis set. \cite{Wat13,Har14,Kan14}
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This diffuse-less basis set offers clear computational advantages and avoids some state mixing. However, it has a clear tendency to overestimate transition energies, especially for Rydberg states, and it makes comparisons
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between methods more difficult as basis set dependencies are significantly different in wave function-based and density-based methods. \cite{Lau15}
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Let us now briefly review these efforts. In 2013, Watson \emph{et al.}~obtained with the TZVP basis set and {\CCSDT}-3 --- a method employing an iterative approximation of the triples --- transition energies very similar to the {\CC{3}} values. \cite{Wat13}
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In 2014, Dreuw and co-workers performed {\ADC{3}} calculations on Thiel's set and concluded that \emph{``based on the quality of the existing benchmark set it is practically not possible to judge whether {\ADC{3}} or {\CC{3}} is more accurate''}.
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The same year, Kannar and Szalay, revisited Thiel's set and proposed {\CCSDT}/TZVP reference energies for 17 singlet states of six molecules, \cite{Kan14} which are, to the best of our knowledge, the highest-level values reported to date.
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However, it remains difficult to know if these {\CCSDT} transition energies are significantly more accurate than their {\CC{3}} or {\ADC{3}} counterparts. Indeed, for the $\pi \rightarrow \pis$ valence singlet excited state of ethylene, the {\CC{3}}/TZVP,
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{\CCSDT}/TZVP and {\CCSDTQ}/TZVP estimates of \IneV{$8.37$}, \IneV{$8.38$}, and \IneV{$8.36$} (respectively) are nearly identical. \cite{Kan14}
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Herein, we propose to continue the quest for ultra-accurate excited-state reference energies. First, although this prevents direct comparisons with previously-published data, we decided to use more accurate {\CC{3}}/{\AVTZ}
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geometries for all the compounds considered here. Second, we employ only diffuse-containing Dunning basis sets to be reasonably close from the complete basis set limit. Third, we climb the mountain via two faces following:
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i) the {\CC{}} route (up to the highest computationally possible order), and ii) the configuration interaction ({\CI}) route with the help of selected {\CI} ({\sCI}) methods. These two approaches allow us to assess reliably the gap
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from the full CI ({\FCI}) result. Fourth, in order not to limit our investigation to vertical absorption, we also report, in a few cases, fluorescence energies. Of course, such extreme choices impose drastic restrictions on the size of
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the molecules one can treat with such approaches. However, we claim here that they allow to estimate the {\FCI} result within ca.~\IneV{$\pm0.03$} for most excited states.
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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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\subsection{Geometries}
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All geometries are obtained at the {\CC{3}}/{\AVTZ} level without applying the frozen core approximation. These geometries are available in the {\SI} (SI). While several structures are extracted from Ref.~\citenum{Bud17}
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(acetylene, diazomethane, ethylene, formaldehyde, ketene, nitrosomethane, thioformaldehyde and streptocyanine-C1) , additional optimizations are performed here following the same protocol as in that earlier work.
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First, we optimize the structures and compute the vibrational spectra at the CCSD/def2-TZVPP level \cite{Pur82} with Gaussian16. \cite{Gaussian16} These calculations confirm the minima nature of the obtained
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geometries. \cite{zzz-tou-1} We then re-optimize the structures at the {\CC{3}}/{\AVTZ} level \cite{Chr95b,Koc97} using Dalton \cite{dalton} and/or CFOUR, \cite{cfour} depending on the size and symmetry of the molecule.
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CFOUR advantageously provides analytical CC3 gradients for ground-state structures. For the CCSD calculations, the energy and geometry convergence thresholds are systematically tightened to \InAU{$10^{-10}$--$10^{-11}$}~for the
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SCF energy, \InAU{$10^{-8}$--$10^{-9}$}~for the {\CCSD} energy, and \InAU{$10^{-7}$--$10^{-8}$}~for the {\EOMCCSD} energy in the case of excited-state optimizations (when required). To check that the structures correspond
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to genuine minima, the (EOM-){\CCSD} gradients are differentiated numerically to obtain the vibrational frequencies. The {\CC{3}} optimizations are performed with the default convergence thresholds of Dalton or CFOUR
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without applying the frozen core approximation.
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\subsection{Coupled Cluster calculations}
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Unless otherwise stated, the {\CC{}} transition energies \cite{Kal04} are computed in the frozen-core approximation (large cores for \ce{Cl} and \ce{S}). We use several codes to achieve our objectives, namely CFOUR,\cite{cfour}
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Dalton,\cite{dalton} Gaussian16,\cite{Gaussian16} Orca,\cite{Nee12} MRCC,\cite{Rol13,mrcc} and Q-Chem. \cite{Sha15} Globally, we use CFOUR for both CCSDT-3 \cite{Wat96,Pro10} and CCSDT \cite{Nog87} calculations,
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Dalton to perform the CIS(D),\cite{Hea95} {\CC{2}}, \cite{Chr95,Hat00} {\CCSD},\cite{Pur82} CCSDR(3), \cite{Chr96b} and CC3 \cite{Chr95b,Koc97} calculations, Gaussian for the CIS(D) \cite{Hea95} and {\CCSD}, \cite{Pur82}
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Orca for the similarity-transformed EOM-CCSD (STEOM-CCSD)\cite{Noo97,Dut18} calculations, Q-Chem for {\ADC{2}} and {\ADC{3}} calculations, and MRCC for the {\CCSDT}, \cite{Nog87} CCSDTQ, \cite{Kuc91} (and higher)
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calculations. As we mainly report transition energies, it it worth noting that the linear-response (LR) and equation-of-motion (EOM) formalisms provide identical results. Nevertheless, the oscillator strengths characterizing the excited
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states are obtained at the (LR) {\CC{3}} level with Dalton. Default program setting are generally applied, or when modified they are tightened. For the STEOM-CCSD calculations which relies on natural transition orbitals,
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it was checked that each state is characterized by an active character percentage of 98\%\ or larger (states not matching this criterion are not reported). Nevertheless, the obtained results do slightly depend on the number of
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states included in the calculations, and we found typical variations of $\pm$\IneV{0.01--0.05}. For all calculations, we use the well-known Dunning's \emph{aug}-cc-pVXZ (X $=$ D, T, Q and 5) atomic basis sets, as well as some
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doubly- and triply-augmented basis sets of the same series (d-\emph{aug}-cc-pVXZ and t-\emph{aug}-cc-pVXZ).
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\subsection{Selected Configuration Interaction methods}
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Alternatively to {\CC{}}, we also compute transition energies using a selected {\CI} ({\sCI}) approach, an idea that goes back to 1969 in the pioneering works of Bender and Davidson, \cite{Ben69} and Whitten and Hackmeyer. \cite{Whi69}
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Recently, sCI methods have demonstrated their ability to reach near FCI quality energies for small organic and transition metal-containing molecules. \cite{Gin13,Caf14,Gin15,Gar17,Caf16,Hol16,Sha17,Hol17,Chi18,Sce18}
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To avoid the exponential increase of the size of the {\CI} expansion, we employ the {\sCI} algorithm CIPSI \cite{Hur73,Eva83,Gin13} (Configuration Interaction using a Perturbative Selection made Iteratively) to retain only the
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energetically-relevant determinants. To do so, the CIPSI algorithm uses a second-order energetic criterion to select perturbatively determinants in the {\FCI} space. \cite{Gin13,Gin15,Caf16,Sce18} We refer the interested readers to
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Ref.~\citenum{Sce18} for more details about the general philosophy of {\sCI} methods.
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In order to treat the electronic states of a given spin manifold on equal footing, a common set of determinants is used for all states. Moreover, to speed up convergence to the {\FCI} limit, a common set of natural orbitals issued
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from a preliminary (smaller) {\sCI} calculation is employed. For a given basis set, the {\FCI} limit has been reached by the method recently proposed by Holmes, Umrigar and Sharma \cite{Hol17} in the context of the (selected)
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heat-bath {\CI} method. \cite{Hol16,Sha17,Hol17,Chie18} This method has been shown to be robust even for challenging chemical situations. \cite{Sce18,Chi18} In order to obtain {\FCI} results, we linearly extrapolate the {\sCI}
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energy $\EsCI$ as a function of $\EPT$, which is an estimate of the truncation error in the {\sCI} algorithm, i.e., $\EPT \approx \EFCI-\EsCI$. When $\EPT = 0$, the {\FCI} limit has effectively been reached.
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Here, $\EPT$ is efficiently evaluated with a recently-proposed hybrid stochastic-deterministic algorithm. \cite{Gar17b} In practice, the extrapolation is based on the two largest sCI wave functions, i.e., we perform a two-point
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extrapolation. Estimating the extrapolation error is a complicated task with no well-defined method to do so. In practice, we have observed that this extrapolation procedure is robust and provides FCI estimates within \IneV{$\pm 0.02$}.
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When the convergence to the FCI limit is too slow to provide reliable estimates, the number of significant digits reported has been reduced accordingly. From herein, the extrapolated {\FCI} results are simply labeled {\exCI}.
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All the {\sCI} calculations are performed with the electronic structure software \textsc{quantum package}, developed in Toulouse and freely available. \cite{QP}
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%
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% III. Results & Discussion
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%
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\section{Results and Discussion}
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\label{sec-res}
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In the discussion below, we first discuss specific molecules of increasing size and compare the results obtained with {\exCI} and {\CC{}} approaches, starting with the {\CC{3}} method for the latter.
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We next define two series of TBE, one at the frozen-core {\AVTZ} level, and one close to complete basis set limit by applying corrections for frozen-core and basis set effects.
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We finally assess the performances of several popular wave function methods using the former benchmark as reference. In the following, we considered the {\exCI} values as benchmarks, except when noted.
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\subsection{Water, ammonia and hydrogen chloride}
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\begin{sidewaystable}[htp]
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\caption{\small Vertical transition energies for the three lowest singlet and three lowest triplet excited states of water (top), the four lowest singlet and the lowest triplet states of
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ammonia (center), and the lowest singlet state of hydrogen chloride (bottom). All states of water and ammonia have a Rydberg character, whereas the lowest state of hydrogen chloride
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is a charge-transfer state. All values are in eV.}
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\label{Table-1}
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\begin{small}
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\begin{tabular}{l|p{.6cm}p{1.1cm}p{1.4cm}p{1.7cm}p{.7cm}|p{.6cm}p{1.1cm}p{1.4cm}p{.7cm}|p{.6cm}p{1.1cm}p{.7cm}|p{.7cm}p{.7cm}p{.7cm}}
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\hline
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\multicolumn{16}{c}{Water}\\
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& \multicolumn{5}{c}{\AVDZ} & \multicolumn{4}{c}{\AVTZ}& \multicolumn{3}{c}{\AVQZ} & \multicolumn{3}{c}{Litt.}\\
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State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\CCSDTQP} & {\exCI} & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI}& {\CC{3}} & {\CCSDT} & {\exCI} & Exp.$^a$ & Th.$^b$ & Th.$^c$\\
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\hline
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$^1B_1 (n \rightarrow 3s)$ &7.51&7.50&7.53&7.53&7.53 &7.60&7.59&7.62&7.62 &7.65 &7.64 &7.68 &7.41 &7.81&7.57\\%
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$^1A_2 (n \rightarrow 3p)$ &9.29&9.28&9.31&9.32&9.32 &9.38&9.37&9.40&9.41 &9.43 &9.41 &9.46 &9.20 &9.30&9.33\\%
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$^1A_1 (n \rightarrow 3s)$ &9.92&9.90&9.94&9.94&9.94 &9.97&9.95&9.98&9.99 &10.00 &9.98 &10.02 &9.67 &9.91&9.91\\%
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$^3B_1 (n \rightarrow 3s)$ &7.13&7.11&7.14&7.14&7.14 &7.23&7.22&7.24&7.25 &7.28 &7.26 &7.30 &7.20 &7.42&7.21\\%
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$^3A_2 (n \rightarrow 3p)$ &9.12&9.11&9.14&9.14&9.14 &9.22&9.20&9.23&9.24 &9.26 &9.25 &9.28 &8.90 &9.42&9.19\\%
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$^3A_1 (n \rightarrow 3s)$ &9.47&9.45&9.48&9.49&9.49 &9.52&9.50&9.53&9.54 &9.56 &9.54 &9.58 &9.46 &9.78&9.50\\%
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\hline
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\multicolumn{16}{c}{Ammonia}\\
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& \multicolumn{5}{c}{\AVDZ} & \multicolumn{4}{c}{\AVTZ}& \multicolumn{3}{c}{\AVQZ} & \multicolumn{3}{c}{Litt.}\\
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State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\CCSDTQP} & {\exCI} & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI}& {\CC{3}} & {\CCSDT} & {\exCI} & Exp.$^d$ & Exp.$^e$ & Th.$^f$\\
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\hline
|
||||
$^1A_2 (n \rightarrow 3s)$ &6.46 &6.46 & 6.48 &6.48&6.48 &6.57&6.57&6.59 &6.59 &6.61&6.61&6.64 &6.38&6.39 &6.48\\
|
||||
$^1E (n \rightarrow 3p)$ &8.06 &8.06 &8.08 &8.08&8.08 &8.15&8.14& &8.16 &8.18&8.17&8.22 &7.90&7.93 &8.02\\
|
||||
$^1A_1 (n \rightarrow 3p)$ &9.66 &9.66 &9.68 &9.68&9.68 &9.32&9.31& &9.33 &9.11&9.10&9.14 &8.14&8.26 &8.50\\
|
||||
$^1A_2 (n \rightarrow 4s)$ &10.40&10.39&10.41&10.41&10.41 &9.95&9.94& &9.96 &9.77&9.77& & & &9.03\\
|
||||
$^3A_2 (n \rightarrow 3s)$ &6.18 &6.18 &6.19 &6.19&6.19 &6.29&6.29&6.30 &6.31 &6.33&6.33&6.35 &\emph{6.02}$^g$ & &\\
|
||||
\hline
|
||||
\multicolumn{16}{c}{Hydrogen chloride}\\
|
||||
& \multicolumn{5}{c}{\AVDZ} & \multicolumn{4}{c}{\AVTZ}& \multicolumn{3}{c}{\AVQZ} & \multicolumn{1}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\CCSDTQP} & {\exCI} & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI}& {\CC{3}} & {\CCSDT} & {\exCI} & Th.$^h$\\
|
||||
\hline
|
||||
$^1 \Pi (\mathrm{CT})$ &7.82&7.81&7.82&7.82 &7.82 &7.84&7.83&7.84 &7.84 &7.89&7.88$^i$ &7.88 &8.23\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${Energy loss experiment from Ref.~\citenum{Ral13};}
|
||||
$^b${MRCI+Q/{\AVTZ} calculations from Ref.~\citenum{Cai00c};}
|
||||
$^c${MRCC/{\AVTZ} calculations from Ref.~\citenum{Li06b};}
|
||||
$^d${Electron impact experiment from Ref.~\citenum{Ske65};}
|
||||
$^e${Electron impact experiment from Ref.~\citenum{Har71};}
|
||||
$^f${EOM-CCSDç($\tilde{T}$)/{\AVTZ} with extra \emph{diffuse} calculations from Ref.~\citenum{Bar97};}
|
||||
$^g${Deduced from the \IneV{$6.38$} value of the $^1A_2 (n \rightarrow 3s)$ state and the \IneV{$-0.36$} shift reported for the 0-0 energies compared
|
||||
to the corresponding singlet state in Ref.~\citenum{Ben91}, a splitting consistent with an earlier estimate of \IneV{$-0.39$} given in Ref.~\citenum{Abu84};}
|
||||
$^h${CC2/cc-pVTZ from Ref.~\citenum{Pea08};}
|
||||
$^i${The {\CCSDTQ}/{\AVQZ} value is \IneV{7.88} as well.}
|
||||
\end{footnotesize}
|
||||
\end{flushleft}
|
||||
\end{sidewaystable}
|
||||
|
||||
Due to its small size and ubiquitous role in life, water is often used as a test case for Rydberg excitations. Indeed, it is part of Head-Gordon's \cite{Rhe07}, Gordon's \cite{Lea12} and Truhlar-Galiardi's \cite{Hoy16} datasets of
|
||||
compounds, and it has been investigated at many levels of theory. \cite{Cai00c,Li06b,Rub08,Pal08} Our results are collected in Table \ref{Table-1}. With the {\AVDZ} basis, there is an nearly perfect agreement between the
|
||||
{\exCI} values and the transition energies obtained with the two largest {\CC{}} expansions, namely {\CCSDTQ} and {\CCSDTQP}. Indeed, the largest discrepancy is as small as \IneV{$0.01$}, and it is therefore reasonable
|
||||
to state that the {\FCI} limit has been reached with that specific basis set. Compared to the {\exCI} results, the {\CCSDT} values are systematically too low, with an average error of \IneV{$-0.03$}. The same trend of underestimation
|
||||
is found with {\CC{3}}, though with smaller absolute deviations for all states. Unsurprisingly, for Rydberg states, increasing the basis set size has a significant impact, and it tends to increase the computed transition energies in
|
||||
water. However, this effect is very similar for all methods listed in Table \ref{Table-1}. This means that, on the one hand, the tendency of {\CCSDT} to provide slightly too small transition energies pertains with both {\AVTZ} and
|
||||
{\AVQZ}, and, on the other hand, that estimating the basis set effect with a ``cheap'' method is possible. Indeed, adding to the {\exCI}/{\AVDZ} energies, the difference between {\CC{3}}/{\AVQZ} and {\CC{3}}/{\AVDZ} results would
|
||||
deliver estimates systematically within \IneV{$0.01$} of the actual {\exCI}/{\AVQZ} values. Such basis set extrapolation approach was already advocated for lower-order {\CC{}} expansions, \cite{Sil10b,Jac15a} and it is therefore
|
||||
not surprising that it can be applied with refined models. As it can be seen in Table S1 in the {\SI}, further extension of the basis set or correlation of the $1s$ electron have small impacts, except for the Rydberg $^1A_1$ state.
|
||||
Eventually, as evidenced by the data from the rightmost columns of Table \ref{Table-1}, the present estimates are in good agreement with previous MRCC values determined on the experimental geometry, \cite{Li06b} whereas
|
||||
the experimental values offer qualitative comparisons only, for reasons discussed elsewhere. \cite{Ral13} We underline that some of the 2013 measurements reported in Table \ref{Table-1} significantly differ from previous electron
|
||||
impact data, \cite{Chu75} that were used previously as reference, \cite{Lea12} with e.g., a \IneV{$0.2$} discrepancy between the two experiments for the lowest triplet state.
|
||||
|
||||
Ammonia is also a prototype molecule for evaluating Rydberg excitations, and it was previously investigated at several levels of theory. \cite{Cha91b,Bar97,Rhe07,Sch17} As in the case of water, we note a nearly perfect match
|
||||
between the {\CCSDTQ} and {\exCI} estimates with both the {\AVDZ} and {\AVTZ} atomic basis sets, indicating that the {\FCI} limit is reached. Both {\CC{3}} and {\CCSDT} are close to this limit, and the former model slightly
|
||||
outperforms the latter. For ammonia, the basis set effects are particularly strong for the third and fourth singlet excited states but these basis set effects are nearly transferrable from one method to another. In fact, as hinted
|
||||
by the large differences between the {\AVTZ} and {\AVQZ} results in Table \ref{Table-2}, these two high-lying states require the use of additional diffuse orbitals to attain convergence. The {\CC{3}}/{\TAVQZ} values of $8.60$ and
|
||||
\IneV{$9.15$} (see Table S1 in the {\SI}), are close from the previous results of Bartlett and coworkers, \cite{Bar97} who also applied extra diffuse orbitals in their calculations relying on approximate triples (see the footnotes in
|
||||
Table \ref{Table-1}). As in water, the experimental values do not provide sufficiently clear-cut results to ultimately decide which method is the most accurate.
|
||||
|
||||
Hydrogen chloride was less frequently used in previous benchmarks, but is included in Tozer's set as an example of charge-transfer (CT) state. \cite{Pea08} Again, the results listed at the bottom of Table \ref{Table-1}
|
||||
demonstrate a remarkable consistency between the various theories. Though large frozen cores are used during the calculations, this does not strongly impact the results, as can be deduced from the data of Table S1.
|
||||
As expected, the absorption band corresponding to this CT state is very broad experimentally (starting at \IneV{$5.5$} and peaking at \IneV{$8.1$}), \cite{Hub79} making direct comparisons tricky.
|
||||
|
||||
\subsection{Dinitrogen and carbon monoxide}
|
||||
|
||||
\begin{sidewaystable}[htp]
|
||||
\caption{\small Vertical transition energies of a selection of excited states of dinitrogen (top) and carbon monoxide (bottom). R indicates Rydberg states. All values are in eV.}
|
||||
\label{Table-2}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|p{.6cm}p{1.1cm}p{1.4cm}p{1.7cm}p{.7cm}|p{.6cm}p{1.1cm}p{1.4cm}p{.7cm}|p{.6cm}p{1.1cm}p{.7cm}|p{.7cm}p{.7cm}p{.7cm}}
|
||||
\hline
|
||||
\multicolumn{16}{c}{Dinitrogen}\\
|
||||
& \multicolumn{5}{c}{\AVDZ} & \multicolumn{4}{c}{\AVTZ}& \multicolumn{3}{c}{\AVQZ} & \multicolumn{3}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\CCSDTQP} & {\exCI} & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI}& {\CC{3}} & {\CCSDT} & {\exCI} & Exp.$^a$ & Exp.$^b$ & Th.$^c$\\
|
||||
\hline %DZ %TZ %QZ %REF
|
||||
$^1\Pi_g (n \rightarrow \pis)$ &9.44 &9.41 & 9.41 &9.41 & 9.41 &9.34 &9.33 &9.32 &9.34 &9.33 &9.31 &9.34 &9.31 &9.31 &9.27 \\
|
||||
$^1\Sigma_u^- (\pi \rightarrow \pis)$ &10.06 &10.06& 10.06&10.05& 10.05 &9.88 &9.89 &9.88 &9.88 &9.87 &9.88 &9.92 &9.92 &9.92 &10.09 \\
|
||||
$^1\Delta_u (\pi \rightarrow \pis)$ &10.43 &10.44& 10.43&10.43& 10.43 &10.29&10.30 & &10.29 &10.27 &10.28 &10.31 &10.27 &10.27 &10.54 \\
|
||||
$^1\Sigma_g^+ (\Ryd)$ &13.23 &13.20& 13.18&13.18& 13.18 &13.01&13.00 & &12.98 &12.90 &12.89 &12.89 & &12.2 &12.20 \\
|
||||
$^1\Pi_u (\Ryd)$ &13.28 &13.17& 13.13&13.13& 13.12 &13.22&13.14 &13.09 &13.03 &13.17 & &13.1$^d$&12.78 &12.90 &12.84 \\
|
||||
$^1\Sigma_u^+ (\Ryd)$ &13.14 &13.13& 13.11&13.11& 13.11 &13.12&13.12 & &13.09 &13.09 &13.09 &13.2$^d$&12.96 &12.98 &12.82 \\
|
||||
$^1\Pi_u (\Ryd)$ &13.64 &13.59& 13.56&13.56& 13.56 &13.49&13.45 &13.42 &13.46 &13.42 &13.37 &13.7$^d$&13.10 &13.24 &13.61 \\
|
||||
$^3\Sigma_u^+ (\pi \rightarrow \pis)$ &7.67 &7.68& 7.69 &7.70 & 7.70 &7.68 &7.69 &7.70 &7.70 &7.71 &7.71 &7.74 &7.75 &7.75 &7.56 \\
|
||||
$^3\Pi_g (n \rightarrow \pis)$ &8.07 &8.06& 8.05 &8.05 & 8.05 &8.04 &8.03 &8.02 &8.01 &8.04 &8.04 &8.03 &8.04 &8.04 &8.05 \\
|
||||
$^3\Delta_u (\pi \rightarrow \pis)$ &8.97 &8.96& 8.96 &8.96 & 8.96 &8.87 &8.87 &8.87 &8.87 &8.87 &8.87 &8.88 &8.88 &8.88 &8.93 \\
|
||||
$^3\Sigma_u^- (\pi \rightarrow \pis)$ &9.78 &9.76& 9.75 &9.75& 9.75 &9.68 &9.68 &9.66 &9.66 &9.68 & &9.66 &9.67 &9.67 &9.86 \\
|
||||
\hline
|
||||
\multicolumn{16}{c}{Carbon monoxide}\\
|
||||
& \multicolumn{5}{c}{\AVDZ} & \multicolumn{4}{c}{\AVTZ}& \multicolumn{3}{c}{\AVQZ} & \multicolumn{3}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\CCSDTQP} & {\exCI} & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI}& {\CC{3}} & {\CCSDT} & {\exCI} & Exp.$^e$ & Th.$^f$ & Th.$^g$\\
|
||||
\hline %DZ %TZ %QZ
|
||||
$^1\Pi (n \rightarrow \pis)$ &8.57 &8.57 &8.56 &8.56& 8.57 &8.49 &8.49 &8.48& 8.49 &8.47 &8.48 & 8.50 &8.51 &8.54 &8.83 \\
|
||||
$^1\Sigma^- (\pi \rightarrow \pis)$ &10.12&10.06&10.06&10.06&10.05 &9.99 &9.94 & 9.93& 9.92 &9.99 &9.94 & 9.99 &9.88 &10.05 &9.97 \\
|
||||
$^1\Delta (\pi \rightarrow \pis)$ &10.23&10.18&10.17&10.17&10.16 &10.12&10.08& & 10.06 &10.12&10.07& 10.11 &10.23 &10.18 &10.00 \\
|
||||
$^1\Sigma^+ (\Ryd)$ &10.92&10.94&10.93&10.92& 10.94 &10.94&10.99& 10.98&10.95 &10.90&10.95& 10.96 &10.78 &10.98 & \\
|
||||
$^1\Sigma^+ (\Ryd)$ &11.48&11.52&11.51 &11.51& 11.52 &11.49&11.54&11.52 & 11.52 &11.46&11.51& 11.53 &11.40 & & \\
|
||||
$^1\Pi (\Ryd)$ &11.74&11.77&11.76 &11.75& 11.76 &11.69&11.74& & 11.72 &11.63&11.69& 11.70 &11.53 & & \\
|
||||
$^3\Pi (n \rightarrow \pis)$ &6.31 &6.30 &6.29 &6.28& 6.29 &6.30 &6.30 &6.28 & 6.28 &6.30 &6.30 & 6.29 &6.32 & &6.41 \\
|
||||
$^3\Sigma^+ (\pi \rightarrow \pis)$ &8.45 &8.43 &8.44 &8.44& 8.46 &8.45 &8.42 & & 8.45 &8.48 &8.45 & 8.49 &8.51 & &8.39 \\
|
||||
$^3\Delta (\pi \rightarrow \pis)$ &9.37 &9.33 &9.34 &9.34& 9.33 &9.30 &9.26 & 9.26& 9.27 &9.31 &9.26 & 9.29 &9.36 & &9.23 \\
|
||||
$^3\Sigma^- (\pi \rightarrow \pis)$ &9.89 & & & & 9.83 &9.82 & & & 9.80 &9.82 & & 9.78 &9.88 & &9.60 \\
|
||||
$^3\Sigma^+ (\Ryd)$ &10.39&10.42&10.42&10.41&10.41 &10.45&10.50& & 10.47 &10.44&10.49& &10.4$^h$& & \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${Experimental vertical values given in Ref.~\citenum{Odd85} and computed from the spectroscopic constants of Ref.~\citenum{Hub79};}
|
||||
$^b${Experimental vertical values given in Ref.~\citenum{Ben90} and computed from the spectroscopic constants of Ref.~\citenum{Hub79};}
|
||||
$^c${MRCCSD/6-311G with one additional $d$ calculations from Ref.~\citenum{Ben90};}
|
||||
$^d${\titou{{\CI} convergence too slow to provide reliable estimates};}
|
||||
$^e${Experimental vertical values given in Ref.~\citenum{Nie80b} and computed from the spectroscopic constants of Ref.~\citenum{Hub79};}
|
||||
$^f${CCSDT/PVTZ+ results from Ref.~\citenum{Kuc01};}
|
||||
$^g${CASSCF(10,10)/cc-pVTZ results from Ref.~\citenum{Dor16};}
|
||||
$^h${Only one digit reported for that state, see Ref.~\citenum{Nie80b}.}
|
||||
\end{footnotesize}
|
||||
\end{flushleft}
|
||||
\end{sidewaystable}
|
||||
|
||||
Dinitrogen is a simple diatomic compound for which the low-lying valence and Rydberg states have been investigated at several levels of theory. \cite{Odd85,Ben90,Kuc01,Pea08} With a numerical solution of the nuclear Schr{\"o}dinger equation,
|
||||
it is possible to treat the experimental spectroscopic constants, \cite{Hub79} so as to obtain reliable vertical estimates, and this procedure was applied previously. \cite{Sta83,Odd85,Ben90} Whilst such approach is supposedly providing experimental
|
||||
vertical excited-state energies with a ca.~\IneV{0.01} error only, it remains that significant excitation energy differences have been reported for the two lowest $^1\Pi_u$ states (see Table \ref{Table-2}). As in the previous cases, we find a remarkable
|
||||
agreement between the {\CCSDTQ} and {\exCI} estimates for most cases in which both could be determined. The only exceptions are the two $^1\Pi_u$ states with the {\AVTZ} basis, but in these two cases, the {\CC{}} expansion is also
|
||||
converging more slowly than usual, which is consistent with the relatively small degree of single excitation character in these two states (82.9 and 87.4\% according to {\CC{3}}). In contrast to water and ammonia, {\CCSDT} outperforms {\CC{3}}
|
||||
with respective mean absolute deviation (MAD) compared to {\exCI} of \IneV{$0.02$} and \IneV{$0.04$}, when using the {\AVDZ} basis set. As it can be deduced from Table S2 in the {\SI}, the basis set corrections are negligible for all valence states,
|
||||
but significant for some of the Rydberg states, especially, $^1\Sigma_g^+$ that requires two sets of diffuse orbitals to be reasonably close from the basis set limit. Applying {\CC{3}}/{\DAVPZ} corrections to the most accurate {\exCI} data, once
|
||||
can determine TBE values (\emph{vide infra}) that deviate only by \IneV{$0.02$} on (absolute) average compared to the experimental estimates for the seven valence states of dinitrogen. Considering the expected inaccuracy of \IneV{$0.01$}
|
||||
of the reference values, chemical accuracy is obviously reached without any experimental input. The deviations are about twice larger for the Rydberg states. Nevertheless, for the two $^1\Pi_u$ states, our TBE values, determined on the basis
|
||||
of {\exCI}/{\AVTZ} are \IneV{$12.73$} and \IneV{$13.27$} (\emph{vide infra}). This indicates that for the lowest $^1\Pi_u$ state the estimate of Ref.~\citenum{Odd85} (\IneV{$12.78$}) is probably more accurate than the one of
|
||||
Ref.~\citenum{Ben90} (\IneV{$12.90$}), whereas the opposite is likely true for the highest $^1\Pi_u$ state that was reported to be located at \IneV{$13.10$} and \IneV{$13.24$} in Refs.~\citenum{Odd85} and \citenum{Ben90}, respectively.
|
||||
One could argue that reaching agreement between CI and CC is particularly challenging for these two states. However, performing the basis set extrapolation starting from the {\CCSDTQP}/{\AVDZ} results would yield similar TBE of \IneV{12.77}
|
||||
and \IneV{$13.22$}.
|
||||
|
||||
For the isoelectronic carbon monoxide, experimental vertical energies deduced from rovibronic data\cite{Hub79} using a numerical approach are available as well. \cite{Nie80b,Pea08} With the {\AVTZ} ({\AVQZ}) atomic basis set, the {\CCSDT}
|
||||
and {\CC{3}} results are within \IneV{$0.02$} (\IneV{$0.03$}) and \IneV{$0.03$} (\IneV{$0.03$}) of the {\exCI} results, whereas the errors made by both {\CCSDTQ} and {\CCSDTQP} are again trifling. As for dinitrogen, all the valence states are
|
||||
rather close from the basis set limit with {\AVTZ}, whereas larger basis sets are required for the Rydberg states (Table S2). By correcting the {\exCI}/{\AVQZ} ({\exCI}/{\AVTZ} for the highest triplet state) data with basis set effects determined
|
||||
at the {\CC{3}}/{\DAVPZ} level, we obtain TBE values that can be compared to the experimental estimates. The computed MAD is \IneV{$0.05$}, the largest deviations being obtained for the $\Delta$ and $\Sigma^-$ excited states of both
|
||||
spin symmetries. The agreement between theory and experiment is therefore very satisfying though slightly less impressive than for \ce{N2}. We note that the {\CC{3}}/{\AVTZ} \ce{C=O} bond length (\InAA{$1.134$}) is \InAA{0.006} larger than the
|
||||
experimental $r_e$ value of \InAA{1.128},\cite{Hub79} whereas the discrepancy is twice smaller for dinitrogen: \InAA{$1.101$} for {\CC{3}}/{\AVTZ} compared to \InAA{$1.098$} experimentally. This might partially explained the larger deviations
|
||||
noticed for carbon monoxide.
|
||||
|
||||
\subsection{Acetylene and ethylene}
|
||||
|
||||
Acetylene is the smallest conjugated organic molecule possessing stable low-lying excited-state structures, therefore allowing to investigate vertical fluorescence. This molecule has been the subject of previous investigations at the
|
||||
{\CASPT},\cite{Mal98} {\CCSD},\cite{Zyu03} and MR-AQCC\cite{Ven03} levels. Our results are collected in Table \ref{Table-3}. With the double-$\zeta$ basis set, the differences between the {\CC{3}}, {\CCSDT}, and {\CCSDTQ} results
|
||||
are negligible, and the latter estimates are also systematically within \IneV{$0.02$} of the {\exCI} results. In contrast to water and ammonia, both {\CC{3}} and {\CCSDT} provide similar accuracies compared to higher levels of theory.
|
||||
As expected, for valence states, going from double- to triple-$\zeta$ basis set tends to slightly decrease the computed energies (except for the lowest triplet). Nonetheless, as with the smaller basis set, the same near-perfect
|
||||
methodological match pertains with {\AVTZ}. Estimating the {\exCI}/{\AVTZ} results from the {\exCI}/{\AVDZ} values and {\CC{3}} basis set effects would yield estimates with absolute errors of \IneV{$0.00--0.02$}. One also notice that
|
||||
the {\exCI}/{\AVTZ} values are all extremely close to the previous MR-AQCC estimates, whereas the published {\CASPT} values appear to be too low. This underestimating trend of standard {\CASPT} was reported before for other
|
||||
molecules.\cite{Ang05b,Sen11} Because using a larger basis set than {\AVTZ} has an almost negligible impact on all states (see Table S3), we claim that our theoretical vertical energy estimates are probably more trustworthy for further
|
||||
benchmarks than the available experimental values.
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical (absorption) transition energies for the five lowest low-lying valence excited states of acetylene (top) and the three lowest singlet and triplet
|
||||
excited states of ethylene (bottom). For acetylene, we also compare the vertical emission (denoted [F]) obtained from the lowest \emph{trans} and \emph{cis}
|
||||
isomers. All values are in eV.}
|
||||
\label{Table-3}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|cccc|ccc|ccc}
|
||||
\hline
|
||||
\multicolumn{11}{c}{Acetylene}\\
|
||||
& \multicolumn{4}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI} & {\CC{3}} & {\CCSDT} & {\exCI}& Exp.$^a$ & Th.$^b$ & Th.$^c$ \\
|
||||
\hline
|
||||
$^1\Sigma_u^- (\pi \rightarrow \pis)$ &7.21&7.21&7.21&7.20 &7.09&7.09&7.10 &7.1&6.96&7.10\\
|
||||
$^1\Delta_u (\pi \rightarrow \pis)$ &7.51&7.52&7.52&7.51 &7.42&7.43&7.44 &7.2&7.30&7.43\\
|
||||
$^3\Sigma_u^+ (\pi \rightarrow \pis)$ &5.48&5.49&5.50&5.50 &5.50&5.51&5.53 &5.2&5.26&5.58\\
|
||||
$^3\Delta_u (\pi \rightarrow \pis)$ &6.46&6.46&6.46&6.46 &6.40&6.39&6.40 &6.0&6.20&6.41\\
|
||||
$^3\Sigma_u^- (\pi \rightarrow \pis)$ &7.13&7.14&7.14&7.14 &7.07& &7.08 &7.1&6.90&7.05\\
|
||||
$^1A_u [\mathrm{F}] (\pi \rightarrow \pis)$ &3.70&3.72&3.70&3.71 &3.64&3.66&3.64 &&\\
|
||||
$^1A_2 [\mathrm{F}] (\pi \rightarrow \pis)$ &3.92&3.94&3.93&3.93 &3.84&3.86&3.85 &&\\
|
||||
\hline
|
||||
\multicolumn{11}{c}{Ethylene}\\
|
||||
& \multicolumn{4}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI} & {\CC{3}} & {\CCSDT} & {\exCI}& Exp.$^d$ & Th.$^e$ \\
|
||||
\hline
|
||||
$^1B_{3u} (\pi \rightarrow 3s)$ &7.29&7.29&7.30&7.31 &7.35&7.37&7.40 &7.11 &7.45\\
|
||||
$^1B_{1u} (\pi \rightarrow \pis)$ &7.94&7.94&7.93&7.93 &7.91&7.92&$^f$ &7.60 &8.00\\
|
||||
$^1B_{1g} (\pi \rightarrow 3p)$ &7.97&7.98&7.99&8.00 &8.03&8.04&8.07 &7.80 &8.06\\
|
||||
$^3B_{1u} (\pi \rightarrow \pis)$ &4.53&4.54&4.54&4.55 &4.53&4.53&4.54 &4.36 &4.55\\
|
||||
$^3B_{3u} (\pi \rightarrow 3s)$ &7.17&7.18&7.18&7.16 &7.24&7.25&$^f$ &6.98 &7.29\\
|
||||
$^3B_{1g} (\pi \rightarrow 3p)$ &7.93&7.94&7.94&7.93 &7.98&7.99&$^f$ &7.79 &8.02\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${Electron impact experiment from Ref.~\citenum{Dre87}. Note that the \IneV{7.1} value for the $\Sigma_u^-$ singlet and triplet states should be viewed as a tentative assignment;}
|
||||
$^b${LS-CASPT2/\emph{aug}-ANO calculations from Ref.~\citenum{Mal98};}
|
||||
$^c${MR-AQCC/{extrap.}~calculations from Ref.~\citenum{Ven03};}
|
||||
$^d${Experimental values collected from various sources from Ref.~\citenum{Rob85b} (see discussions in Refs.~\citenum{Ser93,Sch08} and \citenum{Fel14});}
|
||||
$^e${Best composite theory from Ref.~\citenum{Fel14}, close to {\FCI};}
|
||||
$^f${{\CI} convergence too slow to provide reliable estimates.}
|
||||
\end{footnotesize}
|
||||
\end{flushleft}
|
||||
\end{table}
|
||||
|
||||
Despite its small size, ethylene remains a challenging molecule and is included in many benchmark sets. \cite{Sch08,She09b,Car10,Lea12,Hoy16} The assignments of the experimental data has been the subject of countless works,
|
||||
and we refer the interested readers to the discussions in Refs.~\citenum{Rob85b,Ser93,Sch08,Ang08,Fel14,Chie18}. On the theoretical side, the most complete and accurate investigation dedicated to the excited states of ethylene
|
||||
is due to Davidson's group who performed refined {\CI} calculations. \cite{Fel14} They indeed obtained highly-accurate transition energies for ethylene, including for the valence yet challenging $^1B_{1u}$ state. From our data, collected
|
||||
in Table \ref{Table-3}, one notices that the differences between {\exCI}/{\AVDZ} and {\CCSDTQ}/{\AVDZ} results are again trifling, the largest deviation being obtained for the $^3B_{3u} (\pi \rightarrow 3s)$ Rydberg state ($\Delta = 0.02$ eV).
|
||||
In addition, given the nice agreement between {\CC{3}}, {\CCSDT} and {\exCI} values, one can directly compare our {\CC{3}}/{\AVPZ} results (Table S3) to the values of reported in Ref.~\citenum{Fel14}: a mean absolute
|
||||
deviation (MAD) of \IneV{$0.03$} is obtained. The fact that our transition energies tend to be slightly smaller than Davidson's is likely due to geometrical effects. Indeed, our {\CC{3}}/{\AVTZ} \ce{C=C} distance is \InAA{1.3338}, i.e.,
|
||||
slightly longer than the best estimate provided in Davidson's work (\InAA{1.3305}). Recently, a stochastic heat-bath {\CI} (SHCI)/ANO-L-pVTZ work reported \IneV{$4.59$} and \IneV{$8.05$} values for the $^3B_{1u}$ and $^1B_{1u}$ states,
|
||||
respectively, \cite{Chie18} and we also ascribe the differences with our results to the use of a MP2 geometry in Ref.~\citenum{Chie18}. Interestingly, these authors found quite large discrepancies between their SHCI and their {\CC{}} results.
|
||||
Indeed, they reported CR-EOMCC(2,3)D estimates significantly larger than their SHCI results with \IneV{$+0.17$} and \IneV{$+0.20$} upshifts for the triplet and singlet states, respectively.
|
||||
This highlights that only high-level {\CC{}} schemes are able to recover the {\exCI} (or SHCI) results for ethylene.
|
||||
|
||||
\subsection{Formaldehyde, methanimine and thioformaldehyde}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical (absorption) transition energies for various excited states of formaldehyde (top), methanimine (center), and thioformaldehyde (bottom).
|
||||
All values are in eV.}
|
||||
\label{Table-4}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|cccc|ccc|ccc}
|
||||
\hline
|
||||
\multicolumn{11}{c}{Formaldehyde}\\
|
||||
& \multicolumn{4}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI} & {\CC{3}} & {\CCSDT} & {\exCI}& Exp.$^a$ & Th.$^b$ & Th.$^c$ \\
|
||||
\hline
|
||||
$^1A_2 (n \rightarrow \pis)$ &4.00&3.99&4.00&3.99 &3.97&3.95&3.98 &4.07 &3.98 &3.88 \\
|
||||
$^1B_2 (n \rightarrow 3s)$ &7.05&7.04&7.09&7.11 &7.18&7.16&7.23 &7.11 &7.12 &\\
|
||||
$^1B_2 (n \rightarrow 3p)$ &8.02&8.00&8.04&8.04 &8.07&8.07&8.13 &7.97 &7.94 &8.11\\
|
||||
$^1A_1 (n \rightarrow 3p)$ &8.08&8.07&8.12&8.12 &8.18&8.16&8.23 &8.14 &8.16 &\\
|
||||
$^1A_2 (n \rightarrow 3p)$ &8.65&8.63&8.68&8.65 &8.64&8.61&8.67 &8.37 &8.38 &\\
|
||||
$^1B_1 (\sigma \rightarrow \pis)$ &9.31&9.29&9.30&9.29 &9.19&9.17&9.22 & &9.32 &9.04\\
|
||||
$^1A_1 (\pi \rightarrow \pis)$ &9.59&9.59&9.54&9.53 &9.48&9.49&9.43 & &9.83 &9.29\\
|
||||
$^3A_2 (n \rightarrow \pis)$ &3.58&3.57&3.58&3.58 &3.57&3.56&3.58 &3.50 & &3.50\\
|
||||
$^3A_1 (\pi \rightarrow \pis)$ &6.09&6.08&6.09&6.10 &6.05&6.05&6.06 &5.86 & &5.87\\
|
||||
$^3B_2 (n \rightarrow 3s)$ &6.91&6.90&6.95&6.95 &7.03&7.02&7.06 &6.83 & &\\
|
||||
$^3B_2 (n \rightarrow 3p)$ &7.84&7.82&7.86&7.87 &7.92&7.90&7.94 &7.79 & &\\
|
||||
$^3A_1 (n \rightarrow 3p)$ &7.97&7.95&8.00&8.01 &8.08&8.06&8.10 &7.96 & &\\
|
||||
$^3B_1 (n \rightarrow 3d)$ &8.48&8.47&8.48&8.48 &8.41&8.40&8.42 & & &\\
|
||||
$^1A'' [\mathrm{F}] (n \rightarrow \pis)$ &2.87&2.84&2.86&2.86 &2.84&2.82&2.80 & & &\\
|
||||
\hline
|
||||
\multicolumn{11}{c}{Methanimine}\\
|
||||
& \multicolumn{4}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI} & {\CC{3}} & {\CCSDT} & {\exCI}& Th.$^d$ \\
|
||||
\hline
|
||||
$^1A''(n \rightarrow \pis)$ &5.26&5.24& &5.25 &5.20&5.19&5.23 & 5.32\\
|
||||
$^3A'' (n \rightarrow \pis)$ &4.63&4.63& &4.63 &4.61&4.61&4.65 & \\
|
||||
\hline
|
||||
\multicolumn{11}{c}{Thioformaldehyde}\\
|
||||
& \multicolumn{4}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||||
State & {\CC{3}} & {\CCSDT} & {\CCSDTQ} & {\exCI} & {\CC{3}} & {\CCSDT} & {\exCI}& Exp.$^a$ & Exp.$^e$ \\
|
||||
\hline
|
||||
$^1A_2 (n \rightarrow \pis)$ &2.27&2.25&2.26&2.26 &2.23&2.21&2.22 & &2.03 &\\
|
||||
$^1B_2 (n \rightarrow 4s)$ &5.80&5.80&5.82&5.83 &5.91&5.89&5.96 &5.85 &5.84 &\\
|
||||
$^1A_1 (\pi \rightarrow \pis)$ &6.62&6.60&6.51&6.5$^f$ &6.48&6.47&6.4$^f$&6.2 &5.54 &\\
|
||||
$^3A_2 (n \rightarrow \pis)$ &1.97&1.96&1.96&1.97 &1.94&1.93&1.94 & &1.80 &\\
|
||||
$^3A_1 (\pi \rightarrow \pis)$ &3.43&3.43&3.44&3.45 &3.38&3.38&3.43 &3.28 & &\\
|
||||
$^3B_2 (n \rightarrow 4s)$ &5.64&5.63&5.65&5.66 &5.72&5.71&5.6$^f$ & & &\\
|
||||
$^1A_2 [\mathrm{F}] (n \rightarrow \pis)$ &2.00&2.00&1.98&1.98 &1.97&1.98&1.95 & & &\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${Various experimental sources, summarized in Ref.~\citenum{Rob85b};}
|
||||
$^b${MR-AQCC-LRT calculations from Ref.~\citenum{Mul01};}
|
||||
$^c${{\CC{3}}/{\AVQZ} calculations from Ref.~\citenum{Sch08};}
|
||||
$^d${DMC results form Ref.~\citenum{Sch04e};}
|
||||
$^e${0-0 energies collected in Ref.~\citenum{Pao84};}
|
||||
$^f${{\CI} convergence too slow to provide reliable estimates.}
|
||||
\end{footnotesize}
|
||||
\end{flushleft}
|
||||
\end{table}
|
||||
|
||||
Similarly to ethylene, formaldehyde is a very popular test molecule, \cite{For92b,Had93,Gwa95,Wib98,Wib02,Pea08,Sch08,She09b,Car10,Li11,Lea12,Hoy16} and stands as the prototype carbonyl dye with a low-lying
|
||||
$n \rightarrow \pis$ transition. Nevertheless, even for this particular valence state, well-separated from higher-lying excited states, the choice of an experimental reference remains difficult. Indeed, values of \IneV{$3.94$},\cite{Pea08}
|
||||
\IneV{$4.00$}, \cite{Had93,Car10,Hoy16} \IneV{$4.07$}, \cite{Gwa95,Lea12} and \IneV{$4.1$}, \cite{For92b,Wib98} have been used in previous theoretical benchmarks. In contrast to their oxygen cousin, both methanimine and
|
||||
thioformaldehyde were the subject of much less attention by the theoretical community. The results obtained for these three molecules are collected in Table \ref{Table-4}. Considering all transitions listed in this Table, one obtains
|
||||
a MAD of \IneV{$0.01$} between the {\CCSDTQ}/{\AVDZ} and {\exCI}/{\AVDZ} results, the largest discrepancies of \IneV{$0.03$} being observed for two states for which the difference between {\CCSDT} and {\CCSDTQ} is also large
|
||||
(\IneV{$0.05$}). As in water, using the {\exCI}/{\AVDZ} values as reference, we found that {\CC{3}} delivers slightly more accurate transition energies (MAD of \IneV{$0.02$}, maximal deviation of \IneV{0.06}) than {\CCSDT}
|
||||
(MAD of \IneV{0.03}, maximal deviation of \IneV{0.07}). By adding the difference between {\CC{3}}/{\AVTZ} and {\CC{3}}/{\AVDZ} results to the {\exCI}/{\AVDZ} values, \titou{we} obtain good estimates of the actual {\exCI}/{\AVTZ}
|
||||
data, with a MAD of \IneV{$0.02$} for formaldehyde. Compared to the {\CC{3}}/{\AVQZ} results of Thiel, \cite{Sch08} the transition energies reported in Table \ref{Table-4} are slightly larger, which is probably due to the influence
|
||||
of the ground-state geometry rather than basis set effects (see Table S4). Indeed, the carbonyl bond is significantly more contracted with {\CC{3}}/{\AVTZ} (\InAA{1.208}) than with MP2/6-31G(d) (\InAA{1.221}). In particular, for
|
||||
the hallmark $n \rightarrow \pis$, our best estimate is {\IneV{$3.97$} (\emph{vide infra}), nicely matching a previous MR-AQCC value of \IneV{$3.98$}, \cite{Mul01} but significantly below the previous DMC/BLYP estimate of
|
||||
\IneV{$4.24$}. \cite{Sch04e} The latter discrepancy is probably due to the use of both different structures and pseudo-potentials within DMC calculations.
