626 lines
25 KiB
TeX
626 lines
25 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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% second quantized operators
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\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal/mol}
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% methods
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\newcommand{\D}{\text{D}}
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\newcommand{\T}{\text{T}}
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\newcommand{\Q}{\text{Q}}
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\newcommand{\X}{\text{X}}
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\newcommand{\UEG}{\text{UEG}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\ROHF}{\text{ROHF}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\PBE}{\text{PBE}}
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CBS}{\text{CBS}}
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\newcommand{\exFCI}{\text{exFCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\Ne}{N}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
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\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
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% basis sets
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\newcommand{\Bas}{\mathcal{B}}
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% operators
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% coordinates
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% frozen core
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\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
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% energies
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\newcommand{\EHF}{E_\text{HF}}
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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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\newcommand{\EDMC}{E_\text{DMC}}
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\newcommand{\EexFCI}{E_\text{exFCI}}
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\newcommand{\EexDMC}{E_\text{exDMC}}
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\newcommand{\Ead}{\Delta E_\text{ad}}
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\newcommand{\Eabs}{\Delta E_\text{abs}}
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\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
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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{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
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\begin{document}
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\title{Excitation Energies Near The Complete Basis Set Limit}
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\author{Emmanuel Giner}
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\affiliation{\LCT}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Julien Toulouse}
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\affiliation{\LCT}
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\author{Pierre-Fran\c{c}ois Loos}
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\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set.
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%
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One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
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In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{GinPraFerAssSavTou-JCP-18}
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%Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
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%Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
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%
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%WFT is attractive as it exists a well-defined path for systematic improvement as well as powerful tools, such as perturbation theory, to guide the development of new WFT \textit{ans\"atze}.
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%The coupled cluster (CC) family of methods is a typical example of the WFT philosophy and is well regarded as the gold standard of quantum chemistry for weakly correlated systems.
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%By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually high.
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%One of the most fundamental drawbacks of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
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%This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
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%To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
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%The resulting F12 methods yield a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
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%For example, at the CCSD(T) level, one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
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%To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
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%
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%Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
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%The attractiveness of DFT originates from its very favorable accuracy/cost ratio as it often provides reasonably accurate energies and properties at a relatively low computational cost.
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%Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
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%Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing Perdew's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
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%In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
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%
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%Progress toward unifying WFT and DFT are on-going.
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%In particular, range-separated DFT (RS-DFT) (see Ref.~\citenum{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
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%As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15}
