475 lines
16 KiB
TeX
475 lines
16 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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\usepackage{mathpazo,libertine}
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\usepackage[normalem]{ulem}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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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{\br}{\mathbf{r}}
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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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\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{\ra}{\rightarrow}
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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{\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., J.~Chem.~Phys.~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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This problem was already noticed thirty years ago by Kutzelnigg \cite{Kutzelnigg_1985} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg_1991, Termath_1991, Klopper_1991a, Klopper_1991b, Noga_1994}
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This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
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This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson_1996, Persson_1997, May_2004, Tenno_2004b, Tew_2005, May_2005}
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The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno_2012a, Tenno_2012b, Hattig_2012, Kong_2012}
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For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew_2007b}
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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{Giner_2018}
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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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%%%%%%%%%%%%%%%%%%%%%%%%
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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 and ammonia for various methods and basis sets.}
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\begin{ruledtabular}{}
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\begin{tabular}{llddddddddddddd}
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& & & \mc{12}{c}{Deviation with respect to TBE}
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\\
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\cline{4-15}
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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{4-6} \cline{7-9} \cline{10-12} \cline{13-15}
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Molecule & Transition & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
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& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
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& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
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& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
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\\
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\hline
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Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & 7.70 & -0.17 & -0.07 & -0.02
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& 0.01 & 0.00 & 0.02
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& -0.02 & -0.01 & 0.00
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& -0.04 & -0.01 & 0.01
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\\
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& $1\,^{1}A_1 \ra 1\,^{1}A_2$ & 9.47 & -0.15 & -0.06 & -0.01
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& 0.03 & 0.01 & 0.03
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& 0.00 & 0.00 & 0.02
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& -0.03 & 0.00 & 0.00
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\\
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& $1\,^{1}A_1 \ra 2\,^{1}A_1$ & 9.97 & -0.03 & 0.02 & 0.06
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& 0.13 & 0.08 & 0.09
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& 0.10 & 0.07 & 0.08
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& 0.09 & 0.07 & 0.03
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\\
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& $1\,^{1}A_1 \ra 1\,^{3}B_1$ & 7.33 & -0.19 & -0.08 & -0.03
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& 0.02 & 0.00 & 0.02
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& 0.05 & 0.01 & 0.02
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& 0.00 & 0.00 & 0.04
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\\
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& $1\,^{1}A_1 \ra 1\,^{3}A_2$ & 9.30 & -0.16 & -0.06 & -0.01
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& 0.04 & 0.02 & 0.04
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& 0.07 & 0.03 & 0.04
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& 0.03 & 0.03 & 0.04
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\\
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& $1\,^{1}A_1 \ra 1\,^{3}A_1$ & 9.59 & -0.11 & -0.05 & -0.01
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& 0.07 & 0.02 & 0.03
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& 0.09 & 0.03 & 0.03
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& 0.06 & 0.03 & 0.04
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\\
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\\
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Carbon dimer & $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ & 2.06 & 0.15 & 0.03 & 0.00
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& & &
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& & &
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& & &
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\\
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& $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & 2.40 & 0.10 & 0.02 & 0.00
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& & &
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& & &
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& & &
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\\
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\\
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Hydrogen sulfide & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & 6.10 & 0.19 & 0.08 & 0.05
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& 0.34 & 0.12 & 0.07
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& 0.33 & 0.11 & 0.07
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& 0.33 & 0.11 & 0.07
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\\
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& $1\,^{1}A_1 \ra 1\,^{1}B_1$ & 6.29 & -0.19 & -0.05 & 0.00
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& -0.12 & 0.01 & 0.03
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& -0.14 & 0.00 & 0.03
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& -0.14 & 0.01 & 0.03
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\\
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& $1\,^{1}A_1 \ra 1\,^{3}A_2$ & 5.74 & 0.16 & 0.07 & 0.05
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& 0.33 & 0.12 & 0.08
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& 0.35 & 0.13 & 0.08
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& 0.34 & 0.34 & 0.08
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\\
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& $1\,^{1}A_1 \ra 1\,^{3}B_1$ & 5.94 & -0.19 & -0.05 & -0.01
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& -0.08 & 0.02 & 0.03
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& -0.06 & 0.03 & 0.03
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& -0.08 & 0.04 & 0.04
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\\
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\\
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Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & 6.66 & -0.18 & -0.07 & -0.02
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& -0.04 & -0.02 & 0.00
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& -0.07 & -0.03 & 0.00
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& -0.07 & -0.03 & 0.00
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\\
