122 lines
4.5 KiB
TeX
122 lines
4.5 KiB
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=0.8\linewidth]{h2_kappa_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 $\kappa = 1$.
|
|
}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\includegraphics[width=0.8\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 $\eta = 1$.
|
|
}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\includegraphics[width=0.8\linewidth]{h2_eta_100_meV}
|
|
\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 $\eta = \SI{100}{\milli\eV}$.
|
|
}
|
|
\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 $\eta = 0.1$ (left), $\eta = 1$ (center), and $\eta = 10$ (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 $\kappa = 0.1$ (left), $\kappa = 1$ (center), and $\kappa = 10$ (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 $\kappa = 0.1$ (left), $\eta = 1$ (center), and $\eta = 10$ (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 $\kappa = 0.1$ (left), $\kappa = 1$ (center), and $\kappa = 10$ (right).
|
|
For $\kappa = 0.1$, 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 $\kappa = 10$, the black and gray curves are superposed.}
|
|
\end{figure}
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\bibliography{ufGW}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\end{document} |