|
||||
|
||||
For methanimine and thioformaldehyde, the basis set effects are rather small for the states considered here (see Table S4) and the data reported in the present work are probably the most accurate vertical transition energies reported to date.
|
||||
For the latter molecule, these vertical estimates are systematically larger than the known experimental 0-0 energies, \cite{Pao84} which is the expected trend.
|
||||
|
||||
\subsection{Larger compounds}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small Vertical (absorption) transition energies for various excited states of diazomethane (top) and ketene (bottom). All values are in eV.}
|
||||
\label{Table-5}
|
||||
\begin{footnotesize}
|
||||
\begin{tabular}{ll|ccc|ccc|cc}
|
||||
\hline
|
||||
& & \multicolumn{3}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{2}{c}{Litt.}\\
|
||||
Molecule &State & {\CC{3}} & {\CCSDT} & {\exCI} & {\CC{3}} & {\CCSDT} & {\exCI}& Exp. & Theo. \\
|
||||
\hline
|
||||
Acetaldehyde &$^1A'' (n \rightarrow \pis)$ &4.34&4.32&4.34& 4.31&4.29&4.31 & 4.27$^a$ &4.29$^b$ \\
|
||||
&$^3A'' (n \rightarrow \pis)$ &3.96&3.95&3.98& 3.95& &4.0$^c$ & 3.97$^a$ &3.97$^b$ \\
|
||||
\hline
|
||||
Cyclopropene &$^1B_1 (\sigma \rightarrow \pis)$ &6.72&6.71&6.7$^c$&6.68&6.68&6.6$^c$& 6.45$^d$ & 6.89$^e$ \\
|
||||
&$^1B_2 (\pi \rightarrow \pis)$ &6.77&6.78&6.82& 6.73&6.75&6.7$^c$ & 7.00$^f$ & 7.11$^e$ \\
|
||||
&$^3B_2 (\pi \rightarrow \pis)$ &4.34&4.35&4.35& 4.34& &4.38 & 4.16$^f$ & 4.28$^g$ \\
|
||||
&$^3B_1 (\sigma \rightarrow \pis)$ &6.43&6.43&6.43& 6.40& &6.45 & & 6.40$^g$ \\
|
||||
\hline%
|
||||
Diazomethane &$^1A_2 (\pi \rightarrow \pis)$ &3.10&3.10&3.09& 3.07&3.07&3.14 & 3.14$^h$ &3.21$^i$ \\
|
||||
&$^1B_1 (\pi \rightarrow 3s)$ &5.32&5.35&5.35& 5.45&5.48&5.54 & &5.33$^i$ \\
|
||||
&$^1A_1 (\pi \rightarrow \pis)$ &5.80&5.82&5.79& 5.84&5.86&5.90 & 5.9$^h$ &5.85$^i$ \\
|
||||
&$^3A_2 (\pi \rightarrow \pis)$ &2.84&2.84&2.81& 2.83&2.82&2.8$^c$ & &2.92$^j$ \\
|
||||
&$^3A_1 (\pi \rightarrow \pis)$ &4.05&4.04&4.03& 4.03&4.02&4.05 & &3.97$^j$ \\
|
||||
&$^3B_1 (\pi \rightarrow 3s)$ &5.17&5.20&5.18& 5.31&5.34&5.35 & & \\
|
||||
&$^3A_1 (\pi \rightarrow 3p)$ &6.83&6.83&6.81& 6.80& &6.82 & &7.02$^j$ \\
|
||||
&$^1A'' [\mathrm{F}] (\pi \rightarrow \pis)$ &0.68&0.67&0.65& 0.68&0.67&0.71 & & \\
|
||||
\hline
|
||||
Formamide &$^1A'' (n \rightarrow \pis)$ &5.71& &5.77 & 5.66 & & &5.8$^k$ &5.63$^l$ \\
|
||||
&$^1A' (n \rightarrow 3s)$ &6.65&6.64& & 6.74 & & &6.35$^k$ &6.62$^l$ \\
|
||||
&$^1A' (\pi \rightarrow \pis)$$^m$ &7.63&7.62&7.66 & 7.62 & & &7.37$^k$ &7.22$^l$ \\
|
||||
&$^1A' (n \rightarrow 3p)$$^m$ &7.31&7.29& & 7.40 & & &7.73$^k$ &7.66$^l$ \\
|
||||
&$^3A'' (n \rightarrow \pis)$ &5.42& &5.42 & 5.38 & & &5.2$^k$ &5.34$^l$ \\
|
||||
&$^3A' (\pi \rightarrow \pis)$ &5.83&5.81&5.82 & 5.82 & & &$\sim$6$^k$ &5.74$^l$ \\
|
||||
\hline%
|
||||
Ketene &$^1A_2 (\pi \rightarrow \pis)$ &3.89&3.88&3.84& 3.88&3.87&3.86 &3.7$^n$ &3.74$^o$ \\
|
||||
&$^1B_1 (n \rightarrow 3s)$ &5.83&5.86&5.88& 5.96&5.99&6.01 &5.86$^n$&5.82$^o$ \\
|
||||
&$^1A_2 (\pi \rightarrow 3p)$ &7.05&7.09&7.08& 7.16&7.20&7.18 & &7.00$^o$ \\
|
||||
&$^3A_2 (n \rightarrow \pis)$ &3.79&3.78&3.79& 3.78&3.78&3.77 &3.8$^p$ &3.62$^q$\\
|
||||
&$^3A_1 (\pi \rightarrow \pis)$ &5.62&5.61&5.64& 5.61&5.60&5.61 &5$^p$ &5.42$^q$\\
|
||||
&$^3B_1 (n \rightarrow 3s)$ &5.63&5.66&5.68& 5.76&5.80&5.79 &5.8$^p$ &5.69$^q$\\
|
||||
&$^3A_2 (\pi \rightarrow 3p)$ &7.01&7.05&7.07& 7.12&7.17&7.12 & & \\
|
||||
&$^1A''[\mathrm{F}] (\pi \rightarrow \pis)$ &1.00&0.99&0.96& 1.00&1.00&1.00 & & \\
|
||||
\hline
|
||||
Nitrosomethane&$^1A'' (n \rightarrow \pis)$ &2.00&1.98&1.99& 1.96&1.95&2.0$^c$ &1.83$^r$&1.76$^s$\\
|
||||
&$^1A' (n,n \rightarrow \pis,\pis)$ &5.75&5.26&4.81& 5.76&5.29 &4.72 & &4.96$^s$\\
|
||||
&$^1A' (n \rightarrow 3s/3p)$ &6.20&6.19&6.29& 6.31&6.30&6.37 & &6.54$^s$\\
|
||||
&$^3A'' (n \rightarrow \pis)$ &1.13&1.12&1.15& 1.14&1.13&1.16 & &1.42$^t$\\
|
||||
&$^3A' (\pi \rightarrow \pis)$ &5.54&5.54&5.56& 5.51& &5.60 & &5.55$^t$\\
|
||||
&$^1A'' [\mathrm{F}] (n \rightarrow \pis)$ &1.70&1.69&1.70& 1.69& & & & \\
|
||||
\hline
|
||||
Streptocyanine-C1&$^1B_2 (\pi \rightarrow \pis)$ &7.14&7.12&7.14& 7.13&7.11&7.1$^c$ & &7.16$^u$\\
|
||||
& $^3B_2 (\pi \rightarrow \pis)$ &5.48&5.47&5.47& 5.48&5.47&5.52 & & \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{footnotesize}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${Electron impact experiment from Ref.~\citenum{Wal87};}
|
||||
$^b${NEVPT-PC from Ref.~\citenum{Ang05b};}
|
||||
$^c${{\CI} convergence too slow to provide reliable estimates;}
|
||||
$^d${Maximum in the gas UV from Ref.~\citenum{Rob69};}
|
||||
$^e${CCSDT/TZVP from Ref.~\citenum{Kan14}; }
|
||||
$^f${Electron impact experiment from Ref.~\citenum{Sau76};}
|
||||
$^g${CC3/{\AVTZ} from Ref.~\citenum{Sil10c}; }
|
||||
$^h${VUV maxima from Ref.~\citenum{McG71};}
|
||||
$^i${{\CCSD}/6-311(3+,+)G(d) calculations from Ref.~\citenum{Fed07};}
|
||||
$^j${MR-CC/DZP calculations from Ref.~\citenum{Rit89};}
|
||||
$^k${EELS (singlet) and trapped electron (triplet) experiments from Ref.~\citenum{Gin97};}
|
||||
$^l${$n$R-SI-CCSD(T) results from Ref.~\citenum{Li11}; }
|
||||
$^m${Strong state mixing;}
|
||||
$^n${Electron impact experiment from Ref.~\citenum{Fru76};}
|
||||
$^o${CASPT2/6-311+G(d) results from Ref.~\citenum{Xia13};}
|
||||
$^p${Electron impact experiment from Ref.~\citenum{Rob85b};}
|
||||
$^q${STEOM-CCSD/Sad+//CCSD/Sad+ results from Ref.~\citenum{Noo03}.}
|
||||
$^r${Maximum in the gas UV from Ref.~\citenum{Dix65};}
|
||||
$^s${CASPT2/ANO results from Ref.~\citenum{Are06};}
|
||||
$^t${CASSCF/cc-pVDZ results from Ref.~\citenum{Dol04};}
|
||||
$^u${exCC3//MP2 result from Ref.~\citenum{Sen11}.}
|
||||
\end{footnotesize}
|
||||
\end{flushleft}
|
||||
\end{table}
|
||||
|
||||
Let us now turn our attention to molecules that encompass three heavy (non-hydrogen) atoms. We have treated seven molecules of that family, and all were previously investigated at several levels of theory: acetaldehyde,
|
||||
\cite{Had93,Gwa95,Wib98,Ang05b,Rei09,Car10,Hoy16,Jac17b} cyclopropene, \cite{Sch08,She09b,Sil10b,Sil10c,Coe13,Kan14} diazomethane, \cite{Rit89,Hab95,Fed07,Rei09} formamide, \cite{Ser96,Bes99,Sch08,Sil10b,Sil10c,Kan14}
|
||||
ketene, \cite{Rit89,Sza96b,Noo03,Xia13} nitrosomethane, \cite{Lac00,Dol04,Dol04b,Are06} and the shortest streptocyanine.\cite{Sen11,Bar13,Bou14,Zhe14,Leg15} The results are collected in Table \ref{Table-5}.
|
||||
|
||||
Experimentally, the lowest singlet and triplet $n \rightarrow \pis$ transitions of acetaldehyde are located \IneV{$0.3$--$0.4$} above their formaldehyde counterparts,\cite{Rob85b,Wal87} and this trend is accurately reproduced
|
||||
by theory, which also delivers estimates very close to the NEVPT2 values given in Ref.~\citenum{Ang05b}.
|
||||
|
||||
For cyclopropene, the lowest singlet $\sigma \rightarrow \pis$ and $\pi \rightarrow \pis$ are close from one another, and both {\CCSDT} and {\exCI} predict the former to be slightly more stabilized, which is consistent with the
|
||||
large basis set {\CC{3}} results obtained previously by Thiel. \cite{Sil10c}
|
||||
|
||||
For the isoelectronic diazomethane and ketene molecules, one notes, yet again, consistent results in Table \ref{Table-5} with, however, differences between the {\exCI}/{\AVTZ} and {\CCSDT}/{\AVTZ} results larger than \IneV{$0.05$}
|
||||
for the two lowest singlet states of diazomethane. There is also a reasonable match between our data and previous theoretical results reported for these two molecules. \cite{Rit89,Noo03,Fed07,Xia13} The basis set effects are
|
||||
significant for the Rydberg transitions, especially for the $\pi \rightarrow 3s$ states of diazomethane (Table S5).
|
||||
|
||||
In formamide, we found strong state mixing between the lowest singlet valence and Rydberg states of $A'$ symmetry. This is consistent with the {\CCSDT}/TZVP analysis of Kannar and Szalay, \cite{Kan14} who reported, for example,
|
||||
a larger oscillator strength for the lowest Rydberg state than for the $\pi \rightarrow \pis$ transition. This state-mixing problem pertains with {\AVTZ}, making unambiguous assignments impossible. We have decided to classify the three
|
||||
lowest $^1A'$ transitions according to their dominant orbital character, which gives a picture consistent with the computed oscillator strengths (\emph{vide infra}) but yields state inversions compared to Thiel's and Szalay's assignments.
|
||||
\cite{Sil10b,Kan14} Despite these uncertainties, we obtained transition energies for the Rydberg states that are much closer from experiment \cite{Gin97} as well as from previous multireference {\CC{}} estimates, \cite{Li11} than
|
||||
the TZVP ones. \cite{Kan14}
|
||||
|
||||
Nitrosomethane is an interesting test molecule for three reasons: i) it presents very low-lying $n \rightarrow \pis$ states of $A''$ symmetry, close to ca.~\IneV{$2.0$} (singlet) and \IneV{$1.2$} (triplet), amongst the smallest absorption
|
||||
energies found in a compact molecule; \cite{Tar54} ii) it changes from an eclipsed to a staggered conformation of the methyl group when going from the ground to the lowest singlet state; \cite{Ern78,Gor79b,Dol04} iii) the lowest-lying
|
||||
singlet $A'$ state corresponds to an almost pure double excitation of $(n,n) \rightarrow (\pis,\pis)$ nature. \cite{Are06} Indeed, {\CC{3}} returns a $2.5$\%\ single excitation character only for this second transition, to be compared to
|
||||
more than $80$\%\ (and generally more than $90$\%) in all other states treated in this work (\emph{vide infra}). For example, the notoriously difficult $A_g$ dark state of butadiene has a $72.8$\%\ single character. \cite{Sch08}
|
||||
For the $A''$ state of nitrosomethane, {\CC{3}}, {\CCSDT} and {\exCI} yield similar results, and the corresponding transition energies are slightly larger than previous {\CASPT} estimates. \cite{Are06} In contrast, the {\CC{}} approaches
|
||||
are expectedly far from the spot for the $(n,n) \rightarrow (\pis,\pis)$ transition: they yield values significantly blue shifted and large discrepancies between the {\CC{3}} and {\CCSDT} values are found. For this particular state, it is not surprising
|
||||
that the {\exCI} results is indeed closer to the {\CASPT} value, \cite{Are06} as modeling double excitations with single-reference {\CC{}} models is certainly not the most effective choice.
|
||||
|
||||
Finally for the shortest model cyanine, a molecule known to be difficult to treat with {\TDDFT}, \cite{Leg15} all the theoretical results given in Table \ref{Table-5} closely match each other for both the singlet and triplet manifolds.
|
||||
For the former, the reported {\CASPT} (with IPEA) value of \IneV{$7.14$} also fits these estimates. \cite{Sen11}
|
||||
|
||||
\subsection{Theoretical best estimates}
|
||||
|
||||
We now turn to the definitions of theoretical best estimates. We decided to provide two sets for these estimates, one obtained in the frozen-core approximation with the {\AVTZ} atomic basis set, and
|
||||
one including further corrections for basis set and ``all electron'' (full) effects. This choice allows further benchmarks to either consider a reasonably compact basis set, therefore allowing to test many levels
|
||||
of theory, or to rely on values closer to the basis set limit. For the basis set corrections (see the {\SI} for complete data), we systematically applied the {\CC{3}} level of theory and used {\DAVPZ} for the five smallest
|
||||
molecules and slightly more compact basis sets for the larger compounds. At least for Rydberg states, the use of {\DAVQZ} apparently delivers results closer to basis set convergence than {\AVPZ}, and the
|
||||
former basis set was used when technically possible. The results are listed in Table \ref{Table-6} and provide a total of 106 transition energies. This set of states is rather diverse with 61 singlet and 45 triplet states,
|
||||
62 valence and 43 Rydberg states, 21 $n \rightarrow \pis$ and 38 $\pi \rightarrow \pis$states, with an energetic span from $0.70$ to \IneV{$13.27$}. Amongst these 106 excitation energies, only 13 are characterized
|
||||
by a single-excitation character smaller than 90\%\ according to {\CC{3}}. As expected,\cite{Sch08} the dominant single-excitation character is particularly pronounced for triplet excited states. Therefore, this set
|
||||
is adequate for evaluating single-reference methods, though a few challenging cases are incorporated. Consequently, we think that the TBE listed in Table \ref{Table-6} contribute to fulfill the need of more
|
||||
accurate reference excited state energies, as pointed out by Thiel one decade ago. \cite{Sch08} However, the focus on small compounds and the lack of charge-transfer states constitute significant biases
|
||||
in that set of transition energies.
|
||||
|
||||
\bigskip
|
||||
|
||||
\renewcommand*{\arraystretch}{.55}
|
||||
\LTcapwidth=\textwidth
|
||||
|
||||
\begin{footnotesize}
|
||||
\begin{longtable}{llcccccc}
|
||||
\caption{\small TBE (in eV) for various states and wave function approaches. For each state, we provide the oscillator strength and percentage of single excitations obtained at the \CC{3}(FC)/{\AVTZ} level.
|
||||
Unless otherwise stated, the TBE(FC)/{\AVTZ} have been obtained directly from {\exCI}. For the basis-set-corrected TBE, we provide the method used to determine the starting value and the basis set
|
||||
used at \CC{3}(full) level to correct that starting value. \CC{3}(full)/{\AVTZ} geometries and abbreviated forms of Dunning's basis set are systematically used.} \label{Table-6}\\
|
||||
\hline
|
||||
& & & & TBE(FC)& \multicolumn{3}{c}{Corrected TBE} \\
|
||||
& State & $f$ & \%$T_1$ & AVTZ & Method & Corr. & Value \\
|
||||
\hline
|
||||
\endfirsthead
|
||||
\hline
|
||||
& & & & TBE(FC)& \multicolumn{3}{c}{Corrected TBE} \\
|
||||
& State & $f$ & \%$T_1$ & AVTZ & Method & Corr. & Value \\
|
||||
\hline
|
||||
\endhead
|
||||
\hline \multicolumn{7}{r}{{Continued on next page}} \\
|
||||
\endfoot
|
||||
\hline
|
||||
\endlastfoot
|
||||
Acetaldehyde &$^1A''(\mathrm{V};n \rightarrow \pis)$ & 0.000 &91.3& 4.31 & {\exCI}/AVTZ & AVQZ & \\
|
||||
&$^3A''(\mathrm{V};n \rightarrow \pis)$ & &97.9& 3.97$^a$ & {\exCI}/AVDZ & AVQZ & \\
|
||||
Acetylene &$^1\Sigma_u^- (\mathrm{V};\pi \rightarrow \pis)$ & &96.5& 7.10 & {\exCI}/AVTZ & dAVQZ &7.10 \\
|
||||
&$^1\Delta_u (\mathrm{V};\pi \rightarrow \pis)$ & &93.3& 7.44 & & &7.44 \\
|
||||
&$^3\Sigma_u^+ (\mathrm{V};\pi \rightarrow \pis)$ & &99.2& 5.53 & & &5.54 \\
|
||||
&$^3\Delta_u (\mathrm{V};\pi \rightarrow \pis)$ & &99.0& 6.40 & & &6.40 \\
|
||||
&$^3\Sigma_u^- (\mathrm{V};\pi \rightarrow \pis)$ & &98.8& 7.08 & & &7.08 \\
|
||||
&$^1A_u [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pis)$ & &95.6& 3.64 & & &3.63 \\
|
||||
&$^1A_2 [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pis)$ & &95.5& 3.85 & & &3.85 \\
|
||||
Ammonia &$^1A_2 (\Ryd;n \rightarrow 3s)$ & 0.086 &93.5& 6.59 &{\exCI}/AVQZ & dAV5Z &6.66 \\
|
||||
&$^1E (\Ryd;n \rightarrow 3p)$ & 0.006 &93.7& 8.16 & & &8.21 \\
|
||||
&$^1A_1 (\Ryd;n \rightarrow 3p)$ & 0.003 &94.0& 9.33 & & &8.65 \\
|
||||
&$^1A_2 (\Ryd;n \rightarrow 4s)$ & 0.008 &93.6& 9.96 &{\exCI}/AVTZ & dAV5Z &9.19 \\
|
||||
&$^3A_2 (\Ryd;n \rightarrow 3s)$ & &98.2& 6.31 &{\exCI}/AVQZ & dAV5Z &6.37 \\
|
||||
Carbon monoxyde &$^1\Pi (\mathrm{V};n \rightarrow \pis)$ & 0.084 &93.1 & 8.49 & {\exCI}/AVQZ& dAV5Z &8.48 \\
|
||||
&$^1\Sigma^- (\mathrm{V};\pi \rightarrow \pis)$ & &93.3 & 9.92 & & &9.98 \\
|
||||
&$^1\Delta (\mathrm{V};\pi \rightarrow \pis)$ & &91.8 &10.06 & & &10.10 \\
|
||||
&$^1\Sigma^+ (\Ryd)$ & 0.003 &91.5 &10.95 & & &10.80 \\
|
||||
&$^1\Sigma^+ (\Ryd)$ & 0.200 &92.9 &11.52 & & &11.42 \\
|
||||
&$^1\Pi (\Ryd)$ & 0.053 &92.4 &11.72 & & &11.55 \\
|
||||
&$^3\Pi (\mathrm{V};n \rightarrow \pis)$ & &98.7 & 6.28 & & &6.28 \\
|
||||
&$^3\Sigma^+ (\mathrm{V};\pi \rightarrow \pis)$ & &98.7 & 8.45 & & &8.49 \\
|
||||
&$^3\Delta (\mathrm{V};\pi \rightarrow \pis)$ & &98.4 & 9.27 & & &9.28 \\
|
||||
&$^3\Sigma^- (\mathrm{V};\pi \rightarrow \pis)$ & &97.5 & 9.80 & & &9.77 \\
|
||||
&$^3\Sigma^+ (\Ryd)$ & &98.0 & 10.47 & {\exCI}/AVTZ &dAV5Z &10.37 \\
|
||||
Cyclopropene &$^1B_1 (\mathrm{V};\sigma \rightarrow \pis)$ & 0.001 &92.8 &6.68$^b$ & {\CCSDT}/AVTZ&AVQZ & 6.68 \\
|
||||
&$^1B_2 (\mathrm{V};\pi \rightarrow \pis)$ & 0.071 &95.1 &6.79$^c$ & {\exCI}/AVDZ &AVQZ & 6.78 \\
|
||||
&$^3B_2 (\mathrm{V};\pi \rightarrow \pis)$ & &98.0 &4.38 & {\exCI}/AVTZ &AVQZ & 4.38 \\
|
||||
&$^3B_1 (\mathrm{V};\sigma \rightarrow \pis)$ & &98.9 &6.45 & & & 6.45 \\
|
||||
Diazomethane &$^1A_2 (\mathrm{V};\pi \rightarrow \pis)$ & &90.1 &3.14 &{\exCI}/AVTZ & dAVQZ &3.13 \\
|
||||
&$^1B_1 (\Ryd;\pi \rightarrow 3s)$ & 0.002 &93.8 &5.54 & & &5.59 \\
|
||||
&$^1A_1 (\mathrm{V};\pi \rightarrow \pis)$ & 0.210 &91.4 &5.90 & & & 5.89 \\
|
||||
&$^3A_2 (\mathrm{V};\pi \rightarrow \pis)$ & &97.7 &2.79$^c$ & {\exCI}/AVDZ & dAVQZ & \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pis)$ & &98.6 &4.05 & {\exCI}/AVTZ & dAVQZ &4.05 \\
|
||||
&$^3B_1 (\Ryd;\pi \rightarrow 3s)$ & &98.0 &5.35 & & &5.40 \\
|
||||
&$^3A_1 (\Ryd;\pi \rightarrow 3p)$ & &98.5 &6.82 & & &6.72 \\
|
||||
&$^1A'' [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pis)$ & &87.4 &0.71 & {\exCI}/AVTZ & AVQZ & 0.70 \\
|
||||
Dinitrogen &$^1\Pi_g (\mathrm{V};n \rightarrow \pis)$ & &92.6 &9.34 & {\exCI}/AVQZ&dAV5Z &9.33 \\
|
||||
&$^1\Sigma_u^- (\mathrm{V};\pi \rightarrow \pis)$ & &97.2 &9.88 & & &9.91 \\
|
||||
&$^1\Delta_u (\mathrm{V};\pi \rightarrow \pis)$ & 0.000 &95.9 &10.29 & & &10.31 \\
|
||||
&$^1\Sigma_g^+ (\Ryd)$ & &92.2 &12.98 & & &12.30 \\
|
||||
&$^1\Pi_u (\Ryd)$ & 0.229 &82.9 &13.03 & {\exCI}/AVTZ&dAV5Z &12.73 \\
|
||||
&$^1\Sigma_u^+ (\Ryd)$ & 0.296 &92.8 &13.09 & & &12.95 \\
|
||||
&$^1\Pi_u (\Ryd)$ & 0.000 &87.4 &13.46 & & &13.27 \\
|
||||
&$^3\Sigma_u^+ (\mathrm{V};\pi \rightarrow \pis)$ & &99.3 &7.70 & {\exCI}/AVQZ&dAV5Z &7.74 \\
|
||||
&$^3\Pi_g (\mathrm{V};n \rightarrow \pis)$ & &98.4 &8.01 & & &8.03 \\
|
||||
&$^3\Delta_u (\mathrm{V};\pi \rightarrow \pis)$ & &99.3 &8.87 & & &8.88 \\
|
||||
&$^3\Sigma_u^- (\mathrm{V};\pi \rightarrow \pis)$ & &98.8 &9.66 & & &9.65 \\
|
||||
Ethylene &$^1B_{3u} (\Ryd;\pi \rightarrow 3s)$ & 0.078 &95.1 &7.40 &{\exCI}/AVTZ & dAVQZ &7.43 \\
|
||||
&$^1B_{1u} (\mathrm{V};\pi \rightarrow \pis)$ & 0.346 &95.8 &7.91$^c$ &{\exCI}/AVDZ& dAVQZ &7.92 \\
|
||||
&$^1B_{1g} (\Ryd;\pi \rightarrow 3p)$ & &95.3 &8.07 & {\exCI}/AVTZ& dAVQZ &8.08 \\
|
||||
&$^3B_{1u} (\mathrm{V};\pi \rightarrow \pis)$ & &99.1 &4.54 & & &4.54 \\
|
||||
&$^3B_{3u} (\Ryd;\pi \rightarrow 3s)$ & &98.5 &7.23$^c$ &{\exCI}/AVDZ& dAVQZ &7.27 \\
|
||||
&$^3B_{1g} (\Ryd;\pi \rightarrow 3p)$ & &98.4 &7.98$^c$ & & &7.99 \\
|
||||
Formaldehyde &$^1A_2 (\mathrm{V}; n \rightarrow \pis)$ & &91.5 &3.98 &{\exCI}/AVTZ+ & dAVQZ &3.97 \\
|
||||
&$^1B_2 (\Ryd;n \rightarrow 3s)$ & 0.021 &91.7 &7.23 & & &7.28 \\
|
||||
&$^1B_2 (\Ryd;n \rightarrow 3p)$ & 0.037 &92.4 &8.13 & & &8.12 \\
|
||||
&$^1A_1 (\Ryd;n \rightarrow 3p)$ & 0.052 &91.9 &8.23 & & &8.25 \\
|
||||
&$^1A_2 (\Ryd;n \rightarrow 3p)$ & &91.7 &8.67 & & &8.64 \\
|
||||
&$^1B_1 (\mathrm{V};\sigma \rightarrow \pis)$ & 0.001 &90.8 &9.22 & & &9.21 \\
|
||||
&$^1A_1 (\mathrm{V};\pi \rightarrow \pis)$ & 0.135 &90.4 &9.43 & & &9.26 \\
|
||||
&$^3A_2 (\mathrm{V};n \rightarrow \pis)$ & &98.1 &3.58 & & &3.58 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pis)$ & &99.0 &6.06 & & &6.07 \\
|
||||
&$^3B_2 (\Ryd;n \rightarrow 3s)$ & &97.1 &7.06 & & &7.12 \\
|
||||
&$^3B_2 (\Ryd;n \rightarrow 3p)$ & &97.4 &7.94 & & &7.98 \\
|
||||
&$^3A_1 (\Ryd;n \rightarrow 3p)$ & &97.2 &8.10 & & &8.13 \\
|
||||
&$^3B_1 (\Ryd;n \rightarrow 3d)$ & &97.9 &8.42 & & &8.42 \\
|
||||
&$^1A'' [\mathrm{F}] (\mathrm{V};n \rightarrow \pis)$ & &87.8 &2.80 & & &2.80 \\
|
||||
Formamide &$^1A'' \mathrm{V};(n \rightarrow \pis)$ &0.000 &90.8 & & \\
|
||||
&$^1A' (\Ryd;n \rightarrow 3s)$ &0.001 &88.6 & & \\
|
||||
&$^1A' (\mathrm{V};\pi \rightarrow \pis)$ &0.251 &89.3 & & \\
|
||||
&$^1A' (\Ryd;n \rightarrow 3p)$ &0.111 &89.6 & & \\
|
||||
&$^3A'' (\mathrm{V};n \rightarrow \pis)$ & &97.7 & & \\
|
||||
&$^3A' (\mathrm{V};\pi \rightarrow \pis)$ & &98.2 & & \\
|
||||
Hydrogen chloride & $^1\Pi (\mathrm{CT})$ &0.056 &94.3 &7.84 & {\exCI}/AVQZ &dAV5Z &7.86 \\
|
||||
Ketene &$^1A_2 (\mathrm{V};\pi \rightarrow \pis)$ & &91.0 &3.86 &{\exCI}/AVTZ & dAVQZ &3.86 \\
|
||||
&$^1B_1 (\Ryd;n \rightarrow 3s)$ & 0.035 &93.9 &6.01 & & &6.06 \\
|
||||
&$^1A_2 (\Ryd;\pi \rightarrow 3p)$ & &94.4 &7.18 & & &7.19 \\
|
||||
&$^3A_2 (\mathrm{V};n \rightarrow \pis)$ & &91.0 &3.77 & & &3.77 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pis)$ & &98.6 &5.61 & & &5.60 \\
|
||||
&$^3B_1 (\Ryd;n \rightarrow 3s)$ & &98.1 &5.79 & & &5.85 \\
|
||||
&$^3A_2 (\Ryd;\pi \rightarrow 3p)$ & &94.4 &7.12 & & &7.14 \\
|
||||
&$^1A'' [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pis)$ & &87.9 &1.00 &{\exCI}/AVTZ & AVQZ &1.00 \\
|
||||
Methanimine &$^1A''(\mathrm{V}; n \rightarrow \pis)$ &0.003 &90.7 &5.23 &{\exCI}/AVTZ & dAVQZ &5.21 \\
|
||||
&$^3A'' (\mathrm{V}; n \rightarrow \pis)$ & &98.1 &4.65 & & &4.64 \\
|
||||
Nitrosomethane&$^1A'' (\mathrm{V};n \rightarrow \pis)$ & 0.000 &93.0 &1.96$^c$ & {\exCI}/AVDZ & AVQZ &1.95 \\
|
||||
&$^1A' (\mathrm{V};n,n \rightarrow \pis,\pis)$ &0.000 &2.5 &4.72 & {\exCI}/AVTZ & AVQZ & 4.69 \\
|
||||
&$^1A' (\Ryd;n \rightarrow 3s/3p)$ &0.006 &90.8 &6.37 & & &6.42 \\
|
||||
&$^3A'' (\mathrm{V};n \rightarrow \pis)$ & &98.4 &1.16 & & &1.16 \\
|
||||
&$^3A' (\mathrm{V};\pi \rightarrow \pis)$ & &98.9 &5.60 & & &5.61 Ê\\
|
||||
&$^1A'' [\mathrm{F}] (\mathrm{V};n \rightarrow \pis)$ & &92.7& & & & \\
|
||||
Streptocyanine-C1&$^1B_2 (\mathrm{V};\pi \rightarrow \pis)$ & 0.347 &88.7&7.13$^c$ & {\exCI}/AVDZ & AVQZ &7.12 \\
|
||||
& $^3B_2 (\mathrm{V};\pi \rightarrow \pis)$ & &98.3 &5.52 & {\exCI}/AVTZ & AVQZ &5.52 \\
|
||||
Thioformaldehyde&$^1A_2 (\mathrm{V};n \rightarrow \pis)$ & &89.3 &2.22 & {\exCI}/AVTZ & dAVQZ &2.20 \\
|
||||
&$^1B_2 (\Ryd;n \rightarrow 4s)$ & 0.012 &92.3 &5.96 & & &5.99 \\
|
||||
&$^1A_1 (\mathrm{V};\pi \rightarrow \pis)$ & 0.178 &90.8 &6.38$^d$ &{\CCSDTQ}/AVDZ& dAVQZ &6.34 \\
|
||||
&$^3A_2 (\mathrm{V};n \rightarrow \pis)$ & &97.7 &1.94 &{\exCI}/AVTZ & dAVQZ &1.94 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pis)$ & &98.9 & 3.43 & & &3.44 \\
|
||||
&$^3B_2 (\Ryd;n \rightarrow 4s)$ & &97.6 &5.72$^c$ &{\exCI}/AVDZ& dAVQZ &5.76 \\
|
||||
&$^1A_2 [\mathrm{F}] (\mathrm{V};n \rightarrow \pis)$ & &87.2 &1.95 &{\exCI}/AVTZ & dAVQZ &1.94 \\
|
||||
Water & $^1B_1 (\Ryd; n \rightarrow 3s)$ & 0.054 &93.4 &7.62 & {\exCI}/AVQZ&dAV5Z &7.70 \\%OK
|
||||
& $^1A_2 (\Ryd; n \rightarrow 3p)$ & &93.6 &9.41 & & &9.47 \\
|
||||
& $^1A_1 (\Ryd; n \rightarrow 3s)$ & 0.100 &93.6 &9.99 & & &9.97 \\
|
||||
& $^3B_1 (\Ryd; n \rightarrow 3s)$ & &98.1 &7.25 & & &7.33 \\
|
||||
& $^3A_2 (\Ryd; n \rightarrow 3p)$ & &98.0 &9.24 & & &9.30 \\
|
||||
& $^3A_1 (\Ryd; n \rightarrow 3s)$ & &98.2 &9.54 & & &9.59 \\
|
||||
\end{longtable}
|
||||
\end{footnotesize}
|
||||
\vspace{-1.0 cm}
|
||||
\begin{flushleft}\begin{footnotesize}\begin{singlespace}
|
||||
$^a${{exCI}/{\AVDZ} data corrected with the difference between {\CC{3}}/{\AVTZ} and {\CC{3}}/{\AVDZ} values;}
|
||||
$^b${{\CCSDT}/{\AVTZ} value;}
|
||||
$^c${{exCI}/{\AVDZ} data corrected with the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ} values;}
|
||||
$^d${{\CCSDTQ}/{\AVDZ} data corrected with the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ} values.}
|
||||
\end{singlespace}\end{footnotesize}\end{flushleft}
|
||||
|
||||
|
||||
\subsection{Benchmarks}
|
||||
|
||||
We have used the TBE(FC)/{\AVTZ} values to assess the performances of eleven wavefunction approaches, namely, {\ADC{2}}, {\ADC{3}}, CIS(D), {\CC{2}}, STEOM-CCSD, {\CCSD},
|
||||
CCSDR(3), CCSDT-3, {\CC{3}}, {\CCSDT} and {\CCSDTQ}. The complete list of results can be found in Table S6 in the {\SI}. As expected, only the approaches including iterative
|
||||
triples, that is, {\ADC{3}}, CCSDT-3, {\CC{3}} and {\CCSDT} are able to predict the presence of the doubly excited $(n,n) \rightarrow (\pis,\pis)$ transition in nitrosomethane, but they all yield
|
||||
large quantitative errors. Indeed, the TBE value of \IneV{4.72} is strongly underestimated by {\ADC{3}} (\IneV{3.00}) and significantly overshot by the three {\CC{}} models with
|
||||
estimates of \IneV{6.02}, \IneV{5.76} and \IneV{5.29} with CCSDT-3, {\CC{3}}, and {\CCSDT}, respectively. This \IneV{0.26} difference between the CCSDT-3 and {\CC{3}} values is
|
||||
also the largest discrepancy between these two models in the tested set. Obviously, one should not use the tested single-reference wavefunction methods to describe this $(n,n) \rightarrow (\pis,\pis)$
|
||||
transition, and it was therefore removed from our statistical analysis. Likewise, for the three lowest $^1A'$ excited states of formamide, strong state mixing -- involving two or three states -- are found
|
||||
with all tested levels of theory, making unambiguous assignments impossible, and they were also excluded from our statistics.
|
||||
|
||||
In Table \ref{Table-7}, we report, for the full set of compounds, the mean signed error (MSE), mean absolute error (MAE) as well as the maximal positive (Max+) and negative (Max-) deviations.
|
||||
{A graphical representation of the errors obtained with all methods can be found in Figure \ref{Fig-1}. Let us underline that only singlet states could be modeled with the programs used for CCSDR(3)
|
||||
and CCSDT-3 theories. As can be seen, {\CCSDTQ} is on the spot with completely trifling MSE and MAE, which is consistent with the analyses carried out for individual molecules. With this method,
|
||||
the largest errors are as small as \IneV{-0.05} (singlet $n \rightarrow 4s$ Rydberg transition of thioformaldehyde) and \IneV{+0.06} ($^1\Sigma_u^+$ Rydberg transition of dinitrogen). The three
|
||||
other {\CC{}} models encompassing iterative contributions from the triples, that is, CCSDT-3, {\CC{3}}, and {\CCSDT}, also deliver extremely accurate transition energies with MAE of \IneV{0.03} only. Consistently
|
||||
with the analysis of Watson and co-workers, we indeed found no significant differences between CCSDT-3 and {\CC{3}}, \cite{Wat13} whereas one can also conclude that {\CCSDT} is, on average,
|
||||
not significantly more accurate than {\CC{3}} nor CCSDT-3, though it gives slightly smaller extreme deviations. In other words, {\CCSDT} is probably not a sufficiently accurate benchmark to estimate
|
||||
{\CC{3}}'s accuracy. The perturbative inclusions of triples [CCSDR(3)] stands as a good compromise between computational cost and accuracy with a MAE of \IneV{0.04}, a conclusion also inline
|
||||
of the benchmark performed by Sauer and coworkers.\cite{Sau09} These very good performances are related to the fact that the majority of our set is constituted of transitions with large
|
||||
single-excitation character (see \%$T_1$ in Table \ref{Table-6}), and one can also reasonably predict that they would slightly deteriorate for larger compounds.
|
||||
|
||||
\renewcommand*{\arraystretch}{1.0}
|
||||
\begin{table}[htp]
|
||||
\caption{Mean signed error (MSE), mean absolute error (MAE), maximal positive (Max+) and negative (Max-) deviations obtained for
|
||||
the transition energies listed in Table S6 considering the TBE(FC)/{\AVTZ} . All values are in eV and have been obtained with {\AVTZ}.}
|
||||
\label{Table-7}
|
||||
\begin{tabular}{lccccc}
|
||||
\hline
|
||||
Method & Nb. States & MSE & MAE & Max+ & Max- \\
|
||||
\hline
|
||||
{\ADC{2}} &102 &-0.01 &0.21 &-0.76 &0.57 \\
|
||||
{\ADC{3}} &102 &-0.16 &0.24 &-0.79 &0.39 \\
|
||||
CIS(D) &102 &0.10 &0.26 &-0.63 &1.06 \\
|
||||
{\CC{2}} &102 &0.03 &0.22 &-0.71 &0.63 \\
|
||||
STEOM-CCSD &98 &0.00 &0.10 &-0.56 &0.40 \\
|
||||
{\CCSD} &102 &0.05 &0.08 &-0.17 &0.40 \\
|
||||
CCSDR(3) &57 &0.00 &0.04 &-0.06 &0.25 \\
|
||||
CCSDT-3 &56 &0.01 &0.03 &-0.07 &0.24 \\
|
||||
{\CC{3}} &102 &-0.01 &0.03 &-0.09 &0.19 \\
|
||||
{\CCSDT} &96 &-0.01 &0.03 &-0.07 &0.14 \\
|
||||
{\CCSDTQ} &67 &0.00 &0.01 &-0.05 &0.06 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{table}
|
||||
|
||||
In the {\CC{}} series, the errors increase when using more approximate models including only singles and doubles. Indeed, the MAE are 0.08, 0.10, and \IneV{0.22} with {\CCSD}, STEOM-CCSD and {\CC{2}}, respectively.
|
||||
The magnitude of the {\CC{2}} average deviation is consistent with previous estimates obtained for Thiel's set (\IneV{0.29} for singlets and \IneV{0.18} for triplets), \cite{Sch08} for fluorescence energies (\IneV{0.21} for
|
||||
12 small compounds),\cite{Jac18a} as well as for larger compounds (\IneV{0.15} for 0-0 energies of conjugated dyes). \cite{Jac15b} Likewise, the fact that {\CCSD} tends to overestimate the transition energies (positive MSE)
|
||||
was also reported previously in several works. \cite{Sch08,Wat13,Kan14,Jac17b,Jac18a} It can be seen that Nooijen's STEOM approach, which was much less benchmarked previously, delivers an accuracy comparable
|
||||
to {\CCSD}, with even a smaller MSE. More surprisingly, we found a MAE smaller with {\CCSD} than with {\CC{2}}, which contrasts with the results reported for Thiel's set, \cite{Sau09} and we attribute this effect to the
|
||||
small size of the compounds treated herein. Indeed, by analyzing the TZVP values of Ref. \citenum{Sch08}, it appears clearly that {\CC{2}} more regularly outperforms {\CCSD} for larger compounds.
|
||||
|
||||
|
||||
\begin{figure}[htp]
|
||||
\includegraphics[scale=0.98,viewport=2cm 10cm 19cm 27.5cm,clip]{Figure-1.pdf}
|
||||
\caption{Historgrams of the error patterns obtained compared to TBE(FC) for all methods. Note the different $Y$ scales.}
|
||||
\label{Fig-1}
|
||||
\end{figure}
|
||||
|
||||
In addition, Table \ref{Table-7} indicates that {\ADC{2}} provides an accuracy completely similar to that of {\CC{2}} for transition energies with the advantage of a smaller computational cose, whereas CIS(D) is slightly less
|
||||
accurate, both outcomes perfectly fitting previous benchmarks. \cite{Win13,Har14,Jac15b,Jac18a} On the contrary, we found that {\ADC{3}} results are rather poor with average deviations larger than the ones obtained with
|
||||
{\ADC{2}} and a clear tendency to provide too small transition energies with a MSE of \IneV{-0.16}. This result is in sharp contrast with a previous investigation that concluded that {\ADC{3}} and {\CC{3}} have very similar
|
||||
performances, \cite{Har14} though in that earlier work the {\ADC{3}} were also found to be on average \IneV{-0.20} smaller than their {\CC{3}} counterparts. At this stage, it is difficult to determine if the large MAE of {\ADC{3}}
|
||||
reported in Table \ref{Table-7} originates from the small size of the compounds treated herein. However, the fact that the {\CCSD} MSE is relatively small compared to previous benchmarks hints that the choice of compact
|
||||
compounds has a non-negligible effect on the statistics. Let us analyze the {\ADC{3}} errors more thoroughly. First, {\ADC{3}}'s deviations are quite large for all considered subsets (\emph{vide infra}). Second, we have
|
||||
found that for the 45 transition energies for which {\ADC{2}} yields an absolute error exceeding \IneV{0.15} compared to our TBE, the signs of the {\ADC{2}} and {\ADC{3}} errors systematically differ (see Figure \ref{Fig-2}),
|
||||
i.e., {\ADC{3}} goes in the good ``direction'' in correcting {\ADC{2}} but has tendency to exaggerate the correction. This is clearly reminiscent of the well-known oscillating behavior of the perturbative MP series for ground state
|
||||
properties. Third, this trend of too large correction pertains for the states in which the {\ADC{2}} absolute error is smaller than \IneV{0.15}: indeed in those 57 cases, there are only 9 molecules for which the {\ADC{3}} values are
|
||||
more accurate than their second-order counterpart. Four, as a consequence, taking the average between the {\ADC{2}} and {\ADC{3}} transition energies allows to obtain rather accurate estimates: indeed this yields a MAE
|
||||
as small as \IneV{0.10} for the full set, half of the MAE obtained with the parent methods.
