modif table
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BSE-PES.tex
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BSE-PES.tex
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@ -73,6 +73,7 @@
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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\newcommand{\IP}{\text{IP}}
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\newcommand{\EA}{\text{EA}}
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\newcommand{\Req}{R_\text{eq}}
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% orbital energies
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\newcommand{\e}[1]{\epsilon_{#1}}
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@ -447,7 +448,7 @@ The computational cost of these methods, in their usual implementation, scale as
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All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
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As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions.
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Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fair comparison between methods.
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However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}).
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However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI} for additional information).
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Because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum for each quadrature point, we perform several complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
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This step is, by far, the computational bottleneck in our current implementation.
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@ -506,50 +507,36 @@ Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various
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%\label{sec:PES}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{HCl}, \ce{N2}, \ce{CO}, \ce{BF}, and \ce{F2}.
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The PES of these molecules for various methods are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances for various basis sets are gathered in Table \ref{tab:Req}.
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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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The PES of these molecules for various methods are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances are gathered in Table \ref{tab:Req}.
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Both of these properties have been computed with Dunning's cc-pVQZ basis set.
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Additional graphs and tables for other basis sets can be found in the {\SI}.
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%%% TABLE I %%%
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\begin{squeezetable}
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\begin{table*}
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\caption{
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Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
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Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained with the cc-pVQZ basis set at various levels of theory.
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When irregularities appear in the PES, the values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
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}
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\label{tab:Req}
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\begin{ruledtabular}
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\begin{tabular}{llcccccccc}
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& & \mc{8}{c}{Molecules} \\
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\cline{3-10}
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Method & Basis & \ce{H2} & \ce{LiH} & \ce{LiF} & \ce{HCl} & \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} \\
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\begin{tabular}{lcccccccc}
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& \mc{8}{c}{Molecules} \\
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\cline{2-9}
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Method & \ce{H2} & \ce{LiH} & \ce{LiF} & \ce{HCl} & \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} \\
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\hline
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CC3 & cc-pVDZ & 1.438 & 3.043 & 3.012 & 2.435 & 2.114 & 2.166 & 2.444 & 2.740 \\
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& cc-pVTZ & 1.403 & 3.011 & 2.961 & 2.413 & 2.079 & 2.143 & 2.392 & 2.669 \\
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& cc-pVQZ &\bb{1.402} &\bb{3.019} &\bb{2.963} &\bb{2.403} &\bb{2.075} &\bb{2.136} &\bb{2.390} &\bb{2.663} \\
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CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.433 & 2.101 & 2.149 & 2.435 & 2.695 \\
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& cc-pVTZ & 1.403 & 3.012 & 2.954 & 2.409 & 2.064 & 2.126 & 2.382 & 2.629 \\
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& cc-pVQZ & 1.402 & 3.020 & 2.953 & 2.398 & 2.059 & 2.118 & 2.118 & 2.621 \\
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CC2 & cc-pVDZ & 1.426 & 3.046 & 3.026 & 2.427 & 2.146 & 2.187 & 2.444 & 2.710 \\
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& cc-pVTZ & 1.393 & 3.008 & 2.995 & 2.406 & 2.109 & 2.163 & 2.394 & 2.664 \\
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& cc-pVQZ & 1.391 & 2.989 & 2.982 & 2.396 & 2.106 & 2.156 & 2.393 & 2.665 \\
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MP2 & cc-pVDZ & 1.426 & 3.041 & 3.010 & 2.426 & 2.133 & 2.166 & 2.431 & 2.681 \\
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& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\
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& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
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BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & (2.698) \\
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& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
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& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)} &\gb{(2.408)} &\gb{(2.070)} &\gb{(2.130)} &\gb{(2.383)} &\rb{(2.640)}\\
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RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & (2.629) \\
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& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 & 2.055 & 2.114 & (2.370) & (2.584) \\
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& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
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RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
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& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
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& cc-pVQZ & 1.394 & 3.011 & 2.944 &\gb{(2.393)} &\gb{(2.041)} &\gb{(2.105)} &\gb{(2.367)} &\gb{(2.563)} \\
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RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\
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& cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\
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& cc-pVQZ & 1.386 & 2.994 & 2.946 &\gb{(2.385)} &\gb{(2.042)} &\gb{(2.104)} &\gb{(2.365)} &\gb{(2.571)} \\
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CC3 & 1.402 & 3.019 & 2.963 & 2.403 & 2.075 & 2.136 & 2.390 & 2.663 \\
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CCSD & 1.402[$+0.00\%$]& 3.020[$+0.03\%$] & 2.953[$-0.34\%$] & 2.398[$-0.21\%$] & 2.059[$-0.77\%$] & 2.118[$-0.84\%$] & 2.380[$-0.42\%$] & 2.621[$-1.58\%$] \\
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CC2 & 1.391[$-0.78\%$]& 2.989[$-0.99\%$] & 2.982[$+0.64\%$] & 2.396[$-0.29\%$] & 2.106[$+1.49\%$] & 2.156[$+0.94\%$] & 2.393[$+0.13\%$] & 2.665[$+0.08\%$] \\
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MP2 & 1.391[$-0.78\%$]& 3.008[$-0.36\%$] & 2.970[$+0.24\%$] & 2.395[$-0.33\%$] & 2.091[$+0.77\%$] & 2.137[$+0.05\%$] & 2.382[$-0.33\%$] & 2.634[$-1.09\%$] \\
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BSE@{\GOWO}@HF & 1.399[$-0.21\%$]& 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & \gb{(2.408)} & \gb{(2.070)} & \gb{(2.130)} & \gb{(2.383)} & (2.640)[$-0.86\%$] \\
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RPA@{\GOWO}@HF & 1.382[$-1.43\%$]& 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & \gb{(2.045)} & \gb{(2.110)} & \gb{(2.367)} & (2.571)[$-3.45\%$] \\
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RPAx@HF & 1.394[$-0.57\%$]& 3.011[$-0.26\%$] & 2.944[$-0.64\%$] & 2.391[$-0.50\%$] & \gb{(2.041)} & \gb{(2.105)} & \gb{(2.367)} & \gb{(2.563)}\\
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RPA@HF & 1.386[$-1.14\%$]& 2.994[$-0.83\%$] & 2.946[$-0.57\%$] & \gb{(2.385)} & \gb{(2.042)} & \gb{(2.104)} & \gb{(2.365)} & \gb{(2.571)}\\
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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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Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds.
