419 lines
18 KiB
TeX
419 lines
18 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,wrapfig}
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\usepackage{natbib}
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\usepackage[extra]{tipa}
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\bibliographystyle{achemso}
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\AtBeginDocument{\nocite{achemso-control}}
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\usepackage{mathpazo,libertine}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\jt}[1]{\textcolor{purple}{#1}}
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\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\toto}[1]{\textcolor{brown}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
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\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
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\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}}
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\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
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\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\AS}[1]{\toto{(\underline{\bf TOTO}: #1)}}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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% methods
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\newcommand{\evGW}{evGW}
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\newcommand{\qsGW}{qsGW}
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\newcommand{\GOWO}{G$_0$W$_0$}
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\newcommand{\GW}{GW}
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\newcommand{\GnWn}[1]{G$_{#1}$W$_{#1}$}
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% operators
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\newcommand{\hH}{\Hat{H}}
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% energies
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\EKS}{E_\text{KS}}
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\newcommand{\EcK}{E_\text{c}^\text{Klein}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcGM}{E_\text{c}^\text{GM}}
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\newcommand{\EcMP}{E_c^\text{MP2}}
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\newcommand{\Egap}{E_\text{gap}}
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\newcommand{\IP}{\text{IP}}
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\newcommand{\EA}{\text{EA}}
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\newcommand{\RH}{R_{\ce{H2}}}
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\newcommand{\RF}{R_{\ce{F2}}}
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\newcommand{\RBeO}{R_{\ce{BeO}}}
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% orbital energies
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\newcommand{\nDIIS}{N^\text{DIIS}}
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\newcommand{\maxDIIS}{N_\text{max}^\text{DIIS}}
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\newcommand{\nSat}[1]{N_{#1}^\text{sat}}
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\newcommand{\eSat}[2]{\epsilon_{#1,#2}}
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\teHF}[1]{\Tilde{\epsilon}^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\de}[1]{\Delta\epsilon_{#1}}
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\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
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\newcommand{\Om}[1]{\Omega_{#1}}
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\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
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\newcommand{\eLUMO}{\epsilon_\text{LUMO}}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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% Matrix elements
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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\newcommand{\tA}{\Tilde{A}}
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\newcommand{\tB}{\Tilde{B}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[1]{W_{#1}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
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\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
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\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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% Matrices
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bvc}{\boldsymbol{v}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}}
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\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}}
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\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bDelta}{\boldsymbol{\Delta}}
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\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}}
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\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}}
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\newcommand{\beGnWn}[1]{\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bdeGnWn}[1]{\Delta\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bde}{\boldsymbol{\Delta\epsilon}}
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\newcommand{\bdeHF}{\boldsymbol{\Delta\epsilon}^\text{HF}}
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\newcommand{\bdeGW}{\boldsymbol{\Delta\epsilon}^\text{GW}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bA}{\boldsymbol{A}}
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\newcommand{\bB}{\boldsymbol{B}}
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\newcommand{\bX}{\boldsymbol{X}}
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\bZ}{\boldsymbol{Z}}
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\newcommand{\fc}{f_\text{c}}
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\newcommand{\Vc}{V_\text{c}}
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\newcommand{\MO}[1]{\phi_{#1}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\renewcommand{\b}[1]{\mathbf{#1}}
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\renewcommand{\d}{\text{d}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}}
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\renewcommand{\ket}[1]{\ensuremath{\vert #1 \rangle}}
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\renewcommand{\braket}[2]{\ensuremath{\langle #1 \vert #2 \rangle}}
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
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\begin{document}
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\title{A Density-Based Basis Set Correction for GW Methods}
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\author{Bath\'elemy Pradines}
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\affiliation{\LCT}
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\affiliation{\ISCD}
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\author{Emmanuel Giner}
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\affiliation{\LCT}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Julien Toulouse}
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\affiliation{\LCT}
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\author{Pierre-Fran\c{c}ois Loos}
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\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
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% \centering
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% \includegraphics[width=\linewidth]{TOC}
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%\end{wrapfigure}
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Many-body Green-function theory with DFT basis-set correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $N$-electron system with nuclei-electron potential $v_\text{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ which are ``representable'' in a finite basis set ${\cal B}$
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\begin{equation}
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E_0^{\cal B} = \min_{n \in {\cal D}^{\cal B}} \left\{ F[n] + \int v_\text{ne}(\b{r}) n(\b{r}) \d\b{r}\right\},
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\label{E0B}
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\end{equation}
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where ${\cal D}^{\cal B}$ is the set of $N$-representable densities which can be extracted from a wave function $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$. In this expression, $F[n]=\min_{\Psi\to n} \bra{\Psi} \hat{T} + \hat{W}_\text{ee}\ket{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hat{T}$ and $\hat{W}_\text{ee}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