|
||||
|
||||
\begin{figure}[htp]
|
||||
\includegraphics[scale=0.45,viewport=6cm 3cm 22cm 16cm,clip]{Figure-2.pdf}
|
||||
\caption{Comparison between the errors obtained with {\ADC{2}} and {\ADC{3}} approaches [compared to TBE(FC)] for the 45 states for which {\ADC{2}} yields an absolute deviation
|
||||
larger than \IneV{0.15}. All values are in eV.}
|
||||
\label{Fig-2}
|
||||
\end{figure}
|
||||
|
||||
In the SI, we provide analyses for several subsets of states (\hl{DJ: je les ai dans le .xls, je ferais la table en SI sur les finaux}) Globally, we found no significant differences between the singlet and triplet transitions, though
|
||||
the all models in the {\CC{}} series (but STEOM-CCSD) provide slightly smaller deviations for the latter transitions, in line with their larger single-excitation character. With the computationally lightest methods, CIS(D),
|
||||
{\ADC{2}}, and {\CC{2}}, the MAE are significantly smaller for the valence transitions (0.20, 0.15, and \IneV{0.18}, respectively) than for the Rydberg transitions (0.34, 0.28, and \IneV{0.31}, respectively), whereas,
|
||||
surprisingly the reverse is found with {\ADC{3}} (0.28 and \IneV{0.18} MAE for valence and Rydberg, respectively). All other tested theories deliver similar deviations for both sets of states. All methods provide smaller
|
||||
MAE for the $n \rightarrow \pis$ than for the $\pi \rightarrow \pis$ transitions, which was already found for Thiel's set.\cite{Sch08} The differences are particularly significant with CIS(D), {\CC{2}}, STEOM-CCSD
|
||||
and {\ADC{3}} with errors twice larger for $\pi \rightarrow \pis$ than $n \rightarrow \pis$ states. Finally, when considering the few states with \%$T_1$ smaller than 90\%\ logically yields larger statistical errors for the most advanced
|
||||
approaches with MAE of, e.g., \IneV{0.03} for {\CCSDTQ}, \IneV{0.04} for {\CC{3}}, and \IneV{0.06} for CCSDT-3.
|
||||
|
||||
\section{Conclusions and outlook}
|
||||
|
||||
We have defined a set of more than 100 vertical transition energies, as close as possible to the {\FCI} limit. To this end, we have used both the coupled-cluster route up to the highest computationally-possible order and the
|
||||
selected configuration interaction route up to the largest technically-affordable number of determinants. These calculations have been performed on 17 compounds encompassing one, two or three non-hydrogen atoms, using
|
||||
geometries optimized at the {\CC{3}} level and a series of diffuse-containing Dunning's basis set of increasing size. It was certainly gratifying to find extremely good agreements between the results reached with these independent
|
||||
approaches with e.g., typical differences as small as \IneV{0.01} between {\CCSDTQ} and {\exCI} transition energies. In fact, during the course of this joint work, the two groups involved in this study were able to detect misprints
|
||||
or incorrect assignments in each other calculations even when the differences were apparently negligible. For the two treated diatomic molecules, N$_2$ and CO, the mean absolute deviation between our theoretical best
|
||||
estimates and the ``experimental'' vertical transition energies deduced from spectroscopic measurements using a numerical solution of the nuclear Schr\"odinger equation is as small as \IneV{0.04}, and it was possible to
|
||||
resolve previous inconsistencies between these ``experimental'' values. A significant share of the remaining error is likely related to the use of theoretically-determined geometries. Although, it is not possible to provide a definitive
|
||||
error bar for the 106 TBE listed in this work, our estimate, based on the differences between the two routes as well as the extrapolations used in the {\sCI} procedure, is $\pm$ \IneV{0.03}.
|
||||
|
||||
In the last part of this work, we have used the TBE(FC)/{\AVTZ} values to benchmark a series of eleven popular wavefunction approaches. For the computationally most effective approaches, CIS(D), {\ADC{2}}, and {\CC{2}},
|
||||
we found average deviations of ca. 0.21--\IneV{0.26} range with large similitudes between the {\ADC{2}} and {\CC{2}} results, and both conclusions fit previous works. Likewise, we obtained the expected trend that {\CCSD}
|
||||
overestimates the transition energies, though with an amplitude that is quite small here, likely due to the size of the investigated compounds. More interestingly, we could demonstrate that STEOM-CCSD is, on average, as accurate
|
||||
as {\CCSD}, and we were also able to benchmark the methods including contributions from triples using reliable theoretical references. Interestingly, we found no significant differences between CCSDT-3, {\CC{3}}, and {\CCSDT},
|
||||
that all yield a MAE of \IneV{0.03}. In other words, we could not demonstrate that {\CCSDT} is statistically more accurate than its approximated (and computationally more effective) forms. The use of perturbative triples,
|
||||
as in CCSDR(3), allows to correct for most of the {\CCSD} error and is this a computationally appealing method as it gives average deviations only slightly larger than with iterative triples. In contrast, for the present set of
|
||||
molecules, {\ADC{3}} was found significantly less accurate than {\CC{3}}, and it was showed that {\ADC{3}} overcorrects {\ADC{2}}. Whether this surprising result is related to the size of the compounds used or is a more
|
||||
general trends remains to be determined.
|
||||
|
||||
As stated several times throughout this work, the size of the considered molecules is certainly one of the main limits of the present effort, as it introduces a significant bias, e.g., charge-transfer over several {\AA} are totally absent
|
||||
of the set. Obviously the respective $\mathcal{O}(N^{10})$ and $\mathcal{O}(N!)$ formal scalings of {\CCSDTQ} and {\FCI} offer no easy pathway to circumvent this limit. Nevertheless, it appears that performing {\exCI} calculations
|
||||
with a relatively compact basis, e.g., {\AVDZ}, and correcting the basis set effects with a more affordable approach, e.g., {\CC{3}}, might be a valuable approach to reach very accurate estimates for larger molecules, at least for
|
||||
the electronic transitions presenting a dominant single excitation character. Indeed, we have shown here that such basis set extrapolation approach is trustworthy. We are currently hiking along that path.
|
||||
|
||||
\begin{suppinfo}
|
||||
Basis set and frozen-core effects. Geometries used. Full list of transition energies for the benchmark Section. Additional statistical analysis.
|
||||
\end{suppinfo}
|
||||
|
||||
\begin{acknowledgement}
|
||||
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});
|
||||
iii) the Troy cluster installed in Nantes; and iv) CALMIP under allocations 2018-0510 and 2018-18005 (Toulouse).
|
||||
\end{acknowledgement}
|
||||
|
||||
\bibliography{biblio-new}
|
||||
\clearpage
|
||||
|
||||
\begin{tocentry}
|
||||
\begin{center}
|
||||
\includegraphics[scale=.17]{TOC.png}
|
||||
\end{center}
|
||||
\end{tocentry}
|
||||
|
||||
|
||||
\end{document}
|
||||
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|
|
@ -0,0 +1,763 @@
|
|||
\documentclass[journal=jctcce,manuscript=article,layout=traditional]{achemso}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amsmath,amssymb,amsfonts,physics,float,lscape,soul,rotating,longtable}
|
||||
\usepackage[version=4]{mhchem}
|
||||
|
||||
|
||||
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\mr}{\multirow}
|
||||
|
||||
\newcommand{\EFCI}{E_\text{FCI}}
|
||||
\newcommand{\EexCI}{E_\text{exCI}}
|
||||
\newcommand{\EsCI}{E_\text{sCI}}
|
||||
\newcommand{\EPT}{E_\text{PT2}}
|
||||
\newcommand{\PsisCI}{\Psi_\text{sCI}}
|
||||
\newcommand{\Ndet}{N_\text{det}}
|
||||
|
||||
\newcommand{\ex}[4]{{#1}\,$^{#2}$#3$_{#4}$}
|
||||
|
||||
% methods
|
||||
\newcommand{\TDDFT}{TD-DFT}
|
||||
\newcommand{\CASSCF}{CASSCF}
|
||||
\newcommand{\CASPT}{CASPT2}
|
||||
\newcommand{\ADC}[1]{ADC(#1)}
|
||||
\newcommand{\CC}[1]{CC#1}
|
||||
\newcommand{\CCSD}{CCSD}
|
||||
\newcommand{\EOMCCSD}{EOM-CCSD}
|
||||
\newcommand{\CCSDT}{CCSDT}
|
||||
\newcommand{\CCSDTQ}{CCSDTQ}
|
||||
\newcommand{\CI}{CI}
|
||||
\newcommand{\sCI}{sCI}
|
||||
\newcommand{\exCI}{exCI}
|
||||
\newcommand{\FCI}{FCI}
|
||||
|
||||
% basis
|
||||
\newcommand{\AVDZ}{\emph{aug}-cc-pVDZ}
|
||||
\newcommand{\AVTZ}{\emph{aug}-cc-pVTZ}
|
||||
\newcommand{\DAVTZ}{d-\emph{aug}-cc-pVTZ}
|
||||
\newcommand{\AVQZ}{\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\DAVQZ}{d-\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\TAVQZ}{t-\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\AVPZ}{\emph{aug}-cc-pV5Z}
|
||||
\newcommand{\DAVPZ}{d-\emph{aug}-cc-pV5Z}
|
||||
|
||||
% units
|
||||
\newcommand{\IneV}[1]{#1 eV}
|
||||
\newcommand{\InAU}[1]{#1 a.u.}
|
||||
|
||||
\newcommand{\SI}{Supporting Information}
|
||||
|
||||
\setcounter{table}{0}
|
||||
\setcounter{figure}{0}
|
||||
\setcounter{page}{1}
|
||||
\setcounter{equation}{0}
|
||||
\renewcommand{\thepage}{S\arabic{page}}
|
||||
\renewcommand{\thefigure}{S\arabic{figure}}
|
||||
\renewcommand{\theequation}{S\arabic{equation}}
|
||||
\renewcommand{\thetable}{S\arabic{table}}
|
||||
\renewcommand{\thesection}{S\arabic{section}}
|
||||
|
||||
\renewcommand\floatpagefraction{.99}
|
||||
\renewcommand\topfraction{.99}
|
||||
\renewcommand\bottomfraction{.99}
|
||||
\renewcommand\textfraction{.01}
|
||||
|
||||
\title{A Mountaineering Strategy to Excited States: Highly-Accurate Reference Energies and Benchmarks\\Supporting Information}
|
||||
|
||||
\author{Pierre-Fran{\c c}ois Loos}
|
||||
\email{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Anthony Scemama}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Aymeric Blondel}
|
||||
\affiliation[UN, Nantes]{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
|
||||
\author{Yann Garniron}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Michel Caffarel}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Denis Jacquemin}
|
||||
\email{Denis.Jacquemin@univ-nantes.fr}
|
||||
\affiliation[UN, Nantes]{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\clearpage
|
||||
|
||||
\section{Basis set and frozen-core effects}
|
||||
|
||||
\subsection{Water, ammonia and hydrogen chloride}
|
||||
|
||||
\begin{sidewaystable}[htp]
|
||||
\caption{\small CC3 vertical transition energies of water (top), ammonia (center), and hydrogen chloride (bottom) using various atomic basis sets.
|
||||
FC stands for frozen core (large frozen core for the latter compound). All values are in eV.}
|
||||
\label{Table-wat-bs}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|ccccccccccc}
|
||||
\hline
|
||||
& {\AVDZ} & {\AVTZ} & {\DAVTZ} & \multicolumn{2}{c}{\AVQZ} & \multicolumn{2}{c}{\DAVQZ} & \multicolumn{2}{c}{\TAVQZ} & {\AVPZ} & {\DAVPZ} \\
|
||||
& FC & FC &FC & FC & Full & FC & Full & FC & Full & Full & Full\\
|
||||
\hline
|
||||
& \multicolumn{11}{c}{Water} \\
|
||||
\hline
|
||||
$^1B_1 (n \rightarrow 3s)$ &7.51 &7.60&7.60 &7.65 &7.66 &7.65 &7.66 &7.64 &7.66 &7.67 &7.67 \\
|
||||
$^1A_2 (n \rightarrow 3p)$ &9.29 &9.38&9.37 &9.43 &9.43 &9.42 &9.42 &9.42 &9.42 &9.44 &9.44 \\
|
||||
$^1A_1 (n \rightarrow 3s)$ &9.92 &9.97&9.89 &10.00 &10.00 &9.94 &9.94 &9.93 &9.94 &10.00 &9.95 \\
|
||||
$^3B_1 (n \rightarrow 3s)$ &7.13 &7.23&7.23 &7.28 &7.29 &7.28 &7.29 &7.28 &7.29 &7.31 &7.31 \\
|
||||
$^3A_2 (n \rightarrow 3p)$ &9.12 &9.22&9.30 &9.26 &9.27 &9.26 &9.26 &9.26 &9.26 &9.28 &9.28 \\
|
||||
$^3A_1 (n \rightarrow 3s)$ &9.47 &9.52&9.52 &9.56 &9.56 &9.56 &9.56 &9.56 &9.56 &9.57 &9.57 \\
|
||||
\hline
|
||||
& \multicolumn{11}{c}{Ammonia} \\
|
||||
\hline
|
||||
$^1A_2 (n \rightarrow 3s)$ &6.46 & 6.57&6.57 &6.61 &6.61 &6.61 &6.61 &6.61 &6.61 &6.63 &6.63 \\
|
||||
$^1E (n \rightarrow 3p)$ &8.06 & 8.15&8.12 &8.18 &8.18 &8.16 &8.16 &8.16 &8.16 &8.18 &8.17 \\
|
||||
$^1A_1 (n \rightarrow 3p)$ &9.66 & 9.32&8.56 &9.11 &9.11 &8.61 &8.61 &8.60 &8.60 &8.91 &8.62 \\
|
||||
$^1A_2 (n \rightarrow 4s)$ &10.40& 9.95&9.12 &9.77 &9.77 &9.16 &9.17 &9.15 &9.16 &9.61 &9.18 \\
|
||||
$^3A_2 (n \rightarrow 3s)$ &6.18 & 6.29&6.29 &6.33 &6.34 &6.33 &6.34 &6.33 &6.34 &6.35 &6.35 \\
|
||||
\hline
|
||||
& \multicolumn{11}{c}{Hydrogen chloride} \\
|
||||
\hline
|
||||
$^1\Pi (\mathrm{CT})$ &7.82 & 7.84 & 7.83 &7.89 &7.87 &7.88 &7.87 &7.88 &7.87 &7.87 &7.87\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\end{sidewaystable}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Dinitrogen and carbon monoxide}
|
||||
|
||||
\begin{sidewaystable}[htp]
|
||||
\caption{\small CC3 vertical transition energies of dinitrogen (top) and carbon monoxide (bottom) using various atomic basis sets.
|
||||
See caption of Table \ref{Table-wat-bs} for more details.}
|
||||
\label{Table-n2co-bs}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|cccccccccc}
|
||||
\hline
|
||||
& {\AVDZ} & {\AVTZ} & \multicolumn{2}{c}{\AVQZ} & \multicolumn{2}{c}{\DAVQZ} & \multicolumn{2}{c}{\TAVQZ} & {\AVPZ} & {\DAVPZ} \\
|
||||
& FC & FC & FC & Full & FC & Full & FC & Full & Full & Full\\
|
||||
\hline
|
||||
& \multicolumn{10}{c}{Dinitrogen} \\
|
||||
\hline
|
||||
$^1\Pi_g (\mathrm{V}; n \rightarrow \pi^\star)$ &9.44 &9.34 &9.33 &9.32 &9.33 &9.32 &9.33 &9.32 &9.32 &9.32 \\%B2G-B3G
|
||||
$^1\Sigma_u^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ &10.06 &9.88 &9.87 &9.87 &9.87 &9.87 &9.87 &9.87 &9.86 &9.86 \\%AU
|
||||
$^1\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ &10.43 &10.29 &10.27 &10.27 &10.27 &10.27 &10.27 &10.27 &10.27 &10.27 \\%B1U/AU
|
||||
$^1\Sigma_g^+ (\mathrm{R}; n \rightarrow \sigma^\star)$ &13.23 &13.01 &12.90 &12.91 &12.27 &12.28 &12.27 &12.28 &12.77 &12.31 \\%AG
|
||||
$^1\Pi_u (\mathrm{R})$ &13.28 &13.22 &13.17 &13.16 &12.89 &12.90 &12.89 &12.90 &13.08 &12.92 \\%B2U-B3U
|
||||
$^1\Sigma_u^+ (\mathrm{R}; n \rightarrow \sigma^\star)$ &13.14 &13.12 &13.09 &13.10 &12.94 &12.96 &12.94 &12.96 &13.06 &12.98 \\%B1IU
|
||||
$^1\Pi_u (\mathrm{R})$ &13.64 &13.49 &13.42 &13.40 &13.34 &13.31 &13.34 &13.31 &13.34 &13.30 \\%B2U-B3U
|
||||
$^3\Sigma_u^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ &7.67 &7.68 &7.71 &7.70 &7.71 &7.70 &7.71 &7.70 &7.71 &7.71 \\%B1IU
|
||||
$^3\Pi_g (\mathrm{V}; n \rightarrow \pi^\star)$ &8.07 &8.04 &8.04 &8.03 &8.04 &8.03 &8.04 &8.03 &8.04 &8.04 \\%B2G-B3G
|
||||
$^3\Delta_u (\mathrm{V}; \pi \rightarrow \pi^\star)$ &8.97 &8.87 &8.87 &8.86 &8.87 &8.86 &8.87 &8.86 &8.87 &8.87 \\%B1U/AU
|
||||
$^3\Sigma_u^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ &9.78 &9.68 &9.68 &9.67 &9.68 &9.67 &9.68 &9.67 &9.67 &9.67 \\%AU
|
||||
\hline
|
||||
& \multicolumn{10}{c}{Carbon monoxide} \\
|
||||
\hline
|
||||
$^1\Pi (\mathrm{V}; n \rightarrow \pi^\star)$ &8.57 &8.49 &8.47 &8.46 &8.47 &8.45 &8.47 &8.45 &8.45 &8.45 \\%B1/B2
|
||||
$^1\Sigma^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ &10.12 &9.99 &9.99 &9.98 &9.99 &9.98 &9.99 &9.98 &9.98 &9.98 \\%A2
|
||||
$^1\Delta (\mathrm{V}; \pi \rightarrow \pi^\star)$ &10.23 &10.12 &10.12 &10.11 &10.12 &10.11 &10.11 &10.11 &10.11 &10.11 \\%A1/A2
|
||||
$^1\Sigma^+ (\mathrm{R})$ &10.92 &10.94 &10.90 &10.91 &10.72 &10.72 &10.72 &10.72 &10.85 &10.74 \\%A1
|
||||
$^1\Sigma^+ (\mathrm{R})$ &11.48 &11.49 &11.46 &11.47 &11.33 &11.34 &11.33 &11.34 &11.42 &11.35 \\%A1
|
||||
$^1\Pi (\mathrm{R})$ &11.74 &11.69 &11.63 &11.64 &11.46 &11.46 &11.46 &11.46 &11.57 &11.48 \\%B1/B2
|
||||
$^3\Pi (\mathrm{V}; n \rightarrow \pi^\star)$ &6.31 &6.30 &6.30 &6.29 &6.30 &6.29 &6.30 &6.29 &6.29 &6.29 \\%B1/B2
|
||||
$^3\Sigma^+ (\mathrm{V}; \pi \rightarrow \pi^\star)$ &8.45 &8.45 &8.48 &8.47 &8.48 &8.47 &8.48 &8.47 &8.48 &8.48 \\%A1
|
||||
$^3\Delta (\mathrm{V}; \pi \rightarrow \pi^\star)$ &9.37 &9.30 &9.31 &9.30 &9.31 &9.30 &9.31 &9.30 &9.30 &9.30 \\%A1/A2
|
||||
$^3\Sigma^- (\mathrm{V}; \pi \rightarrow \pi^\star)$ &9.89 &9.82 &9.82 &9.81 &9.81 &9.81 &9.82 &9.81 &9.81 &9.81 \\%A2
|
||||
$^3\Sigma^- (\mathrm{R})$ &10.39 &10.45 &10.44 &10.45 &10.33 &10.33 &10.33 &10.34 &10.42 &10.35 \\%A1
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\end{sidewaystable}
|
||||
\clearpage
|
||||
|
||||
\subsection{Acetylene and ethylene}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small CC3 vertical transition energies determined of acetylene (top) and ethylene (bottom) using various atomic basis sets.
|
||||
See caption of Table \ref{Table-wat-bs} for more details.}
|
||||
\label{Table-acetethy-bs}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|ccccccc}
|
||||
\hline
|
||||
& {\AVDZ} & {\AVTZ} & \multicolumn{2}{c}{\AVQZ} & \multicolumn{2}{c}{\DAVQZ} & {\AVPZ} \\
|
||||
& FC & FC & FC & Full & FC & Full & Full \\
|
||||
\hline
|
||||
& \multicolumn{7}{c}{Acetylene} \\
|
||||
\hline
|
||||
$^1\Sigma_u^- (\pi \rightarrow \pi^\star)$ &7.21&7.09 &7.09 &7.09 &7.09 &7.09 &7.09 \\
|
||||
$^1\Delta_u (\pi \rightarrow \pi^\star)$ &7.51&7.42 &7.41 &7.42 &7.41 &7.42 &7.42 \\
|
||||
$^3\Sigma_u^+ (\pi \rightarrow \pi^\star)$ &5.48&5.50 &5.53 &5.52 &5.53 &5.52 &5.53 \\
|
||||
$^3\Delta_u (\pi \rightarrow \pi^\star)$ &6.46&6.40 &6.40 &6.40 &6.40 &6.40 &6.40 \\
|
||||
$^3\Sigma_u^- (\pi \rightarrow \pi^\star)$ &7.13&7.07 &7.07 &7.07 &7.07 &7.07 &7.08 \\
|
||||
$^1A_u [\mathrm{F}] (\pi \rightarrow \pi^\star)$ &3.70&3.64 &3.63 &3.63 &3.63 &3.63 &3.63 \\
|
||||
$^1A_2 [\mathrm{F}] (\pi \rightarrow \pi^\star)$ &3.92&3.84 &3.83 &3.84 &3.84 &3.84 &3.84 \\
|
||||
|
||||
\hline
|
||||
& \multicolumn{7}{c}{Ethylene} \\
|
||||
\hline
|
||||
$^1B_{3u} (\pi \rightarrow 3s)$ &7.29&7.35 &7.38 &7.39 &7.37 &7.38 &7.39 \\
|
||||
$^1B_{1u} (\pi \rightarrow \pi^\star)$ &7.94&7.91 &7.90 &7.91 &7.90 &7.93 &7.91 \\
|
||||
$^1B_{1g} (\pi \rightarrow 3p)$ &7.97&8.03 &8.04 &8.05 &8.03 &8.04 &8.05 \\
|
||||
$^3B_{1u} (\pi \rightarrow \pi^\star)$ &4.53&4.53 &4.54 &4.53 &4.54 &4.53 &4.53 \\
|
||||
$^3B_{3u} (\pi \rightarrow 3s)$ &7.17&7.24 &7.27 &7.28 &7.26 &7.28 &7.29 \\
|
||||
$^3B_{1g} (\pi \rightarrow 3p)$ &7.93&7.98 &8.00 &8.00 &7.99 &7.99 &8.01 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\end{table}
|
||||
\clearpage
|
||||
|
||||
\subsection{Formaldehyde, methanimine and thioformaldehyde}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small CC3 vertical transition energies determined of formaldehyde (top), methanimine (center), and thioformaldehyde (bottom) using various atomic basis sets.
|
||||
See caption of Table \ref{Table-wat-bs} for more details.}
|
||||
\label{Table-forma-bs}
|
||||
\begin{small}
|
||||
\begin{tabular}{l|ccccccc}
|
||||
\hline
|
||||
& {\AVDZ} & {\AVTZ} & \multicolumn{2}{c}{\AVQZ} & \multicolumn{2}{c}{\DAVQZ} & {\AVPZ} \\
|
||||
& FC & FC & FC & Full & FC & Full & Full \\
|
||||
\hline
|
||||
& \multicolumn{7}{c}{Formaldehyde} \\
|
||||
\hline
|
||||
$^1A_2 (n \rightarrow \pi^\star)$ &4.00&3.97& 3.97& 3.96& 3.97 & 3.96 & 3.96 \\
|
||||
$^1B_2 (n \rightarrow 3s)$ &7.05&7.18& 7.23& 7.23& 7.22 & 7.23 & 7.25 \\
|
||||
$^1B_2 (n \rightarrow 3p)$ &8.02&8.07& 8.10& 8.11& 8.05 & 8.06 & 8.10 \\
|
||||
$^1A_1 (n \rightarrow 3p)$ &8.08&8.18& 8.22& 8.23& 8.20 & 8.20 & 8.23 \\
|
||||
$^1A_2 (n \rightarrow 3p)$ &8.65&8.64& 8.60& 8.61& & 8.61 & 8.54 \\
|
||||
$^1B_1 (\sigma \rightarrow \pi^\star)$ &9.31&9.19& 9.19& 9.18& 9.19 & 9.18 & 9.18 \\
|
||||
$^1A_1 (\pi \rightarrow \pi^\star)$ &9.59&9.48& 9.46& 9.46& 9.30 & 9.31 & 9.43 \\
|
||||
$^3A_2 (n \rightarrow \pi^\star)$ &3.58&3.57& 3.58& 3.57& 3.58 & 3.57 & 3.57 \\
|
||||
$^3A_1 (\pi \rightarrow \pi^\star)$ &6.09&6.05& 6.06& 6.06& 6.07 & 6.06 & 6.06 \\
|
||||
$^3B_2 (n \rightarrow 3s)$ &6.91&7.03& 7.08& 7.09& 7.08 & 7.09 & 7.11 \\
|
||||
$^3B_2 (n \rightarrow 3p)$ &7.84&7.92& 7.95& 7.96& 7.91 & 7.96 & 7.95 \\
|
||||
$^3A_1 (n \rightarrow 3p)$ &7.97&8.08& 8.12& 8.12& 8.10 & 8.11 & 8.13 \\
|
||||
$^3B_1 (n \rightarrow 3d)$ &8.48&8.41& 8.42& 8.41& 8.42 & 8.41 & 8.41 \\
|
||||
$^1A^" [\mathrm{F}] (n \rightarrow \pi^\star)$ &2.87&2.84& 2.85& 2.84& 2.85 & 2.84 & 2.84 \\
|
||||
\hline
|
||||
& \multicolumn{7}{c}{Methanimine} \\
|
||||
\hline
|
||||
$^1A^"(n \rightarrow \pi^\star)$ &5.26&5.20& 5.20& 5.18& 5.20 & 5.18 & 5.18 \\
|
||||
$^3A^" (n \rightarrow \pi^\star)$ &4.63&4.61& 4.62& 4.60& 4.62 & 4.60 & 4.60 \\
|
||||
\hline
|
||||
& \multicolumn{7}{c}{Thioformaldehyde} \\
|
||||
\hline
|
||||
$^1A_2 (n \rightarrow \pi^\star)$ &2.27&2.23& 2.23& 2.21& 2.23& 2.21 & 2.21 \\
|
||||
$^1B_2 (n \rightarrow 4s)$ &5.80&5.91& 5.95& 5.95& 5.95& 5.94 & 5.96 \\
|
||||
$^1A_1 (\pi \rightarrow \pi^\star)$ &6.62&6.48& 6.46& 6.45& 6.45& 6.45 & 6.43 \\
|
||||
$^3A_2 (n \rightarrow \pi^\star)$ &1.97&1.94& 1.95& 1.94& 1.95& 1.94 & \\
|
||||
$^3A_1 (\pi \rightarrow \pi^\star)$ &3.43&3.38& 3.40& 3.39& 3.40& 3.39 & 3.39 \\
|
||||
$^3B_2 (n \rightarrow 4s)$ &5.64&5.72& 5.75& 5.75& 5.75& 5.74 & \\
|
||||
$^1A_2 [\mathrm{F}] (n \rightarrow \pi^\star)$ &2.00&1.97& 1.98& 1.96& 1.97& 1.96 & 1.95 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\end{table}
|
||||
\clearpage
|
||||
|
||||
\subsection{Larger compounds}
|
||||
|
||||
\begin{table}[htp]
|
||||
\caption{\small CC3 vertical transition energies determined of six compounds incorporating three non-hydrogen atoms.
|
||||
See caption of Table \ref{Table-wat-bs} for more details.}
|
||||
\label{Table-3at-bs}
|
||||
\begin{small}
|
||||
\begin{tabular}{ll|ccccc}
|
||||
\hline
|
||||
&& {\AVDZ} & {\AVTZ} & \multicolumn{2}{c}{\AVQZ} & {\DAVQZ} \\
|
||||
Molecule state && FC & FC & FC & Full & Full \\
|
||||
\hline
|
||||
Acetaldehyde &$^1A''(n \rightarrow \pi^\star)$ &4.34 &4.31 &4.32& & \\
|
||||
&$^3A''(n \rightarrow \pi^\star)$ &3.96 &3.95 &3.97& & \\
|
||||
\hline
|
||||
Cyclopropene &$^1B_1 (\sigma \rightarrow \pi^\star)$ &6.72 &6.68 &6.68 &6.68\\
|
||||
&$^1B_2 (\pi \rightarrow \pi^\star)$ &6.77 &6.73 &6.73 &6.73\\
|
||||
&$^3B_2 (\pi \rightarrow \pi^\star)$ &4.34 &4.34 &4.35 &4.34\\
|
||||
&$^3B_1 (\sigma \rightarrow \pi^\star)$ &6.43 &6.40 &6.41 &6.40\\
|
||||
\hline%
|
||||
Diazomethane &$^1A_2 (\pi \rightarrow \pi^\star)$ &3.10 &3.07 &3.07 &3.06 &3.06 \\
|
||||
&$^1B_1 (\pi \rightarrow 3s)$ &5.32 &5.45 &5.49 &5.51 &5.50 \\
|
||||
&$^1A_1 (\pi \rightarrow \pi^\star)$ &5.80 &5.84 &5.85 &5.85 &5.83 \\
|
||||
&$^3A_2 (\pi \rightarrow \pi^\star)$ &2.84 &2.83 &2.82 &2.82 & \\
|
||||
&$^3A_1 (\pi \rightarrow \pi^\star)$ &4.05 &4.03 &4.04 &4.03 &4.03 \\
|
||||
&$^3B_1 (\pi \rightarrow 3s)$ &5.17 &5.31 &5.35 &5.37 &5.36 \\
|
||||
&$^3A_1 (\pi \rightarrow 3p)$ &6.83 &6.80 &6.81 &6.80 &6.70 \\
|
||||
&$^1A'' [\mathrm{F}] (\pi \rightarrow \pi^\star)$ &0.68 &0.68 &0.68 &0.67 & \\
|
||||
\hline
|
||||
Formamide &$^1A'' (n \rightarrow \pi^\star)$ &5.71 &5.66 & &\\
|
||||
&$^1A' (n \rightarrow 3s)$ &6.65 &6.74 & &\\
|
||||
&$^1A' (\pi \rightarrow \pi^\star)$$^a$ &7.63 &7.62 & &\\
|
||||
&$^1A' (n \rightarrow 3p)$$^a$ &7.31 &7.40 & &\\
|
||||
&$^3A'' (n \rightarrow \pi^\star)$ &5.42 &5.38 & &\\
|
||||
&$^3A' (\pi \rightarrow \pi^\star)$ &5.83 &5.82 &5.83 &\\
|
||||
\hline
|
||||
Ketene &$^1A_2 (\pi \rightarrow \pi^\star)$ &3.89 &3.88 &3.88 &3.88 &3.88 \\
|
||||
&$^1B_1 (n \rightarrow 3s)$ &5.83 &5.96 &6.00 &6.01 &6.01 \\
|
||||
&$^1A_2 (\pi \rightarrow 3p)$ &7.05 &7.16 &7.19 &7.20 &7.17 \\
|
||||
&$^3A_2 (n \rightarrow \pi^\star)$ &3.79 &3.78 &3.79 &3.78 &3.78 \\
|
||||
&$^3A_1 (\pi \rightarrow \pi^\star)$ &5.62 &5.61 &5.62 &5.60 &5.60 \\
|
||||
&$^3B_1 (n \rightarrow 3s)$ &5.63 &5.76 &5.81 &5.82 &5.82 \\
|
||||
&$^3A_2 (\pi \rightarrow 3p)$ &7.01 &7.12 &7.15 &7.16 &7.14 \\
|
||||
&$^1A'' [\mathrm{F}] (\pi \rightarrow \pi^\star)$ &1.00 &1.00 &1.00 &1.00 & \\
|
||||
\hline
|
||||
Nitrosomethane&$^1A'' (n \rightarrow \pi^\star)$ &2.00 &1.96 &1.96 &1.96\\
|
||||
&$^1A' (n,n \rightarrow \pi^\star,\pi^\star)$ &5.75 &5.76 &5.74 &5.73\\
|
||||
&$^1A' (n \rightarrow 3s/3p)$ &6.20 &6.31 &6.35 &6.36\\
|
||||
&$^3A'' (n \rightarrow \pi^\star)$ &1.13 &1.14 &1.15 &1.14\\
|
||||
&$^3A' (\pi \rightarrow \pi^\star)$ &5.54 &5.51 &5.52 &5.52\\
|
||||
&$^1A'' [\mathrm{F}] (n \rightarrow \pi^\star)$ &1.70 &1.69 & & \\
|
||||
|
||||
\hline
|
||||
Streptocyanine-C1&$^1B_2 (\pi \rightarrow \pi^\star)$ &7.14 &7.13 &7.13 &7.12\\
|
||||
& $^3B_2 (\pi \rightarrow \pi^\star)$ &5.48 &5.48 &5.49 &5.48\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{small}
|
||||
\begin{flushleft}
|
||||
\begin{footnotesize}
|
||||
$^a${Strong state mixing.}
|
||||
\end{footnotesize}
|
||||
\end{flushleft}
|
||||
\end{table}
|
||||
|
||||
\clearpage
|
||||
|
||||
\section{Geometries}
|
||||
|
||||
Below are given the cartesian coordinates of the compounds investigated in this study.
|
||||
These are provided in atomic units (bohr) and they have been obtained at the \CC{3}(full)/{\AVTZ} level of theory.
|
||||
|
||||
\subsection{Acetaldehyde}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C -0.00234503 0.00000000 0.87125063
|
||||
C -1.75847785 0.00000000 -1.34973671
|
||||
O 2.27947397 0.00000000 0.71968028
|
||||
H -0.92904537 0.00000000 2.73929404
|
||||
H -2.97955463 1.66046488 -1.25209463
|
||||
H -2.97955463 -1.66046488 -1.25209463
|
||||
H -0.70043433 0.00000000 -3.11066412
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Acetylene}
|
||||
|
||||
\begin{singlespace}
|
||||
Ground state
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 1.14048351
|
||||
C 0.00000000 0.00000000 -1.14048351
|
||||
H 0.00000000 0.00000000 3.14009043
|
||||
H 0.00000000 0.00000000 -3.14009043
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\begin{singlespace}
|
||||
\emph{Trans} excited state ($^1A_u$ state in the $C_{2h}$ point group)
|
||||
\begin{verbatim}
|
||||
C 1.29567779 0.00000000 -0.01846047
|
||||
C -1.29567779 0.00000000 0.01846047
|
||||
H 2.41938674 0.00000000 1.70881682
|
||||
H -2.41938674 0.00000000 -1.70881682
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\begin{singlespace}
|
||||
\emph{Cis} excited state ($^1A_2$ state in the $C_{2v}$ point group)
|
||||
\begin{verbatim}
|
||||
C 0.00000000 1.26834508 -0.11726146
|
||||
C 0.00000000 -1.26834508 -0.11726146
|
||||
H 0.00000000 2.67282325 1.39629264
|
||||
H 0.00000000 -2.67282325 1.39629264
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Ammonia}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
N 0.12804615 -0.00000000 0.00000000
|
||||
H -0.59303935 0.88580079 -1.53425197
|
||||
H -0.59303935 -1.77160157 -0.00000000
|
||||
H -0.59303935 0.88580079 1.53425197
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Carbon monoxide}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.24942055
|
||||
O 0.00000000 0.00000000 0.89266692
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Cyclopropene}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.66820880
|
||||
C 0.00000000 1.22523906 0.90681419
|
||||
C 0.00000000 -1.22523906 0.90681419
|
||||
H 1.72255446 0.00000000 -2.77881149
|
||||
H -1.72255446 0.00000000 -2.77881149
|
||||
H 0.00000000 2.97844519 1.92076771
|
||||
H 0.00000000 -2.97844519 1.92076771
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsection{Diazomethane}
|
||||
|
||||
\begin{singlespace}
|
||||
Ground state
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.30830005
|
||||
N 0.00000000 0.00000000 0.14457890
|
||||
N 0.00000000 0.00000000 2.29923216
|
||||
H 0.00000000 1.79875201 -3.24272317
|
||||
H 0.00000000 -1.79875201 -3.24272317
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\begin{singlespace}
|
||||
Excited state
|
||||
\begin{verbatim}
|
||||
C 1.80206107 0.00000000 -1.03389466
|
||||
N -0.01743713 0.00000000 0.84742344
|
||||
N -2.25203764 0.00000000 0.54034983
|
||||
H 3.74280590 0.00000000 -0.44375913
|
||||
H 1.20115546 0.00000000 -2.98380249
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Dinitrogen}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
N 0.00000000 0.00000000 1.04008632
|
||||
N 0.00000000 0.00000000 -1.04008632
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Ethylene}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 1.26026583 0.00000000
|
||||
C 0.00000000 -1.26026583 0.00000000
|
||||
H 0.00000000 2.32345976 1.74287672
|
||||
H 0.00000000 -2.32345976 1.74287672
|
||||
H 0.00000000 2.32345976 -1.74287672
|
||||
H 0.00000000 -2.32345976 -1.74287672
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Formaldehyde}
|
||||
|
||||
\begin{singlespace}
|
||||
Ground state
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.13947666
|
||||
O 0.00000000 0.00000000 1.14402883
|
||||
H 0.00000000 1.76627623 -2.23398653
|
||||
H 0.00000000 -1.76627623 -2.23398653
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\begin{singlespace}
|
||||
Excited state
|
||||
\begin{verbatim}
|
||||
C -0.09942705 0.00000000 1.27071070
|
||||
O 0.01987299 0.00000000 -1.23280536
|
||||
H 0.42778855 1.76729629 2.18470884
|
||||
H 0.42778855 -1.76729629 2.18470884
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Formamide}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00183118 0.00000000 0.79313299
|
||||
O 2.26817156 0.00000000 0.43918824
|
||||
N -1.76886033 0.00000000 -1.06219243
|
||||
H -0.84133459 0.00000000 2.68872485
|
||||
H -1.21254414 0.00000000 -2.87596907
|
||||
H -3.61627502 0.00000000 -0.65031317
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Hydrogen chloride}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
Cl 0.00000000 0.00000000 -0.02489783
|
||||
H 0.00000000 0.00000000 2.38483140
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Ketene}
|
||||
|
||||
\begin{singlespace}
|
||||
Ground state
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.44810151
|
||||
C 0.00000000 0.00000000 0.03498545
|
||||
O 0.00000000 0.00000000 2.23663914
|
||||
H 0.00000000 1.77432079 -3.43705988
|
||||
H 0.00000000 -1.77432079 -3.43705988
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\begin{singlespace}
|
||||
Excited state
|
||||
\begin{verbatim}
|
||||
C 2.04306304 0.00000000 -0.93056721
|
||||
C 0.00400918 0.00000000 0.83531393
|
||||
O -2.23710378 0.00000000 0.46984584
|
||||
H 1.63603518 0.00000000 -2.93687368
|
||||
H 3.96212800 0.00000000 -0.26649149
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Methanimine}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.10696646 0.00000000 1.11091130
|
||||
N 0.10764012 0.00000000 -1.29677742
|
||||
H -1.59140953 0.00000000 2.27296652
|
||||
H 1.90475160 0.00000000 2.09393982
|
||||
H -1.69956184 0.00000000 -1.96217482
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Nitrosomethane}
|
||||
|
||||
\begin{singlespace}
|
||||
Ground state
|
||||
\begin{verbatim}
|
||||
C -1.78426612 0.00000000 -1.07224050
|
||||
N -0.00541753 0.00000000 1.08060391
|
||||
O 2.18814985 0.00000000 0.43452135
|
||||
H -0.77343975 0.00000000 -2.86415606
|
||||
H -2.97471478 1.66801808 -0.86424584
|
||||
H -2.97471478 -1.66801808 -0.86424584
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\begin{singlespace}
|
||||
Excited state
|
||||
\begin{verbatim}
|
||||
C 1.86306273 0.00000000 -1.06035094
|
||||
N 0.00638693 0.00000000 1.02546010
|
||||
O -2.26923072 0.00000000 0.47699489
|
||||
H 3.72600129 0.00000000 -0.21094854
|
||||
H 1.58491147 1.68964774 -2.20977225
|
||||
H 1.58491147 -1.68964774 -2.20977225
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsection{Streptocyanine-C1}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 0.80488833
|
||||
N 0.00000000 2.19423463 -0.33580561
|
||||
N 0.00000000 -2.19423463 -0.33580561
|
||||
H 0.00000000 0.00000000 2.84436959
|
||||
H 0.00000000 2.36978315 -2.23371976
|
||||
H 0.00000000 -2.36978315 -2.23371976
|
||||
H 0.00000000 3.79412648 0.69399206
|
||||
H 0.00000000 -3.79412648 0.69399206
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsection{Thioformaldehyde}
|
||||
|
||||
\begin{singlespace}
|
||||
Ground state
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.08677304
|
||||
S 0.00000000 0.00000000 0.97251194
|
||||
H 0.00000000 1.73657773 -3.17013507
|
||||
H 0.00000000 -1.73657773 -3.17013507
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\clearpage
|
||||
|
||||
\begin{singlespace}
|
||||
Excited state
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.20256705
|
||||
S 0.00000000 0.00000000 1.02717172
|
||||
H 0.00000000 1.76634191 -3.21909384
|
||||
H 0.00000000 -1.76634191 -3.21909384
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsection{Water}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
O 0.00000000 0.00000000 -0.13209669
|
||||
H 0.00000000 1.43152878 0.97970006
|
||||
H 0.00000000 -1.43152878 0.97970006
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\clearpage
|
||||
|
||||
\section{Benchmark}
|
||||
|
||||
|
||||
\clearpage
|
||||
\begin{landscape}
|
||||
\renewcommand*{\arraystretch}{.55}
|
||||
\LTcapwidth=\textwidth
|
||||
\begin{footnotesize}
|
||||
\begin{longtable}{p{2.73cm}p{3.3cm}p{.55cm}|p{.75cm}p{.55cm}p{.95cm}p{.9cm}p{.9cm}p{1.6cm}p{.55cm}p{1.1cm}p{1.2cm}p{1.0cm}p{1.0 cm}}
|
||||
\caption{Comparisons between TBE obtained at the {\AVTZ} basis set in the frozen-core approximation (see Table 6) and the results obtained with various
|
||||
computational approaches, using the same basis set and approximation.
