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BSE-PES.tex
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BSE-PES.tex
@ -107,6 +107,7 @@
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%% bold in Table
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%% bold in Table
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\newcommand{\bb}[1]{\textbf{#1}}
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\newcommand{\bb}[1]{\textbf{#1}}
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\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
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% excitation energies
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% excitation energies
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\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
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\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
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@ -431,47 +432,47 @@ However, we are currently pursuing different avenues to lower this cost by compu
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%\section{Potential energy surfaces}
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%\section{Potential energy surfaces}
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%\label{sec:PES}
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%\label{sec:PES}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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{N2}, \ce{CO}, \ce{BF}, \ce{F2}, and \ce{HCl}.
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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 and Dunning's triple-$\zeta$ basis cc-pVTZ 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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The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ 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 can be found in the {\SI}.
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Additional graphs for other basis sets can be found in the {\SI}.
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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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Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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Equilibrium distances (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@$G_0W_0$@HF data are highlighted in bold for visual convenience.}
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The reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in black and red bold for visual convenience, respectively.}
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\label{tab:Req}
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\label{tab:Req}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{llcccccccc}
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\begin{tabular}{llcccccccc}
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& & \mc{8}{c}{Molecules} \\
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& & \mc{8}{c}{Molecules} \\
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\cline{3-10}
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\cline{3-10}
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Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
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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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\hline
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CC3 & cc-pVDZ & 1.438 & 3.043 & 3.012 & 2.114 & 2.166 & 2.444 & 2.740 & 2.435 \\
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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 & \bb{3.011}& 2.961 & 2.079 & 2.143 & 2.392 & 2.669 & 2.413 \\
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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}& 3.019 & 2.963 & 2.075 & 2.136 & 2.390 & 2.663 & 2.403 \\
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& cc-pVQZ &\bb{1.402} &\bb{3.019} & 2.963 & 2.403 & 2.075 & 2.136 & 2.390 & 2.663 \\
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CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.101 & 2.149 & 2.435 & 2.695 & 2.433 \\
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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 & 2.064 & 2.126 & 2.382 & 2.629 & 2.409 \\
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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 & 2.059 & 2.118 & 2.118 & 2.621 & 2.398 \\
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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 & 2.146 & 2.187 & 2.444 & 2.710 & 2.427 \\
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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 & 2.109 & 2.163 & 2.394 & 2.664 & 2.406 \\
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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 & 2.106 & 2.156 & 2.393 & 2.665 & 2.396 \\
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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 & 2.133 & 2.166 & 2.431 & 2.681 & 2.426 \\
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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 & 2.095 & 2.144 & 2.383 & 2.636 & 2.405 \\
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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 & 2.091 & 2.137 & 2.382 & 2.634 & 2.395 \\
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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 & 2.107 & 2.153 & 2.407 & 2.700 & >2.440 \\
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BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & 2.700 \\
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& cc-pVTZ & 1.404 & \bb{3.023}& glitch & & & <2.420 & & <2.410 \\
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& cc-pVTZ & 1.404 & 3.023 & glitch & <2.410 & 2.068 & & <2.420 & \\
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& cc-pVQZ & \bb{1.399}& & & & & & & \\
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& cc-pVQZ &\rb{1.399} &\rb{3.017} & & & & & & \\
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RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.083 & 2.144 & 2.403 & 2.691 & 2.436 \\
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RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & 2.691 \\
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& cc-pVTZ & 1.388 & 3.013 & glitch & & & <2.420 & & <2.410 \\
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& cc-pVTZ & 1.388 & 3.013 & glitch & <2.410 & 2.065 & & <2.420 & \\
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& cc-pVQZ & 1.382 & & & & & & & \\
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& cc-pVQZ & 1.382 & 3.013 & & & & & & \\
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RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.077 & 2.130 & 2.417 & NaN & 2.424 \\
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RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & NaN \\
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& cc-pVTZ & 1.395 & 3.003 & <2.990 & & & <2.420 & & <2.410 \\
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& cc-pVTZ & 1.395 & 3.003 & <2.990 & <2.410 & <2.060 & & <2.420 & \\
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& cc-pVQZ & 1.394 & & & & & & & \\
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& cc-pVQZ & 1.394 & 3.011 & & & & & & \\
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RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.083 & 2.134 & & 2.623 & 2.424 \\
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RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & & 2.623 \\
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& cc-pVTZ & 1.388 & 2.978 & <2.990 & & & 2.416 & & <2.410 \\
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& cc-pVTZ & 1.388 & 2.978 & <2.990 & <2.410 & <2.060 & & 2.416 & \\
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& cc-pVQZ & 1.386 & & & & & <2.420 & & \\
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& cc-pVQZ & 1.386 & 2.994 & & & & & <2.420 & \\
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% FROZEN CORE VERSION
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% FROZEN CORE VERSION
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% Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
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% Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
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% \hline
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% \hline
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@ -504,15 +505,16 @@ The reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in bold
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\end{ruledtabular}
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\end{ruledtabular}
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\end{table*}
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\end{table*}
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Let us first start with the two smallest molecules, \ce{H2} and \ce{LiH} which are both linked by covalent bonds (see Fig.~\ref{fig:PES-H2-LiH}).
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Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds (see Fig.~\ref{fig:PES-H2-LiH}).
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For \ce{H2}, we take as reference the full configuration interaction (FCI) energies and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
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For \ce{H2}, we take as reference the full configuration interaction (FCI) energies \cite{QP2} and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
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RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly overestimate (in absolute value) the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly underestimate and overestimate the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
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RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over and undershoot the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
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Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
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Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
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For example, with the cc-pVQZ basis set, BSE@{\GOWO}@HF is only off by $0.003$ bohr compared to FCI, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
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For example, with the cc-pVQZ basis set, BSE@{\GOWO}@HF is only off by $0.003$ bohr compared to FCI, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
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The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
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The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
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The scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
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The scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
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In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, and they are almost perfectly parallel.
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In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, and they are almost perfectly parallel.
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Here again, the BSE@{\GOWO}@HF equilibrium bond length is extremely accurate ($3.017$ bohr) as compared to FCI ($3.019$ bohr).
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%%% FIG 1 %%%
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%%% FIG 1 %%%
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\begin{figure*}
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\begin{figure*}
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@ -534,8 +536,8 @@ However, in the case of \ce{LiF}, the attentive reader would have observed a sma
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%%% FIG 2 %%%
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%%% FIG 2 %%%
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\begin{figure*}
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\begin{figure*}
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\includegraphics[width=0.49\linewidth]{LiF_GS_VTZ}
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\includegraphics[height=0.35\linewidth]{LiF_GS_VTZ}
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\includegraphics[width=0.49\linewidth]{HCl_GS_VTZ}
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\includegraphics[height=0.35\linewidth]{HCl_GS_VTZ}
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\caption{
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\caption{
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Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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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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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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@ -550,9 +552,9 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding as shown i
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%%% FIG 3 %%%
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%%% FIG 3 %%%
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\begin{figure*}
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\begin{figure*}
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\includegraphics[width=0.33\linewidth]{N2_GS_VTZ}
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\includegraphics[height=0.26\linewidth]{N2_GS_VTZ}
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\includegraphics[width=0.33\linewidth]{CO_GS_VTZ}
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\includegraphics[height=0.26\linewidth]{CO_GS_VTZ}
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\includegraphics[width=0.33\linewidth]{BF_GS_VTZ}
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\includegraphics[height=0.26\linewidth]{BF_GS_VTZ}
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\caption{
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\caption{
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Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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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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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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