\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress,onecolumn]{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[
	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}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}

\newcommand{\mc}{\multicolumn}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}

% coordinates
\newcommand{\br}{\boldsymbol{r}}
\newcommand{\bx}{\boldsymbol{x}}
\newcommand{\dbr}{d\br}
\newcommand{\dbx}{d\bx}

% 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}}

% 
\newcommand{\Ne}{N}
\newcommand{\Norb}{K}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}

% operators
\newcommand{\hH}{\Hat{H}}
\newcommand{\hS}{\Hat{S}}

% energies
\newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}[1]{E_\text{c}^{#1}}
\newcommand{\EHF}{E^\text{HF}}

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

% Matrix elements
\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
\newcommand{\rSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
\newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\SO}[1]{\psi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\rbra}[1]{(#1|}
\newcommand{\rket}[1]{|#1)}


% Matrices
\newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bvc}{\boldsymbol{v}}
\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bB}[2]{\boldsymbol{B}_{#1}^{#2}}
\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
\newcommand{\bD}[2]{\boldsymbol{D}_{#1}^{#2}}
\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
\newcommand{\bW}[2]{\boldsymbol{W}_{#1}^{#2}}
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\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}}
\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
\newcommand{\bc}[2]{\boldsymbol{c}_{#1}^{#2}}

% orbitals, gaps, etc
\newcommand{\IP}{I}
\newcommand{\EA}{A}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
\newcommand{\Eg}{E_\text{g}}
\newcommand{\EgFun}{\Eg^\text{fund}}
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}

\newcommand{\RHH}{R_{\ce{H-H}}}
\newcommand{\ii}{\mathrm{i}}

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

\begin{document}	

\title{Undressing $GW$ one determinant at a time}

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

\begin{abstract}
Here comes the abstract.
%\bigskip
%\begin{center}
%	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
Here comes the introduction.

%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%
In the case of {\GOWO}, the quasiparticle equation reads
\begin{equation}
\label{eq:qp_eq}
	\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
\end{equation}
where $\eps{p}{\HF}$ is the HF one-electron energy of the spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is
\begin{equation}
\label{eq:SigC}
	\SigC{p}(\omega) 
	= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}}
	+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
\end{equation}
where 
\begin{equation}
	\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
\end{equation}
are the screened two-electron repulsion integrals where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system:
\begin{equation}
	\bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA}
\end{equation}
with
\begin{equation}
	A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
\end{equation}
and
\begin{equation}
	\ERI{pq}{ia} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{i}(\br_2) \MO{a}(\br_2) d\br_1 \dbr_2
\end{equation}

The spectral weight of the solution $\eps{p,s}{\GW}$ is given by
\begin{equation}
\label{eq:Z}
	0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
\end{equation} 
with the following sum rules:
\begin{align}
	\sum_{s} Z_{p,s} & = 1
	&
	\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
\end{align}

As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
\begin{equation}
	\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
\end{equation}
with
\begin{equation}
\label{eq:Hp}
	\bH^{(p)} = 
	\begin{pmatrix}
		\eps{p}{\HF}		&	\bV{p}{\text{2h1p}}	&	\bV{p}{\text{2p1h}}
		\\
		\T{(\bV{p}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}			&	\bO
		\\
		\T{(\bV{p}{\text{2p1h}})}	&	\bO				&	\bC{}{\text{2p1h}}	
	\end{pmatrix}
\end{equation}
and where the expressions of the 2h1p and 2p1h blocks reads
\begin{align}
	C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps{I}{\HF} + \eps{J}{\HF} - \eps{A}{\HF}) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK} 
	\\
	C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps{A}{\HF} + \eps{B}{\HF} - \eps{I}{\HF}) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD} 
\end{align}
with the following expressions for the coupling blocks:
\begin{align}
	V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
	&
	V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
\end{align}
By solving the secular equation
\begin{equation}
	\det[ \bH^{(p)} - \omega \bI ] = 0 
\end{equation}
we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie, 
\begin{equation}
	\SigC{p}(\omega) 
	= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} 
	+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} 
\end{equation}
with
\begin{equation}
\label{eq:Z_proj}
	Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
\end{equation}

In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$)
\begin{equation}
\label{eq:Hp_qia}
	\bH^{(p,qia)} = 
	\begin{pmatrix}
		\eps{p}{\HF}				&	V_{p,qia}					&	\bV{p}{\text{2h1p}}		&	\bV{p}{\text{2p1h}}
		\\
		V_{qia,p}					&	C_{qia,qia}					&	\bC{qia}{\text{2h1p}}	&	\bC{qia}{\text{2p1h}}
		\\
		\T{(\bV{p}{\text{2h1p}})}	&	\T{(\bC{qia}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}		&	\bO
		\\
		\T{(\bV{p}{\text{2p1h}})}	&	\T{(\bC{qia}{\text{2p1h}})}	&	\bO						&	\bC{}{\text{2p1h}}	
	\end{pmatrix}
\end{equation}
with new blocks defined as
\begin{gather}
	V_{p,qia} = V_{qia,p} \sqrt{2} = \ERI{pq}{ia}
	\\
	C_{qia,qia} = \text{sgn}(\eps{q}{\HF} - \mu) \qty[ \qty(\eps{q}{\HF} + \eps{a}{\HF} - \eps{i}{\HF} ) + 2 \ERI{ia}{ia} ]
	\\
	C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
	\\
	C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
\end{gather}

Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy
\begin{equation}
\label{eq:Hp}
	\bSigC{p,qia}(\omega)  = 
	\begin{pmatrix}
		\eps{p}{\HF} + \SigC{p}(\omega)				&	V_{p,qia} + \SigC{p,qia}(\omega)		
		\\
		V_{qia,p} + \SigC{qia,p}(\omega)			&	C_{qia,qia}	+ \SigC{qia}(\omega)				
		\\
	\end{pmatrix}
\end{equation}
with
\begin{gather}
	\SigC{p}(\omega) 
	  = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} 
	  + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} 
	\\
	\SigC{qia}(\omega) 
	  = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} 
	  + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} 	
	\\
	\SigC{p,qia}(\omega) 
	  = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} 
	  + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} 
	\\
	\SigC{qia,p}(\omega) 
	  = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} 
	  + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} 
\end{gather}
Of course, the present procedure can be generalized to any number of states.

%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%%
Here comes the conclusion.
d
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article.% and its supplementary material.

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\bibliography{MRGW}
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\end{document}