|
||||
STEOM stands for STEOM-CCSD and CC(3) for CCSDR(3).} \label{Table-benchmark}\\
|
||||
\hline
|
||||
Compound & State & TBE & CIS(D) & CC2 & STEOM & CCSD & CC(3) &CCSDT-3& CC3& CCSDT& CCSDTQ&ADC(2)& ADC(3) \\
|
||||
\hline
|
||||
\endfirsthead
|
||||
\hline
|
||||
Compound & State & TBE & CIS(D) & CC2 & STEOM & CCSD & CC(3) &CCSDT-3& CC3& CCSDT& CCSDTQ&ADC(2)& ADC(3) \\
|
||||
\hline
|
||||
\endhead
|
||||
\hline \multicolumn{13}{r}{{Continued on next page}} \\
|
||||
\endfoot
|
||||
\hline
|
||||
\endlastfoot
|
||||
Acetaldehyde &$^1A''(\mathrm{V};n \rightarrow \pi^\star)$ &4.31 &4.36 &4.41 &4.25 &4.36 &4.31 &4.32 &4.31 &4.29 & &4.24 &4.29 \\
|
||||
&$^3A''(\mathrm{V};n \rightarrow \pi^\star)$ &3.97 &3.96 &3.98 &3.95 &3.95 & & &3.95 &3.94$^a$& &3.83 &3.89 \\
|
||||
Acetylene &$^1\Sigma_u^- (\mathrm{V};\pi \rightarrow \pi^\star)$ &7.10 & 7.28 &7.26 &7.08 &7.15 &7.09 &7.09 &7.09 &7.09 &7.09$^b$&7.24 &6.72 \\
|
||||
&$^1\Delta_u (\mathrm{V};\pi \rightarrow \pi^\star)$ &7.44 & 7.62 &7.59 &7.42 &7.48 &7.43 &7.42 &7.42 &7.43 &7.43$^b$&7.56 &7.06 \\
|
||||
&$^3\Sigma_u^+ (\mathrm{V};\pi \rightarrow \pi^\star)$ &5.53 & 5.79 &5.76 &5.20 &5.45 & & &5.50 &5.51 &5.52$^b$&5.75 &5.24 \\
|
||||
&$^3\Delta_u (\mathrm{V};\pi \rightarrow \pi^\star)$ &6.40 & 6.62 &6.60 &6.13 &6.41 & & &6.40 &6.39 &6.39$^b$&6.57 &6.06 \\
|
||||
&$^3\Sigma_u^- (\mathrm{V};\pi \rightarrow \pi^\star)$ &7.08 & 7.31 &7.29 &6.84 &7.12 & & &7.07 &7.08$^a$&7.08$^c$&7.27 &6.72 \\
|
||||
&$^1A_u [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pi^\star)$ &3.64 & 3.85 &3.94 &3.65 &3.70 &3.66 &3.64 &3.64 &3.66 &3.64$^b$&3.78 &2.85 \\
|
||||
&$^1A_2 [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pi^\star)$ &3.85 & 4.06 &4.11 &3.85 &3.92 &3.85 &3.84 &3.84 &3.86 &3.85$^b$&3.99 &3.08 \\
|
||||
Ammonia &$^1A_2 (\mathrm{R};n \rightarrow 3s)$ &6.59 &6.37 &6.39 &6.55 &6.60 &6.57 &6.58 &6.57 &6.57 &6.59 &6.40 &6.63 \\
|
||||
&$^1E (\mathrm{R};n \rightarrow 3p)$ &8.16 &7.86 &7.85 &8.14 &8.15 &8.15 &8.15 &8.15 &8.14 &8.16$^b$&7.87 &8.21 \\
|
||||
&$^1A_1 (\mathrm{R};n \rightarrow 3p)$ &9.33 &9.04 &9.05 &9.33 &9.33 &9.32 &9.32 &9.32 &9.31 &9.34$^b$&9.05 &9.38 \\
|
||||
&$^1A_2 (\mathrm{R};n \rightarrow 4s)$ &9.96 &9.59 &9.65 &9.98 &9.95 &9.94 &9.95 &9.95 &9.94 &9.96$^b$&9.67 &10.00 \\
|
||||
&$^3A_2 (\mathrm{R};n \rightarrow 3s)$ &6.31 &6.18 &6.14 &6.31 &6.30 & & &6.29 &6.29 &6.30 &6.16 &6.31 \\
|
||||
Carbon monoxyde &$^1\Pi (\mathrm{V};n \rightarrow \pi^\star)$ & 8.49 &8.78 &8.64 &8.55 &8.59 &8.52 &8.51 &8.49 &8.57 &8.48 &8.69 &8.24 \\
|
||||
&$^1\Sigma^- (\mathrm{V};\pi \rightarrow \pi^\star)$ & 9.92 &10.13 &10.30 &9.90 &9.99 &9.98 &9.98 &9.99 &10.06 &9.93 &10.03 &9.73 \\
|
||||
&$^1\Delta (\mathrm{V};\pi \rightarrow \pi^\star)$ &10.06 &10.41 &10.60 &10.07 &10.12 &10.12 &10.11 &10.12 &10.18 &10.07$^b$&10.30 &9.82 \\
|
||||
&$^1\Sigma^+ (\mathrm{R})$ &10.95 &11.48 &11.11 &11.14 &11.22 &10.99 &11.02 &10.94 &10.94 &10.98 &11.32 &10.79 \\
|
||||
&$^1\Sigma^+ (\mathrm{R})$ &11.52 &11.71 &11.63 &11.75 &11.75 &11.53 &11.55 &11.49 &11.52 &11.52 &11.83 &11.33 \\
|
||||
&$^1\Pi (\mathrm{R})$ &11.72 &12.06 &11.83 &12.00 &11.96 &11.73 &11.76 &11.69 &11.77 &11.73$^b$&12.03 &11.56 \\
|
||||
&$^3\Pi (\mathrm{V};n \rightarrow \pi^\star)$ & 6.28 &6.51 &6.42 &6.32 &6.36 & & &6.30 &6.30 &6.28 &6.45 &5.97 \\
|
||||
&$^3\Sigma^+ (\mathrm{V};\pi \rightarrow \pi^\star)$ & 8.45 &8.63 &8.72 &8.37 &8.34 & & &8.45 &8.43 &8.43$^b$&8.54 &8.21 \\
|
||||
&$^3\Delta (\mathrm{V};\pi \rightarrow \pi^\star)$ & 9.27 &9.44 &9.56 &9.21 &9.23 & & &9.30 &9.33 &9.26 &9.33 &9.03 \\
|
||||
&$^3\Sigma^- (\mathrm{V};\pi \rightarrow \pi^\star)$ & 9.80 &10.10 &10.27 &9.83 &9.81 & & &9.82 & & &10.01 &9.53 \\
|
||||
&$^3\Sigma^+ (\mathrm{R})$ & 10.47 &10.98 &10.60 &10.73 &10.71 & & &10.45 &10.42 &10.50$^b$&10.83 &10.29 \\
|
||||
Cyclopropene &$^1B_1 (\mathrm{V};\sigma \rightarrow \pi^\star)$ &6.68 &6.90 &6.73 & &6.76 &6.68 &6.70 &6.68 &6.68 & &6.75 &6.56 \\
|
||||
&$^1B_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ &6.79 &6.90 &6.78 &6.94 &6.86 &6.73 &6.76 &6.73 &6.75 & &6.86 &6.56 \\
|
||||
&$^3B_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ &4.38 &4.55 &4.46 &4.36 &4.30 & & &4.34 &4.35$^a$& &4.45 &4.09 \\
|
||||
&$^3B_1 (\mathrm{V};\sigma \rightarrow \pi^\star)$ &6.45 &6.49 &6.44 &6.57 &6.46 & & &6.40 &6.40$^a$& &6.45 &6.26 \\
|
||||
Diazomethane &$^1A_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ &3.14 &3.55 &3.37 &3.20 &3.19 &3.12 &3.10 &3.07 &3.07 & &3.34 &2.74 \\
|
||||
&$^1B_1 (\mathrm{R};\pi \rightarrow 3s)$ &5.54 &5.65 &5.53 &5.57 &5.57 &5.48 &5.47 &5.45 &5.48 & &5.63 &5.23 \\
|
||||
&$^1A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ &5.90 &6.03 &6.00 &5.75 &5.94 &5.87 &5.86 &5.84 &5.86 & &5.97 &5.48 \\
|
||||
&$^3A_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ &2.79 &3.21 &3.08 &2.85 &3.19 & & &2.83 &2.82 & &3.01 &2.44 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ &4.05 &4.28 &4.25 &3.91 &3.95 & & &4.03 &4.02 & &4.20 &3.64 \\
|
||||
&$^3B_1 (\mathrm{R};\pi \rightarrow 3s)$ &5.35 &5.53 &5.53 &5.43 &5.42 & & &5.31 &5.34 & &5.50 &5.08 \\
|
||||
&$^3A_1 (\mathrm{R};\pi \rightarrow 3p)$ &6.82 &7.37 &7.04 & &6.85 & & &6.80 &6.80$^a$& &7.09 &6.36 \\
|
||||
&$^1A'' [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pi^\star)$ &0.71 &1.06 &0.90 &0.88 &0.81 &0.73 &0.70 &0.68 &0.67 & &0.81 &0.24 \\
|
||||
Dinitrogen &$^1\Pi_g (\mathrm{V};n \rightarrow \pi^\star)$ &9.34 &9.66 &9.44 &9.37 &9.41 &9.36 &9.35 &9.34 &9.33 &9.32 &9.48 &9.16 \\
|
||||
&$^1\Sigma_u^- (\mathrm{V};\pi \rightarrow \pi^\star)$ &9.88 &10.31 &10.32 &10.09 &10.00 &9.90 &9.89 &9.88 &9.89 &9.88 &10.26 &9.33 \\
|
||||
&$^1\Delta_u (\mathrm{V};\pi \rightarrow \pi^\star)$ &10.29 &10.85 &10.86 &10.56 &10.44 &10.33 &10.31 &10.29 &10.30 &10.29$^b$&10.79 &9.74 \\
|
||||
&$^1\Sigma_g^+ (\mathrm{R})$ &12.98 &13.67 &12.83 &13.13 &13.15 &13.04 &13.06 &13.01 &13.00 &12.98$^b$&12.99 &13.01 \\
|
||||
&$^1\Pi_u (\mathrm{R})$ &13.03 &13.64 &13.15 &13.43 &13.43 &13.28 &13.27 &13.22 &13.14 &13.09 &13.32 &12.98 \\
|
||||
&$^1\Sigma_u^+ (\mathrm{R})$ &13.09 &13.75 &12.89 &13.22 &13.26 &13.14 &13.16 &13.12 &13.12 &13.10$^b$&13.07 &13.09 \\
|
||||
&$^1\Pi_u (\mathrm{R})$ &13.46 &14.52 &13.96 &13.73 &13.67 &13.52 & &13.49 &13.45 &13.42 &14.00 &13.40 \\
|
||||
&$^3\Sigma_u^+ (\mathrm{V};\pi \rightarrow \pi^\star)$ &7.70 &8.20 &8.19 &7.70 &7.66 & & &7.68 &7.69 &7.70 &8.15 &7.25 \\
|
||||
&$^3\Pi_g (\mathrm{V};n \rightarrow \pi^\star)$ &8.01 &8.33 &8.19 &8.16 &8.09 & & &8.04 &8.03 &8.02 &8.20 &7.77 \\
|
||||
&$^3\Delta_u (\mathrm{V};\pi \rightarrow \pi^\star)$ &8.87 &9.30 &9.30 &8.94 &8.91 & & &8.87 &8.87 &8.87 &9.25 &8.36 \\
|
||||
&$^3\Sigma_u^- (\mathrm{V};\pi \rightarrow \pi^\star)$ &9.66 &10.29 &10.29 &9.90 &9.83 & & &9.68 &9.68 &9.66 &10.23 &9.14 \\
|
||||
Ethylene &$^1B_{3u} (\mathrm{R};\pi \rightarrow 3s)$ &7.40 &7.35 &7.29 &7.42 &7.42 &7.35 &7.36 &7.35 &7.37 &7.38$^b$&7.34 &7.17 \\
|
||||
&$^1B_{1u} (\mathrm{V};\pi \rightarrow \pi^\star)$ &7.91 &7.95 &7.92 & &8.02 &7.89 &7.92 &7.91 &7.92 &7.91$^b$&7.91 &7.69 \\
|
||||
&$^1B_{1g} (\mathrm{R};\pi \rightarrow 3p)$ &8.07 &8.01 &7.95 &8.10 &8.08 &8.02 &8.03 &8.03 &8.04 &8.05$^b$&7.99 &7.84 \\
|
||||
&$^3B_{1u} (\mathrm{V};\pi \rightarrow \pi^\star)$ &4.54 &4.62 &4.59 &4.36 &4.46 & & &4.53 &4.53 &4.53$^b$&4.59 &4.28 \\
|
||||
&$^3B_{3u} (\mathrm{R};\pi \rightarrow 3s)$ &7.23 &7.26 &7.19 &7.31 &7.29 & & &7.24 &7.25 &7.25$^b$&7.23 &7.05 \\
|
||||
&$^3B_{1g} (\mathrm{R};\pi \rightarrow 3p)$ &7.98 &7.97 &7.91 &8.08 &8.03 & & &7.98 &7.99 &7.99$^b$&7.95 &7.80 \\
|
||||
Formaldehyde &$^1A_2 (\mathrm{V}; n \rightarrow \pi^\star)$ &3.98 &4.04 &4.07 &3.91 &4.01 &3.97 &3.98 &3.97 &3.95 &3.96$^b$&3.92 &3.90 \\
|
||||
&$^1B_2 (\mathrm{R};n \rightarrow 3s)$ &7.23 &6.64 &6.56 &7.19 &7.23 &7.18 &7.21 &7.18 &7.16 &7.21$^b$&6.50 &7.62 \\
|
||||
&$^1B_2 (\mathrm{R};n \rightarrow 3p)$ &8.13 &7.56 &7.57 &8.05 &8.12 &8.08 &8.11 &8.07 &8.07 &8.11$^b$&7.53 &8.45 \\
|
||||
&$^1A_1 (\mathrm{R};n \rightarrow 3p)$ &8.23 &8.16 &7.52 &8.18 &8.21 &8.17 &8.21 &8.18 &8.16 &8.21$^b$&7.47 &8.61 \\
|
||||
&$^1A_2 (\mathrm{R};n \rightarrow 3p)$ &8.67 &8.04 &8.04 &8.68 &8.65 &8.63 &8.66 &8.64 &8.61 &8.66$^b$&7.99 &9.02 \\
|
||||
&$^1B_1 (\mathrm{V};\sigma \rightarrow \pi^\star)$ &9.22 &9.38 &9.32 &9.08 &9.28 &9.20 &9.20 &9.19 &9.17 &9.18$^b$&9.17 &9.17 \\
|
||||
&$^1A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ &9.43 &9.08 &9.54 & &9.67 &9.51 &9.51 &9.48 &9.49 &9.44$^b$&9.46 &9.05 \\
|
||||
&$^3A_2 (\mathrm{V};n \rightarrow \pi^\star)$ &3.58 &3.58 &3.59 &3.54 &3.56 & & &3.57 &3.56 &3.57$^b$&3.46 &3.48 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ &6.06 &6.27 &6.30 &5.89 &5.97 & & &6.05 &6.05 &6.06$^b$&6.20 &5.71 \\
|
||||
&$^3B_2 (\mathrm{R};n \rightarrow 3s)$ &7.06 &6.66 &6.44 &7.07 &7.08 & & &7.03 &7.02 &7.07$^b$&6.39 &7.44 \\
|
||||
&$^3B_2 (\mathrm{R};n \rightarrow 3p)$ &7.94 &7.52 &7.45 &7.98 &7.94 & & &7.92 &7.90 &7.94$^b$&7.41 &8.23 \\
|
||||
&$^3A_1 (\mathrm{R};n \rightarrow 3p)$ &8.10 &7.68 &7.44 &8.15 &8.09 & & &8.08 &8.06 &8.11$^b$&7.40 &8.46 \\
|
||||
&$^3B_1 (\mathrm{R};n \rightarrow 3d)$ &8.42 &8.57 &8.52 &8.36 &8.43 & & &8.41 &8.40 &8.41$^b$&8.39 &8.32 \\
|
||||
&$^1A^" [\mathrm{F}] (\mathrm{V};n \rightarrow \pi^\star)$ &2.80 &2.90 &2.97 &2.81 &2.93 &2.86 &2.86 &2.84 &2.82 &2.84$^b$&2.71 &2.77 \\
|
||||
Formamide &$^1A'' \mathrm{V};(n \rightarrow \pi^\star)$ &\hl{xxx} &5.58 &5.69 &5.72 &5.69 &5.66 &5.67 &5.66 &\hl{xxx} & &5.45 &5.75 \\
|
||||
&$^1A' (\mathrm{R};n \rightarrow 3s)$ &\hl{xxx} &6.82$^c$&6.31$^c$&6.94 &6.99 &6.83 &6.83 &6.74 &\hl{xxx} & &6.26$^c$&7.20 \\
|
||||
&$^1A' (\mathrm{V};\pi \rightarrow \pi^\star)$ &\hl{xxx} &6.84$^c$&7.55$^c$& &7.55$^c$&7.44 &7.68$^c$&7.62$^c$&\hl{xxx} & &7.39$^c$&7.80$^c$\\
|
||||
&$^1A' (\mathrm{R};n \rightarrow 3p)$ &\hl{xxx} &6.89$^c$&6.89$^c$& &7.78$^c$&7.65 &7.46$^c$&7.40$^c$&\hl{xxx} & &6.83$^c$&8.12$^c$\\
|
||||
&$^3A'' (\mathrm{V};n \rightarrow \pi^\star)$ &\hl{xxx} &5.31 &5.36 &5.29 &5.36 & & &5.38 &\hl{xxx} & &5.15 &5.42 \\
|
||||
&$^3A' (\mathrm{V};\pi \rightarrow \pi^\star)$ &\hl{xxx} &6.07 &5.99 &5.74 &5.77 & & &5.82 &\hl{xxx} & &5.88 &5.63 \\
|
||||
Hydrogen chloride & $^1\Pi (\mathrm{CT})$ &7.84 &7.98 &7.96 &7.91 &7.91 &7.84 &7.85 &7.84 &7.83 &7.84 &7.97 &7.79 \\
|
||||
Ketene &$^1A_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ &3.86 &4.18 &4.17 &3.84 &3.97 &3.92 &3.90 &3.88 &3.87 & &4.11 &3.67 \\
|
||||
&$^1B_1 (\mathrm{R};n \rightarrow 3s)$ &6.01 &6.09 &5.94 &6.08 &6.09 &5.99 &5.99 &5.96 &5.99 & &6.03 &5.87 \\
|
||||
&$^1A_2 (\mathrm{R};\pi \rightarrow 3p)$ &7.18 &7.25 &7.09 &7.29 &7.29 &7.19 &7.20 &7.16 &7.20 & &7.18 &7.07 \\
|
||||
&$^3A_2 (\mathrm{V};n \rightarrow \pi^\star)$ &3.77 &4.00 &3.98 &3.82 &3.83 & & &3.78 &3.78 & &3.92 &3.56 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ &5.61 &5.79 &5.72 &5.53 &5.55 & & &5.61 &5.60 & &5.67 &5.39 \\
|
||||
&$^3B_1 (\mathrm{R};n \rightarrow 3s)$ &5.79 &5.94 &5.77 &5.91 &5.89 & & &5.76 &5.80 & &5.85 &5.67 \\
|
||||
&$^3A_2 (\mathrm{R};\pi \rightarrow 3p)$ &7.12 &7.24 &7.06 &7.32 &7.25 & & &7.12 &7.17 & &7.15 &7.03 \\
|
||||
&$^1A^" [\mathrm{F}] (\mathrm{V};\pi \rightarrow \pi^\star)$ &1.00 &1.28 &1.26 &1.03 &1.13 &1.06 &1.03 &1.00 &1.00 & &1.19 &0.67 \\
|
||||
Methanimine &$^1A^"(\mathrm{V}; n \rightarrow \pi^\star)$ &5.23 &5.38 &5.32 &5.20 &5.28 &5.20 &5.22 &5.20 &5.19 & &5.29 &5.05 \\
|
||||
&$^3A^" (\mathrm{V}; n \rightarrow \pi^\star)$ &4.65 &4.71 &4.65 &4.62 &4.63 & & &4.61 &4.61 & &4.61 &4.44 \\
|
||||
Nitrosomethane&$^1A'' (\mathrm{V};n \rightarrow \pi^\star)$ &1.96 &2.03 &1.98 &1.80 &1.98 &1.96 &1.96 &1.96 &1.95 & &1.88 &1.72 \\
|
||||
&$^1A' (\mathrm{V};n,n \rightarrow \pi^\star,\pi^\star)$ &4.72 & & & & & &6.02 &5.76 &5.29 & & &3.00 \\
|
||||
&$^1A' (\mathrm{R};n \rightarrow 3s/3p)$ &6.37 &5.89 &5.84 &6.51 &6.43 &6.33 &6.38 &6.31 &6.30 & &5.86 &6.48 \\
|
||||
&$^3A'' (\mathrm{V};n \rightarrow \pi^\star)$ &1.16 &1.18 &1.12 &0.99 &1.11 & & &1.14 &1.13 & &1.03 &0.84 \\
|
||||
&$^3A' (\mathrm{V};\pi \rightarrow \pi^\star)$ &5.60 &5.89 &5.74 &5.04 &5.43 & & &5.51 &5.51$^a$& &5.75 &5.04 \\
|
||||
&$^1A'' [\mathrm{F}] (\mathrm{V};n \rightarrow \pi^\star)$ &\hl{xxx} &1.73 &1.68 &1.49 &1.68 &1.67 &1.67 &1.69 &\hl{xxx} & &1.55 &1.40 \\
|
||||
Streptocyanine &$^1B_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ &7.13 &6.99 &7.20 &6.76 &7.24 &7.12 &7.16 &7.13 &7.11 & &7.00 &7.16 \\
|
||||
& $^3B_2 (\mathrm{V};\pi \rightarrow \pi^\star)$ & 5.47 &5.61 &5.60 &5.40 &5.45 & & &5.48 &5.47 & &5.55 &5.33 \\
|
||||
Thioformaldehyde&$^1A_2 (\mathrm{V};n \rightarrow \pi^\star)$ &2.22 &2.30 &2.34 &2.17 &2.29 &2.22 &2.24 &2.23 &2.21 &2.22$^b$&2.24 &2.05 \\
|
||||
&$^1B_2 (\mathrm{R};n \rightarrow 4s)$ &5.96 &5.87 &5.82 &5.92 &5.97 &5.90 &5.94 &5.91 &5.89 &5.91$^b$&5.80 &5.94 \\
|
||||
&$^1A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ &6.38 &6.65 &6.71 &6.48 &6.63 &6.50 &6.51 &6.48 &6.47 &6.38$^b$&6.57 &5.98 \\
|
||||
&$^3A_2 (\mathrm{V};n \rightarrow \pi^\star)$ &1.94 &1.94 &1.94 &1.91 &1.95 & & &1.94 &1.93 &1.93$^b$&1.86 &1.77 \\
|
||||
&$^3A_1 (\mathrm{V};\pi \rightarrow \pi^\star)$ & 3.43 &3.49 &3.48 &3.18 &3.28 & & &3.38 &3.38 &3.39$^b$&3.45 &3.07 \\
|
||||
&$^3B_2 (\mathrm{R};n \rightarrow 4s)$ &5.72 &5.78 &5.64 &5.71 &5.76 & & &5.72 &5.71 &5.73$^b$&5.62 &5.71 \\
|
||||
&$^1A_2 [\mathrm{F}] (\mathrm{V};n \rightarrow \pi^\star)$ &1.95 &2.00 &2.09 &1.92 &2.05 &1.97 &1.98 &1.97 &1.98 &1.96$^b$&1.92 &1.80 \\
|
||||
Water & $^1B_1 (\mathrm{R}; n \rightarrow 3s)$ &7.62 &7.17 &7.23 &7.56 &7.60 &7.60 &7.61 &7.65 &7.65 &7.62 &7.18 &7.84 \\
|
||||
& $^1A_2 (\mathrm{R}; n \rightarrow 3p)$ &9.41 &8.92 &8.89 &9.37 &9.36 &9.38 &9.38 &9.43 &9.42 &9.40 &8.84 &9.63 \\
|
||||
& $^1A_1 (\mathrm{R}; n \rightarrow 3s)$ &9.99 &9.52 &9.58 &9.92 &9.96 &9.96 &9.97 &10.00 &9.98 &9.98 &9.52 &10.22 \\
|
||||
& $^3B_1 (\mathrm{R}; n \rightarrow 3s)$ &7.25 &6.92 &6.91 &7.24 &7.20 & & &7.28 &7.28 &7.24 &6.86 &7.41 \\
|
||||
& $^3A_2 (\mathrm{R}; n \rightarrow 3p)$ &9.24 &8.91 &8.77 &9.21 &9.20 & & &9.26 &9.25 &9.23 &8.72 &9.43 \\
|
||||
& $^3A_1 (\mathrm{R}; n \rightarrow 3s)$ &9.54 &9.30 &9.20 &9.51 &9.49 & & &9.56 &9.54 &9.53 &9.15 &9.70 \\
|
||||
\end{longtable}
|
||||
\end{footnotesize}
|
||||
\begin{flushleft}\begin{footnotesize}\begin{singlespace}
|
||||
$^a${{\CCSDT}/{\AVDZ} value corrected with the difference between {\CC{3}}/{\AVTZ} and {\CC{3}}/{\AVDZ} values;}
|
||||
$^b${{\CCSDTQ}/{\AVDZ} value corrected with the difference between {\CCSDT}/{\AVTZ} and {\CCSDT}/{\AVDZ} values;}
|
||||
$^c${{\CCSDTQ}/{\AVDZ} value corrected with the difference between {\CC{3}}/{\AVTZ} and {\CC{3}}/{\AVDZ} values;}
|
||||
$^d${Strong state mixing.}
|
||||
\end{singlespace}\end{footnotesize}\end{flushleft}
|
||||
\end{landscape}
|
||||
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
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\newcommand{\mr}{\multirow}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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% energies
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\EFCI}{E_\text{FCI}}
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\newcommand{\EsCI}{E_\text{sCI}}
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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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||||
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||||
\newcommand{\ra}{\rightarrow}
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\newcommand{\pis}{\pi^\star}
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\newcommand{\si}{\sigma}
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||||
\newcommand{\sis}{\sigma^\star}
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\newcommand{\nO}{n_\text{O}}
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\newcommand{\nN}{n_\text{N}}
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\newcommand{\piCC}{\pi_{\ce{CC}}}
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\newcommand{\pisCC}{\pi_{\ce{CC}}^\star}
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\newcommand{\piCO}{\pi_{\ce{CO}}}
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\newcommand{\pisCO}{\pi_{\ce{CO}}^\star}
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\newcommand{\siCC}{\si_{\ce{CC}}}
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\newcommand{\sisCC}{\si_{\ce{CC}}^\star}
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\newcommand{\siCO}{\si_{\ce{CO}}}
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\newcommand{\sisCO}{\si_{\ce{CO}}^\star}
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||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\newcommand{\CEISAM}{Laboratoire CEISAM (UMR 6230), CNRS, Universit\'e de Nantes, Nantes, France}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\title{Reference Energies for Double Excitations}
|
||||
|
||||
\author{Pierre-Fran\c{c}ois Loos}
|
||||
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Martial Boggio-Pasqua}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Anthony Scemama}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Michel Caffarel}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Denis Jacquemin}
|
||||
\affiliation{\CEISAM}
|
||||
|
||||
\begin{abstract}
|
||||
Excited states exhibiting double excitation character are notoriously difficult to model using conventional single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC).
|
||||
In addition, these states are typical experimentally ``dark'' making their detection in photo-absorption spectra very challenging.
|
||||
Nonetheless, they play a key role in the faithful description of many physical, chemical, and biological processes.
|
||||
In the present work, we provide accurate reference excitation energies for transitions involving a substantial amount of double excitation using a series of increasingly large diffuse-containing atomic basis sets.
|
||||
Our set gathers 20 vertical transitions from 14 small- and medium-size molecules (acrolein, benzene, beryllium atom, butadiene, carbon dimer and trimer, ethylene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine).
|
||||
Depending on the size of the molecule, selected configuration interaction (sCI) and/or multiconfigurational (CASSCF, CASPT2, (X)MS-CASPT2 and NEVPT2) calculations are performed in order to obtain reliable estimates of the vertical transition energies.
|
||||
In addition, coupled cluster approaches including at least contributions from iterative triples (such as CC3, CCSDT, CCSDTQ, and CCSDTQP) are assessed.
|
||||
Our results clearly evidence that the error in CC methods is intimately related to the amount of double excitation character of the transition.
|
||||
For ``pure'' double excitations (i.e.~for transitions which do not mix with single excitations), the error in CC3 can easily reach \IneV{$1$}, while it goes down to few tenths of an eV for more common transitions (like in \emph{trans}-butadiene) involving a significant amount of singles.
|
||||
As expected, CC approaches including quadruples yield highly accurate results for any type of transitions.
|
||||
The quality of the excitation energies obtained with multiconfigurational methods is harder to predict.
|
||||
We have found that the overall accuracy of these methods is highly dependent of both the system and the selected active space.
|
||||
The inclusion of the $\si$ and $\sis$ orbitals in the active space, even for transitions involving mostly $\pi$ and $\pis$ orbitals, is mandatory in order to reach high accuracy.
|
||||
A theoretical best estimate (TBE) is reported for each transition.
|
||||
We believe that these reference data will be valuable for future methodological developments aiming at accurately describing double excitations.
|
||||
\\
|
||||
\begin{center}
|
||||
\includegraphics[width=0.4\linewidth]{TOC}
|
||||
\\
|
||||
\bf TOC graphical abstract
|
||||
\end{center}
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{
|
||||
Introduction
|
||||
\label{sec:intro}
|
||||
}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Within the theoretical and computational quantum chemistry community, the term \textit{``double excitation''} commonly refers to a state whose configuration interaction (CI) or coupled cluster (CC) expansion includes \textit{significant} coefficients or amplitudes associated to doubly-excited Slater determinants, i.e., determinants in which two electrons have been promoted from occupied to virtual orbitals of the chosen \textit{reference} determinant.
|
||||
Obviously, this definition is fairly ambiguous as it is highly dependent on the actual reference Slater determinant, and on the magnitude associated with the term ``\textit{significant}''.
|
||||
Moreover, such a picture of placing electrons in orbitals only really applies to one-electron theories, e.g., Hartree-Fock \cite{SzaboBook} or Kohn-Sham. \cite{ParrBook}
|
||||
In contrast, in a many-electron picture, an excited state is a linear combination of Slater determinants usually built from an intricate mixture of single, double and higher excitations.
|
||||
In other words, the definition of a double excitation remains fuzzy, and this has led to controversies regarding the nature of the 2\,\ex{1}{A}{1g}{} and 1\,\ex{1}{E}{2g}{} excited states of butadiene \cite{Shu_2017,Barca_2018a,Barca_2018b} and benzene, \cite{Barca_2018a,Barca_2018b} respectively, to mention two well-known examples.
|
||||
Although these two states have been classified as doubly-excited states in the past, Barca et al.~have argued that they can be seen as singly-excited states if one allows sufficient orbital relaxation in the excited state. \cite{Barca_2018a,Barca_2018b}
|
||||
Nonetheless, in the \alert{remainder} of this paper, we will follow one of the common definition and define a double excitation as an excited state with a significant amount of double excitation character in the multideterminant expansion.
|
||||
|
||||
Double excitations do play a significant role in the proper description of several key physical, chemical and biological processes, e.g., in photovoltaic devices, \cite{Delgado_2010} in the photophysics of vision, \cite{Palczewski_2006} and in photochemistry in general \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} involving ubiquitous conical intersections. \cite{Levine_2006}
|
||||
The second example is intimately linked to the correct location of the excited states of polyenes, \cite{Serrano-Andres_1993,Cave_1988b,Lappe_2000,Boggio-Pasqua_2004,Maitra_2004,Cave_2004,Wanko_2005,Starcke_2006,Catalan_2006,Mazur_2009,Angeli_2010,Mazur_2011,Huix-Rotllant_2011}
|
||||
that are closely related to rhodopsin which is involved in visual phototransduction. \cite{Gozem_2012,Gozem_2013,Gozem_2013a,Gozem_2014,Huix-Rotllant_2010,Xu_2013,Schapiro_2014,Tuna_2015,Manathunga_2016}
|
||||
Though doubly-excited states do not appear directly in photo-absorption spectra, these dark states strongly mix with the bright singly-excited states leading to the formation of satellite peaks. \cite{Helbig_2011,Elliott_2011}
|
||||
|
||||
From a theoretical point of view, double excitations are notoriously difficult to model using conventional single-reference methods. \cite{Sundstrom_2014}
|
||||
For example, the adiabatic approximation of time-dependent density-functional theory (TD-DFT) \cite{Casida} yields reliable excitation spectra with great efficiency in many cases.
|
||||
Nevertheless, fundamental deficiencies in TD-DFT have been reported for the computation of extended conjugated systems, \cite{Woodcock_2002,Tozer_2003} charge-transfer states, \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004} Rydberg states,\cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} conical intersections, \cite{Tapavicza_2008,Levine_2006} and, more importantly here, for states with double excitation character. \cite{Levine_2006,Tozer_2000,Elliott_2011,Loos_2018}
|
||||
Although, using range-separated hybrids \cite{Tawada_2004,Yanai_2004} provides an effective solution to the first three cases, one must go beyond the ubiquitous adiabatic approximation to capture the latter two. \alert{(However, this is only true for some types of charge-transfer excitations, as recently discussed by Maitra. \cite{Maitra_2017})}
|
||||
One possible solution is provided by spin-flip TD-DFT which describes double excitations as single excitations from the lowest triplet state. \cite{Huix-Rotllant_2010,Krylov_2001,Shao_2003,Wang_2004,Wang_2006,Minezawa_2009}
|
||||
However, major limitations pertain. \cite{Huix-Rotllant_2010} In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has been proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} (see also Refs.~\onlinecite{Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012}).
|
||||
In this approach the exchange-correlation kernel is made frequency dependent, \cite{Romaniello_2009a,Sangalli_2011} which allows to treat doubly-excited states.
|
||||
Albeit far from being a mature black-box approach, ensemble DFT \cite{Theophilou_1979,Gross_1988,Gross_1988a,Oliveira_1988} is another viable alternative currently under active development. \cite{Kazaryan_2008,Filatov_2015,Senjean_2015,Filatov_2015b,Filatov_2015c,Deur_2017,Gould_2018,Sagredo_2018}
|
||||
|
||||
As shown by Watson and Chan, \cite{Watson_2012} one can also rely on high-level truncation of the equation-of-motion (EOM) formalism of CC theory in order to capture double excitations. \cite{Hirata_2000,Sundstrom_2014}
|
||||
However, in order to provide a satisfactory level of correlation for a doubly-excited state, one must, at least, introduce contributions from the triple excitations in the CC expansion.
|
||||
In practice, this is often difficult as the scalings of CC3, \cite{Christiansen_1995b,Koch_1997} CCSDT, \cite{Noga_1987} and CCSDTQ \cite{Kucharski_1991} are $N^7$, $N^8$ and $N^{10}$, respectively (where $N$ is the number of basis functions), obviously limiting the applicability of
|
||||
this strategy to small molecules.
|
||||
|
||||
Multiconfigurational methods constitute a more natural class of methods to properly treat double excitations.
|
||||
Amongst these approaches, one finds complete active space self-consistent field (CASSCF), \cite{Roos} its second-order perturbation-corrected variant (CASPT2), \cite{Andersson_1990} as well as the second-order $n$-electron valence state perturbation theory (NEVPT2). \cite{Angeli_2001a, Angeli_2001b, Angeli_2002}
|
||||
However, the exponential scaling of such methods with the number of active electrons and orbitals also limits their application to small active spaces in their traditional implementation, although using sCI as an active-space solver allows to target much larger active spaces. \cite{Smith_2017}
|
||||
|
||||
|
||||
Alternatively to CC and multiconfigurational methods, one can also compute transition energies for both singly- and doubly-excited states using selected configuration interaction (sCI) methods
|
||||
\cite{Bender_1969,Whitten_1969,Huron_1973,Evangelisti_1983, Cimiraglia_1985, Cimiraglia_1987, Illas_1988, Povill_1992} which have recently demonstrated their ability to reach near full CI (FCI) quality energies for small molecules.
|
||||
\cite{Abrams_2005,Bunge_2006,Bytautas_2009,Giner_2013,Caffarel_2014,Giner_2015,Garniron_2017b,Caffarel_2016,Holmes_2016,Sharma_2017,Holmes_2017,Chien_2018,Scemama_2018a,Scemama_2018b,Loos_2018,Garniron_2018,Evangelista_2014,Schriber_2016,Tubman_2016,Liu_2016,Per_2017,Ohtsuka_2017,Zimmerman_2017}
|
||||
The idea behind such methods is to avoid the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively
|
||||
determinants in the FCI space. \cite{Giner_2013,Giner_2015,Caffarel_2016,Scemama_2018a,Scemama_2018b,Garniron_2018,Sharma_2017,Blunt_2018}
|
||||
|
||||
By systematically increasing the order of the CC expansion, the number of determinants in the sCI expansion as well as the size of the one-electron basis set, some of us have recently defined a reference series of more than 100 very
|
||||
accurate vertical transition energies in 18 small compounds. \cite{Loos_2018} However, this set is constituted almost exclusively of single excitations. Here, we report accurate reference excitation energies for double excitations obtained
|
||||
with both sCI and multiconfigurational methods for a significant number of small- and medium-size molecules using various diffuse-containing basis sets. Moreover, the accuracy obtained with several coupled cluster approaches including,
|
||||
at least, triple excitations are assessed. We believe that these reference data are particularly valuable for future developments of methods aiming at accurately describing double excitations.
|
||||
|
||||
This manuscript is organized as follows. Computational details are reported in Sec.~\ref{sec:comp} for EOM-CC (Sec.~\ref{sec:CC}), multiconfigurational (Sec.~\ref{sec:CAS}) and sCI (Sec.~\ref{sec:sCI}) methods.
|
||||
In Section \ref{sec:res}, we discuss our results for each compound and report a list of theoretical best estimates (TBEs) for each transition.
|
||||
We further discuss the overall performance of the different methods and draw our conclusions in Sec.~\ref{sec:ccl}.
|
||||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{fig1}
|
||||
\caption{
|
||||
Structure of the various molecules considered in the present set.
|
||||
\label{fig:mol}
|
||||
}
|
||||
\end{figure}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{
|
||||
Computational details
|
||||
\label{sec:comp}
|
||||
}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
All geometries used in the present study are available in {\SI}. They have been obtained at the CC3/aug-cc-pVTZ level (except for hexatriene where the geometry has been optimized at the CCSD(T)/aug-cc-pVTZ level)
|
||||
without applying the frozen core approximation following the same protocol as in earlier works where additional details can be found. \cite{Budzak_2017,Loos_2018} These geometry optimizations were performed with
|
||||
DALTON \cite{dalton} or CFOUR. \cite{cfour}
|
||||
The so-called $\%T_1$ metric giving the percentage of single excitation calculated at the CC3 level in DALTON is employed to characterize the various states.
|
||||
For all calculations, we use the well-known Pople's 6-31+G(d) \alert{(in its ``5D'' spherical version as implemented by default in MOLPRO and DALTON)} \cite{Francl_1982} and Dunning's aug-cc-pVXZ (X $=$ D, T and Q) \cite{Dunning_1989} atomic basis sets.
|
||||
In the following, we employ the AVXZ shorthand notations for Dunning's basis sets.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Coupled cluster calculations
|
||||
\label{sec:CC}
|
||||
}
|
||||
%---------------------------------------------
|
||||
|
||||
Unless otherwise stated, the CC transition energies \cite{Kallay_2004} were computed in the frozen-core approximation.
|
||||
Globally, we used DALTON \cite{dalton} to perform the CC3 calculations, \cite{Christiansen_1995b,Koch_1997} CFOUR \cite{cfour} for the CCSDT \cite{Noga_1987} calculations, and MRCC \cite{mrcc} for CCSDT, \cite{Noga_1987} CCSDTQ, \cite{Kucharski_1991} (and higher) calculations.
|
||||
Because CFOUR and MRCC rely on different algorithms to locate excited states, we have interchangeably used these two softwares for the CCSDT calculations depending on the targeted transition.
|
||||
Default program setting were generally applied, and when modified they have been tightened.
|
||||
Note that transition energies are identical in the EOM and linear response (LR) CC formalisms.
|
||||
Consequently, for the sake of brevity, we do not specify the EOM and LR terms in the remaining of this study.
|
||||
The total energies of all CC calculations are available in {\SI}.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Multiconfigurational calculations
|
||||
\label{sec:CAS}
|
||||
}
|
||||
%---------------------------------------------
|
||||
|
||||
State-averaged (SA) CASSCF and CASPT2 \cite{Roos, Andersson_1990} have been performed with MOLPRO \alert{(RS2 contraction level)}. \cite{molpro}
|
||||
Concerning the NEVPT2 calculations, the partially-contracted (PC) and strongly-contracted (SC) variants have been systematically tested. \cite{Angeli_2001a, Angeli_2001b, Angeli_2002}
|
||||
From a strict theoretical point of view, we point out that PC-NEVPT2 is supposed to be more accurate than SC-NEVPT2 given that it has a larger number of perturbers and greater flexibility.
|
||||
Additional information and technical details about the CASSCF (as well as CASSCF excitation energies), CASPT2 and NEVPT2 calculations can be found in {\SI}.
|
||||
When there is a strong mixing between states with same spin and spatial symmetries, we have also performed calculations with multi-state (MS) CASPT2 \alert{(MS-MR formalism)}, \cite{Finley_1998} and its extended variant (XMS-CASPT2). \cite{Shiozaki_2011}
|
||||
Unless otherwise stated, all CASPT2 calculations have been performed with level shift and IPEA parameters set to the standard values of $0.3$ and \InAU{$0.25$}, respectively.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Selected configuration interaction calculations
|
||||
\label{sec:sCI}
|
||||
}
|
||||
%---------------------------------------------
|
||||
|
||||
The sCI calculations reported here employ the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) \cite{Huron_1973, Evangelisti_1983, Giner_2013} algorithm.
|
||||
We refer the interested reader to Refs.~\onlinecite{Caffarel_2016b, Scemama_2018a, Scemama_2018b, Garniron_2018, Loos_2018} for more details about sCI methods, and the CIPSI algorithm in particular.
|
||||
|
||||
In order to treat the electronic states on equal footing, a common set of determinants is selected for the ground state and excited states.
|
||||
These calculations can then be classified as ``state-averaged'' sCI. Moreover, to speed up convergence to the FCI limit, a common set of natural orbitals issued from a preliminary sCI calculation is employed.
|
||||
For the largest systems, few iterations might be required to obtain a well-behaved convergence of the excitation energies with respect to the number of determinants.
|
||||
For a given atomic basis set, we estimate the FCI limit by linearly extrapolating the sCI energy $\EsCI$ as a function of the second-order perturbative correction $\EPT$ which is an estimate of the truncation error in the sCI algorithm, i.e., $\EPT \approx \EFCI - \EsCI$.
|
||||
When $\EPT = 0$, the FCI limit has effectively been reached.
|
||||
To provide an estimate of the extrapolation error, we report the energy difference between the excitation energies obtained with two- and three-point linear fits.
|
||||
It is, however, a rough estimate as there is no univocal method to quantitatively measure the extrapolation error.
|
||||
This extrapolation procedure has nevertheless been shown to be robust, even for challenging chemical situations. \cite{Holmes_2017, Sharma_2017, Scemama_2018a, Scemama_2018b, Chien_2018, Garniron_2018, Loos_2018}
|
||||
In the following, these extrapolated sCI results are labeled exFCI.
|
||||
Here, $\EPT$ has been efficiently evaluated with a recently proposed hybrid stochastic-deterministic algorithm. \cite{Garniron_2017b}
|
||||
Note that we do not report error bars associated with $\EPT$ because the statistical errors originating from this algorithm are orders of magnitude smaller than the extrapolation errors.
|
||||
|
||||
All the sCI calculations have been performed in the frozen core approximation with the electronic structure software QUANTUM PACKAGE, developed in Toulouse and freely available. \cite{QP}
|
||||
For the largest molecules considered here, our sCI wave functions contain up to $2 \times 10^8$ determinants which corresponds to an increase of two orders of magnitude compared to our previous study. \cite{Loos_2018}
|
||||
Additional information about the sCI wave functions, excitations energies as well as their extrapolated values can be found in {\SI}.
|
||||
|
||||
%%% TABLE 1 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{longtable*}{lllddddc}
|
||||
\caption{
|
||||
\label{tab:2Ex}
|
||||
Vertical transition energies (in eV) for excited states with significant double excitation character in various molecules obtained with various methods and basis sets.
|
||||
$\%T_1$ is the percentage of single excitation calculated at the CC3 level.
|
||||
For exFCI, an estimate of the extrapolation error is reported in parenthesis (not a statistical error bar, see text for details).