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Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
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@ -577,7 +564,7 @@ As observed in Refs.~\onlinecite{vanSetten_2015,Maggio_2017,Loos_2018} and expla
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Including a broadening via the increasing the value of $\eta$ in the $GW$ self-energy and the screened Coulomb operator soften the problem, but does not remove it completely.
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Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
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When irregularities are present in the PES, we have fitted a Morse potential of the form $M(r) = D_e\{1-\exp[-\alpha(r-r_0)]\}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
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These value are reported in parenthesis in Table \ref{tab:Req}.
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These values are reported in parenthesis in Table \ref{tab:Req}.
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For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases.
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Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple to single bond).
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@ -616,7 +603,7 @@ This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supporting Information}
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%%%%%%%%%%%%%%%%%%%%%%%%
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See {\SI} for additional potential energy curves with other basis sets and within the frozen-core approximation.
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See {\SI} for additional potential energy curves computed with other basis sets and within the frozen-core approximation, as well as tables gathering equilibrium distances for smaller basis sets (cc-pVDZ and cc-pVTZ).
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\bibliography{BSE-PES,BSE-PES-control}
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@ -180,6 +180,48 @@
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\maketitle
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%%% TABLE I %%%
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\begin{table*}
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\caption{
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Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
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When irregularities appear in the PES, the values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
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}
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\label{tab:Req}
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\begin{ruledtabular}
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\begin{tabular}{llcccccccc}
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& & \mc{8}{c}{Molecules} \\
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\cline{3-10}
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Method & Basis & \ce{H2} & \ce{LiH} & \ce{LiF} & \ce{HCl} & \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} \\
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\hline
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CC3 & cc-pVDZ & 1.438 & 3.043 & 3.012 & 2.435 & 2.114 & 2.166 & 2.444 & 2.740 \\
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& cc-pVTZ & 1.403 & 3.011 & 2.961 & 2.413 & 2.079 & 2.143 & 2.392 & 2.669 \\
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& cc-pVQZ &\bb{1.402} &\bb{3.019} &\bb{2.963} &\bb{2.403} &\bb{2.075} &\bb{2.136} &\bb{2.390} &\bb{2.663} \\
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CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.433 & 2.101 & 2.149 & 2.435 & 2.695 \\
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& cc-pVTZ & 1.403 & 3.012 & 2.954 & 2.409 & 2.064 & 2.126 & 2.382 & 2.629 \\
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& cc-pVQZ & 1.402 & 3.020 & 2.953 & 2.398 & 2.059 & 2.118 & 2.118 & 2.621 \\
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CC2 & cc-pVDZ & 1.426 & 3.046 & 3.026 & 2.427 & 2.146 & 2.187 & 2.444 & 2.710 \\
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& cc-pVTZ & 1.393 & 3.008 & 2.995 & 2.406 & 2.109 & 2.163 & 2.394 & 2.664 \\
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& cc-pVQZ & 1.391 & 2.989 & 2.982 & 2.396 & 2.106 & 2.156 & 2.393 & 2.665 \\
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MP2 & cc-pVDZ & 1.426 & 3.041 & 3.010 & 2.426 & 2.133 & 2.166 & 2.431 & 2.681 \\
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& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\
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& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
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BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & (2.698) \\
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& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
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& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)} &\gb{(2.408)} &\gb{(2.070)} &\gb{(2.130)} &\gb{(2.383)} &\rb{(2.640)}\\
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RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & (2.629) \\
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& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 & 2.055 & 2.114 & (2.370) & (2.584) \\
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& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
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RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
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& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
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& cc-pVQZ & 1.394 & 3.011 & 2.944 & 2.391 &\gb{(2.041)} &\gb{(2.105)} &\gb{(2.367)} &\gb{(2.563)} \\
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RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\
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& cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\
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& cc-pVQZ & 1.386 & 2.994 & 2.946 & 2.382 &\gb{(2.042)} &\gb{(2.104)} &\gb{(2.365)} &\gb{(2.571)} \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%%% FIG 1 %%%
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\begin{figure*}
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% H2
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