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%Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
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%Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis-set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
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%The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdetails}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The present basis-set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
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The present methodology is identical to the one described in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} where the main working equation are reported and discussed.
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
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We refer the interested reader to Refs.~\citenum{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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The one-electron density and on-top density is computed from a very large CIPSI expansion containing several million determinants.
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All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
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For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
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The geometries have been extracted from Refs.~\citenum{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
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They are also reported in the {\SI}.
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Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
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The FC density-based correction is used consistently with the FC approximation in WFT methods.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%=======================
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\subsection{Water}
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\label{sec:H2O}
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%=======================
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%=======================
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\subsection{Formaldehyde}
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\label{sec:CH2O}
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%=======================
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%=======================
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\subsection{Methylene}
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\label{sec:CH2}
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%=======================
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%%% TABLE 1 %%%
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\begin{squeezetable}
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\begin{table*}
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\caption{
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Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.}
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\begin{ruledtabular}{}
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\begin{tabular}{llddddddd}
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& & \mc{1}{c}{$1\,^{3}B_1$}
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& \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}A_1$}
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& \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}B_1$}
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& \mc{2}{c}{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\
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\cline{3-3} \cline{4-5}
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\cline{6-7} \cline{8-9}
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Method & Basis set & \tabc{$E$ (a.u.)}
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& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)}
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& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)}
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& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} \\
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\hline
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exFCI & AVDZ & -39.04846(1)
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& -39.03225(1) & 0.441
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& -38.99203(1) & 1.536
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& -38.95076(1) & 2.659 \\
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& AVTZ & -39.08064(3)
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& -39.06565(2) & 0.408
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& -39.02833(1) & 1.423
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& -38.98709(1) & 2.546 \\
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& AVQZ & -39.08854(1)
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& -39.07402(2) & 0.395
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& -39.03711(1) & 1.399
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& -38.99607(1) & 2.516 \\
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& AV5Z & -39.09079(1)
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& -39.07647(1) & 0.390
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& -39.03964(3) & 1.392
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& -38.99867(1) & 2.507 \\
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& CBS & -39.09111
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& -39.07682 & 0.389
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& -39.04000 & 1.391
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& -38.99904 & 2.505 \\
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\\
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exFCI+LDA & AVDZ & -39.07450(1)
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& -39.06213(1) & 0.337
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& -39.02233(1) & 1.420
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& -38.97946(1) & 2.586 \\
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& AVTZ & -39.09099(3)
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& -39.07779(2) & 0.359
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& -39.04051(1) & 1.374
