\documentclass[aps,prb,reprint,noshowkeys,onecolumn,superscriptaddress]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts} \usepackage[version=4]{mhchem} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{txfonts} \usepackage[ colorlinks=true, citecolor=blue, breaklinks=true ]{hyperref} \urlstyle{same} \newcommand{\ie}{\textit{i.e.}} \newcommand{\eg}{\textit{e.g.}} \newcommand{\alert}[1]{\textcolor{red}{#1}} \definecolor{darkgreen}{HTML}{009900} \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\denis}[1]{\textcolor{purple}{#1}} \newcommand{\xavier}[1]{\textcolor{darkgreen}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashDJ}[1]{\textcolor{purple}{\sout{#1}}} \newcommand{\trashXB}[1]{\textcolor{darkgreen}{\sout{#1}}} \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} \renewcommand{\DJ}[1]{\denis{(\underline{\bf DJ}: #1)}} \newcommand{\XB}[1]{\xavier{(\underline{\bf XB}: #1)}} \newcommand{\mc}{\multicolumn} \newcommand{\fnm}{\footnotemark} \newcommand{\fnt}{\footnotetext} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\SI}{\textcolor{blue}{supporting information}} \newcommand{\QP}{\textsc{quantum package}} \newcommand{\T}[1]{#1^{\intercal}} % coordinates \newcommand{\br}[1]{\mathbf{r}_{#1}} \newcommand{\dbr}[1]{d\br{#1}} % methods \newcommand{\evGW}{ev$GW$} \newcommand{\qsGW}{qs$GW$} \newcommand{\GOWO}{$G_0W_0$} \newcommand{\Hxc}{\text{Hxc}} \newcommand{\xc}{\text{xc}} \newcommand{\Ha}{\text{H}} \newcommand{\co}{\text{x}} % \newcommand{\Norb}{N} \newcommand{\Nocc}{O} \newcommand{\Nvir}{V} \newcommand{\IS}{\lambda} % operators \newcommand{\hH}{\Hat{H}} % energies \newcommand{\Enuc}{E^\text{nuc}} \newcommand{\Ec}{E_\text{c}} \newcommand{\EHF}{E^\text{HF}} \newcommand{\EBSE}{E^\text{BSE}} \newcommand{\EcRPA}{E_\text{c}^\text{RPA}} \newcommand{\EcRPAx}{E_\text{c}^\text{RPAx}} \newcommand{\EcBSE}{E_\text{c}^\text{BSE}} \newcommand{\IP}{\text{IP}} \newcommand{\EA}{\text{EA}} % orbital energies \newcommand{\e}[1]{\epsilon_{#1}} \newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}} \newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}} \newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}} \newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}} \newcommand{\eGW}[1]{\epsilon^{GW}_{#1}} \newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}} \newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}} \newcommand{\Om}[2]{\Omega_{#1}^{#2}} % Matrix elements \newcommand{\A}[2]{A_{#1}^{#2}} \newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}} \newcommand{\B}[2]{B_{#1}^{#2}} \renewcommand{\S}[1]{S_{#1}} \newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}} \newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}} \newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}} \newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}} \newcommand{\ARPAx}[2]{A_{#1}^{#2,\text{RPAx}}} \newcommand{\BRPAx}[2]{B_{#1}^{#2,\text{RPAx}}} \newcommand{\G}[1]{G_{#1}} \newcommand{\LBSE}[1]{L_{#1}} \newcommand{\XiBSE}[1]{\Xi_{#1}} \newcommand{\Po}[1]{P_{#1}} \newcommand{\W}[2]{W_{#1}^{#2}} \newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\vc}[1]{v_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}} \newcommand{\Z}[1]{Z_{#1}} \newcommand{\MO}[1]{\phi_{#1}} \newcommand{\ERI}[2]{(#1|#2)} \newcommand{\sERI}[2]{[#1|#2]} %% bold in Table \newcommand{\bb}[1]{\textbf{#1}} \newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} \newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} % excitation energies \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}} \newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}} \newcommand{\OmBSE}[2]{\Omega_{#1}^{#2,\text{BSE}}} \newcommand{\spinup}{\downarrow} \newcommand{\spindw}{\uparrow} \newcommand{\singlet}{\uparrow\downarrow} \newcommand{\triplet}{\uparrow\uparrow} % Matrices \newcommand{\bO}{\mathbf{0}} \newcommand{\bI}{\mathbf{1}} \newcommand{\bvc}{\mathbf{v}} \newcommand{\bSig}{\mathbf{\Sigma}} \newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}} \newcommand{\bSigC}{\mathbf{\Sigma}^\text{c}} \newcommand{\bSigGW}{\mathbf{\Sigma}^{GW}} \newcommand{\be}{\mathbf{\epsilon}} \newcommand{\beGW}{\mathbf{\epsilon}^{GW}} \newcommand{\beGnWn}[1]{\mathbf{\epsilon}^\text{\GnWn{#1}}} \newcommand{\bde}{\mathbf{\Delta\epsilon}} \newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}} \newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}} \newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}} \newcommand{\bA}[1]{\mathbf{A}^{#1}} \newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}} \newcommand{\bB}[1]{\mathbf{B}^{#1}} \newcommand{\bX}[1]{\mathbf{X}^{#1}} \newcommand{\bY}[1]{\mathbf{Y}^{#1}} \newcommand{\bZ}[1]{\mathbf{Z}^{#1}} \newcommand{\bK}{\mathbf{K}} \newcommand{\bP}[1]{\mathbf{P}^{#1}} % units \newcommand{\IneV}[1]{#1 eV} \newcommand{\InAU}[1]{#1 a.u.} \newcommand{\InAA}[1]{#1 \AA} \newcommand{\kcal}{kcal/mol} \newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France} \newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\CEA}{Universit\'e Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France} \begin{document} \title{Supporting Information for ``Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Potential Energy Surfaces''} \author{Pierre-Fran\c{c}ois \surname{Loos}} \email{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \author{Anthony \surname{Scemama}} \email{scemama@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \author{Ivan \surname{Duchemin}} \email{ivan.duchemin@cea.fr} \affiliation{\CEA} \author{Denis \surname{Jacquemin}} \email{denis.jacquemin@univ-nantes.fr} \affiliation{\CEISAM} \author{Xavier \surname{Blase}} \email{xavier.blase@neel.cnrs.fr } \affiliation{\NEEL} \begin{abstract} \end{abstract} \maketitle %%% FIG 1 %%% \begin{figure*} % H2 \includegraphics[width=0.49\linewidth]{../Data/H2_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/H2_GS_VTZ} \caption{ Ground-state potential energy surfaces of \ce{H2} around its respective equilibrium geometry obtained at various levels of theory and basis sets. \label{fig:PES-H2} } \end{figure*} %%% %%% %%% %%% FIG 2 %%% \begin{figure*} % LiH \includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{LiH} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-LiH} } \end{figure*} %%% %%% %%% %%% FIG 3 %%% \begin{figure*} % LiF \includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{LiF} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-LiF} } \end{figure*} %%% %%% %%% %%% FIG 5 %%% \begin{figure*} % HCl \includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{HCl} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-HCl} } \end{figure*} %%% %%% %%% %%% FIG 6 %%% \begin{figure*} % N2 \includegraphics[width=0.49\linewidth]{../Data/N2_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/N2_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/N2_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/N2_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/N2_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{N2} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-N2} } \end{figure*} %%% %%% %%% %%% FIG 