|
||||
Values from the literature are provided when available \alert{together with their respective reference and level of theory as footnote.}}
|
||||
\\
|
||||
\hline\hline
|
||||
Molecule & Transition & Method & \mc{4}{c}{Basis set} & \mcc{Lit.} \\
|
||||
\cline{4-7}
|
||||
& & &\mcc{6-31+G(d)}& \mcc{AVDZ}& \mcc{AVTZ}& \mcc{AVQZ} \\
|
||||
\hline
|
||||
\endfirsthead
|
||||
\hline\hline
|
||||
Molecule & Transition & Method & \mc{4}{c}{Basis set} & \mcc{Lit.} \\
|
||||
\cline{4-7}
|
||||
& & &\mcc{6-31+G(d)}& \mcc{AVDZ}& \mcc{AVTZ}& \mcc{AVQZ} \\
|
||||
\hline
|
||||
\endhead
|
||||
\hline \multicolumn{8}{r}{{Continued on next page}} \\
|
||||
\endfoot
|
||||
\hline\hline
|
||||
\multicolumn{8}{l}{$^a$Reference \onlinecite{Saha_2006}: SAC-CI results using $[4s2p1d/2s] + [2s2p2d]$ basis.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^b$Reference \onlinecite{Christiansen_1996}: CC3 results using ANO1 basis (see footnote of Table V in Ref.~\onlinecite{Christiansen_1996} for more details about the basis set).}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^c$Reference \onlinecite{Barca_2018b}: Maximum overlap method (MOM) calculations at the BLYP/cc-pVTZ level.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^d$Reference \onlinecite{Galvez_2002}: Multideterminant explicitly-correlated calculations with 17 variational nonlinear parameters in the correlation factor.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^e$Reference \onlinecite{Dallos_2004}:RCA3-F/MR-CISD+Q results with aug'-cc-pVTZ.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^f$Reference \onlinecite{Watson_2012}: Incremental EOM-CC procedure (up to EOM-CCSDTQ) with CBS extrapolation.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^g$Reference \onlinecite{Chien_2018}: Heat-bath CI results using AVDZ basis.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^h$Reference \onlinecite{Boschen_2014}: CEEIS extrapolation procedure (up to sextuple excitations) with CBS extrapolation.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^i$Reference \onlinecite{Holmes_2017}: Heat-bath CI results cc-pV5Z basis.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^j$Reference \onlinecite{Barbatti_2004}: MRCISD+Q/SA3-CAS(2,2) results with AVDZ.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^k$Reference \onlinecite{Saha_2006}: SAC-CI results using $[4s2p1d/2s] + [2s2p2d]+ [2s2p]$ basis.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^l$Reference \onlinecite{Loos_2018}: exFCI/AVTZ data corrected with the difference between CC3/AVQZ and exFCI/AVTZ values.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^m$Reference \onlinecite{Angeli_2009}: State-specific PC-NEVPT2 results using ANO basis.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^n$Reference \onlinecite{Silva-Junior_2010c}: SA-CASSCF/MS-CASPT2 results using AVTZ basis.}
|
||||
\\
|
||||
\multicolumn{8}{l}{$^o$Reference \onlinecite{Schreiber_2008}: SA-CASSCF/MS-CASPT2 results using TZVP basis.}
|
||||
\endlastfoot
|
||||
Acrolein & 1\,\ex{1}{A}{}{'} $\ra$ 3\,\ex{1}{A}{}{'}
|
||||
& exFCI & 8.00(3) & & & & 8.16$^a$ \\
|
||||
& \tr{\pi,\pi}{\pis,\pis}
|
||||
& CC3($\%T_1$)& 8.21(73\%) & 8.11(75\%) & 8.08(75\%) & & \\
|
||||
& & CASPT2 & 7.93 & 7.93 & 7.85 & 7.84 & \\
|
||||
& & MS-CASPT2 & 8.36 & 8.30 & 8.28 & 8.30 & \\
|
||||
& & XMS-CASPT2 & 8.18 & 8.12 & 8.07 & 8.07 & \\
|
||||
& & PC-NEVPT2 & 7.91 & 7.93 & 7.85 & 7.84 & \\
|
||||
& & SC-NEVPT2 & 8.08 & 8.09 & 8.01 & 8.00 & \\
|
||||
\\
|
||||
Benzene & 1\,\ex{1}{A}{1g}{} $\ra$ 1\,\ex{1}{E}{2g}{}
|
||||
& exFCI & 8.40(3) & & & & 8.41$^b$ \\
|
||||
& \tr{\pi,\pi}{\pis,\pis}
|
||||
& CCSDT & 8.42 & 8.38 & & & \\
|
||||
& & CC3($\%T_1$)& 8.50(72\%) & 8.44(72\%) & 8.38(73\%) & & \\
|
||||
& & CASPT2 & 8.43 & 8.40 & 8.34 & 8.34 & \\
|
||||
& & PC-NEVPT2 & 8.58 & 8.56 & 8.51 & 8.52 & \\
|
||||
& & SC-NEVPT2 & 8.62 & 8.61 & 8.56 & 8.56 & \\
|
||||
\\
|
||||
& 1\,\ex{1}{A}{1g}{} $\ra$ 2\,\ex{1}{A}{1g}{}
|
||||
& CASPT2 & 10.54 & 10.38 & 10.28 & 10.27 & 10.20$^c$ \\
|
||||
& \tr{\pi,\pi}{\pis,\pis}
|
||||
& MS-CASPT2 & 11.08 & 11.00 & 10.96 & 10.97 & \\
|
||||
& & XMS-CASPT2 & 10.77 & 10.64 & 10.55 & 10.54 & \\
|
||||
& & PC-NEVPT2 & 10.35 & 10.18 & 10.00 & & \\
|
||||
& & SC-NEVPT2 & 10.63 & 10.48 & 10.38 & 10.36 & \\
|
||||
\\
|
||||
Beryllium & 1\,\ex{1}{S}{}{} $\ra$ 1\,\ex{1}{D}{}{}
|
||||
& exFCI & 8.04(0) & 7.22(0) & 7.15(0) & 7.11(0) & 7.06$^d$ \\
|
||||
& \tr{2s,2s}{2p,2p}
|
||||
& CCSDTQ & 8.04 & 7.23 & 7.15 & 7.11 & \\
|
||||
& & CCSDT & 8.04 & 7.22 & 7.15 & 7.11 & \\
|
||||
& & CC3($\%T_1$)& 8.04(2\%) & 7.23(29\%) & 7.17(32\%) & 7.12(34\%) & \\
|
||||
& & CASPT2 & 8.02 & 7.21 & 7.12 & 7.10 & \\
|
||||
& & NEVPT2 & 8.01 & 7.20 & 7.11 & 7.10 & \\
|
||||
\\
|
||||
Butadiene & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{}
|
||||
& exFCI & 6.55(3) & 6.51(12) & & & 6.55$^e$, 6.39$^f$, 6.58$^g$ \\
|
||||
& \tr{\pi,\pi}{\pi,\pi}
|
||||
& CCSDT & 6.63 & 6.59 & & & \\
|
||||
& & CC3($\%T_1$)& 6.73(74\%) & 6.68(76\%) & 6.67(75\%) & 6.67(75\%) & \\
|
||||
& & CASPT2 & 6.80 & 6.78 & 6.74 & 6.75 & \\
|
||||
& & PC-NEVPT2 & 6.75 & 6.74 & 6.70 & 6.70 & \\
|
||||
& & SC-NEVPT2 & 6.83 & 6.82 & 6.78 & 6.78 & \\
|
||||
\\
|
||||
Carbon dimer & 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{}
|
||||
& exFCI & 2.29(0) & 2.21(0) & 2.09(0) & 2.06(0) & 2.11$^h$ \\
|
||||
& \tr{\pi,\pi}{\si,\si}
|
||||
& CCSDTQP & 2.29 & 2.21 & & & \\
|
||||
& & CCSDTQ & 2.32 & 2.24 & 2.13 & & \\
|
||||
& & CCSDT & 2.69 & 2.63 & 2.57 & 2.57 & \\
|
||||
& & CC3($\%T_1$)& 3.10(0\%) & 3.11(0\%) & 3.05(0\%) & 3.03(0\%) & \\
|
||||
& & CASPT2 & 2.40 & 2.36 & 2.24 & 2.21 & \\
|
||||
& & PC-NEVPT2 & 2.33 & 2.26 & 2.12 & 2.08 & \\
|
||||
& & SC-NEVPT2 & 2.35 & 2.28 & 2.14 & 2.11 & \\
|
||||
\\
|
||||
& 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+}
|
||||
& exFCI & 2.51(0) & 2.50(0) & 2.42(0) & 2.40(0) & 2.43$^h$, 2.46$^i$ \\
|
||||
& \tr{\pi,\pi}{\si,\si}
|
||||
& CCSDTQP & 2.51 & 2.50 & & & \\
|
||||
& & CCSDTQ & 2.52 & 2.52 & 2.45 & & \\
|
||||
& & CCSDT & 2.86 & 2.87 & 2.86 & 2.87 & \\
|
||||
& & CC3($\%T_1$)& 3.23(0\%) & 3.28(0\%) & 3.26(0\%) & 3.24(0\%) & \\
|
||||
& & CASPT2 & 2.62 & 2.65 & 2.53 & 2.50 & \\
|
||||
& & PC-NEVPT2 & 2.54 & 2.54 & 2.42 & 2.39 & \\
|
||||
& & SC-NEVPT2 & 2.58 & 2.60 & 2.48 & 2.44 & \\
|
||||
\\
|
||||
Carbon trimer & 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{}
|
||||
& exFCI & 5.27(1) & 5.21(0) & 5.22(4) & 5.23(5) & \\
|
||||
& \tr{\pi,\pi}{\si,\si}
|
||||
& CCSDTQ & 5.35 & 5.31 & & & \\
|
||||
& & CCSDT & 5.85 & 5.82 & 5.90 & 5.92 & \\
|
||||
& & CC3($\%T_1$)& 6.65(0\%) & 6.65(0\%) & 6.68(1\%) & 6.66(1\%) & \\
|
||||
& & CASPT2 & 5.13 & 5.06 & 5.08 & 5.08 & \\
|
||||
& & PC-NEVPT2 & 5.26 & 5.24 & 5.25 & 5.26 & \\
|
||||
& & SC-NEVPT2 & 5.21 & 5.19 & 5.21 & 5.22 & \\
|
||||
\\
|
||||
& 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+}
|
||||
& exFCI & 5.93(1) & 5.88(0) & 5.91(2) & 5.86(1) & \\
|
||||
& \tr{\pi,\pi}{\si,\si}
|
||||
& CCSDTQ & 6.02 & 6.00 & & & \\
|
||||
& & CCSDT & 6.52 & 6.49 & 6.57 & 6.58 & \\
|
||||
& & CC3($\%T_1$)& 7.20(1\%) & 7.20(1\%) & 7.24(1\%) & 7.22(1\%) & \\
|
||||
& & CASPT2 & 5.86 & 5.81 & 5.82 & 5.82 & \\
|
||||
& & PC-NEVPT2 & 5.97 & 5.97 & 5.99 & 5.99 & \\
|
||||
& & SC-NEVPT2 & 5.98 & 5.97 & 5.99 & 6.00 & \\
|
||||
\\
|
||||
Ethylene & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{}
|
||||
& exFCI & 13.38(6) & 13.07(1) & 12.92(6) & & 12.15$^j$ \\
|
||||
& \tr{\pi,\pi}{\pis,\pis}
|
||||
& CCSDTQP & 13.39 & & & & \\
|
||||
& & CCSDTQ & 13.39 & 13.07 & & & \\
|
||||
& & CCSDT & 13.50 & 13.20 & & & \\
|
||||
& & CC3($\%T_1$)& 13.82(4\%) & 13.57(15\%) & 13.42(20\%) & 13.06(61\%) & \\
|
||||
& & CASPT2 & 13.49 & 13.23 & 13.17 & 13.17 & \\
|
||||
& & MS-CASPT2 & 13.51 & 13.26 & 13.21 & 13.21 & \\
|
||||
& & XMS-CASPT2 & 13.50 & 13.25 & 13.20 & 13.20 & \\
|
||||
& & PC-NEVPT2 & 14.35 & 13.42 & 13.11 & 13.04 & \\
|
||||
& & SC-NEVPT2 & 13.57 & 13.33 & 13.26 & 13.26 & \\
|
||||
\\
|
||||
Formaldehyde & 1\,\ex{1}{A}{1}{} $\ra$ 3\,\ex{1}{A}{1}{}
|
||||
& exFCI & 10.86(1) & 10.45(1) & 10.35(3) & & 9.82$^c$ \\
|
||||
& \tr{n,n}{\pis,\pis}
|
||||
& CCSDTQP & 10.86 & & & & \\
|
||||
& & CCSDTQ & 10.87 & 10.44 & & & \\
|
||||
& & CCSDT & 11.10 & 10.78 & 10.79 & 10.80 & \\
|
||||
& & CC3($\%T_1$)& 11.49(5\%) & 11.22(4\%) & 11.20(5\%) & 11.19(34\%) & \\
|
||||
& & CASPT2 & 10.80 & 10.38 & 10.27 & 10.26 & \\
|
||||
& & MS-CASPT2 & 10.86 & 10.45 & 10.35 & 10.34 & \\
|
||||
& & XMS-CASPT2 & 10.87 & 10.47 & 10.36 & 10.34 & \\
|
||||
& & PC-NEVPT2 & 10.84 & 10.37 & 10.26 & 10.25 & \\
|
||||
& & SC-NEVPT2 & 10.87 & 10.40 & 10.30 & 10.29 & \\
|
||||
\\
|
||||
Glyoxal & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{}
|
||||
& exFCI & 5.60(1) & 5.48(0) & & & 5.66$^k$ \\
|
||||
& \tr{n,n}{\pis,\pis}
|
||||
& CCSDT & 6.24 & 6.22 & 6.35 & & \\
|
||||
& & CC3($\%T_1$)& 6.74(0\%) & 6.70(1\%) & 6.76(1\%) & 6.76(1\%) & \\
|
||||
& & CASPT2 & 5.58 & 5.47 & 5.42 & 5.43 & \\
|
||||
& & PC-NEVPT2 & 5.66 & 5.56 & 5.52 & 5.52 & \\
|
||||
& & SC-NEVPT2 & 5.68 & 5.58 & 5.55 & 5.55 & \\
|
||||
\\
|
||||
Hexatriene & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{}
|
||||
% & exFCI & olympe & & & & 5.58$^g$ \\
|
||||
& CC3($\%T_1$)& 5.78(65\%) & 5.77(67\%) & & & 5.58$^g$ \\
|
||||
& \tr{\pi,\pi}{\pis,\pis}
|
||||
& CCSDT & 5.64 & 5.65 & & & \\
|
||||
& & CASPT2 & 5.62 & 5.61 & 5.58 & 5.58 & \\
|
||||
& & PC-NEVPT2 & 5.67 & 5.66 & 5.64 & 5.64 & \\
|
||||
& & SC-NEVPT2 & 5.70 & 5.69 & 5.67 & 5.67 & \\
|
||||
\\
|
||||
Nitrosomethane & 1\,\ex{1}{A}{}{'} $\ra$ 2\,\ex{1}{A}{}{'}
|
||||
& exFCI & 4.86(1) & 4.84(2) & 4.76(4) & & 4.72$^l$ \\
|
||||
& \tr{n,n}{\pis,\pis}
|
||||
& CCSDT & 5.26 & 5.26 & 5.29 & & \\
|
||||
& & CC3($\%T_1$)& 5.73(2\%) & 5.75(4\%) & 5.76(3\%) & 5.74(2\%) & \\
|
||||
& & CASPT2 & 4.93 & 4.88 & 4.79 & 4.78 & \\
|
||||
& & PC-NEVPT2 & 4.92 & 4.88 & 4.79 & 4.78 & \\
|
||||
& & SC-NEVPT2 & 4.94 & 4.90 & 4.81 & 4.80 & \\
|
||||
\\
|
||||
Nitroxyl & 1\,\ex{1}{A}{}{'} $\ra$ 2\,\ex{1}{A}{}{'}
|
||||
& exFCI & 4.51(0) & 4.40(1) & 4.33(0) & 4.32(0) & \\
|
||||
& \tr{n,n}{\pis,\pis}
|
||||
& CCSDTQP & 4.51 & & & & \\
|
||||
& & CCSDTQ & 4.54 & 4.42 & & & \\
|
||||
& & CCSDT & 4.81 & 4.76 & 4.79 & 4.80 & \\
|
||||
& & CC3($\%T_1$)& 5.28(0\%) & 5.25(0\%) & 5.26(0\%) & 5.23(0\%) & \\
|
||||
& & CASPT2 & 4.55 & 4.46 & 4.36 & 4.34 & \\
|
||||
& & PC-NEVPT2 & 4.56 & 4.46 & 4.37 & 4.35 & \\
|
||||
& & SC-NEVPT2 & 4.58 & 4.48 & 4.40 & 4.38 & \\
|
||||
\\
|
||||
Pyrazine & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{}
|
||||
% % exFCI & 17279
|
||||
& CC3($\%T_1$)& 9.27(7\%) & 9.17(28\%) & 9.17(12\%) & & \\
|
||||
& \tr{n,n}{\pis,\pis}
|
||||
& CASPT2 & 8.06 & 7.91 & 7.81 & 7.80 & \\
|
||||
& & PC-NEVPT2 & 8.25 & 8.12 & 8.04 & 8.04 & \\
|
||||
& & SC-NEVPT2 & 8.27 & 8.15 & 8.07 & 8.07 & \\
|
||||
\\
|
||||
& 1\,\ex{1}{A}{g}{} $\ra$ 3\,\ex{1}{A}{g}{}
|
||||
& CC3($\%T_1$)& 8.88(73\%) & 8.77(72\%) & 8.69(71\%) & & \\
|
||||
& \tr{\pi,\pi}{\pis,\pis}
|
||||
& CASPT2 & 8.91 & 8.85 & 8.77 & 8.77 & \\
|
||||
& & PC-NEVPT2 & 9.12 & 9.07 & 9.00 & 9.00 & \\
|
||||
& & SC-NEVPT2 & 9.16 & 9.12 & 9.05 & 9.05 & \\
|
||||
\\
|
||||
Tetrazine & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{}
|
||||
% & exFCI & 16904 & & & & 4.66$^m$ \\
|
||||
& CCSDT & 5.86 & 5.86 & & & 4.66$^m$ \\
|
||||
& \tr{n,n}{\pis,\pis}
|
||||
& CC3($\%T_1$)& 6.22(1\%) & 6.22(1\%) & 6.21(1\%) & 6.19(1\%) & \\
|
||||
& & CASPT2 & 4.86 & 4.79 & 4.69 & 4.68 & \\
|
||||
& & PC-NEVPT2 & 4.75 & 4.70 & 4.61 & 4.60 & \\
|
||||
& & SC-NEVPT2 & 4.82 & 4.78 & 4.69 & 4.68 & \\
|
||||
\\
|
||||
& 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{1}{B}{3g}{}
|
||||
& CC3($\%T_1$)& 7.64(0\%) & 7.62(2\%) & 7.62(1\%) & 7.60(1\%) & 5.76$^n$,6.01$^m$ \\
|
||||
& \tr{n,n}{\pis_1,\pis_2}
|
||||
& CASPT2 & 6.00 & 5.95 & 5.85 & 5.85 & \\
|
||||
& & PC-NEVPT2 & 6.25 & 6.22 & 6.15 & 6.14 & \\
|
||||
& & SC-NEVPT2 & 6.30 & 6.27 & 6.20 & 6.20 & \\
|
||||
\\
|
||||
& 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{3}{B}{3g}{}
|
||||
& CC3($\%T_1$)& 7.35(5\%) & 7.33(5\%) & 7.35(6\%) & 7.34(6\%) & 5.50$^o$ \\
|
||||
& \tr{n,n}{\pis_1,\pis_2}
|
||||
& CASPT2 & 5.54 & 5.47 & 5.39 & 5.39 & \\
|
||||
& & PC-NEVPT2 & 5.63 & 5.58 & 5.51 & 5.51 & \\
|
||||
& & SC-NEVPT2 & 5.69 & 5.64 & 5.57 & 5.57 & \\
|
||||
\end{longtable*}
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% FIG 2 %%%
|
||||
\begin{figure*}
|
||||
\begin{tabular}{ccc}
|
||||
\includegraphics[height=0.31\linewidth]{fig2a}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2b}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2c}
|
||||
\\
|
||||
\includegraphics[height=0.31\linewidth]{fig2d}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2e}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2f}
|
||||
\\
|
||||
\includegraphics[height=0.31\linewidth]{fig2g}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2h}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2i}
|
||||
\\
|
||||
\includegraphics[height=0.31\linewidth]{fig2j}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2k}
|
||||
&
|
||||
\includegraphics[height=0.31\linewidth]{fig2l}
|
||||
\\
|
||||
\end{tabular}
|
||||
\caption{
|
||||
Error in excitation energies (for a given basis and compared to exFCI) for various chemical systems, methods and basis sets.
|
||||
\label{fig:chart}
|
||||
}
|
||||
\end{figure*}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{
|
||||
Results and discussion
|
||||
\label{sec:res}
|
||||
}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The molecules considered in the present set are depicted in Fig.~\ref{fig:mol}.
|
||||
Vertical transition energies (in eV) obtained with various methods and basis sets are reported in Table \ref{tab:2Ex}, together with the nature of the transition.
|
||||
The percentage of single excitation, $\%T_1$, calculated at the CC3 level, is also reported to assess the amount of double excitation character.
|
||||
\alert{Although most of the double excitations are a complicated mixture of singly-, doubly- (and higher-) excited determinants, we have observed that, overall, the $\%T_1$ values obtained at the CC3 level provide a similar qualitative picture than the weights of the CI and multiconfigurational wave functions.}
|
||||
Reference values taken from the literature are also reported when available.
|
||||
Total energies for each states, additional information as well as CASSCF excitation energies can be found in {\SI}.
|
||||
Finally, the error in excitation energies (for a given atomic basis set and compared to exFCI) for each system is plotted in Fig.~\ref{fig:chart}.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Beryllium
|
||||
\label{sec:Be}
|
||||
}
|
||||
%---------------------------------------------
|
||||
The beryllium atom (\ce{Be}) is the smallest system we have considered, and in this specific case, the core electrons have been correlated in all calculations.
|
||||
The lowest double excitation corresponds to the $1s^2 2s^2 (^1S) \ra 1s^2 2p^2 (^1D)$ transition.
|
||||
The $\%T_1$ values which provide an estimate of the weight of the single excitations in the CC3 calculation shows that it is mostly a double excitation with a contribution of roughly (only) $30\%$ from the singles.
|
||||
|
||||
The energy of the ground and excited states of \ce{Be} have been computed by G{\`a}lves et al. \cite{Galvez_2002} using explicitly correlated wave functions, and one can extract a value of \IneV{$7.06$} for the ${}^1S \ra {}^1D$
|
||||
transition from their study.
|
||||
This value is in good agreement with our best estimate of \IneV{$7.11$} obtained using the AVQZ basis, the difference being a consequence of the basis set incompleteness.
|
||||
Due to the small number of electrons in \ce{Be}, exFCI, CCSDT and CCSDTQ(=FCI) yield identical values for this transition for any of the basis set considered here.
|
||||
Although slightly different, the CC3 values are close to these reference values with a trifling maximum deviation of \IneV{$0.02$}.
|
||||
Irrespectively of the method, we note a significant energy difference between the results obtained with Pople's 6-31+G(d) basis and the ones obtained with Dunning's basis sets.
|
||||
|
||||
We have also performed multiconfigurational calculations with an active space of $2$ electrons in $12$ orbitals [CAS(2,12)] constituted by the $2s$, $2p$, $3p$ and $3d$ orbitals.
|
||||
Due to the diffuse nature of the excited state, it is compulsory to take into account the $n=3$ shell to reach high accuracy.
|
||||
Excitation energies computed with CASPT2 and NEVPT2 deviate by a maximum of \IneV{$0.01$} and are in excellent agreement with the exFCI numbers.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Carbon dimer and trimer
|
||||
\label{sec:C2-C3}
|
||||
}
|
||||
%---------------------------------------------
|
||||
The second system we wish to discuss is the carbon dimer (\ce{C2}) which is a prototype system for strongly correlated and multireference systems. \cite{Mulliken_1939,Clementi_1962} Thanks to its small size, its ground and excited states
|
||||
have been previously scrutinized using highly-accurate methods.
|
||||
\cite{Abrams_2004,Sherrill_2005,Angeli_2012,Blunt_2015,Boggio-Pasqua_2000,Sharma_2015b,Booth_2011,Boschen_2014,Mahapatra_2008,Purwanto_2009a,Shi_2011,Su_2011,Toulouse_2008,Varandas_2008,Sokolov_2016a,Wouters_2014}
|
||||
Here, we study two double excitations of different symmetries which are, nonetheless, close in energy: 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{} and 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+}.
|
||||
These two excitations --- both involving excitations from the occupied $\piCC$ orbitals to the vacant $\siCC$ orbital --- can be classified as ``pure'' double excitations, as they involve an insignificant amount of single excitations
|
||||
(see Table \ref{tab:2Ex}). For the transition 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+}, the theoretical best estimate is most probably the \IneV{$2.46$} value reported by Holmes et al.~using the heat-bath CI method
|
||||
and the cc-pV5Z basis set at the experimental geometry. \cite{Holmes_2017} For the 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{} transition, the value of \IneV{$2.11$} obtained by Boschen et al.\cite{Boschen_2014} (also at
|
||||
the experimental geometry) can be been taken as reference. We emphasize that the value for the 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+} transition taken from this previous investigation is only \IneV{$0.03$} from the value
|
||||
reported in Ref.~\onlinecite{Holmes_2017}.
|
||||
|
||||
The carbon dimer constitutes a nice playground in order to illustrate the convergence of the various methods with respect to the excitation level. For example, we have been able to perform CCSDTQP calculations for the two smallest basis sets,
|
||||
and these results perfectly agree, for each basis set, with the reference exFCI results obtained on the same (CC3) geometry.
|
||||
For all basis sets except the largest one, the CCSDTQ excitation energies are in good agreement with the exFCI results with a maximum deviation of \IneV{$0.04$}.
|
||||
With CCSDT, the error compared to exFCI ranges from $0.35$ up to half an eV, while this error keeps rising for CC3 with a deviation of the order of \IneV{$0.7$--$1.0$}.
|
||||
|
||||
Concerning multiconfigurational methods, we have used an active space containing 8 electrons in 8 orbitals [CAS(8,8)], which corresponds to the valence space.
|
||||
NEVPT2 is, by far, the most accurate method with errors below \IneV{$0.05$} compared to exFCI.
|
||||
As expected, the partially-contracted version of NEVPT2 yields slightly more accurate results compared to its (cheaper) strongly contracted version. CASPT2 excitation energies are consistently higher than exFCI by $0.10$--$0.15$ eV for both transitions.
|
||||
Additional calculations indicate that this bias is due to the IPEA parameter and lowering its value yields substantial improvements.
|
||||
Although CASPT2 is known to generally underestimate excitation energies for single excitations, this rule of thumb does not
|
||||
seem to apply to double excitations.
|
||||
|
||||
Due to its relevance in space as well as in terrestrial sooting flames and combustion processes, the carbon trimer \ce{C3} (also known as tricarbon) has motivated numerous theoretical studies. \cite{Ahmed_2004,Carter_1980,Carter_1984,Chabalowski_1986,Clementi_1962a,Hoffmann_1966,Jensen_1989,Jensen_1992,Jorgensen_1989,Kraemer_1984,Liskow_1972,Mebel_2002,Mladenovic_1994,Monninger_2002,Murrell_1990,Peric-Radic_1977,Pitzer_1959,Rocha_2015,Rocha_2016,Romelt_1978,Saha_2006b,Schroder_2016,Spirko_1997,Terentyev_2004,Varandas_2008a,Varandas_2009,Varandas_2018,Yousaf_2008}
|
||||
However, its doubly-excited states have, to the best of our knowledge, never been studied. Here, we consider the linear geometry which has been found to be the most stable isomer, although the potential energy surface
|
||||
around this minimum is known to be particularly flat. \cite{Varandas_2018}
|
||||
|
||||
Similarly to \ce{C2}, we have studied two transitions --- 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{} and 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+} --- which also both involve excitations from the occupied $\piCC$ orbitals to the vacant $\siCC$ orbitals.
|
||||
These lie higher in energy than in the dimer but remain energetically close to each other.
|
||||
Again, due to the ``pure double'' nature of the transitions, CC3 very strongly overestimates the reference values (error up to \IneV{$1.5$}).
|
||||
Interestingly, CCSDT reduces this error by roughly a factor two, bringing the deviation between CCSDT and exFCI in the \IneV{$0.6$-$0.7$} range.
|
||||
This outcome deserves to be highlighted, as for transitions dominated by single excitations, CC3 and CCSDT have very similar accuracies as compared to exFCI. \cite{Loos_2018}
|
||||
Although very expensive, CCSDTQ brings down the error even further to a quite acceptable value of \IneV{$0.1$}.
|
||||
|
||||
Consistently to \ce{C2}, we have defined a $(12,12)$ active space for the trimer in order to perform multiconfigurational calculations, and we found that the CASPT2 excitation energies are consistently below exFCI by ca.~\IneV{$0.15$}.
|
||||
Again, NEVPT2 calculations are very accurate with a small preference for SC-NEVPT2, probably due to error compensation.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Nitroxyl and nitrosomethane
|
||||
\label{sec:nitro}
|
||||
}
|
||||
%---------------------------------------------
|
||||
|
||||
Nitroxyl (\ce{H-N=O}) is an important molecule in biochemistry, \cite{Fukuto_2005, Miranda_2005} but only a limited number of theoretical studies of its excited states have been reported to date. \cite{Williams_1975, Luna_1995, Ehara_2011}
|
||||
For this molecule, the 1\,\ex{1}{A}{}{'} $\ra$ 2\,\ex{1}{A}{}{'} transition is a genuine double excitation of $(n,n) \ra (\pis,\pis)$ nature.
|
||||
This system is small enough to perform high-order CC calculations and we have been able to push up to CCSDTQP with the 6-31+G(d) basis.
|
||||
This particular value is in perfect agreement with its exFCI analog in the same basis. For CCSDTQ, we have found that, again, the vertical excitation energies are extremely accurate, with a significant reduction of computational cost compared to CCSDTQP.
|
||||
CCSDT calculations are, as usual, significantly less accurate with an overestimation around \IneV{$0.3$}.
|
||||
CC3 adds up half an eV to this consistent overshooting of the transition energies.
|
||||
|
||||
Multiconfigurational calculations have been performed with a $(12,9)$ active space corresponding to the valence space of the nitroso (\ce{-N=O}) fragment.
|
||||
In the case of nitroxyl, NEVPT2 and CASPT2 yield almost identical excitation energies, also very close to the exFCI target.
|
||||
|
||||
Nitrosomethane (\ce{CH3-N=O}) is an interesting test molecule \cite{Lacombe_2000, Dolgov_2004, Dolgov_2004b, Arenas_2006} and it was included in our previous study. \cite{Loos_2018}
|
||||
Similar to nitroxyl, its lowest-lying singlet $A'$ excited state corresponds to an almost pure double excitation of $(n,n) \ra (\pis,\pis)$ nature. \cite{Arenas_2006}
|
||||
Indeed, CC3/AVTZ calculations return a $3$\%\ single excitation character for this transition. Compared to nitroxyl, a clear impact of the methyl group on the double excitation energy can be noted, but overall, the same conclusions as in nitroxyl can be drawn for both CC and CAS methods.
|
||||
Therefore, we eschew discussing this case further for the sake of conciseness.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Ethylene and formaldehyde
|
||||
\label{sec:ethyform}
|
||||
}
|
||||
%---------------------------------------------
|
||||
|
||||
Despite its small size, ethylene remains a challenging molecule that has received much attention from the theoretical chemistry community, \cite{Robin_1985, Serrano-Andres_1993, Watts_1996, Schreiber_2008, Angeli_2008, Feller_2014, Chien_2018}
|
||||
and is included in many benchmark sets. \cite{Head-Gordon_1994, Schreiber_2008, Shen_2009b, Caricato_2010, Leang_2012, Hoyer_2016, Loos_2018} In particular, we refer the interested readers to the work of Davidson and coworkers\cite{Feller_2014}
|
||||
for, what we believe, is the most complete and accurate investigation dedicated to the excited states of ethylene.
|
||||
|
||||
In ethylene, the double excitation 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} is of \tr{\pi,\pi}{\pis,\pis} nature. Unsurprisingly, it has been much less studied than the single excited states due to its fairly high energy and the absence of experimental
|
||||
value. Nevertheless, in 2004, Barbatti et al.~have reported a value of \IneV{$12.15$} at the MRCISD+Q/AVDZ level of theory. \cite{Barbatti_2004} We have found that this state has a fairly high degree of double excitation which, at the CC3 level,
|
||||
decreases with the size of the basis set, with $\%T_1$ going from $4\%$ with 6-31+G(d) to $61\%$ with AVQZ. Due to its Rydberg character, there is obviously a large basis set effect for this transition, with a magnitude that is additionally strongly
|
||||
method dependent.
|
||||
|
||||
Here again, thanks to the small size of this molecule, we have been able to perform high-order CC calculations, and, once more, we have found that CCSDTQP and CCSDTQ yield very accurate excitation energies.
|
||||
Removing the quadruples has the effect of blue-shifting the transition by at least \IneV{0.1}, while CC3 is off by half an eV independently of the basis set.
|
||||
|
||||
In the case of ethylene, we have studied two types of active spaces: a $(2,2)$ active space which includes the $\piCC$ and $\pisCC$ orbitals and a $(4,4)$ active space obtained by adding the $\siCC$ and $\sisCC$ orbitals.
|
||||
Table \ref{tab:2Ex} only reports the results for the largest active space; the values determined with the smallest active space can be found in {\SI}. In accordance with previous studies, \cite{Angeli_2008, Feller_2014, Giner_2015b} we have
|
||||
found that it is essential to take into account the bonding and anti-bonding $\si$ orbitals in the active space due to the strong coupling between the $\si$ and $\pi$ spaces. CASPT2 and NEVPT2 are overestimating the transition energy
|
||||
by at least \IneV{$0.2$} with Dunning's bases, while CASPT2 and MS-CASPT2 yield similar excitation energies. We note that the PC-NEVPT2 energies seem to become more accurate when the quality of the atomic basis
|
||||
set improves, whereas the opposite trend is observed for SC-NEVPT2.
|
||||
|
||||
From a computational point of view, formaldehyde is similar to ethylene and it has also been extensively studied at various levels of theory.
|
||||
\cite{Merchan_1995, Paterson_2006, Muller_2001, Foresman_1992b, Hadad_1993, Head-Gordon_1994, Head-Gordon_1995, Gwaltney_1995, Wiberg_1998, Wiberg_2002, Peach_2008, Schreiber_2008, Shen_2009b, Caricato_2010, Li_2011, Leang_2012, Hoyer_2016,Loos_2018}
|
||||
However, the 1\,\ex{1}{A}{1}{} $\ra$ 3\,\ex{1}{A}{1}{} transition in \ce{CH2=O} is rather chemically different from its \ce{H2C=CH2} counterpart, as it is a transition from the ground state to the second excited state of \ex{1}{A}{1}{} symmetry with
|
||||
a \tr{n,n}{\pis,\pis} character. For this transition, Barca et al.\cite{Barca_2018a} have reported a value of \IneV{$9.82$} at the BLYP/cc-pVTZ level [using the maximum overlap method (MOM) to locate the excited state] in qualitative agreement with our reference
|
||||
energies. The lack of diffuse functions may have, however, a substantial effect on this value.
|
||||
|
||||
In terms of the performance of the CC-based methods, the conclusion that we have drawn in ethylene can be almost perfectly transposed to formaldehyde. For the CAS-type calculations, two active spaces were tested: a $(4,3)$ active space
|
||||
that includes the $\piCO$ and $\pisCO$ orbitals as well as the lone pair $\nO$ on the oxygen atom, and the $(6,5)$ active space that adds the $\siCO$ and $\sisCO$ orbitals.
|
||||
Again, Table \ref{tab:2Ex} only reports the results obtained with the largest active space whereas the values for the smallest active space can be found in {\SI}.
|
||||
The performance of multiconfigurational calculations are fairly consistent and there are no significant differences between the various methods, although, due to the strong mixing between the first three \ex{1}{A}{1}{} states, the results obtained with CASPT2, MS-CASPT2, and XMS-CASPT2 differ slightly.
|
||||
The excitation energies obtained with the multi-state variants (extended or not) almost perfectly match the exFCI values, thanks to a small blueshift of the energies compared to the CASPT2 results.
|
||||
Note that the same methods would return excitation energies with errors consistently red-shifted by \IneV{$0.15$} with the small active space, highlighting once more that $\sigma$ orbitals should be included if high accuracy is desired.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Butadiene, glyoxal, and acrolein
|
||||
\label{sec:butadiene}
|
||||
}
|
||||
%---------------------------------------------
|
||||
|
||||
The excited states of (\emph{trans}-)butadiene have been thoroughly studied during the past thirty years.
|
||||
\cite{Watts_1996, Boggio-Pasqua_2004, Cave_2004, Chien_2018, Daday_2012, Dallos_2004, Hsu_2001, Kitao_1988, McDiarmid_1988, Mosher_1973, Ostojic_2001, Saha_2006, Serrano-Andres_1993, Strodel_2002, Watson_2012, Sundstrom_2014, Shu_2017}
|
||||
In 2012, Watson and Chan \cite{Watson_2012} have studied the hallmark singlet bright (1\,\ex{1}{B}{u}{}) and dark (2\,\ex{1}{A}{g}{}) states. They reported best estimates of $6.21 \pm 0.02$ eV and $6.39 \pm 0.07$ eV, respectively, settling
|
||||
down the controversy about the ordering of these two states. \cite{Strodel_2002} While the bright 1\,\ex{1}{B}{u}{} state has a clear (HOMO $\ra$ LUMO) single excitation character, the dark 2\,\ex{1}{A}{g}{} state includes a substantial fraction
|
||||
of doubly-excited character from the HOMO $\ra$ LUMO double excitation (roughly $30\%$), yet dominant contributions from the HOMO-1$\ra$LUMO and HOMO$\ra$LUMO+1 single excitations. Butadiene (as well as hexatriene, see below) has
|
||||
been also studied at the dressed TD-DFT level. \cite{Maitra_2004, Cave_2004, Huix-Rotllant_2011}
|
||||
|
||||
For butadiene (and the two other molecules considered in this Section), exFCI results are only reported for the two double-$\zeta$ basis sets, as it was not possible to converge the excitation energies with larger basis sets. Our exFCI estimates agree nicely with the
|
||||
reference values obtained by Dallos and Lischka, \cite{Dallos_2004} Watson and Chan, \cite{Watson_2012} and Chien et al.\cite{Chien_2018} at the MR-CI, incremental CC and heat-bath CI levels, respectively (see Table \ref{tab:2Ex}).
|
||||
|
||||
Concerning the multiconfigurational calculations, the $(4,4)$ active space includes the $\piCC$ and $\pisCC$ orbitals while the $(10,10)$ active space adds the $\siCC$ and $\sisCC$ orbitals. Expanding the active space has a non-negligible
|
||||
impact on the NEVPT2 excitation energies with a neat improvement by ca.~$0.1$ eV, whereas CASPT2 results are less sensitive to this active space expansion (see {\SI}). As previously mentioned, this effect is reminiscent of the strong
|
||||
coupling between the $\si$ and $\pi$ spaces in compounds like butadiene, \cite{Watson_2012, Dash_2018} ethylene, \cite{Angeli_2008, Giner_2015b} and cyanines. \cite{Send_2011, Boulanger_2014, LeGuennic_2015, Garniron_2018}
|
||||
Here, it is important to note that both CC3 and CCSDT provide more accurate excitation energies than any multiconfigurational method. This clearly illustrates the strength of CC approaches when there is a dominant ``single'' nature in the
|
||||
considered transition as discussed in previous works. \cite{Shu_2017, Barca_2018a, Barca_2018b}
|
||||
|
||||
The genuine double excitation 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} in glyoxal, \cite{Dykstra_1977, Ha_1972, Hirao_1983, Hollauer_1991, Pincellt_1971, Saha_2006} which corresponds to a \tr{n,n}{\pis,\pis} transition, has been studied by Saha et al.~at the SAC-CI level \cite{Saha_2006} (see Table \ref{tab:2Ex} for additional information). They reported a value of \IneV{$5.66$} in very good agreement with our exFCI reference.
|
||||
As expected now, given the ``pure'' double excitation character, CC3 and CCSDT are off by the usual margin (more than \IneV{$1$} for CC3). Due to the nature of the considered transition, the lone pairs of the two oxygen atoms are included in both the small $(8,6)$ and large $(14,12)$ active spaces.
|
||||
In glyoxal, we have logically found that the lone pairs of both oxygen atoms equally contribute to the double excitation.
|
||||
The $(8,6)$ active space also contains the $\piCC$, $\piCO$, $\pisCC$ and $\pisCO$ orbitals, while the $(14,12)$ active space adds up the $\siCC$, $\siCO$, $\sisCC$ and $\sisCO$ orbitals.
|
||||
CASPT2 excitation energies are particularly close to our exFCI energies while PC- and SC-NEVPT2 energies are slightly blue-shifted but remain in very good agreement with the exFCI benchmark.
|
||||
|
||||
The 1\,\ex{1}{A}{}{'} $\ra$ 3\,\ex{1}{A}{}{'} excitation in acrolein \cite{Aquilante_2003, Bouabca_2009, Guareschi_2013, Saha_2006} has the same nature as the one in butadiene.
|
||||
However, there is a 1\,\ex{1}{A}{}{'} $\ra$ 2\,\ex{1}{A}{}{'} transition of $\pi \ra \pis$ nature slightly below in energy and these two transitions are strongly coupled. From a computational point of view, it means that the 1\,\ex{1}{A}{}{'} $\ra$ 3\,\ex{1}{A}{}{'} transition is, from a technical point of view, tricky to get, and this explains why we have not been able to obtain reliable exFCI estimates except for the smallest 6-31+G(d) basis.
|
||||
|
||||
The (small) $(4,4)$ active space contains the $\piCC$, $\piCO$, $\pisCC$ and $\pisCO$ orbitals, while the (larger) $(10,10)$ active space adds up the $\siCC$, $\siCO$, $\sisCC$ and $\sisCO$ orbitals.
|
||||
Due to the nature of the transitions involved, it was not necessary to include the lone pair of the oxygen atom in the active space, and this has been confirmed by preliminary calculations.
|
||||
Moreover, CASSCF predicts the $\pi \ra \pis$ transition higher in energy than the \tr{\pi,\pi}{\pis,\pis} transition, and CASPT2 and NEVPT2 correct this erroneous ordering via the introduction of dynamic correlation.
|
||||
The CAS(4,4) calculations clearly show that the multi-state treatment of CASPT2 strongly mix these two transitions, while its extended variant mitigates this trend.
|
||||
Consequently, because of the strong mixing of the three \ex{1}{A}{}{'} states in acrolein, CASPT2, MS-CASPT2 and XMS-CASPT2 deviate by several tenths of eV.
|
||||
|
||||
For the 1\,\ex{1}{A}{}{'} $\ra$ 3\,\ex{1}{A}{}{'} excitation of acrolein, Saha et al.\cite{Saha_2006} provided an estimate of \IneV{$8.16$} at the SAC-CI level as compared to our exFCI/6-31+G(d) value of \IneV{$8.00$}, which nestles between the PC- and SC-NEVPT2 values.
|
||||
The CC3 excitation energy in the same basis is off by ca.~\IneV{$0.2$}, so is the XMS-CASPT2 energy.
|
||||
|
||||
%---------------------------------------------
|
||||
\subsection{
|
||||
Benzene, pyrazine, tetrazine, and hexatriene
|
||||
\label{sec:BigMol}
|
||||
}
|
||||
%---------------------------------------------
|
||||
In this last section, we report excitation energies for four larger molecules containing 6 heavy atoms (see Fig.~\ref{fig:mol}).
|
||||
Due to their size, we have not been able to provide reliable exFCI results (except for benzene, see below).
|
||||
Therefore, we mainly restrict ourselves to multiconfigurational calculations with valence $\pi$ active space as well as with CC3 and CCSDT (when technically possible).
|
||||
For the nitrogen-containing molecules, the lone pairs have been included in the active space as we have found that they are always involved in double excitations.
|
||||
We refer the reader to the {\SI} for details about the active spaces.
|
||||
|
||||
Thanks to the high degree of symmetry of benzene, we have been able to obtain a reliable estimate of the excitation energy at the exFCI/6-31+G(d) for the lowest double excitation of 1\,\ex{1}{A}{1g}{} $\ra$ 1\,\ex{1}{E}{2g}{}
|
||||
character. \cite{Buenker_1968, Peyerimhoff_1970, Hay_1974, Palmer_1989, Kitao_1987, Lorentzon_1995, Hashimoto_1996, Christiansen_1996, Christiansen_1998, Handy_1999, Hald_2002, Barca_2018a, Barca_2018b}
|
||||
Our value of \IneV{$8.40$} is in almost perfect agreement with the one reported by Christiansen et al.\cite{Christiansen_1996} at the CC3 level (\IneV{$8.41$}).
|
||||
Indeed, as this particular transition has a rather small double excitation character, CC3 and CCSDT provide high-quality results.
|
||||
This contrasts with the 1\,\ex{1}{A}{1g}{} $\ra$ 2\,\ex{1}{A}{1g}{} transition which has almost a pure double excitation nature.
|
||||
This genuine double excitation has received less attention but Gill and coworkers reported a value of \IneV{$10.20$} at the BLYP(MOM)/cc-pVTZ level in nice agreement with our CASPT2 results.
|
||||
However, we observe that depending on the flavor of post-CASSCF treatment, we have an important variation (by ca.~\IneV{$0.6$--$0.9$}) of the excitation energies, the lower and upper bounds being respectively provided by PC-NEVPT2 and MS-CASPT2.
|
||||
|
||||
For pyrazine, \cite{Palmer_1991, Fulscher_1992, Durig_1984, Flscher_1994, Weber_1999, Nooijen_1999} we have studied the three lowest states of \ex{1}{A}{g}{} symmetry and their corresponding excitation energies.
|
||||
The 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} transition of \tr{n,n}{\pis,\pis} nature has a large fraction of double excitation, while the 1\,\ex{1}{A}{g}{} $\ra$ 3\,\ex{1}{A}{g}{} transition has a \tr{\pi,\pi}{\pis,\pis} nature, and is dominated by single excitations, similar to the one studied in butadiene and acrolein.
|
||||
In pyrazine, both lone pairs contribute to the second excitation.
|
||||
One can note an interesting methodological inversion between these two transitions. Indeed, due to the contrasted quality of CC3 excitation energies for the \tr{n,n}{\pis,\pis} and \tr{\pi,\pi}{\pis,\pis} transitions, the latter
|
||||
is (incorrectly) found below the former at the CC3 level while the opposite is observed with CASPT2 or NEVPT2.
|
||||
|
||||
Tetrazine (or s-tetrazine) \cite{Mason_1959, Innes_1988, Fridh_1972, Palmer_1997, Livak_1971, Rubio_1999, Nooijen_2000, Schreiber_2008, Harbach_2014} is a particularly ``rich'' molecule in terms of double excitations thanks to the
|
||||
presence of four lone pairs. Here, we have studied three transitions: two singlet-singlet and one singlet-triplet excitations. In these three transitions, electrons from the nitrogen lone pairs $\nN$ are excited to $\pis$ orbitals.
|
||||
As expected, they can be labeled as genuine double excitations as they have very small $\%T_1$ values.
|
||||
For the 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{1}{B}{3g}{} and 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{3}{B}{3g}{} transitions, we note that the two
|
||||
excited electrons end up in different $\pis$ orbitals, contrary to most cases encountered in the present study. The basis set effect is pretty much inexistent for these three excitations with a maximum difference of \IneV{$0.04$} between
|
||||
the smallest and the largest basis sets. For tetrazine, previous high-accuracy reference values are: i) \IneV{$4.66$} for the 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} transition reported by Angeli et al.\cite{Angeli_2009} with
|
||||
NEVPT2, ii) \IneV{$5.76$} for the 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{1}{B}{3g}{} transition reported by Silva-Junior et al. \cite{Silva-Junior_2010c} at the MS-CASPT2/AVTZ level and, iii) \IneV{$5.50$} for the 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{3}{B}{3g}{}
|
||||
transition reported by Schreiber et al. \cite{Schreiber_2008} at the MS-CASPT2/TZVP level. In comparison, for the second transition, Angeli et al.\cite{Angeli_2009} have obtained a value of \IneV{$6.01$} at the NEVPT2 level.
|
||||
For the first transition, the CCSDT results indicate that the CC3 excitation energies are, again, fairly inaccurate and pushing up to CCSDT does not seem to significantly improve the results as the deviations between CCSDT and CASPT2/NEVPT2 results are still substantial.
|
||||
However, it is hard to determine which method is the most reliable in this case.
|
||||
Finally, we note that, for the second and third transitions, there is an important gap between CASPT2 and NEVPT2 energies.
|
||||
|
||||
For hexatriene, \cite{Serrano-Andres_1993, Flicker_1977, Nakayama_1998, Maitra_2004, Cave_2004} the accurate energy of the 2\,\ex{1}{A}{g}{} state is not known experimentally, illustrating the difficulty to observe these states via conventional spectroscopy techniques.
|
||||
For this molecule, we have unfortunately not been able to provide reliable exFCI results, even for the smallest basis sets.
|
||||
However, Chien et al.~have recently reported a value of \IneV{5.58} at the heat-bath CI/AVDZ level with a MP2/cc-pVQZ geometry. \cite{Chien_2018}
|
||||
This reference value indicates that our CASPT2 and NEVPT2 calculations are particularly accurate even with a minimal valence $\pi$ active space, the coupling between $\si$ and $\pi$ spaces becoming weaker for larger polyenes. \cite{Garniron_2018}
|
||||
Because the 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} transition is of \tr{\pi,\pi}{\pis,\pis} nature (and very similar to its butadiene analog), the CC3 transition energies are not far off the reference values.
|
||||
|
||||
%%% TABLE 2 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{
|
||||
\label{tab:TBE}
|
||||
Theoretical best estimates (TBEs) of vertical transition energies (in eV) for excited states with significant double excitation character in various molecules (see Table \ref{tab:2Ex} for details).
|
||||
TBEs are computed as $\Delta E_\text{R/SB} + \Delta E_\text{C/LB} - \Delta E_\text{C/SB}$, where $\Delta E_\text{R/SB}$ is the excitation energy computed with a reference (R) method in a small basis (SB), and $\Delta E_\text{C/SB}$ and $\Delta E_\text{C/LB}$ are excitation energies computed with a correction (C) method in the small and large basis (LB), respectively.