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\begin{equation}
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F[n] = F^{\cal B}[n] + \bar{E}^{\cal B}[n],
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\label{Fn}
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\end{equation}
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where $F^{\cal B}[n]$ is the Levy-Lieb density functional with wave functions $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$
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\begin{equation}
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F^{\cal B}[n] = \min_{\Psi^{\cal B}\to n} \bra{\Psi^{\cal B}} \hat{T} + \hat{W}_\text{ee}\ket{\Psi^{\cal B}},
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\end{equation}
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and $\bar{E}^{\cal B}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^{\cal B}[n]$, we reexpress it with a contrained search over $N$-representable one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
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\begin{equation}
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F^{\cal B}[n] = \min_{G^{\cal B}\to n} \Omega^{\cal B}[G^{\cal B}],
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\label{FBn}
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\end{equation}
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where $\Omega^{\cal B}[G]$ is chosen to be a Klein-like energy functional of the Green function (see, e.g., Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
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\begin{equation}
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\Omega^{\cal B}[G] = \Tr \left[\ln ( - G ) \right] - \Tr \left[ (G_\text{s}^{\cal B})^{-1} G -1 \right] + \Phi_\text{Hxc}^{\cal B}[G],
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\label{OmegaB}
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\end{equation}
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where $(G_\text{s}^{\cal B})^{-1}$ is the projection into ${\cal B}$ of the inverse free-particle Green function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^{\cal B}[G]$ is a Hartree-exchange-correlation (Hxc) functional of the Green functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\text{Hxc}^{\cal B}[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^{\cal B}[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
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\begin{equation}
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E_0^{\cal B} = \min_{G^{\cal B}} \left\{ \Omega^{\cal B}[G^{\cal B}] + \int v_\text{ne}(\b{r}) n_{G^{\cal B}}(\b{r}) \d\b{r} + \bar{E}^{\cal B}[n_{G^{\cal B}}] \right\},
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\label{E0BGB}
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\end{equation}
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where the minimization is over $N$-representable one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$.
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The stationary condition from Eq.~(\ref{E0BGB}) gives the following Dyson equation
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\begin{equation}
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(G^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1}- \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
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\label{Dyson}
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\end{equation}
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where $(G_\text{0}^{\cal B})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $v_\text{ne}(\b{r})$,
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$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^{\cal B}$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
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\begin{equation}
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\bar{\Sigma}^{\cal B}[n](\b{r},\b{r}') = \bar{v}^{\cal B}[n](\b{r}) \delta(\b{r}-\b{r}'),
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\end{equation}
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with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
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%From Julien:
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%\begin{equation}
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%\fdv{E[n_G]}{G(r,r',\omega)} = \int \fdv{E[n_G]}{n(r'')}] \fdv{n_G(r'')}{G(r,r',w)} dr''
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%\end{equation}
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%
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%\begin{equation}
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%n_G(r'') = i \int G(r'',r'',w) d\omega
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%\end{equation}
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%
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%
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%\begin{equation}
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%\fdv{n_G(r'')}{G(r,r',w)} = \delta(r -r') \delta (r'-r'')
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%\end{equation}
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%
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%
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%\begin{equation}
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%\begin{split}
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% \fdv{E[n_G]}{G(r,r',w)}
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% & = \int \fdv{E[n_G]}{n(r'')} \delta(r -r') \delta (r'-r'') dr''
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% \\
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% & = \fdv{E[n_G]}{n(r)} \delta(r -r')
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% \\
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% & = v[n_G](r) \delta(r -r')
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%\end{split}
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%\end{equation}
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\subsection{The GW Approximation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The Dyson equation can be written with an arbitrary reference
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\begin{equation}
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(G^{\cal B})^{-1} = (G_\text{ref}^{\cal B})^{-1}- \left( \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \Sigma_\text{ref}^{\cal B} \right) - \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
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\end{equation}
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where $(G_\text{ref}^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1} - \Sigma_\text{ref}^{\cal B}$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^{\cal B}(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = v_\text{Hxc}^{\cal B}(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.
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Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
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More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
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For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy is conveniently split in its hole (h) and particle (p) contributions
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\begin{equation}
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\label{eq:SigC}
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\SigC{p}(\omega) = \SigCp{p}(\omega) + \SigCh{p}(\omega),
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\end{equation}
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which, within the GW approximation, read
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\begin{subequations}
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\begin{align}
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\label{eq:SigCh}
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\SigCh{p}(\omega)
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& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta},
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\\
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\label{eq:SigCp}
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\SigCp{p}(\omega)
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& = 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
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\end{align}
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\end{subequations}
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where $\eta$ is a positive infinitesimal.