6 %%% \begin{figure*} % CO \includegraphics[width=0.49\linewidth]{../Data/CO_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/CO_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/CO_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/CO_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/CO_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{CO} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-CO} } \end{figure*} %%% %%% %%% %%% FIG 6 %%% \begin{figure*} % N2 \includegraphics[width=0.49\linewidth]{../Data/BF_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/BF_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/BF_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/BF_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/BF_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{BF} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-BF} } \end{figure*} %%% %%% %%% %%% FIG 6 %%% \begin{figure*} % N2 \includegraphics[width=0.49\linewidth]{../Data/F2_GS_VDZ} \includegraphics[width=0.49\linewidth]{../Data/F2_GS_VTZ} \includegraphics[width=0.49\linewidth]{../Data/F2_GS_VDZ_FC} \includegraphics[width=0.49\linewidth]{../Data/F2_GS_VTZ_FC} \includegraphics[width=0.49\linewidth]{../Data/F2_GS_VQZ_FC} \caption{ Ground-state potential energy surfaces of \ce{F2} around its respective equilibrium geometry obtained at various levels of theory and basis sets. FC stands for frozen core. \label{fig:PES-F2} } \end{figure*} %%% %%% %%% %%% %%% %%% %%% TABLE I %%% %\begin{table*} %\caption{ %Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. %All these values have been obtained within the frozen-core approximation. %The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively. %The values in parenthesis have been obtained by fitting a Morse potential to the PES. %} %\label{tab:Req-FC} % % \begin{ruledtabular} % \begin{tabular}{llcccccccc} % & & \mc{8}{c}{Molecules} \\ % \cline{3-10} % Method & Basis & \ce{H2} & \ce{LiH} & \ce{LiF} & \ce{HCl} & \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} \\ % \hline % CC3 & cc-pVDZ & 1.438 & 3.052 & 3.014 & 2.115 & 2.167 & 2.447 & 2.741 & 2.438 \\ % & cc-pVTZ & 1.403 & 3.036 & 2.985 & 2.087 & 2.150 & 2.405 & 2.672 & 2.414 \\ % & cc-pVQZ & 1.402 & 3.037 & 2.985 & 2.080 & 2.142 & 2.398 & 2.667 & 2.413 \\ % CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.101 & 2.149 & 2.435 & 2.695 & 2.433 \\ % & cc-pVTZ & 1.403 & 3.012 & 2.954 & 2.064 & 2.126 & 2.382 & 2.629 & 2.409 \\ % & cc-pVQZ & 1.402 & 3.020 & 2.953 & 2.059 & 2.118 & 2.380 & 2.621 & 2.398 \\ % CC2 & cc-pVDZ & 1.426 & & & & & & & \\ % & cc-pVTZ & 1.393 & & & & & & & \\ % & cc-pVQZ & 1.391 & & & & & & & \\ % MP2 & cc-pVDZ & 1.426 & 3.049 & 3.012 & 2.134 & 2.167 & 2.433 & 2.681 & 2.429 \\ % & cc-pVTZ & 1.393 & 3.026 & 2.990 & 2.104 & 2.151 & 2.395 & 2.640 & 2.407 \\ % & cc-pVQZ & 1.391 & 3.026 & 2.990 & 2.098 & 2.144 & 2.389 & 2.638 & 2.405 \\ % BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & & & & & & & \\ % & cc-pVTZ & 1.404 & & & & & & & \\ % & cc-pVQZ & 1.399 & & & & & & & \\ % RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & & & & & & & \\ % & cc-pVTZ & 1.388 & & & & & & & \\ % & cc-pVQZ & 1.382 & & & & & & & \\ % RPAx@HF & cc-pVDZ & 1.428 & & & & & & & \\ % & cc-pVTZ & 1.395 & & & & & & & \\ % & cc-pVQZ & 1.394 & & & & & & & \\ % RPA@HF & cc-pVDZ & 1.431 & & & & & & & \\ % & cc-pVTZ & 1.388 & & & & & & & \\ % & cc-pVQZ & 1.386 & & & & & & & \\ % \end{tabular} % \end{ruledtabular} %\end{table*} \bibliography{../BSE-PES,../BSE-PES-control} \end{document}