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& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & 8.65 & 1.03 & 0.68 & 0.49
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& 1.17 & 0.73 & 0.75
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& 1.13 & 0.72 & 0.74
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& 1.13 & 0.71 & 0.78
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\\
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& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & 6.37 & -0.18 & -0.06 & -0.02
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& -0.03 & 0.00 & 0.03
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& -0.07 & 0.02 & 0.00
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& -0.07 & -0.01 & 0.00
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\\
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\\
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Hydrogen chloride& ${}^1\Sigma \ra {}^1\Pi$ & 7.86 & -0.04 & -0.02 & 0.02
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& 0.13 & 0.06 & 0.06
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& 0.11 & 0.04 & 0.05
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& 0.10 & 0.05 & 0.06
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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 3 %%%
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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 ethylene and formaldehyde for various methods and basis sets.}
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\begin{ruledtabular}{}
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\begin{tabular}{llddddddddd}
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& & & \mc{8}{c}{Deviation with respect to TBE}
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\\
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\cline{4-11}
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& & & \mc{2}{c}{exFCI}
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& \mc{2}{c}{exFCI+PBEot}
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& \mc{2}{c}{exFCI+PBE}
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& \mc{2}{c}{exFCI+LDA}
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\\
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\cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11}
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Molecule & Transition & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ}
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& \tabc{AVDZ} & \tabc{AVTZ}
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& \tabc{AVDZ} & \tabc{AVTZ}
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& \tabc{AVDZ} & \tabc{AVTZ}
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\\
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\hline
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Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & 7.10 & 0.10 & 0.00
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& &
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& &
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& &
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\\
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& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & 7.44 & 0.07 & 0.00
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& &
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& &
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& &
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\\
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& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & 5.56 & -0.06 & -0.03
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& &
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& &
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& &
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\\
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& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & 6.40 & 0.06 & 0.00
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& &
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& &
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& &
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\\
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& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & 7.09 & 0.05 & -0.01
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& &
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& &
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& &
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\\
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\\
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Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & 7.43 & -0.12 & -0.04
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& -0.05 & -0.01
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& -0.04 & -0.01
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& -0.02 & 0.00
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\\
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& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & 7.92 & 0.01 & 0.01
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& 0.00 & 0.00
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& 0.06 & 0.03
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& 0.06 & 0.03
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\\
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& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & 8.10 & -0.1 & -0.02
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& -0.03 & 0.00
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& -0.02 & 0.00
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& 0.00 & 0.01
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\\
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& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & 4.54 & 0.01 & 0.00
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& 0.07 & 0.03
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& 0.10 & 0.04
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& 0.08 & 0.04
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\\
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\\
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Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & 3.97 & 0.02 & 0.01
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& 0.05 & 0.02
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& 0.03 & 0.02
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& 0.02 & 0.01
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\\
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& $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & 7.30 & -0.19 & -0.07
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& 0.00 & 0.00
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& -0.02 & 0.00
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& -0.04 & 0.00
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & 8.14 & -0.10 & -0.01
|
|
& 0.09 & 0.07
|
|
& 0.08 & 0.06
|
|
& 0.05 & 0.06
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & 8.27 & -0.15 & -0.04
|
|
& 0.03 & 0.04
|
|
& 0.02 & 0.03
|
|
& 0.00 & 0.03
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & 3.58 & 0.00 & 0.00
|
|
& 0.09 & 0.05
|
|
& 0.11 & 0.06
|
|
& 0.07 & 0.04
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & 6.07 & 0.03 & 0.01
|
|
& 0.13 & 0.04
|
|
& 0.15 & 0.05
|
|
& 0.11 & 0.04
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & 7.14 & -0.19 & -0.08
|
|
& 0.01 & 0.01
|
|
& 0.02 & 0.01
|
|
& -0.01 & 0.00
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & 7.96 & -0.09 & -0.02
|
|
& 0.13 & 0.08
|
|
& 0.14 & 0.08
|
|
& 0.10 & 0.07
|
|
\\
|
|
& $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & 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}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
See {\SI} for geometries and additional information (including total energies).
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{acknowledgements}
|
|
This work was performed using HPC resources from
|
|
i) GENCI-TGCC (Grant No. 2018-A0040801738),
|
|
ii) CALMIP (Toulouse) under allocations 2018-0510 and 2018-12158.
|
|
\end{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
|
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\bibliography{Ex-srDFT}
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\end{document}
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