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The screened two-electron integrals
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\begin{equation}
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[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x}
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\end{equation}
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are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} $(pq|rs)$ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
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\begin{equation}
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\label{eq:LR}
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\begin{pmatrix}
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\bA & \bB \\
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\bB & \bA \\
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\end{pmatrix}
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\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix}
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=
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\bOm
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\begin{pmatrix}
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\boldsymbol{1} & \boldsymbol{0} \\
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\boldsymbol{0} & \boldsymbol{-1} \\
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\end{pmatrix}
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\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix},
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\end{equation}
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with
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\begin{align}
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\label{eq:RPA}
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A_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb),
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&
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B_{ia,jb} & = 2 (ia|bj),
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\end{align}
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and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
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The one-electron energies $\epsilon_p$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} are either the HF or the GW quasiparticle energies.
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Equation \eqref{eq:LR} also provides the neutral excitation energies $\Om{x}$.
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In practice, there exist two ways of determining the {\GOWO} QP energies. \cite{Hybertsen_1985a, vanSetten_2013}
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In its ``graphical'' version, they are provided by one of the many solutions of the (non-linear) QP equation
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\begin{equation}
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\label{eq:QP-G0W0}
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\omega = \eHF{p} + \Re[\SigC{p}(\omega)].
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\end{equation}
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In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution.
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In particular, it is usually worth calculating its renormalization weight (or factor), $\Z{p}(\eHF{p})$, where
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\begin{equation}
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\label{eq:Z}
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\Z{p}(\omega) = \qty[ 1 - \pdv{\Re[\SigC{p}(\omega)]}{\omega} ]^{-1}.
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\end{equation}
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Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
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In a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.
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Within the linearized version of {\GOWO}, one assumes that
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\begin{equation}
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\label{eq:SigC-lin}
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\SigC{p}(\omega) \approx \SigC{p}(\eHF{p}) + (\omega - \eHF{p}) \left. \pdv{\SigC{p}(\omega)}{\omega} \right|_{\omega = \eHF{p}},
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\end{equation}
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that is, the self-energy behaves linearly in the vicinity of $\omega = \eHF{p}$.
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Substituting \eqref{eq:SigC-lin} into \eqref{eq:QP-G0W0} yields
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\begin{equation}
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\label{eq:QP-G0W0-lin}
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\eGOWO{p} = \eHF{p} + \Z{p}(\eHF{p}) \Re[\SigC{p}(\eHF{p})].
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\end{equation}
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Unless otherwise stated, in the remaining of this paper, the {\GOWO} QP energies are determined via the linearized method.
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In the case of {\evGW}, the QP energy, $\eGW{p}$, are obtained via Eq.~\eqref{eq:QP-G0W0}, which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref{eq:SigC}]. \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
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At least in the weakly correlated regime where a clear QP solution exists, we believe that, within {\evGW}, the self-consistent algorithm should select the solution of the QP equation \eqref{eq:QP-G0W0} with the largest renormalization weight $\Z{p}(\eGW{p})$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Basis Set Correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The present basis set correction is a two-level correction.
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First, one has to correct the neutral excitations $\Om{x}$ from the RPA calculation.
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The corrected matrix elements read
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\begin{align}
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\label{eq:RPA}
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\tA_{ia,jb} & = \A{ia,jb} + (ia|\fc|jb),
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&
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\tB_{ia,jb} & = \B{ia,jb} + (ia|\fc|bj),
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\end{align}
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where the elements $\A{ia,jb}$ and $\B{ia,jb}$ are given by Eq.~\eqref{eq:RPA}.
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\begin{equation}
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\fc(\br{1},\br{2})= \frac{\delta^2 \Ec}{\delta n(\br{1})\delta n(\br{2})}
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\end{equation}
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In a second time, we correct the GW energy
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\begin{equation}
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\tSigC{p} = \SigC{p} + (p|\Vc|p)
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\end{equation}
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with
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\begin{equation}
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\Vc(\br{}) = \fdv{\Ec}{n(\br{})}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdetails}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and Discussion}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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\label{sec:ccl}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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%\section*{Supporting Information Available}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), CALMIP (Toulouse) under allocation 2019-18005 and the Jarvis-Alpha cluster from the \textit{Institut Parisien de Chimie Physique et Th\'eorique}.
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{GW-srDFT,GW-srDFT-control,biblio}
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\end{document}
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