He
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@ -283,7 +283,7 @@ The latters are determined by solving the ensemble KS equation
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where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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\begin{equation}
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\begin{equation}
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\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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= \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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= \fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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\end{equation}
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\end{equation}
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is the Hxc potential, with
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is the Hxc potential, with
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\begin{subequations}
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\begin{subequations}
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@ -670,117 +670,79 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
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& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
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& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
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& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
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\\
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\\
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B & LYP & aug-mcc-pV8Z\fnm[1] & & & & 28.42\fnm[2] \\
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B & LYP & aug-mcc-pV8Z & & & & 28.42 \\
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B3 & LYP & aug-mcc-pV8Z\fnm[1] & & & & 27.77\fnm[2] \\
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B3 & LYP & aug-mcc-pV8Z & & & & 27.77 \\
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HF & LYP & aug-mcc-pV8Z\fnm[1] & & & & 29.18\fnm[2] \\
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HF & LYP & aug-mcc-pV8Z & & & & 29.18 \\
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HF & & aug-mcc-pV8Z\fnm[1] & & & & 28.65\fnm[2] \\
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HF & & aug-mcc-pV8Z & & & & 28.65 \\
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\\
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\\
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HF & FCI & aug-mcc-pV8Z\fnm[1] & & & & 28.75\fnm[2] \\
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\hline
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\mc{5}{l}{Accurate (FCI/aug-mcc-pV8Z)\fnm[1]} & 28.75 \\
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\end{tabular}
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\end{tabular}
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\end{ruledtabular}
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\end{ruledtabular}
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\fnt[1]{Reference \onlinecite{Mielke_2005}.}
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\fnt[1]{Reference \onlinecite{Barca_2018a}.}
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\fnt[2]{Reference \onlinecite{Barca_2018a}.}
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\end{table*}
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\end{table*}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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%%% TABLE I %%%
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%%% TABLE I %%%
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\begin{table*}
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\begin{table}
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\caption{
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\caption{
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Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 3.7$ bohr for various methods, combinations of xc functionals, and basis sets.
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Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 3.7$ bohr obtained with the aug-cc-pVTZ basis set for various methods and combinations of xc functionals.
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\label{tab:BigTab_H2st}
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\label{tab:BigTab_H2st}
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}
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}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{llccccc}
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\begin{tabular}{llcccc}
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\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
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\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
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\cline{1-2} \cline{4-5}
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\cline{1-2} \cline{3-4}
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exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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exchange & correlation & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\hline
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\hline
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HF & & aug-cc-pVDZ & 19.08 & 6.58 & 12.92 & 6.52 \\
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HF & & 19.09 & 6.59 & 12.92 & 6.52 \\
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& & aug-cc-pVTZ & 19.09 & 6.59 & 12.92 & 6.52 \\
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S & & 5.31 & 5.60 & 5.46 & 5.56 \\
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\\
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S & VWN5 & 5.34 & 5.57 & 5.46 & 5.52 \\
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S & & aug-cc-pVDZ & 5.31 & 5.60 & 5.46 & 5.56 \\
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S & eVWN5 & 5.53 & 5.76 & 5.56 & 5.72 \\
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& & aug-cc-pVTZ & 5.31 & 5.60 & 5.46 & 5.56 \\
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GIC-S & & 5.55 & 5.56 & 5.56 & 5.56 \\
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\\
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GIC-S & VWN5 & 5.58 & 5.53 & 5.57 & 5.52 \\
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S & VWN5 & aug-cc-pVDZ & 5.34 & 5.57 & 5.46 & 5.53 \\
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GIC-S & eVWN5 & 5.77 & 5.72 & 5.66 & 5.72 \\
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& & aug-cc-pVTZ & 5.34 & 5.57 & 5.46 & 5.52 \\
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B & LYP & & & & 5.28 \\
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\\
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B3 & LYP & & & & 5.55 \\
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S & eVWN5 & aug-cc-pVDZ & 5.53 & 5.76 & 5.56 & 5.72 \\
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HF & LYP & & & & 6.68 \\
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& & aug-cc-pVTZ & 5.53 & 5.76 & 5.56 & 5.72 \\
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\hline
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& & aug-cc-pVQZ & & & & \\
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\mc{5}{l}{Accurate (FCI/aug-cc-pV5Z)\fnm[1]} & 8.69 \\
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\\
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GIC-S & & aug-cc-pVDZ & 5.56 & 5.56 & 5.56 & 5.56 \\
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& & aug-cc-pVTZ & 5.55 & 5.56 & 5.56 & 5.56 \\
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\\
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GIC-S & VWN5 & aug-cc-pVDZ & 5.59 & 5.53 & 5.57 & 5.53 \\
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& & aug-cc-pVTZ & 5.58 & 5.53 & 5.57 & 5.52 \\
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\\
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GIC-S & eVWN5 & aug-cc-pVDZ & 5.78 & 5.72 & 5.66 & 5.72 \\
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& & aug-cc-pVTZ & 5.77 & 5.72 & 5.66 & 5.72 \\
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\\
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B & LYP & aug-cc-pVTZ & & & & 5.28 \\
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B3 & LYP & aug-cc-pVTZ & & & & 5.55 \\
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HF & LYP & aug-cc-pVTZ & & & & 6.68 \\
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\\
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% HF & FCI & aug-cc-pVDZ & & & & 8.78 \\
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% HF & FCI & aug-cc-pVTZ & & & & 8.71 \\
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% HF & FCI & aug-cc-pVQZ & & & & 8.70 \\
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HF & FCI & aug-cc-pV5Z & & & & 8.69 \\
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\end{tabular}
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\end{tabular}
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\end{ruledtabular}
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\end{ruledtabular}
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\end{table*}
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\fnt[1]{FCI calculations performed with QUANTUM PACKAGE. \cite{QP2}}
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\end{table}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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%%% TABLE I %%%
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%%% TABLE I %%%
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\begin{table*}
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\begin{table}
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\caption{
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\caption{
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Excitation energies (in hartree) associated with the lowest double excitation of \ce{He} for various methods, combinations of xc functionals, and basis sets.
