ufGW/Manuscript/ufGW-SI.tex

\documentclass[aip,jcp,reprint,onecolumn,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx}
\usepackage[version=4]{mhchem}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage{siunitx}
\usepackage{soul}
\DeclareSIUnit[number-unit-product = {\,}]
\cal{cal}
\DeclareSIUnit\kcal{\kilo\cal}
\newcommand{\kcalmol}{\si{\kcal\per\mole}}

\usepackage[
	colorlinks=true,
    citecolor=blue,
    breaklinks=true
	]{hyperref}
\urlstyle{same}

\usepackage[normalem]{ulem}

% methods
\newcommand{\GW}{\text{$GW$}}	
\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{c}}
\newcommand{\x}{\text{x}}
\newcommand{\KS}{\text{KS}}
\newcommand{\HF}{\text{HF}}
\newcommand{\RPA}{\text{RPA}}


% orbital energies
\newcommand{\eps}[2]{\epsilon_{#1}^{#2}}
\newcommand{\reps}[2]{\Tilde{\epsilon}_{#1}^{#2}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}

\newcommand{\RHH}{R_{\ce{H-H}}}

% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}

\begin{document}

\title{Supplementary Material for ``Unphysical Discontinuities, Intruder States and Regularization in $GW$ Methods''}


\author{Enzo \surname{Monino}}
	\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\maketitle

\begin{figure}
	\includegraphics[width=0.6\linewidth]{h2_eta_100_meV}
	\includegraphics[width=0.6\linewidth]{h2_kappa_1}
	\includegraphics[width=0.6\linewidth]{h2_eta_1}
	\caption{
	Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = \SI{100}{\milli\eV}$ (top), $\eta = \SI{1}{\hartree}$ (center), and $\kappa = \SI{1}{\hartree}$ (bottom).
}
\end{figure}

\begin{figure}
	\includegraphics[width=0.33\linewidth]{eta_0_1}
	\includegraphics[width=0.33\linewidth]{eta_1}
	\includegraphics[width=0.33\linewidth]{eta_10}
	\caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = \SI{0.1}{\hartree}$ (left), $\eta = \SI{1}{\hartree}$ (center), and  $\eta = \SI{10}{\hartree}$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. }
\end{figure}		

\begin{figure}
	\includegraphics[width=0.33\linewidth]{kappa_0_1}
	\includegraphics[width=0.33\linewidth]{kappa_1}
	\includegraphics[width=0.33\linewidth]{kappa_10}
	\caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with computed with $\kappa = \SI{0.1}{\hartree}$ (left), $\kappa = \SI{1}{\hartree}$ (center), and  $\kappa = \SI{10}{\hartree}$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. }
\end{figure}	

\begin{figure}
	\includegraphics[width=0.3\linewidth]{f2_eta_0_1}
	\hspace{0.03\textwidth}
	\includegraphics[width=0.3\linewidth]{f2_eta_1}
	\hspace{0.03\textwidth}
	\includegraphics[width=0.3\linewidth]{f2_eta_10}
	\caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = \SI{0.1}{\hartree}$ (left), $\eta = \SI{1}{\hartree}$ (center), and $\eta = \SI{10}{\hartree}$ (right).}
\end{figure}		

\begin{figure}
	\includegraphics[width=0.3\linewidth]{f2_kappa_0_1}
	\hspace{0.03\textwidth}
	\includegraphics[width=0.3\linewidth]{f2_kappa_1}
	\hspace{0.03\textwidth}
	\includegraphics[width=0.3\linewidth]{f2_kappa_10}
	\caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = \SI{0.1}{\hartree}$ (left), $\kappa = \SI{1}{\hartree}$ (center), and $\kappa = \SI{10}{\hartree}$ (right).
	For $\kappa = \SI{0.1}{\hartree}$, the BSE@ev$GW$@HF calculations do not converge for numerous values of $R_{\ce{F-F}}$ and are not shown in this figure.
	For $\kappa = \SI{10}{\hartree}$, the black and gray curves are superposed.}
\end{figure}	
	


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\bibliography{ufGW}
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\end{document}