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{llldldd}
|
||||
Molecule & Transition & \mc{2}{c}{Reference} & \mc{2}{c}{Correction} & \mcc{TBE} \\
|
||||
\cline{3-4} \cline{5-6}
|
||||
& & Level R/SB & \mcc{$\Delta E_\text{R/SB}$}
|
||||
& Level C/LB & \mcc{$\Delta E_\text{C/LB} - \Delta E_\text{C/SB}$} \\
|
||||
\hline
|
||||
Acrolein & 1\,\ex{1}{A}{}{'} $\ra$ 3\,\ex{1}{A}{}{'} & exFCI/6-31+G(d) & 8.00 & CC3/AVTZ & -0.13 & 7.87 \\
|
||||
Benzene & 1\,\ex{1}{A}{1g}{} $\ra$ 1\,\ex{1}{E}{2g}{} & exFCI/6-31+G(d) & 8.40 & CC3/AVTZ & -0.12 & 8.28 \\
|
||||
& 1\,\ex{1}{A}{1g}{} $\ra$ 2\,\ex{1}{A}{1g}{} & XMS-CASPT2/AVQZ & 10.54 & \cdash & \cdash & 10.54 \\
|
||||
Beryllium & 1\,\ex{1}{S}{}{} $\ra$ 1\,\ex{1}{D}{}{} & Ref.~\onlinecite{Galvez_2002} & 7.06 & \cdash & \cdash & 7.06 \\
|
||||
Butadiene & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} & exFCI/AVDZ & 6.51 & CC3/AVQZ & -0.01 & 6.50 \\
|
||||
Carbon dimer & 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{} & exFCI/AVQZ & 2.06 & \cdash & \cdash & 2.06 \\
|
||||
& 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+} & exFCI/AVQZ & 2.40 & \cdash & \cdash & 2.40 \\
|
||||
Carbon trimer & 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 1\,\ex{1}{\Delta}{g}{} & exFCI/AVQZ & 5.23 & \cdash & \cdash & 5.23 \\
|
||||
& 1\,\ex{1}{\Sigma}{g}{+} $\ra$ 2\,\ex{1}{\Sigma}{g}{+} & exFCI/AVQZ & 5.86 & \cdash & \cdash & 5.86 \\
|
||||
Ethylene & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} & exFCI/AVTZ & 12.92 & CC3/AVQZ & -0.36 & 12.56 \\
|
||||
Formaldehyde & 1\,\ex{1}{A}{1}{} $\ra$ 3\,\ex{1}{A}{1}{} & exFCI/AVTZ & 10.35 & CC3/AVQZ & -0.01 & 10.34 \\
|
||||
Glyoxal & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} & exFCI/AVDZ & 5.48 & CC3/AVQZ & +0.06 & 5.54 \\
|
||||
Hexatriene & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} & CC3/AVDZ & 5.77 & PC-NEVPT2/AVQZ & -0.02 & 5.75 \\
|
||||
Nitrosomethane & 1\,\ex{1}{A}{}{'} $\ra$ 2\,\ex{1}{A}{}{'} & exFCI/AVTZ & 4.76 & CC3/AVQZ & -0.02 & 4.74 \\
|
||||
Nitroxyl & 1\,\ex{1}{A}{}{'} $\ra$ 2\,\ex{1}{A}{}{'} & exFCI/AVQZ & 4.32 & \cdash & \cdash & 4.32 \\
|
||||
Pyrazine & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} & PC-NEVPT2/AVQZ & 8.04 & \cdash & \cdash & 8.04 \\
|
||||
& 1\,\ex{1}{A}{g}{} $\ra$ 3\,\ex{1}{A}{g}{} & CC3/AVTZ & 8.69 & PC-NEVPT2/AVQZ & +0.00 & 8.69 \\
|
||||
Tetrazine & 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} & PC-NEVPT2/AVQZ & 4.60 & \cdash & \cdash & 4.60 \\
|
||||
& 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{1}{B}{3g}{} & PC-NEVPT2/AVQZ & 6.14 & \cdash & \cdash & 6.14 \\
|
||||
& 1\,\ex{1}{A}{g}{} $\ra$ 1\,\ex{3}{B}{3g}{} & PC-NEVPT2/AVQZ & 5.51 & \cdash & \cdash & 5.51 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{
|
||||
Theoretical best estimates
|
||||
\label{sec:TBE}
|
||||
}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In Table \ref{tab:TBE}, we report TBEs for the vertical excitations considered in Table \ref{tab:2Ex}.
|
||||
These TBEs are computed as $\Delta E_\text{R/SB} + \Delta E_\text{C/LB} - \Delta E_\text{C/SB}$, where $\Delta E_\text{R/SB}$ is the excitation energy computed with a reference (R) method in a small basis (SB), and $\Delta E_\text{C/SB}$ and $\Delta E_\text{C/LB}$ are excitation energies computed with a correction (C) method in the small and large basis (LB), respectively.
|
||||
By default, we have taken as reference the exFCI excitation energies ($\Delta E_\text{R/SB}$) computed in the present study, while the basis set correction ($\Delta E_\text{C/LB} - \Delta E_\text{C/SB}$) is calculated at the CC3 level.
|
||||
When the exFCI result is unavailable, we have selected, for each excitation separately, what we believe is the most reliable reference method.
|
||||
For most excitations (except the 1\,\ex{1}{A}{g}{} $\ra$ 2\,\ex{1}{A}{g}{} transition in ethylene), the basis set correction is small.
|
||||
In the case of \ce{Be}, the value of Ref.~\onlinecite{Galvez_2002} is indisputably more accurate than ours.
|
||||
For \ce{C2}, butadiene and hexatriene, we have not chosen the heat-bath CI results \cite{Holmes_2017, Chien_2018} as reference because these calculations were not performed at the same CC3 geometry.
|
||||
However, these values are certainly outstanding references for their corresponding geometry.
|
||||
|
||||
%%% TABLE 3 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{
|
||||
Mean absolute error (MAE), root mean square error (RMSE), as well as minimum (Min.) and maximum (Max.) absolute errors (with respect to exFCI) of CC3, CCSDT, CCSDTQ, CASPT2, PC-NEVPT2 and SC-NEVPT2 excitation energies.
|
||||
All quantities are given in eV.
|
||||
``Count'' refers to the number of transitions considered for each method.
|
||||
\label{tab:stat}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lddddd}
|
||||
Method & \mcc{Count} & \mcc{MAE} & \mcc{RMSE} & \mcc{Min.} & \mcc{Max.} \\
|
||||
\hline
|
||||
\mc{6}{l}{All excitations} \\
|
||||
CC3 & 39 & 0.78 & 0.90 & 0.00 & 1.46 \\
|
||||
CCSDT & 37 & 0.40 & 0.46 & 0.00 & 0.74 \\
|
||||
CCSDTQ & 19 & 0.03 & 0.05 & 0.00 & 0.12 \\
|
||||
CASPT2 & 39 & 0.03 & 0.11 & 0.01 & 0.27 \\
|
||||
PC-NEVPT2 & 39 & 0.07 & 0.18 & 0.00 & 0.97 \\
|
||||
SC-NEVPT2 & 39 & 0.07 & 0.12 & 0.01 & 0.34 \\
|
||||
\hline
|
||||
\mc{6}{l}{Excitations with $\%T_1 > 50\%$} \\
|
||||
CC3 & 4 & 0.11 & 0.13 & 0.00 & 0.18 \\
|
||||
CCSDT & 3 & 0.06 & 0.07 & 0.00 & 0.08 \\
|
||||
CCSDTQ & 0 & \cdash & \cdash & \cdash & \cdash \\
|
||||
CASPT2 & 4 & 0.12 & 0.19 & 0.03 & 0.27 \\
|
||||
PC-NEVPT2 & 4 & 0.13 & 0.18 & 0.18 & 0.23 \\
|
||||
SC-NEVPT2 & 4 & 0.22 & 0.24 & 0.08 & 0.31 \\
|
||||
\hline
|
||||
\mc{6}{l}{Excitations with $\%T_1 < 50\%$} \\
|
||||
CC3 & 35 & 0.86 & 0.95 & 0.00 & 1.46 \\
|
||||
CCSDT & 34 & 0.42 & 0.48 & 0.00 & 0.74 \\
|
||||
CCSDTQ & 19 & 0.03 & 0.05 & 0.00 & 0.12 \\
|
||||
CASPT2 & 35 & 0.02 & 0.10 & 0.01 & 0.25 \\
|
||||
PC-NEVPT2 & 35 & 0.07 & 0.18 & 0.00 & 0.97 \\
|
||||
SC-NEVPT2 & 35 & 0.06 & 0.10 & 0.01 & 0.34 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
\end{squeezetable}
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[height=0.33\linewidth]{fig3a}
|
||||
\includegraphics[height=0.33\linewidth]{fig3b}
|
||||
\caption{
|
||||
Error in excitation energies (in eV) with respect to exFCI as a function of the percentage of single excitation $\%T_1$ (computed at the CC3 level) for various molecules and basis sets.
|
||||
Left: CC3 (blue), CCSDT (red) and CCSDTQ (black).
|
||||
Right: CASPT2 (green), PC-NEVPT2 (orange) and SC-NEVPT2 (pink).
|
||||
Note the difference in scaling of the vertical axes.
|
||||
\label{fig:ExvsT1}
|
||||
}
|
||||
\end{figure*}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{
|
||||
Conclusion
|
||||
\label{sec:ccl}
|
||||
}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
We have reported reference vertical excitation energies for 20 transitions with significant double excitation character in a set of 14 small- and medium-size compounds using a series of increasingly large diffuse-containing atomic basis sets (from Pople's 6-31+G(d) to Dunning's aug-cc-pVQZ basis).
|
||||
Depending on the size of the molecule, selected configuration interaction (sCI) and/or multiconfigurational (CASSCF, CASPT2, (X)MS-CASPT2 and NEVPT2) calculations have been performed in order to obtain reliable estimates of the vertical transition energies.
|
||||
|
||||
We have shown that the error obtained with CC methods including iterative triples can significantly vary with the exact nature of the transition.
|
||||
For ``pure'' double excitations (i.e.~for transition which do not mix with single excitations), the error in CC3 can easily reach \IneV{$1$} (and up to \IneV{$1.5$}), while it goes down to few tenths of an eV for more common transitions (like in butadiene, acrolein and benzene) involving a significant amount of singles.
|
||||
This analysis is corroborated by Fig.~\ref{fig:ExvsT1} which reports the CC3, CCSDT, and CCSDTQ excitation energy errors with respect to exFCI as a function of the percentage of single excitation $\%T_1$ (computed at the CC3 level).
|
||||
A statistical analysis of these data is also provided in Table \ref{tab:stat} where one can find the mean absolute error (MAE), root mean square error (RMSE), as well as the minimum and maximum absolute errors associated with the CC3, CCSDT, and CCSDTQ excitation energies.
|
||||
For CC3, one can see a clear correlation between the magnitude of the error and the degree of double excitation of the corresponding transition.
|
||||
CC3 returns an overall MAE of \IneV{$0.78$} which drops to \IneV{$0.11$} when one considers solely excitations with $\%T_1 > 50\%$ (with a maximum error as small as \IneV{$0.18$}), but raises to \IneV{$0.86$} for excitations with $\%T_1 < 50\%$.
|
||||
Therefore, one can conclude that CC3 is a particularly accurate method for excitations dominated by single excitations which are ubiquitous, for instance, in compounds like butadiene, acrolein, hexatriene, and benzene derivatives.
|
||||
Indeed, according to our results, CC3 outperforms CASPT2 and NEVPT2 for these transitions (see below).
|
||||
This corroborates the conclusions drawn in our previous investigation where we evidenced that CC3 delivers very small errors with respect to FCI estimates for small compounds. \cite{Loos_2018}
|
||||
A similar trend is observed with CCSDT at a lower scale: the overall MAE is \IneV{$0.40$} (a two-fold reduction compared to CC3), but $0.06$ and \IneV{$0.42$} for transitions with $\%T_1 > 50\%$ and $\%T_1 < 50\%$, respectively.
|
||||
As expected, more computationally demanding approaches like CCSDTQ (and beyond) yield highly accurate results even for genuine double excitations.
|
||||
For CCSDTQ, we have not been able to perform calculations on single-dominant excitations as such type of excitations does not seem to appear in small molecules.
|
||||
From a general point of view, CC methods consistently overestimate excitation energies compared to exFCI.
|
||||
|
||||
The quality of the excitation energies obtained with multiconfigurational methods such as CASPT2, (X)MS-CASPT2, and NEVPT2 is harder to predict.
|
||||
We have found that the overall accuracy of these methods is highly dependent of the system and the selected active space.
|
||||
Note, however, that including the $\si$ and $\sis$ orbitals in the active space, even for transitions involving mostly $\pi$ and $\pis$ orbitals, can significantly improve the excitation energies.
|
||||
The statistics associated with the CASPT2, PC-NEVPT2 and SC-NEVPT2 data are also provided in Table \ref{tab:stat} and depicted in Fig.~\ref{fig:ExvsT1}.
|
||||
The overall MAE of CASPT2 is \IneV{$0.03$}, i.e., identical to CCSDTQ, while it is slightly larger for the two NEVPT2 variants (\IneV{$0.07$} for both of them).
|
||||
However, their RMSE (which gives a bigger weight to large errors) is much larger.
|
||||
Similar observations can be made for excitations with $\%T_1 < 50\%$, while for single-dominant excitations (i.e.~$\%T_1 > 50\%$), the MAEs in multiconfigurational methods are higher than in CC-based methods.
|
||||
As a final comment, we note that the consistent overestimation of the exFCI excitation energies observed in CC methods does not apply to multiconfigurational methods.
|
||||
|
||||
We believe that the reference data reported in the present study will be particularly valuable for the future development of methods trying to accurately describe double excitations.
|
||||
\alert{Although the oscillator strength associated with a double excitation is usually zero or extremely small (dark state), we believe that it would be valuable to study their sensitivity with respect to the level of theory.}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supporting Information}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for geometries and additional information (including total energies) on the CC, multiconfigurational and sCI calculations.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
P.F.L.~would like to thank Emmanuel Giner and Jean-Paul Malrieu for NEVPT2-related discussions.
|
||||
P.F.L.~and A.S.~would like to thank Nicolas Renon and Pierrette Barbaresco (CALMIP, Toulouse) for technical assistance.
|
||||
D.J.~acknowledges the \emph{R\'egion des Pays de la Loire} for financial support.
|
||||
This work was performed using HPC resources from
|
||||
i) 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 (Toulouse) under allocations 2018-0510, 2018-18005 and 2018-12158.
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\bibliography{QUEST2,QUEST2-control}
|
||||
|
||||
|
||||
\end{document}
|
||||
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\usepackage{mathtools}
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|
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\usepackage{qcircuit}
|
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|
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|
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\usepackage{graphicx,longtable,dcolumn,mhchem}
|
||||
\usepackage{rotating,color}
|
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\usepackage{lscape}
|
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\usepackage{amsmath}
|
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\usepackage{dsfont}
|
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\usepackage{soul}
|
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\usepackage{physics}
|
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\newcolumntype{d}{D{.}{.}{-1}}
|
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\newcommand{\ctab}[1]{\multicolumn{1}{c}{#1}}
|
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\newcommand{\CheA}{\%{\text{CA}}}
|
||||
\newcommand{\AccE}{\%{\text{AE}}}
|
||||
\newcommand{\ie}{\textit{i.e}}
|
||||
\newcommand{\eg}{\textit{e.g}}
|
||||
\newcommand{\ra}{\rightarrow}
|
||||
\newcommand{\pis}{\pi^\star}
|
||||
|
||||
\newcommand{\Pop}{6-31+G(d)}
|
||||
\newcommand{\AVDZ}{\emph{aug}-cc-pVDZ}
|
||||
\newcommand{\AVTZ}{\emph{aug}-cc-pVTZ}
|
||||
\newcommand{\AVQZ}{\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\AVFZ}{\emph{aug}-cc-pV5Z}
|
||||
\newcommand{\Td}{\%T_1}
|
||||
|
||||
\usepackage[colorlinks = true,
|
||||
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|
||||
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|
||||
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|
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|
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|
||||
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|
||||
\ifthenelse{\equal{#1}{R}}{
|
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|
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|
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\ifthenelse{\equal{#1}{N}}{
|
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|
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|
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|
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|
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|
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|
||||
%\usepackage{footnote}
|
||||
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|
||||
\let\titlefont\undefined
|
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|
||||
\captionsetup{font={sf,footnotesize}}
|
||||
%
|
||||
\titleformat{\section}
|
||||
{\normalfont\sffamily\bfseries\color{Blue}}
|
||||
{\thesection.}{0.25em}{\uppercase}
|
||||
|
||||
\titleformat{\subsection}
|
||||
{\normalfont\sffamily\bfseries}
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{\thesubsection}{0.25em}{}
|
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
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|
||||
|
||||
\titlespacing*{\section}{0pt}{0.5\baselineskip}{0.01\baselineskip}
|
||||
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|
||||
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|
||||
|
||||
\renewcommand{\refname}{\normalfont\sffamily\bfseries\color{Blue}{\normalsize REFERENCES}}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\author{Pierre-Fran\c{c}ois Loos}
|
||||
\email{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Anthony Scemama}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Martial Boggio-Pasqua}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Denis Jacquemin}
|
||||
\email{Denis.Jacquemin@univ-nantes.fr}
|
||||
\affiliation[CEISAM, Nantes]{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
|
||||
|
||||
\setlength{\bibsep}{0pt plus 0.3ex}
|
||||
|
||||
% bibliography print title or not
|
||||
%\setkeys{acs}{usetitle=true}
|
||||
|
||||
|
||||
\let\oldmaketitle\maketitle
|
||||
\let\maketitle\relax
|
||||
|
||||
\title{A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Exotic Molecules and Radicals}
|
||||
|
||||
\date{\today}
|
||||
|
||||
|
||||
|
||||
\begin{tocentry}
|
||||
\centering
|
||||
\includegraphics[width=.8\textwidth]{TOC.png}
|
||||
\end{tocentry}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\ifpreprint
|
||||
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|
||||
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|
||||
\begin{@twocolumnfalse}
|
||||
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|
||||
\oldmaketitle
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%
|
||||
%%% ABSTRACT %%%
|
||||
%%%%%%%%%%%%%%%%
|
||||
\begin{abstract}
|
||||
Aiming at completing the sets of FCI-quality transition energies that we recently developed (\textit{J.~Chem.~Theory Comput.} \textbf{14} (2018) 4360--4379, \textit{ibid.}~\textbf{15} (2019) 1939--1956, and \textit{ibid.}~\textbf{16} (2020)
|
||||
1711--1741), we provide, in the present contribution, ultra-accurate vertical excitation energies for a series of ``exotic'' closed-shell molecules containing F, Cl, P, and Si atoms and small radicals, such as CON and its variants,
|
||||
that were not considered to date in such investigations. This represents a total of 81 high-quality transitions obtained with a series of diffuse-containing basis sets of various sizes. For the exotic compounds, these transitions are used to
|
||||
perform benchmarks with a vast array of lower-level models, \textit{i.e.}, CIS(D), EOM-MP2, (SOS/SCS)-CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, (SOS-)ADC(2), and ADC(3). Additional comparisons are made with literature data.
|
||||
For the open-shell compounds, we have compared the performances of both the unrestricted and restricted open-shell CCSD and CC3 formalisms.
|
||||
\end{abstract}
|
||||
|
||||
\ifpreprint
|
||||
\else
|
||||
\end{@twocolumnfalse}
|
||||
]
|
||||
\fi
|
||||
|
||||
\ifpreprint
|
||||
\else
|
||||
\small
|
||||
\fi
|
||||
|
||||
\noindent
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
%%% INTRODUCTION %%%
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Introduction}
|
||||
|
||||
The increase of computational ressources coupled to the emergence of more advanced algorithms has led to a resurgence of the selected configuration interaction (SCI) approaches \cite{Ben69,Whi69,Hur73} as an effective strategy to rapidly
|
||||
reach the full CI (FCI) limit at a fraction of the cost of a genuine FCI calculation thanks to a sparse exploration of the FCI space.\cite{Gin13,Caf14,Eva14,Gin15,Gar17b,Caf16,Caf16b,Sch16,Hol16,Liu16b,Sha17,Hol17,Chi18,Gar18,Sce18,Gar19}
|
||||
This revival is especially beneficial for the calculation of transition energies between electronic states, \cite{Eva14,Hol17,Gar18,Chi18,Sce18,Gar19,Loo18a,Loo19c,Gin19,Loo20a} as the accurate determination of these energies remains one of
|
||||
the great challenges faced by theoretical chemists.
|
||||
|
||||
Recently, we have developed two sets of theoretical best estimates (TBEs) of FCI quality for the vertical transition energies of small closed-shell compounds. \cite{Loo18a,Loo19c} (See Ref.~\citenum{Loo20c} for a recent review.)
|
||||
In our first work, \cite{Loo18a} we reported TBEs for more than 100 electronic transitions of single-excitation character in organic compounds containing from one to three non-hydrogen atoms, namely \ce{C}, \ce{N}, \ce{O}, and \ce{S}.
|
||||
These TBEs have been obtained thanks to an efficient implementation of the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) SCI algorithm, \cite{Gar19} which selects the most important determinants
|
||||
in the FCI space using a second-order perturbative criterion. \cite{Gar17b} Their quality was further confirmed by equation-of-motion coupled-cluster (CC) calculations performed up to high excitation degrees. It turned out that CC
|
||||
including contributions up to the quadruples (CCSDTQ) \cite{Kuc91} yields transition energies almost systematically equal to FCI, with a mean absolue error (MAE) as small as $0.01$ eV, whereas the three tested CC approaches
|
||||
including perturbative triples, namely, CC3, \cite{Chr95b,Koc97} CCSDT-3, \cite{Wat96,Pro10} and CCSDT \cite{Nog87} are also very effective with MAEs of $0.03$ eV. \cite{Loo18a} This means that these four CC models are
|
||||
(on average) chemically accurate (error smaller than $1$ kcal.mol$^{-1}$ or $0.043$ eV) for these single excitation transitions.
|
||||
|
||||
Our second set encompasses 20 transitions characterized by a large and/or dominant double excitation nature. \cite{Loo19c} These types of electronic excitations are known to be much more challenging for single-reference methods.
|
||||
For this set, we relied again on SCI methods to determine TBEs and we evaluated the performances of various multi-reference approaches, such as the second-order complete active space perturbation theory (CASPT2), \cite{Roo96,And90}
|
||||
and the second-order $n$-electron valence state perturbation theory (NEVPT2) \cite{Ang01,Ang01b,Ang02} methods. Interestingly, for excitations with a large but not dominant double excitation character, such as the first $^1A_g$
|
||||
excited state of \textit{trans}-butadiene, it turns out that the accuracy obtained with CC3 and NEVPT2 are rather similar with MAEs of ca.~$0.12$ eV. \cite{Loo19c} In contrast, for genuine double excitations (\ie, excitations with an insignificant
|
||||
amount of single excitation character) in which one photon effectively promotes two electrons, the CC3 error becomes extremely large (of the order of $1$ eV) and multi-reference approaches have clearly the edge (for example, the MAE of
|
||||
NEVPT2 is $0.07$ eV). \cite{Loo19c}
|
||||
|
||||
To the very best of our knowledge, these two sets taken together constitute the largest ensemble of chemically-accurate vertical transition energies published to date with roughly $130$ transition energies of FCI quality. Despite their
|
||||
decent sizes and the consideration of both valence and Rydberg excited states, these sets have obvious limitations. Let us point out four of these biases: (i) only small compounds are included; (ii) some important classes of transitions,
|
||||
such as charge-transfer (CT) excitations, are absent; (iii) compounds including only \ce{C}, \ce{N}, \ce{O}, \ce{S}, and \ce{H} atoms have been considered; (iv) these sets include only singlet-singlet and singlet-triplet
|
||||
excitations in closed-shell molecules.
|
||||
|
||||
Very recently, we have made extensive efforts in order to solve the first limitation. \cite{Loo20a} However, performing SCI or high-level CC calculations rapidly becomes extremely tedious when one increases the system size as one
|
||||
hits the exponential wall inherently linked to these methods. At this stage, we believe that circumventing the second limitation is beyond reach as clear intramolecular CT transitions only occur in (very) large molecules for which CCSDTQ
|
||||
or SCI calculations remain clearly out of reach with current technologies. We note, however, that intermolecular CT energies were recently obtained at the CCSDT level by Kozma and coworkers. \cite{Koz20} Therefore, the aim of the present
|
||||
contribution is to get rid of the two latter biases. To this end, we consider here (i) a series of closed-shell compounds including (at least) one of the following atoms: \ce{F}, \ce{Cl}, \ce{Si}, or \ce{P}; (ii) a series of radicals characterized by
|
||||
open-shell electronic configurations and an unpaired electron. For the sake of simplicity, we denote the first additional set as ``exotic'' because it includes a series of chemical species that are rather unusual for organic chemistry, \eg, \ce{H-P=S}
|
||||
and \ce{H2C=Si}. Similar compounds were included in a benchmark set by the Ortiz group. \cite{Hah14} They were, however, using experimental data as reference, which often precludes straightforward comparisons with theoretical vertical transition
|
||||
energies. \cite{Loo18b,Loo19b} On the other hand, the second set, simply labeled as ``radical'', encompasses doublet-doublet transitions in radicals. We believe that the additional FCI-quality estimates that we provide in the present study
|
||||
for both types of compounds nicely complete our previous works and will be valuable for the electronic structure community.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
%%% METHODS %%%
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational methods}
|
||||
|
||||
Our computational protocol closely follows the one of Ref.~\citenum{Loo18a}. Consequently, we only report key elements below. We refer the reader to our previous work for further information about the methodology and
|
||||
the technical details. \cite{Loo18a}
|
||||
In the following, we report several statistical indicators: the mean signed error (MSE), mean absolute error (MAE), root-mean square error (RMSE), and standard deviation of the errors (SDE).
|
||||
|
||||
\subsection{Geometries and basis sets}
|
||||
|
||||
For the exotic set, we use CC3/{\AVTZ} ground-state geometries obtained without frozen-core (FC) approximation (\ie, correlating all electrons) to be consistent with our previously-published geometries.
|
||||
\cite{Bud17,Jac18a,Bre18a,Loo18a} These optimizations have been performed using DALTON 2017 \cite{dalton} and CFOUR 2.1, \cite{cfour} applying default parameters. For the open-shell derivatives, the geometries
|
||||
are optimized at the UCCSD(T)/{\AVTZ} level using the GAUSSIAN16 program \cite{Gaussian16} and applying the \textsc{tight} convergence threshold. The Cartesian coordinates of each compound are available in the
|
||||
Supporting Information (SI).
|
||||
|
||||
Throughout this paper, we use either the diffuse-containing Pople {\Pop} basis set, or the Dunning \emph{aug}-cc-pVXZ (X $=$ D, T, Q, and 5) correlation-consistent family of atomic bases.
|
||||
|
||||
\subsection{CC reference calculations}
|
||||
|
||||
The CC calculations are performed with several codes. For closed-shell molecules, CC3 \cite{Chr95b,Koc97} calculations are achieved with DALTON \cite{dalton} and CFOUR; \cite{cfour} CCSDT calculations are performed
|
||||
with CFOUR \cite{cfour} and MRCC 2017;\cite{Rol13,mrcc} the latter code being also used for CCSDTQ and CCSDTQP. Note that all our excited-state CC calculations are performed within the equation-of-motion (EOM)
|
||||
or linear-response (LR) formalism that yield equivalent excited-state energies. The reported oscillator strengths have been computed in the LR-CC3 formalism only. For open-shell molecules, the CCSDT, CCSDTQ, and
|
||||
CCSDTQP calculations performed with MRCC \cite{Rol13,mrcc} do consider an unrestricted Hartree-Fock (UHF) wave function as reference. All excited-state calculations are performed, except when explicitly mentioned, in
|
||||
the FC approximation using large cores for the third-row atoms. All electrons are correlated for the \ce{Be} atom, for which we systematically applied the basis set as included in MRCC. \cite{Pra10} (We have noted
|
||||
differences in the definition of the Dunning bases for this particular atom depending on the software that one considers.)
|
||||
|
||||
\subsection{Selected Configuration Interaction}
|
||||
|
||||
All the SCI calculations are performed within the FC approximation using QUANTUM PACKAGE \cite{Gar19} where the CIPSI algorithm \cite{Hur73} is implemented. Details regarding this specific CIPSI implementation
|
||||
can be found in Refs.~\citenum{Gar19} and \citenum{Sce19}. We use a state-averaged formalism which means that the ground and excited states are described with the same number and same set of determinants, but
|
||||
different CI coefficients. The SCI energy is defined as the sum of the variational energy (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction which estimates the
|
||||
contribution of the determinants not included in the CI space. \cite{Gar17b} By extrapolating this second-order correction to zero, one can efficiently estimate the FCI limit for the total energies, and hence, compute the
|
||||
corresponding transition energies. We estimate the extrapolation error by the difference between the transition energies obtained with the largest SCI wave function and the FCI extrapolated value. These errors are
|
||||
systematically reported in the Tables below. Although this cannot be viewed as a true error bar, it provides a rough idea of the quality of the FCI extrapolation and estimate.
|
||||
|
||||
\subsection{Other wave function calculations}
|
||||
|
||||
Our benchmark effort consists in evaluating the accuracy of vertical transition energies obtained at lower levels of theory. These calculations are performed with a variety of codes. For the exotic set, we
|
||||
rely on: GAUSSIAN \cite{Gaussian16} and TURBOMOLE 7.3 \cite{Turbomole} for CIS(D); \cite{Hea94,Hea95} Q-CHEM 5.2 \cite{Kry13} for EOM-MP2 [CCSD(2)] \cite{Sta95c} and ADC(3); \cite{Tro02,Har14,Dre15}
|
||||
Q-CHEM \cite{Kry13} and TURBOMOLE \cite{Turbomole} for ADC(2); \cite{Tro97,Dre15} DALTON \cite{dalton} and TURBOMOLE \cite{Turbomole} for CC2; \cite{Chr95,Hat00} DALTON \cite{dalton} and GAUSSIAN
|
||||
for CCSD;\cite{Pur82} DALTON \cite{dalton} for CCSDR(3); \cite{Chr96b} CFOUR \cite{cfour} for CCSDT-3; \cite{Wat96,Pro10} and ORCA \cite{Nee12} for similarity-transformed EOM-CCSD (STEOM-CCSD). \cite{Noo97,Dut18}
|
||||
In addition, we evaluate the spin-opposite scaling (SOS) variants of ADC(2), SOS-ADC(2), as implemented in both Q-CHEM, \cite{Kra13} and TURBOMOLE. \cite{Hel08} Note that these two codes have distinct SOS
|
||||
implementations, as explained in Ref.~\citenum{Kra13}. We also test the SOS and spin-component scaled (SCS) versions of CC2, as implemented in TURBOMOLE. \cite{Hel08,Turbomole} Discussion of various spin-scaling
|
||||
schemes can be found elsewhere. \cite{Goe10a} When available, we take advantage of the resolution-of-the-identity (RI) approximation in TURBOMOLE and Q-CHEM. For the STEOM-CCSD calculations, it was checked that the
|
||||
active character percentage was, at least, $98\%$. When comparisons between various codes/implementations were possible, we could not detect variations in the transition energies larger than $0.01$ eV. For the radical set
|
||||
molecules, we applied both the U (unrestricted) and RO (restricted open-shell) versions of CCSD and CC3 as implemented in the PSI4 code, \cite{Psi4} to perform our benchmarks.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
%%% METHODS %%%
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Results and Discussion}
|
||||
|
||||
\subsection{Exotic set}
|
||||
|
||||
\subsubsection{Reference values and comparison to literature}
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{table*}[htp]
|
||||
\scriptsize
|
||||
\caption{Excitation energies (in eV) of the exotic set obtained within the FC approximation. For each transition, we also report, on the left hand side, the LR-CC3/{\AVTZ} oscillator strength, the CC3 single excitation
|
||||
character ($\Td)$, and the TBE/{\AVTZ} excitation energy. Except otherwise stated, the latter has been obtained directly at FCI/{\AVTZ} level. We also provide the TBE/CBS estimate obtained by correcting the
|
||||
TBE/{\AVTZ} value by the difference between CC3/\emph{aug}-cc-pV5Z and CC3/{\AVTZ}. On the right hand side, one finds the transition energies computed at various levels of theory. T, TQ, and TQP
|
||||
stand for CCSDT, CCSDTQ, and CCSDTQP, respectively.}
|
||||
\label{Table-1}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{tabular}{llp{1.0cm}p{.3cm}p{.5cm}p{.5cm}|p{.4cm}p{.4cm}p{.4cm}p{.4cm}|p{.4cm}p{.4cm}p{.4cm}p{1.1cm}|p{.4cm}p{.4cm}p{.4cm}p{1.1cm}}
|
||||
\hline
|
||||
& &\multicolumn{3}{c}{\AVTZ} & CBS & \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ} & \multicolumn{4}{c}{\AVTZ} \\
|
||||
& & $f$ [CC3] &$\Td$ & TBE & TBE &CC3 & T & TQ & TQP & CC3 & T & TQ & FCI &CC3 & T & TQ & FCI \\
|
||||
\hline
|
||||
Carbonylfluoride& $^1A_2$ & &91.1&7.31$^a$ &7.31 &7.33 &7.30 & & &7.34 &7.31 & &7.30$\pm$0.04 &7.31 &7.28 & &7.32$\pm$0.05\\
|
||||
& $^3A_2$ & &97.8&7.06$^b$ &7.07 &7.03 &7.00 & & &7.05 &7.02 & &7.08$\pm$0.01 &7.03 &7.00 & &7.04$\pm$0.10 \\
|
||||
\ce{CCl2} & $^1B_1$ &0.002 &93.7&2.59$^b$ &2.57 &2.71 &2.70 &2.70 & &2.69 &2.69 & &2.68$\pm$0.02 &2.61 &2.60 & & \\
|
||||
& $^1A_2$ & &88.3&4.40$^b$ &4.41 &4.46 &4.44 &4.47 & &4.40 &4.39 & &4.46$\pm$0.01 &4.35 &4.33 & &\\
|
||||
& $^3B_1$ & &98.6&1.22$^b$ &1.23 &1.10 &1.09 &1.11 & &1.20 &1.19 & &1.22$\pm$0.03 &1.20 &1.19 & &1.22$\pm$0.05 \\
|
||||
& $^3A_2$ & &96.1&4.31$^b$ &4.32 &4.41 &4.38 &4.42 & &4.34 &4.31 & &4.36$\pm$0.01 &4.28 &4.26 & & \\
|
||||
|
||||
\ce{CClF} & $^1A''$ &0.007 &93.9&3.55$^b$ &3.54 &3.66 &3.66 &3.66 & &3.63 &3.62 & &3.62$\pm$0.01 &3.56 &3.55 & &3.63$\pm$0.06 \\
|
||||
\ce{CF2} & $^1B_1$ &0.034 &94.7&5.09 &5.07 &5.18 &5.18 &5.18 & &5.12 &5.11 &5.11 &5.12$\pm$0.00 &5.07 &5.06 & &5.09$\pm$0.01 \\
|
||||
& $^3B_1$ & &99.1&2.77 &2.78 &2.71 &2.70 &2.71 & &2.71 &2.70 &2.71 &2.71$\pm$0.01 &2.76 &2.75 & &2.77$\pm$0.01 \\
|
||||
Difluorodiazirine&$^1B_1$ &0.002 &93.1&3.74$^c$ &3.73$^d$&3.83 &3.83 & & &3.80 &3.80 & & &3.74 &3.74\\
|
||||
&$^1A_2$ & &91.4&7.00$^c$ &6.98$^d$&7.13 &7.11 & & &7.11 &7.08 & & &7.02 &7.00\\
|
||||
&$^1B_2$ &0.026 &93.3&8.52$^c$ &8.54$^d$&8.51 &8.52 & & &8.45 &8.46 & & &8.50 &8.52\\
|
||||
&$^3B_1$ & &98.2&3.03$^e$ &3.03$^d$&3.09 &3.09 & & &3.06 &3.06 & & &3.03 &\\
|
||||
&$^3B_2$ & &98.9&5.44$^e$ &5.46$^d$&5.48 &5.48 & & &5.47 &5.46 & & &5.45 &\\
|
||||
&$^3B_1$ & &98.4&5.80$^e$ &5.81$^d$&5.86 &5.85 & & &5.83 &5.82 & & &5.81 &\\
|
||||
Formylfluoride & $^1A''$ &0.001 &91.2&5.96$^b$ &5.97 &6.09 &6.06 &6.07 & &6.03 &6.00 & &6.00$\pm$0.03 &5.99 &5.96 & &\\
|
||||
& $^3A''$ & &97.9&5.73$^b$ &5.75 &5.72 &5.70 &5.71 & &5.65 &5.62 & &5.65$\pm$0.01 &5.62 &5.60 & &\\
|
||||
\ce{HCCl} & $^1A''$ &0.003 &94.5&1.98 &1.97 &2.05 &2.04 &2.05 &2.05 &2.02 &2.02 &2.02 &2.04$\pm$0.01 &1.97 &1.97 & &1.98$\pm$0.00\\
|
||||
\ce{HCF} &$^1A''$ &0.006 &95.4&2.49 &2.49 &2.58 &2.57 &2.58 &2.58 &2.53 &2.53 &2.53 &2.54$\pm$0.00 &2.49 &2.49 & &2.49$\pm$0.02\\
|
||||
\ce{HCP} & $^1\Sigma^-$ & &94.9&4.84 &4.81 &5.19 &5.19 &5.18 &5.18 &5.06 &5.05 &5.04 &5.04$\pm$0.00 &4.85 &4.85 &4.84 &4.84$\pm$0.00\\
|
||||
& $^1\Delta$ & &94.0&5.15 &5.10 &5.48 &5.48 &5.48 &5.48 &5.33 &5.33 &5.32 &5.32$\pm$0.00 &5.15 &5.15 & &5.15$\pm$0.00\\
|
||||
& $^3\Sigma^+$& &98.9&3.47 &3.49 &3.44 &3.45 &3.46 &3.46 &3.47 &3.47 &3.49 &3.49$\pm$0.00 &3.45 &3.45 &3.46 &3.47$\pm$0.00\\
|
||||
& $^3\Delta$ & &98.8&4.22 &4.20 &4.40 &4.39 &4.39 &4.39 &4.35 &4.34 &4.34 &4.34$\pm$0.00 &4.22 &4.21 &4.21 &4.22$\pm$0.00\\
|
||||
\ce{HPO} & $^1A''$ &0.003 &90.9&2.47 &2.49 &2.49 &2.47 &2.48 &2.48 &2.47 &2.45 &2.46 &2.46$\pm$0.00 &2.46 &2.46 & &2.47$\pm$0.00\\
|
||||
\ce{HPS} & $^1A''$ &0.001 &90.3&1.59 &1.61 &1.57 &1.55 &1.56 &1.56 &1.60 &1.59 &1.59 &1.60$\pm$0.00 &1.59 &1.58 & &1.59$\pm$0.00\\
|
||||
\ce{HSiF} & $^1A''$ &0.024 &93.1&3.05 &3.05 &3.09 &3.08 &3.08 &3.08 &3.08 &3.07 &3.07 &3.06$\pm$0.00 &3.07 &3.06 & &3.05$\pm$0.00\\
|
||||
\ce{SiCl2} &$^1B_1$ &0.031 &92.1&3.91$^b$ &3.93 &3.94 &3.94 &3.94 & &3.93 &3.92 & &3.95$\pm$0.02 &3.90 &3.88 & &3.88$\pm$0.03\\
|
||||
&$^3B_1$ & &98.7&2.48$^f$ &2.50 &2.39 &2.39 &2.40 & &2.45 &2.44 & &2.47$\pm$0.05 &2.48 &2.47 & &2.49$\pm$0.04\\
|
||||
Silylidene &$^1A_2$ & &92.3&2.11 &2.12 &2.14 &2.11 &2.10 &2.10 &2.18 &2.15 &2.14 &2.14$\pm$0.00 &2.15 &2.13 &2.12 &2.11$\pm$0.01\\
|
||||
&$^1B_2$ &0.033 &88.0&3.78 &3.80 &3.88 &3.87 &3.88 &3.88 &3.81 &3.80 &3.80 &3.79$\pm$0.01 &3.78 &3.78 &3.78 &3.78$\pm$0.01\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{flushleft}
|
||||
$^a${FCI/{\Pop} value of 7.33$\pm$0.02 eV corrected by the difference between CCSDT/{\AVTZ} and CCSDT/{\Pop};}
|
||||
$^b${FCI/{\AVDZ} value corrected by the difference between CCSDT/{\AVTZ} and CCSDT/{\AVDZ};}
|
||||
$^c${CCSDT/{\AVTZ} value;}
|
||||
$^d${Corrected with the quadruple-$\zeta$ basis rather than the quintuple-$\zeta$ basis;}
|
||||
$^e${CCSDT/{\AVDZ} value corrected by the difference between CC3/{\AVTZ} and CC3/{\AVDZ};}
|
||||
$^f${CCSDTQ/{\Pop} value corrected by the difference between CCSDT/{\AVTZ} and CCSDT/{\Pop}.}
|
||||
\end{flushleft}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
Our main results are listed in Table \ref{Table-1} for the exotic set that encompasses 30 electronic transitions (19 singlets and 11 triplets) in 14 molecules containing between two and five non-hydrogen atoms. Before briefly discussing the
|
||||
compounds individually, let us review some general trends. First, as one could expect for rather low-lying excitations, the {\AVTZ} basis set is sufficient large to provide excitation energies close to the complete basis set (CBS) limit \cite{Gin19}
|
||||
and the FC approximation is rather unimportant. Indeed, CC3 calculations performed with quadruple- and quintuple-$\zeta$ basis sets, with and without correlating the core electrons for the former basis, yield negligible changes as compared to
|
||||
the {\AVTZ} results. As more quantitatively illustrated by the results gathered in Table S1 \hl{and Figure S1} in the SI, the maximal variation between CC3/{\AVTZ} and CC3/{\AVQZ} excitation energies is $0.03$ eV ($^1\Delta$ state of \ce{HCP}),
|
||||
and the MAE between the two basis sets is as small as $0.01$ eV. The same observation applies to the FC approximation with a mean absolute variation of $0.02$ eV between the CC3(full)/\emph{aug}-cc-pCVQZ and CC3(FC)/\emph{aug}-cc-pVQZ
|
||||
excitation energies. \hl{Of course, using a smaller basis set than \emph{aug}-cc-pVTZ, e.g., Pople's 6-31+G(d) or Dunning's \emph{aug}-cc-pVDZ would induce larger errors with an overestimation trend (see Figure S1).}
|
||||
\hl{As \emph{aug}-cc-pVTZ is sufficient,} we do not discuss further the quadruple- and quintuple-$\zeta$ results in the following, although basis set corrected TBEs can be found in Table \ref{Table-1}. Secondly, it can be seen,
|
||||
from the CC3 $\Td$ values (which provides a measure of the amount single excitation character of the considered transition) listed in Table \ref{Table-1}, that all the transitions considered here are largely dominated by single excitations,
|
||||
the smallest $\Td$ being $88\%$ (the second transition of silylidene). Such character is favorable to ensure a rapid convergence of the CC series. This is clearly exemplified by the convergence behavior of the {\Pop} excitation energies
|
||||
for which the CCSDTQ and the CCSDTQP transition energies are equal for the 11 cases for which the latter level of theory was achievable. Likewise, one notices that the CCSDTQ estimate systematically falls within $0.01$ eV of the FCI
|
||||
value that comes with a very small error bar for most transitions. It is also reassuring to see that, for a given basis set, we could not detect variations larger than $0.04$ eV between CCSDTQ results and their CC3 and CCSDT counterparts,
|
||||
the changes being typically of ca.~$0.01$--$0.02$ eV. All these facts indicate that one can trust the FCI estimates, and hence the TBEs listed in Table \ref{Table-1} (for the larger difluorodiazirine molecule, see discussion below).
|
||||
|
||||
In the spirit of the famous Thiel paper, \cite{Sch08} let us now briefly discuss each compound and compare the results to available data. We do not intend here to provide an exhaustive review of previous calculations, which would lead to
|
||||
a gigantic list of references for the triatomic systems, but rather to pinpoint the ``best'' published excitation energies to date.
|
||||
|
||||
\emph{Carbonylfluoride.} For this compound encompassing four heavy atoms, the convergence of the SCI approach is rather slow and one notices a $0.03$ eV drop of the transition energies between CC3 and CCSDT.