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& -38.99859(1) & 2.514 \\
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& AVQZ & -39.09319(1)
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& -39.07959(2) & 0.370
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& -39.04267(1) & 1.375
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& -39.00135(1) & 2.499 \\
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\\
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exFCI+PBE & AVDZ & -39.07282(1)
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& -39.06150(1) & 0.308
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& -39.02181(1) & 1.388
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& -38.97873(1) & 2.560 \\
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& AVTZ & -39.08948(3)
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& -39.07639(2) & 0.356
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& -39.03911(1) & 1.371
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& -38.99724(1) & 2.510 \\
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& AVQZ & -39.09247(1)
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& -39.07885(2) & 0.371
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& -39.04193(1) & 1.375
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& -39.00066(1) & 2.498 \\
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\\
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exFCI+PBEot & AVDZ & -39.06924(1)
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& -39.05651(1) & 0.347
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& -39.01777(1) & 1.401
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& -38.97698(1) & 2.511 \\
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& AVTZ & -39.08805(3)
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& -39.07430(2) & 0.374
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& -39.03742(1) & 1.378
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& -38.99652(1) & 2.491 \\
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& AVQZ & -39.09189(1)
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& -39.07795(2) & 0.379
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& -39.04124(1) & 1.378
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& -39.00044(1) & 2.489 \\
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\\
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SHCI & AVQZ & -39.08849(1)
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& -39.07404(1) & 0.393
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& -39.03711(1) & 1.398
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& -38.99603(1) & 2.516 \\
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CR-EOMCC (2,3)D& AVQZ & -39.08817
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& -39.07303 & 0.412
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& -39.03450 & 1.460
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& -38.99457 & 2.547 \\
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FCI & TZ2P & -39.066738
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& -39.048984 & 0.483
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& -39.010059 & 1.542
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& -38.968471 & 2.674 \\
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DMC & &
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& & 0.406
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& & 1.416
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& & 2.524 \\
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Exp. & &
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& & 0.400
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& & 1.411
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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\end{squeezetable}
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%%% %%% %%%
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%%% TABLE 2 %%%
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\begin{squeezetable}
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\begin{table*}
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\caption{
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Vertical absorption energies $\Eabs$ (in eV) of excited states of water, carbon dimer and ammonia for various methods and basis sets.}
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\begin{ruledtabular}{}
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\begin{tabular}{lllddddddddddddd}
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& & & & \mc{12}{c}{Deviation with respect to TBE}
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\\
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\cline{5-16}
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& & & & \mc{3}{c}{exFCI}
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& \mc{3}{c}{exFCI+PBEot}
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& \mc{3}{c}{exFCI+PBE}
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& \mc{3}{c}{exFCI+LDA}
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\\
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\cline{5-7} \cline{8-10} \cline{11-13} \cline{14-16}
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Molecule & Transition & Nature & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
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& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
|
& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
|
& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
|
\\
|
|
\hline
|
|
Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Ryd. & 6.66 & -0.18 & -0.07 & -0.02
|
|
& -0.04 & -0.02 & 0.00
|
|
& -0.07 & -0.03 & 0.00
|
|
& -0.07 & -0.03 & 0.00
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{1}E$ & Ryd. & 8.21 & -0.13 & -0.05 & 0.01
|
|
& 0.01 & 0.00 & 0.04
|
|
& -0.03 & -0.01 & 0.03
|
|
& -0.03 & 0.00 & 0.03
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.65 & 1.03 & 0.68 & 0.49
|
|
& 1.17 & 0.73 & 0.75
|
|
& 1.13 & 0.72 & 0.74
|
|
& 1.13 & 0.71 & 0.78
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{1}A_{2}$ & Ryd. & 8.65 & 1.22 & 0.77 & 0.60
|
|
& 1.17 & 0.73 & 0.75
|
|
& 1.13 & 0.72 & 0.74
|
|
& 1.13 & 0.71 & 0.78
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Ryd. & 9.19 & -0.18 & -0.06 & -0.02
|
|
& 1.36 & 0.83 & 0.63
|
|
& 1.33 & 0.81 & 0.62
|
|