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Excitation energies (in hartree) associated with the lowest double excitation of \ce{He} obtained with the d-aug-cc-pVQZ basis set for various methods and combinations of xc functionals.
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\label{tab:BigTab_He}
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\label{tab:BigTab_He}
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}
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}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{llccccc}
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\begin{tabular}{llcccc}
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\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
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\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
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\cline{1-2} \cline{4-5}
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\cline{1-2} \cline{3-4}
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exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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exchange & correlation & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\hline
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\hline
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HF & & d-aug-cc-pVDZ & 1.897 & 2.209 & 2.078 & 2.143 \\
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HF & & 1.874 & 2.212 & 2.080 & 2.142 \\
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& & d-aug-cc-pVTZ & 1.874 & 2.213 & 2.080 & 2.143 \\
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S & & 1.062 & 2.056 & 1.547 & 2.030 \\
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& & d-aug-cc-pVQZ & 1.874 & 2.212 & 2.080 & 2.142 \\
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S & VWN5 & 1.163 & 2.104 & 1.612 & 2.079 \\
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\\
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S & eVWN5 & 1.174 & 2.108 & 1.615 & 2.083 \\
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S & & d-aug-cc-pVDZ & 1.075 & 2.055 & 1.546 & 2.030 \\
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GIC-S & & 1.996 & 2.044 & 1.988 & 2.030 \\
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& & d-aug-cc-pVTZ & 1.062 & 2.057 & 1.547 & 2.031 \\
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GIC-S & VWN5 & 2.107 & 2.097 & 2.060 & 2.079 \\
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& & d-aug-cc-pVQZ & 1.062 & 2.056 & 1.547 & 2.030 \\
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GIC-S & eVWN5 & 2.118 & 2.100 & 2.063 & 2.083 \\
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\\
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B & LYP & & & & 2.147 \\
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S & VWN5 & d-aug-cc-pVDZ & 1.172 & 2.102 & 1.612 & \\
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B3 & LYP & & & & 2.150 \\
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& & d-aug-cc-pVTZ & 1.163 & & & \\
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HF & LYP & & & & 2.171 \\
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& & d-aug-cc-pVQZ & & & & \\
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\hline
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\\
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\mc{5}{l}{Exact (explicitly-correlated method)\fnm[1] } & 2.126 \\
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S & eVWN5 & d-aug-cc-pVDZ & & & & \\
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& & d-aug-cc-pVTZ & & & & \\
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& & d-aug-cc-pVQZ & & & & \\
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\\
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GIC-S & & d-aug-cc-pVDZ & & & & \\
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& & d-aug-cc-pVTZ & & & & \\
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& & d-aug-cc-pVQZ & & & & \\
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\\
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GIC-S & VWN5 & d-aug-cc-pVDZ & & & & \\
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& & d-aug-cc-pVTZ & & & & \\
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& & d-aug-cc-pVQZ & & & & \\
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\\
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GIC-S & eVWN5 & d-aug-cc-pVDZ & & & & \\
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& & d-aug-cc-pVTZ & & & & \\
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& & d-aug-cc-pVQZ & & & & \\
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\\
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B & LYP & d-aug-cc-pVQZ & & & & \\
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B3 & LYP & d-aug-cc-pVQZ & & & & \\
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HF & LYP & d-aug-cc-pVQZ & & & & \\
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\\
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\mc{2}{l}{Exact} & & & & & 2.126\fnm[1] \\
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\end{tabular}
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\end{tabular}
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\end{ruledtabular}
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\end{ruledtabular}
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\fnt[1]{Reference \onlinecite{Burges_1995}}
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\fnt[1]{Reference \onlinecite{Burges_1995}.}
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\end{table*}
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\end{table}
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%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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%%% CONCLUSION %%%
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