|
||||
We therefore used FCI estimates determined with small bases, corrected for basis set effects to generate our TBEs. For the lowest singlet, that is heavily blueshifted as compared to the parent formaldehyde, the most advanced previous
|
||||
theoretical studies reported vertical transition energies of $7.31$ eV [CCSDR(3)], \cite{Lav11} and $7.31$ eV [MRCI+Q]. \cite{Kat11} The measured EEL value is ca.~$7.3$ eV, \cite{Kat11} whereas the UV spectrum shows a peak at $7.34$
|
||||
eV. \cite{Wor70} All these values are obviously compatible with the current result. Note that the interpretation of the measured 0-0 values for \ce{F2C=O} \cite{Jud83b} is challenging, as discussed elsewhere. \cite{Loo18b} For the triplet,
|
||||
the previous TBE is likely a $7.07$ eV MRCI+Q result, \cite{Kat11} also very close to our present value, whereas there also exists estimates of the triplet adiabatic energies. \cite{Bok09}
|
||||
|
||||
\emph{CCl$_2$, CClF, and CF$_2$.} Dichlorocarbene is large enough to make the convergence of the SCI calculations difficult with the triple-$\zeta$ basis, and our TBEs are based on the FCI/\emph{aug}-cc-pVDZ values corrected for basis set effects
|
||||
determined at the CC level. While both CC3 and CCSDT almost perfectly reproduce the FCI results for the singlet and triplet $B_1$ states, more significant differences are noted for the higher-lying $A_2$ states that seem slightly too low with
|
||||
CCSDT. This is also confirmed by the CCSDTQ results obtained with the Pople basis set. Previous calculations are available at CCSD, \cite{Cze07} and MRCI \cite{Cai93,Sun15b} levels. The most recent MRCI+Q values, obtained with a large atomic basis
|
||||
set are $2.61$, $4.49$, $1.25$ and $4.43$ eV for the $^1B_1$, $^1A_2$, $^3B_1$, and $^3A_2$ transitions, respectively. These values are reasonably close to the present TBEs. For CClF, the most accurate literature value is probably the
|
||||
MRCI+Q/triple-$\zeta$ estimate of $3.59$ eV, \cite{Sun15c} within $0.03$ eV of our current TBE. For this compound, we are also aware of three previous experimental investigations focussing on its vibronic spectra. \cite{Sch91,Kar93,Gus01} For \ce{CF2},
|
||||
the SCI calculations converge rapidly even with the {\AVTZ} basis and yield TBEs of $5.09$ and $2.77$ eV for the lowest singlet and triplet transitions. There has been countless experimental and theoretical investigations for this stable carbene,
|
||||
but the most accurate previous estimates of the vertical transition energies are likely the $5.12$ and $2.83$ eV values, obtained at the MRCI+Q/{\AVTZ} level of theort. \cite{Sun16b}
|
||||
|
||||
\emph{Difluorodiazirine.} This cyclopropene analogue is the largest derivative considered herein. There is a remarkable agreement between CC3 and CCSDT values, and the $\Td$ value is very large for each transition, so that we consider the CC values
|
||||
to obtain our TBEs. For the $^1B_1$ and $^3B_1$ transitions, FCI/{\Pop} calculations deliver transition energies of $3.81\pm0.01$ and $3.09\pm0.01$ eV, perfectly consistent with the present CC values. Our TBEs are likely the most accurate to date for vertical
|
||||
transitions. At the GVVPT2/cc-pVTZ level, the transition energies reported in Ref.~\citenum{Pan04} are $2.25$ eV ($^3B_1$), $2.95$ eV ($^1B_1$), $4.86$ eV ($^3B_2$), $5.21$ eV ($^3A_2$), $6.63$ eV ($^1A_2$), and $8.23$ eV ($^1B_2$), which follows exactly the
|
||||
same state ordering as the present CCSDT values. More recently, QCISD/{\AVTZ} estimates of $2.81$ and $3.99$ eV for the lowest triplet and singlet vertical transitions have been reported, which are respectively
|
||||
slightly smaller and larger than the present data. There are also quite a few studies of the 0-0 energies of various states for this derivative, both experimentally \cite{Lom69,Hep74,Sie90} and theoretically. \cite{Pan04,Ter16,Loo19a}
|
||||
|
||||
\emph{Formylfluoride.} For this formal intermediate between carbonylfluoride and formaldehyde, we note that the CCSDTQ/{\Pop} values are bracketed by their CC3 and CCSDT counterparts. The previous best estimates are likely the very
|
||||
recent MRCI-F12 results of Pradhan and Brown who reported vertical transition energies of $6.03$ eV and $5.68$ eV for the $^1A''$ and $^3A''$ states, respectively. These energies obtained on the CCSD(T)-F12 ground-state geometries are only ca.~$0.05$ eV
|
||||
larger than the present TBEs. Most other previous studies focussed on 0-0 energies of the lowest singlet state, \cite{Gid62,Fis69,Sta95b,Cra97,Fan01,Bok09,Loo18b,Loo19a,Pra19} and it is noteworthy that CC3 reproduces the experimental 0-0 energies
|
||||
with high accuracy. \cite{Loo18b,Loo19a} Our TBE for the singlet state ($5.96$ eV) is much larger than the measured 0-0 peak ($4.64$ eV) \cite{Cra97} which is expected for a molecule undergoing an important geometrical relaxation after excitation. \cite{Sta95b}
|
||||
|
||||
\emph{HCCl, HCF, and HSiF.} For these three compounds, the SCI calculations deliver values very close to the CC estimates. For \ce{HCCl}, a MRCI+Q/quintuple-$\zeta$ vertical transition energy, corrected for ground-state ZPVE effects, of $1.68$ eV was recently
|
||||
reported. \cite{Sha15c} Given that the ZPVE energy at the MP2/{\AVTZ} level is $0.31$ eV, our TBE is basically equivalent to this recent result. For \ce{HSiF}, the most accurate previous estimate of the excitation energy is likely the CC3/{\AVTZ} $3.07$ eV value,
|
||||
\cite{Chr02} which is extremely close to our TBE. For the records, Ehara and coworkers also investigated the 0-0 energies and excited-state geometries of these three systems at the SAC-CI level, \cite{Eha11} and experimental 0-0 energies of
|
||||
$1.52$ eV (\ce{HCCl}), \cite{Cha95c} $2.14$ eV (\ce{HCF}), \cite{Kak81,Sch99} and $2.88$ eV (\ce{HSiF}), \cite{Har95} have been measured.
|
||||
|
||||
\emph{HCP.} Phosphaethyne is a linear compound for which the CC series and the SCI values do converge rapidly and give equivalent results. Consequently, one can trust the TBEs listed in Table \ref{Table-1}. We nevertheless note that there is a
|
||||
significant basis set effect for the $^1\Delta$ excited state that is downshifted by $0.05$ eV from {\AVTZ} to \emph{aug}-cc-pV5Z (see Table S1 in the SI). The two most refined previous theoretical works we are aware of have been performed at the
|
||||
MRCI/double-$\zeta$ \cite{Nan00} and CC3/cc-pVQZ \cite{Ing06} levels of theory and respectively focussed on reproducing the experimental vibronic couplings and understanding the \ce{HCP -> HPC} isomerization process. However,
|
||||
somehow surprisingly, we could not find recent estimates of the vertical transition energies for phosphaethyne, the previously published data being apparently of CASSCF quality. \cite{Gol93} There are, of course, experimental characterizations
|
||||
of the 0-0 energies for several excited states of this compound. \cite{Her66}
|
||||
|
||||
\emph{HPO and HPS.} The lowest excited state of HPO has been studied several times in the last twenty years, \cite{Lun95,Tac02,Lee07b,Eha11,Loo19a} whereas its sulfur analogue has only been considered more recently. \cite{Gri13b,Mok14,Meh18,Loo19a} In both cases,
|
||||
refined MRCI calculations of the vibronic spectra have been performed \cite{Lee07b,Eha11,Gri13b,Mok14,Meh18} but few reported vertical transition energies. We are aware of a quite old CASPT2 estimate of $2.25$ eV for \ce{HPO}, \cite{Lun95}
|
||||
and a recent MRCI vertical transition energy of $1.69$ eV (obtained with a very large basis set) for \ce{HPS}. \cite{Meh18}
|
||||
|
||||
\emph{SiCl$_2$.} In this heavier analogue of dichlorocarbene, there are no strong methodological effects but the SCI convergence is shaky, especially for the triplet and we used a basis set extrapolated CCSDTQ value as TBE for this state.
|
||||
Advanced calculations of the adiabatic energies \cite{Cha99} as well as experimental 0-0 energies \cite{Du91,Kar93b} can be found in the literature, the latter being $3.72$ and $2.35$ eV for the lowest singlet and triplet states, respectively. These
|
||||
values are sightly larger than our vertical estimates. For the vertical singlet excitation, there is also a recent $4.06$ eV CCSD//CAM-B3LYP estimate, \cite{Ran16b} which slightly overshoots ours, consistent with the expected error
|
||||
sign of CCSD. \cite{Sch08,Kan17,Loo18a}
|
||||
|
||||
\emph{Silylidene.} One notes an excellent agreement between CCSDT, CCSDTQ, and FCI for this derivative. Our TBEs of $2.11$ eV and $3.78$ eV are again exceeding the experimental 0-0 energies of $1.88$ eV \cite{Smi03}
|
||||
and $3.63$ eV, \cite{Har97b} as it should. The previous theoretical studies we are aware of have been performed with CISD(+Q), \cite{Hil97,Smi03} and CC3 \cite{Loo18b,Loo19a} methods and mainly discussed the 0-0 energies, for which an
|
||||
excellent agreement with experiment was obtained by both approaches.
|
||||
|
||||
\subsubsection{Benchmarks}
|
||||
|
||||
Benchmarks using the TBEs obtained in the previous Section can be naturally done. As we consider closed-shell compounds, there is a large number of methods that one can evaluate. Here, we have chosen 15 popular wave function
|
||||
methods for excited states (see \textit{Computational Details} and Table S2 in the SI for the raw data). The statistical result can be found in Figure \ref{Fig-1} and Table \ref{Table-2}.
|
||||
|
||||
\renewcommand*{\arraystretch}{1.0}
|
||||
\begin{table}[htp]
|
||||
\scriptsize
|
||||
\caption{Statistical values obtained by comparing the results of various methods to the TBE/{\AVTZ} values listed in Table \ref{Table-1}. We report the
|
||||
mean signed error (MSE), mean absolute error (MAE), root-mean square error (RMSE), and standard deviation of the errors (SDE). All
|
||||
quantities are given in eV and have been obtained with the {\AVTZ} basis set. TM and QC stand for the TURBOMOLE and Q-CHEM
|
||||
definitions of the scaling factors, respectively. ADC(2.5) is the simple average of the ADC(2) and ADC(3) transition energies, as defined in Ref.~\citenum{Loo20b}.
|
||||
``Count'' refers to the number of transitions computed for each method.}
|
||||
\label{Table-2}
|
||||
\begin{tabular}{lccccc}
|
||||
\hline
|
||||
Method & Count & MSE &MAE &RMSE &SDE \\
|
||||
\hline
|
||||
CIS(D) &30&0.09 &0.14 &0.19 &0.16 \\
|
||||
EOM-MP2 &30&-0.06 &0.17 &0.22 &0.21 \\
|
||||
STEOM-CCSD &25&-0.10 &0.12 &0.14 &0.10 \\
|
||||
CC2 &30&0.07 &0.12 &0.15 &0.14 \\
|
||||
SOS-CC2 [TM] &30&0.17 &0.18 &0.20 &0.10 \\
|
||||
SCS-CC2 [TM] &30&0.14 &0.14 &0.16 &0.09 \\
|
||||
ADC(2) &30&-0.02 &0.15 &0.16 &0.17 \\
|
||||
SOS-ADC(2) [TM] &30&0.11 &0.13 &0.17 &0.14 \\
|
||||
SOS-ADC(2) [QC] &30&-0.04 &0.12 &0.14 &0.14 \\
|
||||
CCSD &30&0.03 &0.07 &0.08 &0.08 \\
|
||||
ADC(3) &30&-0.19 &0.24 &0.27 &0.19 \\
|
||||
ADC(2.5) &30&-0.11 &0.11 &0.13 &0.07 \\
|
||||
CCSDR(3) &19&0.01 &0.02 &0.02 &0.02 \\
|
||||
CCSDT-3 &19&0.01 &0.02 &0.02 &0.02 \\
|
||||
CC3 &30&0.00 &0.01 &0.02 &0.02 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{table}
|
||||
|
||||
\begin{figure*}[htp]
|
||||
\includegraphics[scale=0.85,viewport=2.8cm 5.5cm 18.3cm 27.5cm,clip]{Figure-1.pdf}
|
||||
\caption{Histograms of the error distribution (in eV) obtained with 15 theoretical methods, choosing the TBE/{\AVTZ} of Table \ref{Table-1} as references.
|
||||
TM and QC stand for the TURBOMOLE and Q-CHEM definitions of the scaling factors, respectively.
|
||||
Note the difference of scaling in the vertical axes.}
|
||||
\label{Fig-1}
|
||||
\end{figure*}
|
||||
|
||||
Most of the conclusions that can be extracted from these benchmarks are consistent with recent analyses made in the field, \cite{Kan14,Taj16,Kan17,Loo18a,Loo19a,Loo20a,Taj20a,Loo20b} and we will therefore only briefly comment on the most
|
||||
significant outcomes. First, one notes that CC3, which is an expensive approach, is superbly accurate and consistent with a trifling MSE and a tiny SDE, whereas both CCSDT-3 and CCSDR(3), for which only singlet excited states can be
|
||||
evaluated with the current implementations, are also extremely satisfying with average errors well below the chemical accuracy threshold. This is unsurprisingly inline with the trends obtained for more ``standard'' organic compounds: CC
|
||||
methods including (at least partially) contributions from the triples are trustworthy for the description of single excitations. \cite{Hat05c,Sau09,Wat13,Kan17,Loo18a,Loo19a,Sue19} Going down in the CC hierarchy, we find that CCSD slightly
|
||||
overestimates the transition energies, but nevertheless provides very consistent estimates (SDE of $0.08$ eV), whereas CC2 is clearly less satisfying in terms of consistency (SDE of $0.14$ eV). Comparing with previous benchmarks,
|
||||
\cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a,Loo20b} we can foresee that the CCSD overestimation will likely grow in larger compounds, whereas the CC2 accuracy should remain less affected by the system size. The SOS and SCS
|
||||
variants of CC2 deliver larger MAE, with a clear overestimation (see Figure \ref{Fig-1}), but a smaller error dispersion than the standard CC2 method. The accuracy deterioration and the improved consistency of the spin-scaled CC2
|
||||
versions (w.r.t. standard CC2) is known, \cite{Goe10a,Jac15b,Taj20a} though some works reported that SOS-CC2 and SCS-CC2 can also improve the accuracy.\cite{Win13} STEOM-CCSD delivers results of roughly CC2 quality for the present set, whereas patterns
|
||||
more alike the ones of CCSD have been previously obtained. \cite{Loo18a,Dut18,Loo20a} In the present case, both CIS(D) and EOM-MP2 [also denoted CCSD(2)], which are the two computationally lightest approaches, are also the ones
|
||||
yielding the largest dispersions alongside quite significant MAEs. For EOM-MP2, similar outcomes were observed for valence excited states by Tajti and Szalay, \cite{Taj16} whereas the relatively poor performance of CIS(D) is well
|
||||
documented. \cite{Goe10a,Jac15b,Loo18a,Loo20a} In the ADC series, we note that ADC(2) yields results only slightly less accurate than CC2 for a smaller computational cost, which is consistent with the conclusions of Dreuw's group, \cite{Har14}
|
||||
whereas the SOS variant developed by the same group \cite{Kra13} has a slight edge over its TURBOMOLE variant. ADC(3) provides rather poor excitation energies, a trend we recently evidenced in other molecular sets.
|
||||
\cite{Loo18a,Sue19,Loo20b} Finally, the very recently introduced ADC(2.5) scheme, which corresponds to the simple average of the ADC(2) and ADC(3) excitation energies, \cite{Loo20b} provides significantly more consistent estimates
|
||||
than both ADC(2) or ADC(3), with a SDE of $0.07$ eV only compared to ca.~$0.18$ eV for the ``parent'' methods. ADC(2.5) can then be seen as a cost effective approach to improve upon ADC(3), at least for small compounds.
|
||||
|
||||
\subsection{Radical set}
|
||||
|
||||
\subsubsection{Reference values and comparison to literature}
|
||||
|
||||
Let us now turn to radicals. As nicely summarized by Crawford fifteen years ago, \cite{Smi05b} electronic transitions in open-shell systems are more challenging, not only due to the more limited number of methods and codes available for treating them
|
||||
(as compared to closed-shell molecules), but also because: (i) strong spin contamination can take place with ``low''-level methods; (ii) large contributions from doubly-excited configurations are quite common; and (iii) basis set effects can be
|
||||
very large, meaning that reaching the CBS limit can be laborious. At the CCSD level for instance, significant differences between U and RO transition energies can sometimes be observed. \cite{Smi05b} This is why our results, listed in Table \ref{Table-3},
|
||||
use as computationally-lightest approach (U)CCSDT, so that the wave function is robust enough in order to mitigate the two former issues for most of the considered transitions. As can be seen in the {\Pop} and {\AVDZ} columns of Table \ref{Table-3},
|
||||
one generally finds an excellent agreement between the various CC estimates and their FCI counterparts, UCCSDT being already extremely accurate except in specific cases (such as the $^2\Sigma^+$ excited state of \ce{CO+}). This overall consistency
|
||||
indicates yet again that one can trust the present TBEs. \hl{Figure S2 in the SI provides histograms of the errors obtained when comparing CCSDT results obtained with 6-31+G(d), \emph{aug}-cc-pVDZ and \emph{aug}-cc-pVTZ to their \emph{aug}-cc-pVQZ
|
||||
counterparts. While some large errors can be noticed with the Pople basis set, one notes a relatively satisfying behavior of \emph{aug}-cc-pVDZ. More importantly, the accuracy of \emph{aug}-cc-pVTZ is clearly confirmed.}
|
||||
We also underline that, except for diatomics, UCCSDT calculations performed with diffuse basis sets on open-shell molecules are quite rare in the literature (see below), and the same obviously holds for higher-order CC. As for the exotic set,
|
||||
we do not intend here to provide an exhaustive list of previous works, but rather to pinpoint a few interesting comparisons with earlier accurate estimates.
|
||||
|
||||
%%% TABLE 3 %%%
|
||||
\begin{table*}[htp]
|
||||
\scriptsize
|
||||
\caption{Excitation energies (in eV) of the radical set obtained within the FC approximation. For each state, we report, on the left hand side, the TBE/{\AVTZ} excitation energy obtained directly at the FCI level (except otherwise stated). The TBE/CBS excitation energy is obtained with the largest affordable basis set (see footnotes). On the right hand side, one finds the transition energies computed at various levels of theory. T, TQ, and TQP stand for UCCSDT, UCCSDTQ, and UCCSDTQP, respectively.}
|
||||
\label{Table-3}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{tabular}{ll|ll|p{.36cm}p{.36cm}p{.36cm}p{1.0cm}|p{.36cm}p{.36cm}p{.36cm}p{1.0cm}|p{.36cm}p{.36cm}p{1.0cm}|p{.36cm}p{.36cm}p{1.0cm}}
|
||||
\hline
|
||||
& & AVTZ & CBS & \multicolumn{4}{c}{\Pop} & \multicolumn{4}{c}{\AVDZ} & \multicolumn{3}{c}{\AVTZ} & \multicolumn{3}{c}{\AVQZ} \\
|
||||
& & TBE & TBE & T & TQ & TQP & FCI & T & TQ & TQP & FCI & T & TQ & FCI & T & TQ & FCI \\
|
||||
\hline
|
||||
Allyl &$^2B_1$ &3.39$^a$& & 3.46& 3.44& & 3.42$\pm$0.02 &3.46& & & 3.44$\pm$0.04&3.43 & & & & \\%Valence
|
||||
&$^2A_1$ &4.99$^a$& & 5.16& 5.14& & 5.18$\pm$0.01 &4.88& & & 4.91$\pm$0.04&4.97 & & & & \\%Rydberg
|
||||
\ce{BeF} &$^2\Pi$ &4.14 &4.13$^b$& 4.29& 4.28& 4.28& 4.28$\pm$0.00 &4.21& 4.20& 4.20 & 4.20$\pm$0.09 &4.15& 4.15& 4.14$\pm$0.01 & 4.14& & 4.13$\pm$0.01 \\%Vale
|
||||
&$^2\Sigma^+$ &6.21 & & 6.32& 6.31& 6.31& 6.32$\pm$0.00 &6.30& 6.29& 6.29 & 6.29$\pm$0.00 &6.23& 6.22& 6.21$\pm$0.02 & & & \\%Rydb
|
||||
\ce{BeH} &$^2\Pi$ &2.49 &2.48$^b$& 2.53& 2.53& 2.53& 2.53$\pm$0.00 &2.52& 2.52& 2.52& 2.52$\pm$0.00 &2.49& 2.49& 2.49$\pm$0.00 & 2.48& 2.48& 2.48$\pm$0.00 \\%Valence
|
||||
&$^2\Pi$ &6.46 &6.46$^b$& 6.42& 6.42& 6.42& 6.42$\pm$0.00 &6.43& 6.43& 6.43& 6.43$\pm$0.00 &6.45& 6.46& 6.46$\pm$0.00 & 6.46& 6.46& 6.46$\pm$0.00 \\%Rydberg
|
||||
\ce{BH2} &$^2B_1$ &1.18 &1.18$^c$& 1.19& 1.19& 1.19& 1.19$\pm$0.00 &1.21& 1.21& 1.21& 1.21$\pm$0.00 &1.18& 1.18& 1.18$\pm$0.01 & 1.18& 1.18& 1.18$\pm$0.00 \\%V
|
||||
\ce{CH} &$^2\Delta$ &2.91 &2.90$^c$& 3.07& 3.05& 3.05& 3.05$\pm$0.00 &3.01& 2.99& 2.99& 2.99$\pm$0.00 &2.94& 2.91& 2.91$\pm$0.00 & 2.93& 2.90& 2.90$\pm$0.00 \\%V
|
||||
&$^2\Sigma^-$ &3.29 &3.28$^c$& 3.36& 3.35& 3.35& 3.35$\pm$0.00 &3.34& 3.32& 3.32& 3.32$\pm$0.00 &3.31& 3.29& 3.29$\pm$0.00 & 3.30& 3.29& 3.28$\pm$0.01 \\%???
|
||||
&$^2\Sigma^+$ &3.98 &3.96$^c$& 4.12& 4.10& 4.10& 4.09$\pm$0.00 &4.07& 4.04& 4.04& 4.03$\pm$0.00 &4.03& 3.98& 3.98$\pm$0.00 & 4.01& 3.97& 3.96$\pm$0.01 \\%V
|
||||
\ce{CH3} &$^2A_1'$ &5.85 &5.88$^c$& 6.00& 6.00& 6.00& 6.00$\pm$0.00 &5.78& 5.79& 5.79& 5.79$\pm$0.00 &5.86& 5.86& 5.85$\pm$0.01 & 5.88& & 5.88$\pm$0.00 \\%R/3s ??
|
||||
&$^2E'$ &6.96 &6.96$^c$& 7.28& 7.28& 7.28& 7.28$\pm$0.00 &7.01& 7.02& 7.02& 7.01$\pm$0.00 &6.97& 6.97& 6.96$\pm$0.01 & 6.96& & 6.96$\pm$0.00 \\%R 3p ? Ou Val ?
|
||||
&$^2E'$ &7.18 &7.17$^d$& 7.43& 7.43& 7.43& 7.43$\pm$0.00 &7.17& 7.18& 7.18& 7.18$\pm$0.00 &7.19& 7.19& 7.18$\pm$0.02 & 7.19& & \\%Val ou R 3p ?
|
||||
&$^2A_2''$ &7.65 &7.48$^d$& 7.81& 7.81& 7.81& 7.81$\pm$0.00 &7.76& 7.76& 7.76& 7.76$\pm$0.00 &7.65& 7.66& 7.65$\pm$0.01 & 7.57& & \\%R 3p
|
||||
\ce{CN} &$^2\Pi$ &1.34 &1.33$^b$& 1.44& 1.41& 1.41& 1.40$\pm$0.00 &1.42& 1.39& 1.38 & 1.38$\pm$0.00 &1.38& 1.35 & 1.34$\pm$0.01 & 1.38& & 1.33$\pm$0.01 \\%V
|
||||
&$^2\Sigma^+$ &3.22 &3.21$^b$& 3.24& 3.23& 3.23& 3.23$\pm$0.01 &3.25& 3.23& 3.23 & 3.23$\pm$0.00 &3.25& 3.22 & 3.22$\pm$0.00 & 3.25& & 3.21$\pm$0.00 \\%V
|
||||
\ce{CNO} &$^2\Sigma^+$ &1.61 &1.61$^e$& 1.66& 1.59& & 1.57$\pm$0.00 &1.66& 1.59 & & 1.58$\pm$0.00 &1.71& & 1.61$\pm$0.01 & 1.71 & & \\%
|
||||
&$^2\Pi$ &5.49$^a$&5.50$^f$& 5.62& 5.56& & 5.54$\pm$0.02 &5.59& 5.53& & 5.49$\pm$0.05 &5.57& & & 5.58 & & \\%
|
||||
\ce{CON} &$^2\Pi$$^g$ &3.53$^h$& & 3.54 & 3.50 & & 3.53$\pm$0.00 &3.55& & & 3.54$\pm$0.01 &3.54& & & & \\%pi-pi
|
||||
&$^2\Sigma^+$ $^g$ &\hl{3.86}$^i$& & 4.04 & 3.79& & &4.05& & & &\hl{4.12}& & & & \\%n-pi
|
||||
\ce{CO+} &$^2\Pi$ &3.28 &3.26$^b$& 3.30& 3.33& 3.33& 3.33$\pm$0.00 &3.30& 3.33& 3.33 & 3.33$\pm$0.00 &3.26& 3.28 & 3.28$\pm$0.00 & 3.27& & 3.26$\pm$0.00 \\%V
|
||||
&$^2\Sigma^+$ &5.81 &5.80$^b$& 5.69& 5.79& 5.82& 5.82$\pm$0.00 &5.78& 5.87& 5.89 & 5.90$\pm$0.00 &5.70& 5.78 & 5.81$\pm$0.00 & 5.72& & 5.80$\pm$0.00 \\%R
|
||||
\ce{F2BO} &$^2B_1$ &0.73$^h$& & 0.73& & & 0.72$\pm$0.00 &0.72& & & 0.74$\pm$0.02 &0.71& & & & & \\%V p-n
|
||||
&$^2A_1$ &2.80$^h$& & 2.85& & & 2.87$\pm$0.00 &2.86& & & 2.88$\pm$0.00 &2.78& & & & & \\%V sigma-n
|
||||
\ce{F2BS} &$^2B_1$ &0.51$^h$& & 0.49& & & 0.48$\pm$0.00 &0.50& & & 0.53$\pm$0.00 &0.48& & & & & \\%V p-n
|
||||
&$^2A_1$ &2.99$^h$& & 3.07& & & 3.06$\pm$0.03 &2.96& & & 3.02$\pm$0.01 &2.93& & & & & \\%V sigma-n
|
||||
\ce{H2BO} &$^2B_1$ &2.15 &2.14$^e$& 2.27& 2.28& 2.28& 2.28$\pm$0.00 &2.23& 2.23& & 2.23$\pm$0.00 &2.17& & 2.15$\pm$0.01 & 2.16& & \\%V p-n
|
||||
&$^2A_1$ &3.49 &3.49$^e$& 3.62& 3.62& 3.62& 3.62$\pm$0.00 &3.61& 3.61& & 3.60$\pm$0.01 &3.51& & 3.49$\pm$0.01 & 3.51& & \\%V sigma-n
|
||||
\ce{HCO} &$^2A''$ &2.09 &2.09$^e$& 2.18& 2.17& 2.18& 2.17$\pm$0.01 &2.12& 2.12& & 2.13$\pm$0.01 &2.10& & 2.09$\pm$0.01 & 2.10& & \\ %Valence, n-pi
|
||||
&$^2A'$ &5.45$^h$&5.49$^j$& 5.45& 5.47& 5.47& 5.47$\pm$0.00 &5.32& 5.33& & 5.33$\pm$0.01 &5.44& & 5.42$\pm$0.07 & 5.48& & \\%Rydberg 3s
|
||||
\ce{HOC} &$^2A''$ &0.92 &0.91$^e$& 0.99& 0.99& 0.99& 0.99$\pm$0.00 &0.96& 0.96& & 0.96$\pm$0.00 &0.93& & 0.92$\pm$0.00 & 0.92& & \\%??, cfr papier
|
||||
\ce{H2PO} &$^2A''$ &2.80 &2.83$^e$& 2.85& 2.86& 2.86 & 2.88$\pm$0.01 &2.80& 2.82 & & 2.82$\pm$0.02 &2.81& & 2.80$\pm$0.02 & 2.84& & \\%Val, n-pi*
|
||||
&$^2A'$ &4.21$^h$&4.22$^j$& 4.30& 4.30& 4.30& 4.31$\pm$0.00 &4.28& 4.28& & 4.28$\pm$0.02 &4.21& & 4.19$\pm$0.04 & 4.22& & \\%Val, pi-pi*
|
||||
\ce{H2PS} &$^2A''$ &1.16 &1.18$^e$ & 1.10& 1.10& 1.10& 1.11$\pm$0.00 &1.16& 1.16 & & 1.17$\pm$0.00 &1.15& & 1.16$\pm$0.01 & 1.17& & \\%Val, n-pi*
|
||||
&$^2A'$ &2.72 &2.71$^e$ & 2.88& 2.87& 2.87& 2.87$\pm$0.00 &2.81& 2.80 & & 2.80$\pm$0.00 &2.75& & 2.72$\pm$0.02 & 2.74& & \\%Val, pi-pi*
|
||||
\ce{NCO} &$^2\Sigma^+$ &2.89$^h$&2.89$^j$& 2.87& 2.87& & 2.88$\pm$0.00 &2.87& 2.86 & & 2.89$\pm$0.02 &2.87& & 2.83$\pm$0.05 & 2.87 & & \\
|
||||
&$^2\Pi$ &4.73$^h$&4.74$^j$& 4.80& 4.76& & 4.76$\pm$0.00 &4.80& 4.76 & & 4.76$\pm$0.01 &4.77& & 4.70$\pm$0.04 & 4.78 & & \\
|
||||
\ce{NH2} &$^2A_1$ &2.12 &2.11$^c$ & 2.19& 2.18& 2.18& 2.18$\pm$0.00 &2.15& 2.15& 2.15& 2.14$\pm$0.00 &2.12& 2.12& 2.12$\pm$0.00 & 2.11& 2.11 & 2.11$\pm$0.00\\%V
|
||||
Nitromethyl &$^2B_2$ &2.05$^k$& & 2.10& & & &2.04& & & &2.05& & & & & \\
|
||||
&$^2A_2$ &2.38$^k$& & 2.40& & & 2.39$\pm$0.01 &2.39& & & &2.38& & & & & \\
|
||||
&$^2A_1$ &2.56$^k$& & 2.64& & & &2.58& & & &2.56& & & & & \\
|
||||
&$^2B_1$ &5.35$^k$& & 5.48& & & &5.39& & & &5.35& & & & & \\
|
||||
\ce{NO} &$^2\Sigma^+$&6.13 &6.12$^e$& 6.12& 6.11& 6.11& 6.11$\pm$0.00 &6.03& 6.02& 6.02& 6.03$\pm$0.01 &6.13& 6.12& 6.13$\pm$0.02 & 6.12 & & \\%R3s
|
||||
&$^2\Sigma^+$ &7.29$^l$&7.21$^m$& 7.59& 7.59& 7.59& &7.34& 7.34& 7.34& &7.29& 7.29& & 7.21 & & \\%R3p
|
||||
\ce{OH} &$^2\Sigma^+$ &4.10 &4.09$^c$& 4.28& 4.28& 4.28& 4.28$\pm$0.00 &4.16& 4.16& 4.16& 4.16$\pm$0.00 &4.12& 4.12& 4.10$\pm$0.01 & 4.11& 4.10& 4.10$\pm$0.00\\%V
|
||||
&$^2\Sigma^-$ &8.02 &8.11$^c$& 8.83& 8.83& 8.83& 8.83$\pm$0.00 &7.88& 7.88& 7.88& 7.88$\pm$0.00 &8.04& 8.02& 8.02$\pm$0.00 & 8.10& 8.09& 8.09$\pm$0.00\\%VSigma ??
|
||||
\ce{PH2} &$^2A_1$ &2.77 &2.76$^c$& 2.90& 2.90& 2.90& 2.90$\pm$0.00 &2.79& 2.79& 2.79& 2.79$\pm$0.00 &2.77& 2.77& 2.77$\pm$0.00 & 2.76& 2.76 & 2.76$\pm$0.00\\%V
|
||||
Vinyl &$^2A''$ &3.26 & & 3.45& 3.43& 3.43& 3.43$\pm$0.00 &3.36& 3.34& & 3.35$\pm$0.00 &3.31& & 3.26$\pm$0.02 & & & \\%V:pi-n
|
||||
&$^2A''$ &4.69 & & 4.98& 4.96& 4.96& 4.96$\pm$0.00 &4.80& & & 4.78$\pm$0.01 &4.73& & 4.69$\pm$0.02 & \\%V: n-pi*
|
||||
&$^2A'$$^g$ &5.60 & & 5.83 & 5.75 & 5.75& 5.74$\pm$0.01 &5.75& 5.67& & 5.68$\pm$0.00 &5.74 & & 5.60$\pm$0.01 & \\%V: pipi*
|
||||
&$^2A'$ &6.20$^a$& & 6.50 & 6.48 &\hl{6.48} & 6.49$\pm$0.01 &6.15& 6.14& & & 6.21& & & \\%Ryd
|
||||
\hline
|
||||
\end{tabular}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{flushleft}
|
||||
$^a${FCI/{\Pop} value corrected by the difference between CCSDT/{\AVTZ} and CCSDT/{\Pop};}
|
||||
$^b${FCI/{\AVQZ} value;}
|
||||
$^c${FCI/{\AVQZ} value corrected by the difference between CCSDT/{\AVFZ} and CCSDT/{\AVQZ};}
|
||||
$^d${FCI/{\AVTZ} value corrected by the difference between CCSDT/{\AVFZ} and CCSDT/{\AVTZ};}
|
||||
$^e${FCI/{\AVTZ} value corrected by the difference between CCSDT/{\AVQZ} and CCSDT/{\AVTZ};}
|
||||
$^f${FCI/{\Pop} value corrected by the difference between CCSDT/{\AVQZ} and CCSDT/{\Pop};}
|
||||
$^g${For these challenging states, ROCC rather than UCC is used.}
|
||||
$^h${FCI/{\AVDZ} value corrected by the difference between CCSDT/{\AVTZ} and CCSDT/{\AVDZ};}
|
||||
$^i${CCSDTQ/{\Pop} value corrected by the difference between CCSDT/{\AVTZ} and CCSDT/{\Pop};}
|
||||
$^j${FCI/{\AVDZ} value corrected by the difference between CCSDT/{\AVQZ} and CCSDT/{\AVDZ};}
|
||||
$^k${CCSDT/{\AVTZ} value;}
|
||||
$^l${CCSDTQ/{\AVTZ} value;}
|
||||
$^m${CCSDTQ/{\AVTZ} value corrected by the difference between CCSDT/{\AVQZ} and CCSDT/{\AVTZ}.}
|
||||
\end{flushleft}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
\emph{Allyl.} For the lowest valence ($B_1$) and Rydberg ($A_1$) transitions of the allyl radical, the previous TBEs are likely the ROCC3 $3.44$ and $4.94$ eV vertical transition energies obtained by the Crawford group with the {\AVTZ} basis further
|
||||
augmented with molecule-centered functions (mcf). \cite{Mac10} For the lowest state, a very similar value of $3.43$ eV was obtained at the ROCC3 level without mcf. \cite{Loo19a} The present work is the first to report
|
||||
CCSDT and CCSDTQ results. They clearly show that these previous ROCC3 estimates are very accurate. In addition, our TBEs of $3.39$ and $4.99$ eV are reasonably consistent with earlier CASPT2 ($3.32$ and $5.11$ eV)
|
||||
\cite{Aqu03b} and MRCI ($3.32$ and $4.68$ eV) \cite{Gas10} data. The experimental 0-0 energies have been reported to be $3.07$ eV, \cite{Cas06c} and $4.97$ eV \cite{Gas09,Gas10} for the $^2B_1$ and $^2A_1$ states, respectively.
|
||||
The fact that the experimental $T_0$ value is very close to the computed vertical transition energy of the second state is rather surprising, but remains unchanged with the present work.
|
||||
|
||||
\emph{BeF.} In this compound, CCSDT delivers transition energies in very good agreement with FCI (and higher CC levels), but one notices a non-negligible basis set effect for the second transition of Rydberg character. This transition
|
||||
becomes significantly mixed in very large basis sets, making a clear attribution difficult. For this derivative (and other diatomics), experimental vertical transition energies can be calculated by analyzing the experimental spectroscopic
|
||||
constants. \cite{Mau95} Our TBE/{\AVTZ} values of $4.14$ and $6.21$ eV are obviously close to these measured values of $4.14$ and $6.16$ eV. \cite{Mau95} For the lowest state, a previous MRCI value of $4.23$ eV can be
|
||||
found in the literature. \cite{Orn92} There is also a recent evaluation of the adiabatic energies for numerous excited states at the MRCI+Q level. \cite{Elk17}
|
||||
|
||||
\emph{BeH.} The convergences with respect to both the CC excitation order and the basis set size is extremely fast for this five-electron system. A previous study reports FCI values for many excited states \cite{Pit08} and, in particular, excitation energies of $2.53$ and $6.30$ eV
|
||||
for the two $^2\Pi$ states considered herein. The experimental vertical transition energies are $2.48$ and $6.32$ eV. \cite{Mau95} Our larger value associated with the second transition is likely a consequence of the UCCSD(T)
|
||||
geometry, which delivers a slightly shorter bond length ($1.321$ \emph{vs} $1.327$ \AA\ experimentally).
|
||||
|
||||
\emph{BH$_2$, NH$_2$, and PH$_2$.} In these three related compounds, convergence with respect to the CC excitation order and basis set size is also very fast, so that accurate estimates can be easily produced for the lowest-lying transition:
|
||||
With the {\AVTZ} basis set, near-CBS excitation energies of $1.18$, $2.12$ and $2.77$ eV for the boron, nitrogen, and phosphorus derivative are respectively obtained. For \ce{BH2}, a previous MRCI estimate of $1.10$ eV is available in the literature. \cite{Per95}
|
||||
We note that, for \ce{BH2}, the geometry relaxation of the bent ground state structure would lead to a linear geometry in its lowest excited state, \cite{Sun15d} a phenomenon that was extensively studied both experimentally and
|
||||
theoretically (see Ref.~\citenum{Sun15d} and references therein). For \ce{NH2}, a vertical estimate of $2.18$ eV was reported by Szalay and Gauss using a CCSD approach including ``pseudo'' triple excitations, \cite{Sza00} and
|
||||
high-order CC calculations have been latter performed by Kallay and Gauss to investigate the structures and energetics of the ground and excited states. \cite{Kal03,Kal04} For \ce{PH2}, the most detailed \emph{ab initio} studies
|
||||
that are available in the literature focus exclusively on the 0-0 energies and rovibronic spectra, \cite{Woo01,Yur06,Jak06} except for a recent report listing a ROCC3 vertical transition energy of $2.75$ eV, \cite{Loo19a} obviously close to present TBE.
|
||||
|
||||
\emph{CH.} For the three considered transitions, the CCSDT values are slightly too large, whereas the basis set effects are rather usual, with nearly converged results for the {\AVTZ} basis set. Although we consider a theoretical geometry, our basis set corrected
|
||||
TBEs of $2.90$, $3.28$, and $3.96$ eV for the $^2\Delta$, $^2\Sigma^-$, and $^2\Sigma^+$ states are all extremely close to the vertical experimental values of $2.88$, $3.26$ and $3.94$ eV. \cite{Mau95,Sli05}
|
||||
There are many previous works on the \ce{CH} radical and it is interesting to mention that the ROCCSD values are $3.21$, $4.25$, and $5.22$ eV for the same three states, \cite{Sza00} whereas the
|
||||
corresponding ROCC3 results are $3.16$, $3.58$, and $4.47$ eV; \cite{Smi05b} the ROCC(2,3) excitation energies are $2.97$, $3.33$, and $4.06$ eV. \cite{Sli05} This clearly illustrates the challenge of reaching accurate values for
|
||||
the second and third transitions with ``low-order'' methods. For \ce{CH}, high-order CC calculations of the adiabatic energies and other properties are also available in the literature. \cite{Hir04,Fan07}
|
||||
|
||||
\emph{CH$_3$.} For the methyl radical, the convergence of the CC excitation energies and the near-perfect agreement between CC and FCI is worth noting.
|
||||
Nonetheless, large basis set effects are present for these transition energies, especially for the high-lying $^2A_2''$
|
||||
state for which the {\AVTZ} excitation energy is still far from being converged basis set wise. Our TBEs, including corrections up to quintuple-$\zeta$ are: $5.88$, $6.96$, $7.17$, and $7.48$ eV for the four lowest transitions. These values can be compared
|
||||
to the previous MRCI estimates \cite{Meb97b,Zan16} of $5.86$ ($5.91$), $6.95$ ($7.03$), $7.13$ (--) and $7.37$ (7.66) eV reported in Ref.~\citenum{Meb97b} (\citenum{Zan16}). The experimental $T_0$ value is $5.73$ eV for the
|
||||
$^2A_1'$ state, \cite{Her66,Set03b} whereas the experimental $T_e$ value is $7.43$ eV for the $^2A_2''$ state, \cite{Hud83,Fu05} both slightly below our FCI vertical estimates.
|
||||
|
||||
\emph{CN.} Both methodological and basis set effects are firmly under control for the cyano radical, so that our FCI/{\AVTZ} results of $1.34$ and $3.22$ eV for the lowest excited states are likely very accurate for the considered geometry.
|
||||
These values are indeed close to the experimental energies of $1.32$ and $3.22$ eV. \cite{Mau95} One can find careful MRCI studies, \cite{Yaz05,Shi11} as well as an extensive benchmark \cite{Bao17}
|
||||
for the adiabatic energies of the \ce{CN} radical.
|
||||
|
||||
\emph{CNO, CON, and NCO.} Inspired by a previous investigation, \cite{Yaz05} we have evaluated the two lowest doublet transitions in these three linear isomers. For \ce{CNO} --- the second most stable isomer --- one notes non-negligible
|
||||
drops of the transition energies going from CCSDT to CCSDTQ, the latter theory providing data in perfect match with the FCI results. Our TBEs of $1.61$ eV ($^2\Sigma^+$) and $5.50$ eV ($^2\Pi$), do compare very favorably with the
|
||||
corresponding MRCI+Q results of $1.66$ and $5.50$ eV, respectively. \cite{Yaz05} For the former transition, there is also a ROCC3 vertical transition energy of $1.71$ eV \cite{Loo19a} and a detailed rovibronic
|
||||
investigation \cite{Leo08} available in the literature. The data are much scarcer for \ce{CON}, and the only previous work we are aware of reports potential energy surfaces without listing explicitly the transition energies. \cite{Yaz05}
|
||||
For \ce{CON}, we have performed multi-reference calculations to identify the lowest states (see Table S4 in the SI). The NEVPT2 calculations locate the $^2\Pi$ and $^2\Sigma^+$ transitions at $3.52$ and $3.81$ eV, respectively,
|
||||
similar values being obtained with both CASPT2 and MRCI. As can be seen in Table \ref{Table-3} the FCI-based estimate of $3.53$ eV for the former transition is extremely consistent. For the latter transition, the difference
|
||||
between CCSDT and CCSDTQ energies is as large as -0.25 eV, suggesting that further corrections would be required. Nevertheless, our CC-derived TBE of $3.86$ eV is rather consistent with the NEVPT2 and MRCI values.
|
||||
For \ce{NCO}, the most stable of the three isomers, the basis set effects are trifling, but CCSDTQ is again mandatory in order to obtain a very accurate transition energy for the $^2\Pi$ state. This compound was studied previously at the
|
||||
MRCI+Q level, a method which delivers respective vertical transition energies of $2.89$ and $4.68$ eV for the $^2\Sigma^+$ and $^2\Pi$ states, \cite{Yaz05} whereas the ROCC3/{\AVTZ} transition energy of
|
||||
the lowest excited state is $2.83$ eV. \cite{Loo19a} The measured experimental 0-0 energies are $2.82$, \cite{Wu92} and $3.94$ eV. \cite{Dix60} All these data are quite consistent with our new values of $2.89$ and $4.74$ eV.
|
||||
|
||||
\emph{CO$^+$.} Our FCI/{\AVQZ} values for the $^2\Pi$ and $^2\Sigma^+$ transitions, $3.26$ and $5.80$ eV, are clearly matching the experimental values of $3.26$ and $5.81$ eV. \cite{Mau95} While
|
||||
basis set effects are rather standard for this radical cation, it is noteworthy that the CC expansion converges slowly for the Rydberg $^2\Sigma^+$ transition: one needs CCSDTQP to be within $0.01$ eV of the
|
||||
FCI result! Nonetheless, previous ROCC3 ($3.29$ and $5.73$ eV) \cite{Smi05b} and ROCC(2,3) data ($3.35$ and $5.81$ eV), \cite{Sli05} also fall within $\pm0.10$ eV of the present TBEs.
|
||||
|
||||
\emph{F$_2$BO and F$_2$BS.} These two radicals present a very low-lying $\pi-n$ transition, that is described very similarly by all basis sets used in Table \ref{Table-3}. For these transitions our
|
||||
TBEs are $0.73$ (\ce{F2BO}) and $0.51$ (\ce{F2BS}) eV, whereas, for the second transition of $\sigma-n$ nature, our TBEs are $2.80$ (F$_2$BO) and $2.99$ (\ce{F2BS}) eV.
|
||||
For these two compounds, the most advanced previous calculations are likely the ROCC3/{\AVTZ} values of $0.71$ and $2.78$ eV (F$_2$BO), and $0.47$ and $2.93$ eV (F$_2$BS)
|
||||
obtained by some of us in a recent study. \cite{Loo19a} For the former radical, these values are also very close to earlier CASPT2 ($0.70$ and $2.93$ eV) \cite{Bar08b} and SAC-CI ($0.73$ and $2.89$ eV) \cite{Li18b} estimates.
|
||||
The $T_0$ energies of these two states were both measured recently as well: $0.65$ and $2.78$ eV for the oxygen derivative, \cite{Gri14} and $0.44$ and $2.87$ eV for the sulfur radical. \cite{Jin15} These two
|
||||
works and an earlier study by the same group, \cite{Clo14} also provide advanced theoretical studies of both the 0-0 transitions and vibronic couplings.
|
||||
|
||||
\emph{H$_2$BO.} This lighter analogue of \ce{F2BO} remains to be detected experimentally, but its excited states have been studied twice with \emph{ab initio} theoretical methods, \cite{Clo14,Li18b} the most recent SAC-CI estimates for
|
||||
the lowest-lying transitions being $2.08$ and $3.49$ eV. \cite{Li18b} These SAC-CI excitation energies are within $0.10$ eV of our FCI-based TBEs.