& 1.32 & 0.81 & 0.61
|
|
\\
|
|
\\
|
|
Carbon dimer\fnm[1] & $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ & Val. & 2.06 & 0.15 & 0.03 & 0.00
|
|
& 0.02 & -0.02 & -0.02
|
|
& 0.13 & 0.02 & 0.00
|
|
& 0.15 & 0.03 & 0.00
|
|
\\
|
|
& $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & Val. & 2.40 & 0.10 & 0.02 & 0.00
|
|
& 0.02 & -0.03 & -0.02
|
|
& 0.09 & 0.01 & 0.00
|
|
& 0.11 & 0.02 & 0.00
|
|
\\
|
|
\\
|
|
Hydrogen chloride& ${}^1\Sigma \ra {}^1\Pi$ & CT\fnm[2] & 7.86 & -0.04 & -0.02 & 0.02
|
|
& 0.13 & 0.06 & 0.06
|
|
& 0.11 & 0.04 & 0.05
|
|
& 0.10 & 0.05 & 0.06
|
|
\\
|
|
\\
|
|
Hydrogen sulfide & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 6.10 & 0.00 & 0.08 & 0.05
|
|
& 0.15 & 0.12 & 0.07
|
|
& 0.14 & 0.11 & 0.07
|
|
& 0.14 & 0.11 & 0.07
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 6.29 & 0.00 & -0.05 & 0.00
|
|
& -0.12 & 0.01 & 0.03
|
|
& -0.14 & 0.00 & 0.03
|
|
& -0.14 & 0.01 & 0.03
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 5.74 & 0.01 & 0.07 & 0.05
|
|
& 0.18 & 0.12 & 0.08
|
|
& 0.20 & 0.13 & 0.08
|
|
& 0.19 & 0.13 & 0.08
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 5.94 & -0.04 & -0.05 & -0.01
|
|
& 0.07 & 0.02 & 0.03
|
|
& 0.09 & 0.03 & 0.03
|
|
& 0.07 & 0.04 & 0.04
|
|
\\
|
|
\\
|
|
Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 7.70 & -0.17 & -0.07 & -0.02
|
|
& 0.01 & 0.00 & 0.02
|
|
& -0.02 & -0.01 & 0.00
|
|
& -0.04 & -0.01 & 0.01
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 9.47 & -0.15 & -0.06 & -0.01
|
|
& 0.03 & 0.01 & 0.03
|
|
& 0.00 & 0.00 & 0.02
|
|
& -0.03 & 0.00 & 0.00
|
|
\\
|
|
& $1\,^{1}A_1 \ra 2\,^{1}A_1$ & Ryd. & 9.97 & -0.03 & 0.02 & 0.06
|
|
& 0.13 & 0.08 & 0.09
|
|
& 0.10 & 0.07 & 0.08
|
|
& 0.09 & 0.07 & 0.03
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 7.33 & -0.19 & -0.08 & -0.03
|
|
& 0.02 & 0.00 & 0.02
|
|
& 0.05 & 0.01 & 0.02
|
|
& 0.00 & 0.00 & 0.04
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 9.30 & -0.16 & -0.06 & -0.01
|
|
& 0.04 & 0.02 & 0.04
|
|
& 0.07 & 0.03 & 0.04
|
|
& 0.03 & 0.03 & 0.04
|
|
\\
|
|
& $1\,^{1}A_1 \ra 1\,^{3}A_1$ & Ryd. & 9.59 & -0.11 & -0.05 & -0.01
|
|
& 0.07 & 0.02 & 0.03
|
|
& 0.09 & 0.03 & 0.03
|
|
& 0.06 & 0.03 & 0.04
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.}
|
|
\fnt[2]{CT stands for charge transfer.}
|
|
\end{table*}
|
|
\end{squeezetable}
|
|
%%% %%% %%%
|
|
|
|
%%% TABLE 3 %%%
|
|
\begin{squeezetable}
|
|
\begin{table*}
|
|
\caption{
|
|
Vertical absorption energies $\Eabs$ (in eV) of excited states of acetylene, ethylene and formaldehyde for various methods and basis sets.}
|
|
\begin{ruledtabular}{}
|
|
\begin{tabular}{lllddddddddd}
|
|
& & & & \mc{8}{c}{Deviation with respect to TBE}
|
|
\\
|
|
\cline{5-12}
|
|
& & & & \mc{2}{c}{exFCI}
|
|
& \mc{2}{c}{exFCI+PBEot}
|
|
& \mc{2}{c}{exFCI+PBE}
|
|
& \mc{2}{c}{exFCI+LDA}
|
|
\\
|
|
\cline{5-6} \cline{7-8} \cline{9-10} \cline{11-12}
|
|
Molecule & Transition & Nature & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ}
|
|
& \tabc{AVDZ} & \tabc{AVTZ}
|
|
& \tabc{AVDZ} & \tabc{AVTZ}
|
|
& \tabc{AVDZ} & \tabc{AVTZ}
|
|
\\
|
|
\hline
|
|
Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00
|
|
& 0.07 & 0.00
|
|
& 0.11 & 0.00
|
|
& 0.11 & 0.00
|
|
\\
|
|
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00
|
|
& 0.04 & -0.01
|
|
& 0.12 & 0.02
|
|
& 0.11 & 0.02
|
|
\\
|
|
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03
|
|
& 0.07 & 0.02
|
|
& 0.04 & 0.00
|
|
& 0.02 & 0.00
|
|
\\
|
|
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00
|
|
& 0.10 & 0.02
|
|
& 0.14 & 0.03
|
|
& 0.12 & 0.03
|
|
\\
|
|
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01
|
|
& 0.08 & 0.00
|
|
& 0.16 & 0.04
|
|
& 0.14 & 0.04
|
|
\\
|
|
\\
|
|
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43 & -0.12 & -0.04
|
|
& -0.05 & -0.01
|
|
& -0.04 & -0.01
|
|
& -0.02 & 0.00
|
|
\\
|
|
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & Val. & 7.92 & 0.01 & 0.01
|
|
& 0.00 & 0.00
|
|
& 0.06 & 0.03
|
|
& 0.06 & 0.03
|
|
\\
|
|
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & Ryd. & 8.10 & -0.1 & -0.02
|
|
& -0.03 & 0.00
|
|
& -0.02 & 0.00
|
|
& 0.00 & 0.01
|
|
\\
|
|
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54 & 0.01 & 0.00
|
|
& 0.07 & 0.03
|
|
& 0.10 & 0.04
|
|
& 0.08 & 0.04
|
|
\\
|
|
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28 & -0.12 & -0.04
|
|
& -0.03 & 0.00
|
|
& 0.00 & 0.00
|
|
& 0.00 & 0.02
|
|
\\
|
|
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00 & -0.07 & -0.01
|
|
& 0.01 & 0.03
|
|
& 0.04 & 0.03
|
|
& 0.05 & 0.04
|
|
\\
|
|
\\
|
|
Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Val. & 3.97 & 0.02 & 0.01
|
|
& 0.05 & 0.02
|
|
& 0.03 & 0.02
|
|
& 0.02 & 0.01
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & Ryd. & 7.30 & -0.19 & -0.07
|
|
& 0.00 & 0.00
|
|
& -0.02 & 0.00
|
|
& -0.04 & 0.00
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & Ryd. & 8.14 & -0.10 & -0.01
|
|
& 0.09 & 0.07
|
|
& 0.08 & 0.06
|
|
& 0.05 & 0.06
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.27 & -0.15 & -0.04
|
|
& 0.03 & 0.04
|
|
& 0.02 & 0.03
|
|
& 0.00 & 0.03
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Val. & 3.58 & 0.00 & 0.00
|
|
& 0.09 & 0.05
|
|
& 0.11 & 0.06
|
|
& 0.07 & 0.04
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Val. & 6.07 & 0.03 & 0.01
|
|
& 0.13 & 0.04
|
|
& 0.15 & 0.05
|
|
& 0.11 & 0.04
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & Ryd. & 7.14 & -0.19 & -0.08
|
|
& 0.01 & 0.01
|
|
& 0.02 & 0.01
|
|
& -0.01 & 0.00
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & Ryd. & 7.96 & -0.09 & -0.02
|
|
& 0.13 & 0.08
|
|
& 0.14 & 0.08
|
|
& 0.10 & 0.07
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Ryd. & 8.15 & -0.14 & -0.05
|
|
& 0.07 & 0.05
|
|
& 0.07 & 0.04
|
|
& 0.04 & 0.04
|
|
\\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
\end{squeezetable}
|
|
%%% %%% %%%
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Conclusion}
|
|
\label{sec:ccl}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Supporting Information Available}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
See {\SI} for geometries and additional information (including total energies).
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{acknowledgements}
|
|
The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding.
|
|
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
|
|
\end{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
|
|
\bibliography{Ex-srDFT,Ex-srDFT-control}
|
|
|
|
\end{document}
|