|
||||
|
||||
\emph{HCO and HOC (formyl and isoformyl).} For the formyl radical, our TBEs are $2.09$ and $5.49$ eV. Kus and Bartlett reported CCSDT/6-311++G(d,p) transition energies of $2.17$ and $5.29$ eV (likely the best vertical estimates available previously), \cite{Kus08} obviously close to ours
|
||||
for the former valence transition. We are also aware of earlier CASPT2 estimates of $2.07$ and $5.45$ eV for these two states, \cite{Ser98} that happen to be within
|
||||
$\pm0.04$ eV of our TBEs. There are detailed studies of the potential energy surfaces for the ground and lowest excited states of \ce{HCO}. \cite{Nde16} For isoformyl, the convergence with respect to the basis set
|
||||
is fast and the lowest excited state is well converged with our FCI approach. Hence, we propose a safe TBE of $0.91$ eV for the lowest vertical excitation. Most previous studies did not, once more, discuss
|
||||
vertical transition energies. However, we are aware of a recent $0.87$ eV CC estimate for the adiabatic energy obtained with a large basis set. \cite{Mor15}
|
||||
|
||||
\emph{H$_2$PO and H$_2$PS.} These two radical homologues of formaldehyde are puckered in their ground state, and CCSDT is already giving very accurate estimates. Indeed, the CCSDT values are consistent with their FCI counterparts,
|
||||
and one likely needs a triple-$\zeta$ basis set to be close to convergence. The only previous experimental and theoretical studies we are aware of for these two compounds are rather recent. \cite{Gha11,Gri11b,Loo19a}
|
||||
They reported: (i) CCSD/{\AVTZ} adiabatic energies of $1.42$ and $3.32$ eV for \ce{H2PO}, \cite{Gha11} and $0.57$ and $2.58$ eV for its sulfur counterpart; \cite{Gri11b} (ii), ROCC3 vertical transitions
|
||||
to the lowest $^2A'$ states of $4.35$ eV (\ce{H2PO}) and $2.78$ eV (\ce{H2PS}). \cite{Loo19a} The latter are obviously compatible with the present data.
|
||||
|
||||
\emph{Nitromethyl.} For this (comparatively) large derivative, even the UCCSDT/{\AVTZ} calculations are a challenge in terms of computational resources. The calculations converge too slowly with the number of determinants
|
||||
to ensure valuable FCI extrapolations, except for the second state for which the CCSDT estimate falls within the extrapolation error bar. Fortunately, for all transitions, the difference between ROCC3 and UCCSDT estimates are small,
|
||||
and we can safely propose our CCSDT values as references. These values of $2.05$, $2.38$, $2.56$, and $5.35$ eV do agree rather well with the 2005 ROCC3/Sadlej-TZ estimates of $2.03$, $2.41$, $2.53$ and $5.28$ eV, \cite{Smi05b}
|
||||
that remain the most advances carried out previously to the very best of our knowledge. Retrospectively, the MRCI excitation energies of $1.25$ and $1.52$ eV for the two lowest states seem way too low. \cite{Cai94} The measured photoelectron
|
||||
spectrum of the related anion indicates the presence of the $^2A_2$ transition at $1.59$ eV in the radical, \cite{Met91} whereas a rough estimate of $4.25$ eV can also be deduced from experimental data for the $^2B_1$ state.
|
||||
\cite{Cyr93} We trust that the TBEs given in Table \ref{Table-3} are more trustworthy estimates of the vertical transition energies than these indirect experimental transition energies.
|
||||
|
||||
\emph{NO.} This highly reactive radical is unsurprisingly quite difficult to capture with theoretical approaches and our current TBEs of $6.12$ and $7.21$ eV for the two lowest Rydberg states
|
||||
are significantly above the vertical experimental energies of $5.93$ and $7.03$ eV. \cite{Mau95} Our geometry is associated with a \ce{NO} bond distance of $1.149$ \AA\, slightly larger than the experimental value of $1.115$ \AA. Moreover,
|
||||
basis set convergence is slow, so that a quadruple-$\zeta$ basis might still be insufficient to be close to the CBS limit for the second excited state.
|
||||
|
||||
\emph{OH.} For \ce{OH}, the convergence of the CC energy with respect to the excitation degree is extremely fast, but the basis set effects are non-negligible. Our TBEs are $4.09$ and $8.11$ eV for the $^2\Sigma^+$ and $^2\Sigma^-$ transitions, respectively. The
|
||||
former value compares very nicely with the experimental one ($4.08$ eV), \cite{Mau95} and is smaller than previous MRCI estimates of $4.27$ \cite{For91} and $4.22$ eV. \cite{Sza00} In contrast, for the $^2\Sigma^-$
|
||||
transition, our estimate is higher than a previously reported value of $7.87$ eV. \cite{For91}
|
||||
|
||||
\emph{Vinyl.} For this final radical, we considered four states, two in each spatial symmetry. For the lowest transition of $\pi \rightarrow n$ nature, our FCI/{\AVTZ} result is $3.26\pm0.02$ eV, and one can find
|
||||
many previous calculations yielding similar transition energies: $3.17$ (MRCI), \cite{Wan96b} $3.24$ (MRCI), \cite{Meb97} $3.31$ (CCSD), \cite{Koz06} and $3.30$ eV (CC3), \cite{Loo19a} whereas the measured 0-0
|
||||
energy is $2.49$ eV. \cite{Pib99} For the second transition of the same $A''$ symmetry and of $n \rightarrow \pi^\star$ character, the previous theoretical values we are aware of are $4.78$ eV (MRCI) \cite{Meb97} and $4.93$ eV (CCSD). \cite{Koz06}
|
||||
Our TBE of $4.69$ eV is lower. The lowest $^2A'$ transition is a tricky valence excitation of $\pi \rightarrow \pi^\star$ character with a significant multi-excitation character,
|
||||
and we decided to use ROCC for this specific case. It is clear from Table \ref{Table-3} that one needs to go as high as CCSDTQ to be close to FCI. Our TBE of $5.60$ eV can be compared to previous
|
||||
estimates of $5.58$ eV (MRCI) \cite{Meb97} or $5.60$ eV (spin-flip CCSD), \cite{Koz06} which clearly highlights the fantastic accuracy of the spin-flip approach for such transition. Eventually, the last transition of Rydberg character is
|
||||
easier to describe at the CC level, with our TBE of $6.20$ eV again close to previously reported results: $6.25$ (MRCI) \cite{Meb97} and $6.31$ eV (CCSD). \cite{Koz06}
|
||||
|
||||
\subsubsection{Benchmarks}
|
||||
|
||||
As for the exotic set, we have used our TBEs/{\AVTZ} to perform benchmarks of ``lower-order'' methods, and we have especially compared the U and RO versions of CCSD and CC3, considering all transition energies
|
||||
listed in Table \ref{Table-3} (except three particularly challenging ones that have been omitted, see footnote $g$ in the corresponding Table). The raw data are listed in Table S3 of the SI, whereas Table \ref{Table-4} and Figure \ref{Fig-2}
|
||||
gathers the associated statistical data. As expected from previous works, \cite{Smi05b,Koz06,Loo19a} the excitation energy errors associated with these doublet-doublet transitions in open-shell molecules tend to be larger than for closed-shell systems. Indeed, we note that (i) CCSD overshoots by more than 1 eV the
|
||||
transition energies of the second and third excited states of CH; (ii) the MAE obtained with CC3 is $0.05$ eV, five times larger than in the exotic set; and (iii) the error dispersion is obviously larger in Figure \ref{Fig-2}
|
||||
than in Figure \ref{Fig-1}. This confirms that accurately describing doublet-doublet transition energies is very challenging. On a more positive note, we observe that the statistical results are improved by using a RO starting
|
||||
point instead of the usual U approximation, an effect particularly significant at the CCSD level.
|
||||
|
||||
\renewcommand*{\arraystretch}{1.0}
|
||||
\begin{table}[htp]
|
||||
\scriptsize
|
||||
\caption{Statistical values obtained by comparing the results of various methods to the TBE/{\AVTZ} reported in Table S3. See caption of Table \ref{Table-2} for more details.}
|
||||
\label{Table-4}
|
||||
\begin{tabular}{lccccc}
|
||||
\hline
|
||||
Method & Count & MSE &MAE &RMSE &SDE \\
|
||||
\hline
|
||||
UCCSD &48 & 0.19 &0.20 &0.35 &0.30 \\
|
||||
ROCCSD &48 & 0.14 &0.15 &0.30 &0.27 \\
|
||||
UCC3 &48 & 0.03 &0.06 &0.11 &0.11 \\
|
||||
ROCC3 &48 & 0.02 &0.05 &0.10 &0.10 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{table}
|
||||
|
||||
\begin{figure}[htp]
|
||||
\includegraphics[scale=0.8,viewport=2.8cm 18.3cm 13.3cm 27.5cm,clip]{Figure-2.pdf}
|
||||
\caption{Histograms of the error distribution (in eV) obtained with 4 theoretical methods, choosing the TBE/{\AVTZ} of Table \ref{Table-3} as references (raw data in Table S3). For the CCSD cases, even larger errors (out of scale) are observed.}
|
||||
\label{Fig-2}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\section{Conclusions}
|
||||
|
||||
In order to complete our three previous sets of highly-accurate excitation energies, \cite{Loo18b,Loo19b,Loo20a} we have reported here two additional sets of TBEs for: (i) 30 excited states in
|
||||
a series of ``exotic'' closed-shell compounds including (at least) one of the following atoms: \ce{F}, \ce{Cl}, \ce{Si}, or \ce{P}; (ii) 51 doublet-doublet transitions in a series of radicals characterized by
|
||||
an open-shell electronic configuration. In all cases, we have reported at least {\AVTZ} estimates, the vast majority being obtained at the FCI level, and we have applied increasingly accurate
|
||||
CC methods to ascertain these estimates. For most of these transitions, it is very likely that the present TBEs are the most accurate published to date (for a given geometry).
|
||||
|
||||
For the former exotic set, these TBEs have been used to assess the performances of fifteen ``lower-order'' wave function approaches, including several CC and ADC variants. Consistently
|
||||
with our previous works, we found that CC3 is astonishingly accurate with a MAE as small as $0.01$ eV and a SDE of $0.02$ eV, whereas the trends
|
||||
for the other methods are similar to the one obtained on more standard organic compounds. In contrast, for the radical set, even the refined ROCC3 method yields a MAE of $0.05$ eV, and
|
||||
a rather large SDE of $0.10$ eV. Likewise, the excitation energies obtained with CCSD are much less
|
||||
satisfying for open-shell derivatives (MAE of $0.20$ eV with UCCSD and $0.15$ eV with ROCCSD) than for the closed-shell systems (MAE of $0.07$ eV).
|
||||
|
||||
We hope that these two new sets, which provide a fair ground for the assessments of high-level excited-state models, will be an additional valuable asset for the electronic structure community, and will
|
||||
stimulate further developments in the field.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%% ACKNOWLEDGEMENTS %%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Acknowledgements}
|
||||
PFL thanks the \textit{Centre National de la Recherche Scientifique} for funding. This research used resources of (i) the GENCI-TGCC (Grant No.~2019-A0060801738);
|
||||
(ii) CALMIP under allocation 2020-18005 (Toulouse); (iii) CCIPL (\emph{Centre de Calcul Intensif des Pays de Loire}); (iv) a local Troy cluster and (v) HPC resources from ArronaxPlus
|
||||
(grant ANR-11-EQPX-0004 funded by the French National Agency for Research).
|
||||
|
||||
%%%%%%%%%%%%%%%%%
|
||||
%%% SUPP INFO %%%
|
||||
%%%%%%%%%%%%%%%%%
|
||||
%\begin{suppinfo}
|
||||
\section*{Supporting Information Available}
|
||||
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/{doi}.
|
||||
|
||||
Basis set effects at CC3 level for the exotic set. Benchmark data. Multi-reference values for CON. Cartesian coordinates.
|
||||
%\end{suppinfo}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
%%% BIBLIOGRAPHY %%%
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{biblio-new}
|
||||
|
||||
\end{document}
|
||||
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|
|
@ -0,0 +1,724 @@
|
|||
\documentclass[journal=jctcce,manuscript=letter,layout=traditional]{achemso}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amsmath,amssymb,amsfonts,physics,float,lscape,soul,rotating,longtable}
|
||||
\usepackage[version=4]{mhchem}
|
||||
|
||||
|
||||
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\mr}{\multirow}
|
||||
|
||||
\newcommand{\EFCI}{E_\text{FCI}}
|
||||
\newcommand{\EexCI}{E_\text{exCI}}
|
||||
\newcommand{\EsCI}{E_\text{sCI}}
|
||||
\newcommand{\EPT}{E_\text{PT2}}
|
||||
\newcommand{\PsisCI}{\Psi_\text{sCI}}
|
||||
\newcommand{\Ndet}{N_\text{det}}
|
||||
|
||||
\newcommand{\ex}[4]{{#1}\,$^{#2}$#3$_{#4}$}
|
||||
|
||||
% methods
|
||||
\newcommand{\TDDFT}{TD-DFT}
|
||||
\newcommand{\CASSCF}{CASSCF}
|
||||
\newcommand{\CASPT}{CASPT2}
|
||||
\newcommand{\ADC}[1]{ADC(#1)}
|
||||
\newcommand{\CC}[1]{CC#1}
|
||||
\newcommand{\CCSD}{CCSD}
|
||||
\newcommand{\EOMCCSD}{EOM-CCSD}
|
||||
\newcommand{\CCSDT}{CCSDT}
|
||||
\newcommand{\CCSDTQ}{CCSDTQ}
|
||||
\newcommand{\CI}{CI}
|
||||
\newcommand{\sCI}{sCI}
|
||||
\newcommand{\exCI}{exCI}
|
||||
\newcommand{\FCI}{FCI}
|
||||
|
||||
% basis
|
||||
\newcommand{\Pop}{6-31+G(d)}
|
||||
\newcommand{\AVDZ}{\emph{aug}-cc-pVDZ}
|
||||
\newcommand{\AVTZ}{\emph{aug}-cc-pVTZ}
|
||||
\newcommand{\DAVTZ}{d-\emph{aug}-cc-pVTZ}
|
||||
\newcommand{\AVQZ}{\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\ACVQZ}{\emph{aug}-cc-pCVQZ}
|
||||
\newcommand{\DAVQZ}{d-\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\TAVQZ}{t-\emph{aug}-cc-pVQZ}
|
||||
\newcommand{\AVPZ}{\emph{aug}-cc-pV5Z}
|
||||
\newcommand{\DAVPZ}{d-\emph{aug}-cc-pV5Z}
|
||||
|
||||
% units
|
||||
\newcommand{\IneV}[1]{#1 eV}
|
||||
\newcommand{\InAU}[1]{#1 a.u.}
|
||||
\newcommand{\Ryd}{\mathrm{R}}
|
||||
\newcommand{\Val}{\mathrm{V}}
|
||||
\newcommand{\Fl}{\mathrm{F}}
|
||||
\newcommand{\ra}{\rightarrow}
|
||||
\newcommand{\pis}{\pi^\star}
|
||||
\newcommand{\si}{\sigma}
|
||||
\newcommand{\sis}{\sigma^\star}
|
||||
|
||||
\newcommand{\SI}{Supporting Information}
|
||||
|
||||
\setcounter{table}{0}
|
||||
\setcounter{figure}{0}
|
||||
\setcounter{page}{1}
|
||||
\setcounter{equation}{0}
|
||||
\renewcommand{\thepage}{S\arabic{page}}
|
||||
\renewcommand{\thefigure}{S\arabic{figure}}
|
||||
\renewcommand{\theequation}{S\arabic{equation}}
|
||||
\renewcommand{\thetable}{S\arabic{table}}
|
||||
\renewcommand{\thesection}{S\arabic{section}}
|
||||
|
||||
\renewcommand\floatpagefraction{.99}
|
||||
\renewcommand\topfraction{.99}
|
||||
\renewcommand\bottomfraction{.99}
|
||||
\renewcommand\textfraction{.01}
|
||||
|
||||
\title{A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Exotic Molecules and Radicals\\Supporting Information}
|
||||
|
||||
\author{Pierre-Fran{\c c}ois Loos}
|
||||
\email{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Anthony Scemama}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Martial Boggio-Pasqua}
|
||||
\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\author{Denis Jacquemin}
|
||||
\email{Denis.Jacquemin@univ-nantes.fr}
|
||||
\affiliation[UN, Nantes]{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\clearpage
|
||||
|
||||
\section{Basis set and frozen-core effects}
|
||||
|
||||
\begin{sidewaystable}[htp]
|
||||
\scriptsize
|
||||
\caption{CC3 transition energies (in eV) determined with various basis sets. FC, SC, and full stand for frozen-core (large cores), small-core (freezing only the $1s$ electrons) and correlating all electrons, respectively.}
|
||||
\label{Table-S1}
|
||||
\begin{tabular}{cc|cccccccc}
|
||||
\hline
|
||||
& &{\Pop} & {\AVDZ} & {\AVTZ} & {\AVQZ} & {\AVQZ} & {\AVQZ} & {\ACVQZ}& {\AVPZ}\\
|
||||
& &FC & FC & FC & FC & SC & Full & Full & FC \\
|
||||
\hline
|
||||
Carbonylfluoride& $^1A_2$ &7.33 &7.34 &7.31 &7.31 & &7.29 &7.28 &7.31\\
|
||||
& $^3A_2$ &7.03 &7.05 &7.03 &7.03 & &7.01 &7.00 &7.04\\
|
||||
CCl$_2$ & $^1B_1$ &2.71 &2.69 &2.61 &2.60 &2.59 &2.57 &2.57 &2.59\\
|
||||
& $^1A_2$ &4.46 &4.40 &4.35 &4.37 &4.36 &4.34 &4.33 &4.36\\
|
||||
& $^3B_1$ &1.10 &1.20 &1.20 &1.21 &1.21 &1.19 &1.19 &1.21\\
|
||||
& $^3A_2$ &4.41 &4.34 &4.28 &4.30 &4.29 &4.28 &4.26 &4.29\\
|
||||
CClF & $^1A''$ &3.66 &3.63 &3.56 &3.55 &3.55 &3.53 &3.53 &3.55\\
|
||||
CF$_2$ & $^1B_1$ &5.18 &5.12 &5.07 &5.06 & &5.02 &5.02 &5.05\\
|
||||
& $^3B_1$ &2.71 &2.71 &2.76 &2.77 & &2.75 &2.75 &2.77\\
|
||||
Difluorodiazirine&$^1B_1$ &3.83 &3.80 &3.74 &3.73 & &&&\\
|
||||
&$^1A_2$ &7.13 &7.11 &7.02 &7.00 & &&&\\
|
||||
&$^1B_2$ &8.51 &8.45 &8.50 &8.52 & &&&\\
|
||||
&$^3B_1$ &3.09 &3.06 &3.03 &3.03 & &&&\\
|
||||
&$^3B_2$ &5.48 &5.47 &5.45 &5.47 & &&&\\
|
||||
&$^3B_1$ &5.86 &5.83 &5.81 &5.82 & &&&\\
|
||||
Formylfluoride & $^1A''$ &6.09 &6.03 &5.99 &5.99 & &5.98 &5.97 &6.00\\
|
||||
& $^3A''$ &5.72 &5.65 &5.62 &5.63 & &5.62 &5.61 &5.64\\
|
||||
HCCl & $^1A''$ &2.05 &2.02 &1.97 &1.96 &1.96 &1.95 &1.94 &1.96\\
|
||||
HCF &$^1A''$ &2.58 &2.53 &2.49 &2.49 & &2.47 &2.47 &2.49\\
|
||||
HCP & $^1\Sigma^-$ &5.19 &5.06 &4.85 &4.83 &4.83 &4.82 &4.81 &4.82\\
|
||||
& $^1\Delta$ &5.48 &5.33 &5.15 &5.12 &5.11 &5.11 &5.09 &5.10\\
|
||||
& $^3\Sigma^+$&3.44 &3.47 &3.45 &3.46 &3.46 &3.46 &3.44 &3.47\\
|
||||
& $^3\Delta$ &4.40 &4.35 &4.22 &4.21 &4.21 &4.20 &4.20 &4.20\\
|
||||
HPO & $^1A''$ &2.49 &2.47 &2.46 &2.47 &2.47 &2.47 &2.47 &2.48\\
|
||||
HPS & $^1A''$ &1.57 &1.60 &1.59 &1.60 &1.59 &1.59 &1.59 &1.61\\
|
||||
HSiF & $^1A''$ &3.09 &3.08 &3.07 &3.07 &3.06 &3.06 &3.06 &3.07\\
|
||||
SiCl$_2$ &$^1B_1$ &3.94 &3.93 &3.90 &3.91 &3.91 &3.91 &3.90 &3.92\\
|
||||
&$^3B_1$ &2.39 &2.45 &2.48 &2.49 &2.52 &2.52 &2.52 &2.50\\
|
||||
Silylidene &$^1A_2$ &2.14 &2.18 &2.15 &2.16 &2.15 &2.16 &2.15 &2.16\\
|
||||
&$^1B_2$ &3.88 &3.81 &3.78 &3.79 &3.78 &3.78 &3.78 &3.80\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{flushleft}
|
||||
\end{flushleft}
|
||||
\end{sidewaystable}
|
||||
|
||||
\clearpage
|
||||
\begin{figure}[htp]
|
||||
\includegraphics[scale=1.05,viewport=2.8cm 22.3cm 18.3cm 27.5cm,clip]{Figure-S1.pdf}
|
||||
\caption{Histograms of the error distribution (in eV) obtained by comparing the CC3 transitions obtained with three basis set to the corresponding CC3/{\AVQZ} values
|
||||
for the data listed in Table \ref{Table-S1}. Note the different $Y$ scales.}
|
||||
\label{Fig-S1}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{figure}[htp]
|
||||
\includegraphics[scale=1.05,viewport=2.8cm 22.3cm 18.3cm 27.5cm,clip]{Figure-S2.pdf}
|
||||
\caption{Histograms of the error distribution (in eV) obtained by comparing the CCSDT transitions obtained with three basis set to the corresponding CCSDT/{\AVQZ} values
|
||||
for the data listed in Table 3 of the main text. Note the different $Y$ scales.}
|
||||
\label{Fig-S2}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\clearpage
|
||||
|
||||
\section{Benchmark data}
|
||||
\begin{sidewaystable}[htp]
|
||||
\scriptsize
|
||||
\caption{Transition energies determined with various models for the exotic set. All values are in eV and have been obtained with the {\AVTZ} basis set applying the FC approximation.}
|
||||
\label{Table-S2}
|
||||
\begin{tabular}{cc|c|ccccccccccccccc}
|
||||
\hline
|
||||
& &TBE & \rotatebox{90}{CIS(D)} & \rotatebox{90}{CC2} & \rotatebox{90}{EOM-MP2} & \rotatebox{90}{STEOM-CCSD} & \rotatebox{90}{CCSD} & \rotatebox{90}{CCSDR(3)} &
|
||||
\rotatebox{90}{CCSDT-3} & \rotatebox{90}{CC3} & \rotatebox{90}{SOS-ADC(2) [TM]} & \rotatebox{90}{SOS-CC2 [TM]} & \rotatebox{90}{SCS-CC2 [TM]} & \rotatebox{90}{SOS-ADC(2) [QM]} & \rotatebox{90}{ADC(2)}
|
||||
& \rotatebox{90}{ADC(3)} & \rotatebox{90}{ADC(2.5)} \\
|
||||
\hline
|
||||
Carbonylfluoride& $^1A_2$ &7.31 &7.38 &7.47 &7.39 &7.07 &7.36 &7.32 &7.32 &7.31 &7.27 &7.48 &7.47 &7.04 &7.22 &7.32 &7.27\\
|
||||
& $^3A_2$ &7.06 &7.08 &7.14 &7.08 &6.82 &7.03 & & &7.03 &7.05 &7.24 &7.21 &6.81 &6.91 &7.01 &6.96\\
|
||||
CCl$_2$ & $^1B_1$ &2.59 &2.59 &2.58 &2.36 &2.35 &2.61 &2.59 &2.61 &2.61 &2.58 &2.67 &2.64 &2.44 &2.46 &2.41 &2.44\\
|
||||
& $^1A_2$ &4.40 &4.20 &4.27 &4.27 &4.33 &4.57 &4.37 &4.41 &4.35 &4.50 &4.61 &4.50 &4.29 &4.12 &4.76 &4.44\\
|
||||
& $^3B_1$ &1.22 &1.09 &1.15 &0.84 & &1.11 & & &1.20 &1.16 &1.27 &1.23 &1.06 &0.98 &0.91 &0.95\\
|
||||
& $^3A_2$ &4.31 &4.24 &4.20 &4.17 &4.23 &4.45 & & &4.28 &4.48 &4.59 &4.46 &4.29 &4.05 &4.62 &4.34\\
|
||||
CClF & $^1A''$ &3.55 &3.56 &3.57 &3.34 &3.39 &3.57 &3.55 &3.56 &3.56 &3.54 &3.63 &3.61 &3.39 &3.44 &3.35 &3.40\\
|
||||
CF$_2$ & $^1B_1$ &5.09 &5.06 &5.09 &4.90 &4.90 &5.09 &5.07 &5.08 &5.07 &5.05 &5.15 &5.13 &4.89 &4.94 &4.86 &4.90\\
|
||||
& $^3B_1$ &2.77 &2.63 &2.70 &2.47 &2.61 &2.69 & & &2.76 &2.74 &2.84 &2.79 &2.64 &2.54 &2.48 &2.51\\
|
||||
Difluorodiazirine&$^1B_1$ &3.74 &3.89 &3.74 &3.94 &3.56 &3.83 &3.76 &3.75 &3.74 &3.97 &3.97 &3.90 &3.77 &3.74 &3.52 &3.63\\
|
||||
&$^1A_2$ &7.00 &7.46 &7.19 &7.24 & &7.10 &7.05 &7.02 &7.02 &7.29 &7.28 &7.25 &7.10 &7.19 &6.70 &6.95\\
|
||||
&$^1B_2$ &8.52 &8.53 &8.29 &8.90 & &8.69 &8.55 &8.55 &8.50 &8.95 &8.82 &8.65 &8.77 &8.42 &8.50 &8.46\\
|
||||
&$^3B_1$ &3.03 &3.17 &3.03 &3.17 &2.91 &3.07 & & &3.03 &3.32 &3.33 &3.23 &3.14 &3.01 &2.77 &2.89\\
|
||||
&$^3B_2$ &5.44 &5.89 &5.77 &5.97 & &5.40 & & &5.45 &5.53 &5.55 &5.63 &5.41 &5.72 &5.04 &5.38\\
|
||||
&$^3B_1$ &5.80 &6.13 &5.99 &5.71 &5.59 &5.84 & & &5.81 &6.20 &6.21 &6.13 &6.05 &5.97 &5.47 &5.72\\
|
||||
Formylfluoride & $^1A''$ &5.96 &6.03 &6.14 &6.00 &5.88 &6.02 &5.99 &6.00 &5.99 &5.99 &6.19 &6.17 &5.78 &5.91 &5.93 &5.92\\
|
||||
& $^3A''$ &5.73 &5.63 &5.70 &5.60 &5.51 &5.60 & & &5.62 &5.67 &5.85 &5.80 &5.48 &5.50 &5.54 &5.52\\
|
||||
HCCl & $^1A''$ &1.98 &1.95 &1.91 &1.65 &1.80 &1.99 &1.95 &1.98 &1.97 &2.01 &2.06 &2.01 &1.88 &1.84 &1.81 &1.83\\
|
||||
HCF &$^1A''$ &2.49 &2.54 &2.44 &2.19 &2.32 &2.51 &2.48 &2.50 &2.49 &2.51 &2.58 &2.53 &2.38 &2.34 &2.30 &2.32\\
|
||||
HCP & $^1\Sigma^-$ &4.84 &5.07 &5.07 &4.83 &4.90 &4.87 &4.85 &4.84 &4.85 &5.02 &5.07 &5.07 &4.91 &5.02 &4.37 &4.70\\
|
||||
& $^1\Delta$ &5.15 &5.40 &5.41 &5.12 &5.22 &5.16 &5.16 &5.14 &5.15 &5.23 &5.29 &5.33 &5.12 &5.33 &4.66 &5.00\\
|
||||
& $^3\Sigma^+$&3.47 &3.74 &3.73 &3.55 &3.41 &3.36 & & &3.45 &3.37 &3.38 &3.50 &3.30 &3.69 &3.10 &3.40\\
|
||||
& $^3\Delta$ &4.22 &4.44 &4.43 &4.23 &4.20 &4.17 & & &4.22 &4.47 &4.52 &4.49 &4.39 &4.39 &3.79 &4.09\\
|
||||
HPO & $^1A''$ &2.47 &2.54 &2.50 &2.44 &2.45 &2.54 &2.48 &2.48 &2.46 &2.57 &2.68 &2.62 &2.39 &2.35 &2.35 &2.35\\
|
||||
HPS & $^1A''$ &1.59 &1.68 &1.68 &1.39 &1.55 &1.67 &1.59 &1.60 &1.59 &1.74 &1.79 &1.75 &1.60 &1.62 &1.39 &1.51\\
|
||||
HSiF & $^1A''$ &3.05 &3.16 &3.14 &2.78 &3.02 &3.12 &3.07 &3.08 &3.07 &3.22 &3.24 &3.21 &3.12 &3.11 &2.88 &3.00\\
|
||||
SiCl$_2$ &$^1B_1$ &3.91 &3.99 &3.99 &3.70 &3.80 &3.96 &3.89 &3.91 &3.90 &4.01 &4.04 &4.02 &3.89 &3.95 &3.76 &3.86\\
|
||||
&$^3B_1$ &2.48 &2.40 &2.39 &2.18 & &2.45 & & &2.48 &2.51 &2.52 &2.48 &2.44 &2.35 &2.31 &2.33\\
|
||||
Silylidene &$^1A_2$ &2.11 &2.39 &2.37 &2.09 &2.21 &2.29 &2.16 &2.17 &2.15 &2.35 &2.35 &2.35 &2.24 &2.37 &1.87 &2.12\\
|
||||
&$^1B_2$ &3.78 &3.91 &3.85 &3.66 &3.81 &3.88 &3.79 &3.80 &3.78 &3.98 &3.94 &3.91 &3.87 &3.88 &3.40 &3.64\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{flushleft}
|
||||
\end{flushleft}
|
||||
\end{sidewaystable}
|
||||
|
||||
|
||||
\begin{table}[htp]
|
||||
\scriptsize
|
||||
\caption{Transition energies determined with various models for the radical set. All values are in eV and have been obtained with the {\AVTZ} basis set applying the FC approximation.}
|
||||
\label{Table-S3}
|
||||
\begin{tabular}{cc|c|cccc}
|
||||
\hline
|
||||
& &TBE & U-CCSD & RO-CCSD & U-CC3 & RO-CC3 \\
|
||||
\hline
|
||||
Allyl &$^2B_1$ &3.39 &3.70 &3.48 &3.48 &3.44 \\
|
||||
&$^2A_1$ &4.99 &5.12 &5.01 &4.97 &4.95 \\
|
||||
BeF &$^2\Pi$ &4.14 &4.18 &4.18 &4.15 &4.15 \\
|
||||
&$^2\Sigma^+$ &6.21 &6.31 &6.31 &6.21 &6.21 \\
|
||||
BeH &$^2\Pi$ &2.49 &2.51 &2.51 &2.50 &2.50 \\
|
||||
&$^2\Pi$ &6.46 &6.47 &6.47 &6.46 &6.46 \\
|
||||
BH$_2$ &$^2B_1$ &1.18 &1.20 &1.20 &1.19 &1.20 \\
|
||||
CH &$^2\Delta$ &2.91 &3.18 &3.17 &3.11 &3.10 \\
|
||||
&$^2\Sigma^-$ &3.29 &4.58 &4.39 &3.61 &3.55 \\
|
||||
&$^2\Sigma^+$ &3.98 &5.47 &5.36 &4.45 &4.40 \\
|
||||
CH$_3$ &$^2A_1'$ &5.85 &5.89 &5.87 &5.86 &5.85 \\
|
||||
&$^2E'$ &6.96 &7.00 &6.98 &6.97 &6.97 \\
|
||||
&$^2E'$ &7.18 &7.21 &7.20 &7.19 &7.19 \\
|
||||
&$^2A_2''$ &7.65 &7.67 &7.66 &7.65 &7.65 \\
|
||||
CN &$^2\Pi$ &1.34 &1.56 &1.34 &1.40 &1.36 \\
|
||||
&$^2\Sigma^+$ &3.22 &3.54 &3.35 &3.31 &3.26 \\%CHECK
|
||||
CNO &$^2\Sigma^+$ &1.61 &2.24 &2.25 &1.75 &1.77 \\%CHECK
|
||||
&$^2\Pi$ &5.49 &5.68 &5.60 &5.52 &5.51 \\
|
||||
CO$^+$ &$^2\Pi$ &3.28 &3.60 &3.29 &3.33 &3.29 \\
|
||||
&$^2\Sigma^+$ &5.81 &6.21 &6.02 &5.76 &5.68 \\
|
||||
F$_2$BO &$^2B_1$ &0.73 &0.74 &0.73 &0.71 &0.71 \\
|
||||
&$^2A_1$ &2.80 &2.84 &2.83 &2.79 &2.79 \\
|
||||
F$_2$BS &$^2B_1$ &0.51 &0.51 &0.49 &0.48 &0.48 \\
|
||||
&$^2A_1$ &2.99 &3.03 &3.01 &2.94 &2.93 \\
|
||||
H$_2$BO &$^2B_1$ &2.15 &2.14 &2.13 &2.17 &2.17 \\
|
||||
&$^2A_1$ &3.49 &3.53 &3.51 &3.52 &3.52 \\
|
||||
HCO &$^2A''$ &2.09 &2.14 &2.13 &2.10 &2.11 \\
|
||||
&$^2A'$ &5.45 &5.54 &5.53 &5.44 &5.44 \\
|
||||
HOC &$^2A''$ &0.92 &0.95 &0.93 &0.93 &0.93 \\
|
||||
H$_2$PO &$^2A''$ &2.80 &2.91 &2.91 &2.83 &2.83 \\
|
||||
&$^2A'$ &4.21 &4.26 &4.27 &4.21 &4.23 \\
|
||||
H$_2$PS &$^2A''$ &1.16 &1.18 &1.14 &1.16 &1.15 \\
|
||||
&$^2A'$ &2.72 &2.79 &2.77 &2.75 &2.75 \\
|
||||
NCO &$^2\Sigma^+$ &2.89 &3.04 &2.94 &2.94 &2.86 \\
|
||||
&$^2\Pi$ &4.73 &5.01 &5.02 &4.80 &4.81 \\
|
||||
NH$_2$ &$^2A_1$ &2.12 &2.13 &2.12 &2.13 &2.12 \\
|
||||
Nitromethyl &$^2B_2$ &2.05 &2.47 &2.46 &2.06 &2.05 \\
|
||||
&$^2A_2$ &2.38 &2.71 &2.71 &2.47 &2.46 \\
|
||||
&$^2A_1$ &2.56 &2.94 &2.93 &2.56 &2.55 \\
|
||||
&$^2B_1$ &5.35 &5.59 &5.56 &5.38 &5.36 \\
|
||||
NO &$^2\Sigma^+$&6.13 &6.23 &6.21 &6.13 &6.12 \\
|
||||
&$^2\Sigma^+$ &7.29 &7.40 &7.38 &7.30 &7.28 \\
|
||||
OH &$^2\Sigma^+$ &4.10 &4.14 &4.13 &4.13 &4.13 \\
|
||||
&$^2\Sigma^-$ &8.02 &7.75 &7.76 &7.66 &7.66 \\
|
||||
PH$_2$ &$^2A_1$ &2.77 &2.81 &2.78 &2.78 &2.77 \\
|
||||
Vinyl &$^2A''$ &3.26 &3.51 &3.35 &3.34 &3.30 \\
|
||||
&$^2A''$ &4.69 &4.91 &4.80 &4.76 &4.73 \\
|
||||
&$^2A'$ &6.20 &6.38 &6.32 &6.22 &6.24 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{flushleft}
|
||||
\end{flushleft}
|
||||
\end{table}
|
||||
|
||||
|
||||
|
||||
|
||||
\clearpage
|
||||
|
||||
\section{Multi-reference approaches for CON}
|
||||
|
||||
|
||||
|
||||
\begin{table}[htp]
|
||||
|
||||
\caption{Vertical transition energies (eV) of CON. All calculations using a full valence active space of (15e,12o) and the {\AVTZ} basis set. NEVPT2
|
||||
calculations are performed within the partially-contracted scheme whereas CASPT2 calculations use a level shift of 0.3 a.u. and a IPEA of 0.25 a.u.}
|
||||
\label{Table-S4}
|
||||
\begin{tabular}{ccccccc}
|
||||
\hline
|
||||
State & Active space & State-average & CASSCF & NEVPT2 & CASPT2 & MRCI \\
|
||||
& ($a_1,b_1,b_2,a_2$) & ($A_1,B_1,B_2,A2$) \\
|
||||
\hline
|
||||
$^4\Pi(\pi\rightarrow\pi^\star)$ & (6,3,3,0) & (0,2,2,0) &3.01 &2.72 &2.74 &2.81\\
|
||||
$^2\Pi(\pi\rightarrow\pi^\star)$ & (6,3,3,0) & (0,2,2,0) &3.94 &3.52 &3.55 &3.62\\
|
||||
$^2\Sigma^+(n\rightarrow\pi^\star)$ & (6,3,3,0) & (1,1,1,0) &3.85 &3.81 &3.72 &3.83\\
|
||||
$^2\Phi(\pi\rightarrow\pi^\star)$ & (6,3,3,0) & (0,2,2,0) &4.86 &4.32 &4.35 &4.44\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\vspace{-0.3 cm}
|
||||
\begin{flushleft}
|
||||
\end{flushleft}
|
||||
\end{table}
|
||||
|
||||
|
||||
\clearpage
|
||||
\section{Geometries}
|
||||
|
||||
\subsection{Exotic compounds}
|
||||
|
||||
Below, we provide the Cartesian coordinates of the exotic compounds investigated in this study.
|
||||
These are given in atomic units (bohr) and they have been obtained at the \CC{3}(full)/{\AVTZ} level of theory.
|
||||
|
||||
\subsubsection{Carbonylfluoride (F$_2$CO)}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -0.30652633
|
||||
O 0.00000000 0.00000000 -2.52469534
|
||||
F 0.00000000 2.00254958 1.16003038
|
||||
F 0.00000000 -2.00254958 1.16003038
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{CCl$_2$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.60920674
|
||||
Cl 0.00000000 2.65360612 0.27602958
|
||||
Cl 0.00000000 -2.65360612 0.27602958
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{CClF}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.29776085 0.00000000 1.47969075
|
||||
F 2.16980264 0.00000000 -0.10569879
|
||||
Cl -2.46756349 0.00000000 -0.32822320
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsubsection{CF$_2$}
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.14170749
|
||||
F 0.00000000 1.94810617 0.36114458
|
||||
F 0.00000000 -1.94810617 0.36114458
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsubsection{Difluorodiazirine (CF$_2$N$_2$)}
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -0.15283028
|
||||
F 0.00000000 2.06077297 -1.57706828
|
||||
F 0.00000000 -2.06077297 -1.57706828
|
||||
N 1.20382241 0.00000000 2.20566821
|
||||
N -1.20382241 0.00000000 2.20566821
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsubsection{Formylfluoride (FHCO)}
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00536098 0.00000000 0.75320959
|
||||
O 2.17369813 0.00000000 0.22287752
|
||||
H -0.83846350 0.00000000 2.62640974
|
||||
F -1.84051320 0.00000000 -0.99373750
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HCCl}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
H -1.88068369 0.00000000 -0.14323924
|
||||
Cl 2.28559426 0.00000000 -0.43261163
|
||||
C -0.40491057 0.00000000 1.32161964
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HCF}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C -0.13561085 0.00000000 1.20394474
|
||||
F 1.85493976 0.00000000 -0.27610752
|
||||
H -1.71932891 0.00000000 -0.18206846
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HCP}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
H 0.00000000 0.00000000 -4.03090449
|
||||
C 0.00000000 0.00000000 -2.01691641
|
||||
P 0.00000000 0.00000000 0.91401621
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HPO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
H 0.31668637 0.00000000 0.14072725
|
||||
P -0.80573521 0.00000000 2.65136926
|
||||
O 1.43391190 0.00000000 4.38886277
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HPS}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
H -2.56278959 0.00000000 2.36296006
|
||||
P 0.09114182 0.00000000 1.82568543
|
||||
S 0.07946992 0.00000000 -1.85778170
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HSiF}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
Si -0.06438136 0.00000000 1.67253150
|
||||
F 2.24990164 0.00000000 -0.33928119
|
||||
H -2.18552027 0.00000000 -0.28748154
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{SiCl$_2$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
Si 0.00000000 0.00000000 -1.78528322
|
||||
Cl 0.00000000 3.04414528 0.71619419
|
||||
Cl 0.00000000 -3.04414528 0.71619419
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{Silylidene (H$_2$CSi)}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.09539928
|
||||
Si 0.00000000 0.00000000 1.14992930
|
||||
H 0.00000000 1.70929524 -3.22894481
|
||||
H 0.00000000 -1.70929524 -3.22894481
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\clearpage
|
||||
|
||||
|
||||
\subsection{Radicals}
|
||||
|
||||
Below, we provide the Cartesian coordinates of the radical compounds investigated in this study.
|
||||
These are given in atomic units (bohr) and they have been obtained at the UCCSD(T)(full)/{\AVTZ} level of theory,
|
||||
except when noted.
|
||||
|
||||
\subsubsection{Allyl (C$_3$H$_5$)}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 0.83050732
|
||||
C 0.00000000 2.30981224 -0.38722841
|
||||
C 0.00000000 -2.30981224 -0.38722841
|
||||
H 0.00000000 0.00000000 2.87547067
|
||||
H 0.00000000 4.06036949 0.65560561
|
||||
H 0.00000000 -4.06036949 0.65560561
|
||||
H 0.00000000 2.41059890 -2.42703281
|
||||
H 0.00000000 -2.41059890 -2.42703281
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{BeF}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
Be 0.00000000 0.00000000 -1.77936990
|
||||
F 0.00000000 0.00000000 0.79083149
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{BeH}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
Be 0.00000000 0.00000000 0.25103976
|
||||
H 0.00000000 0.00000000 -2.24485003
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsubsection{BH$_2$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
B 0.00000000 0.00000000 0.14984923
|
||||
H 0.00000000 2.01119016 -0.81846345
|
||||
H 0.00000000 -2.01119016 -0.81846345
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{CH}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -0.16245872
|
||||
H 0.00000000 0.00000000 1.93436816
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{CH$_3$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 0.00000000
|
||||
H 0.00000000 0.00000000 2.03379507
|
||||
H 0.00000000 1.76131924 -1.01689753
|
||||
H 0.00000000 -1.76131924 -1.01689753
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{CN}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.18953886
|
||||
N 0.00000000 0.00000000 1.01938091
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsubsection{CNO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.50680714
|
||||
N 0.00000000 0.00000000 -0.22402176
|
||||
O 0.00000000 0.00000000 2.07682752
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
|
||||
\subsubsection{CON}
|
||||
|
||||
Optimized at the U-CCSDT/cc-pVTZ level.
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.44062558
|
||||
O 0.00000000 0.00000000 -0.20455596
|
||||
N 0.00000000 0.00000000 2.32515818
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\clearpage
|
||||
\subsubsection{CO$^+$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -1.20324172
|
||||
O 0.00000000 0.00000000 0.90271821
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{F$_2$BO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
O 0.00000000 0.00000000 2.65260017
|
||||
B 0.00000000 0.00000000 0.07681654
|
||||
F 0.00000000 2.16433924 -1.13888019
|
||||
F 0.00000000 -2.16433924 -1.13888019
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{F$_2$BS}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
S 0.00000000 0.00000000 2.64960984
|
||||
B 0.00000000 0.00000000 -0.74406239
|
||||
F 0.00000000 2.14169276 -2.01390354
|
||||
F 0.00000000 -2.14169276 -2.01390354
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{H$_2$BO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
O 0.00000000 0.00000000 1.17360276
|
||||
B 0.00000000 0.00000000 -1.27133435
|
||||
H 0.00000000 1.98370787 -2.36904602
|
||||
H 0.00000000 -1.98370787 -2.36904602
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HCO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
H 0.00000000 -2.55038496 1.39798104
|
||||
C 0.00000000 -1.17300976 -0.19046167
|
||||
O 0.00000000 1.04073447 0.05480615
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{HOC}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
H 0.00000000 1.82002973 1.50851586
|
||||
O 0.00000000 0.96467865 -0.12887834
|
||||
C 0.00000000 -1.43868535 0.04508983
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{H$_2$PO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
P 0.00000000 0.87766783 -0.10010856
|
||||
O 0.00000000 -1.95912323 0.05701315
|
||||
H 2.08101554 2.05955113 1.08591181
|
||||
H -2.08101554 2.05955113 1.08591181
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{H$_2$PS}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
P 0.00000000 1.81994516 -0.10769248
|
||||
S 0.00000000 -1.93707861 0.02086846
|
||||
H 2.03762554 2.75934101 1.32385757
|
||||
H -2.03762554 2.75934101 1.32385757
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{NCO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
N 0.00000000 0.00000000 -2.39343558
|
||||
C 0.00000000 0.00000000 -0.07238136
|
||||
O 0.00000000 0.00000000 2.14968523
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{NH$_2$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
N 0.00000000 0.00000000 0.15111603
|
||||
H 0.00000000 1.51574744 -1.04982949
|
||||
H 0.00000000 -1.51574744 -1.04982949
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{Nitromethyl (CH$_2$-NO$_2$)}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 0.00000000 -2.58417104
|
||||
N 0.00000000 0.00000000 0.08692471
|
||||
O 0.00000000 -2.06715629 1.15098225
|
||||
O 0.00000000 2.06715629 1.15098225
|
||||
H 0.00000000 1.81656349 -3.48616378
|
||||
H 0.00000000 -1.81656349 -3.48616378
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{NO}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
N 0.00000000 0.00000000 -1.15775086
|
||||
O 0.00000000 0.00000000 1.01357658
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{OH}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
O 0.00000000 0.00000000 -0.10864763
|
||||
H 0.00000000 0.00000000 1.72431679
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{PH$_2$}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
P 0.00000000 0.00000000 0.11427641
|
||||
H 0.00000000 1.91899987 -1.75604411
|
||||
H 0.00000000 -1.91899987 -1.75604411
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\subsubsection{Vinyl (C$_2$H$_3$)}
|
||||
|
||||
\begin{singlespace}
|
||||
\begin{verbatim}
|
||||
C 0.00000000 1.16769663 -0.04303146
|
||||
C 0.00000000 -1.29945364 0.15810072
|
||||
H 0.00000000 2.38429609 1.59801822
|
||||
H 0.00000000 2.08759130 -1.87998309
|
||||
H 0.00000000 -2.90307925 -1.08814513
|
||||
\end{verbatim}
|
||||
\end{singlespace}
|
||||
|
||||
\bibliography{biblio-new}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
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Reference in New Issue