tex
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Pierre-Francois Loos
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Manuscript/ufGW.tex
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222
Manuscript/ufGW.tex
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{txfonts}
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\usepackage[
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colorlinks=true,
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citecolor=blue,
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breaklinks=true
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]{hyperref}
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\urlstyle{same}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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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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\newcommand{\T}[1]{#1^{\intercal}}
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% coordinates
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bx}{\boldsymbol{x}}
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\newcommand{\dbr}{d\br}
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\newcommand{\dbx}{d\bx}
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% methods
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\newcommand{\GW}{\text{$GW$}}
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\Ha}{\text{H}}
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\newcommand{\co}{\text{c}}
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\newcommand{\x}{\text{x}}
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\newcommand{\KS}{\text{KS}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\ppRPA}{\text{pp-RPA}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\dBSE}{\text{dBSE}}
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\newcommand{\stat}{\text{stat}}
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\newcommand{\dyn}{\text{dyn}}
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\newcommand{\TDA}{\text{TDA}}
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%
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\newcommand{\Norb}{N}
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\newcommand{\Nocc}{O}
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\newcommand{\Nvir}{V}
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% operators
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hS}{\Hat{S}}
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% energies
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\newcommand{\Enuc}{E^\text{nuc}}
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\newcommand{\Ec}[1]{E_\text{c}^{#1}}
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\newcommand{\EHF}{E^\text{HF}}
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% orbital energies
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\newcommand{\eps}[2]{\epsilon_{#1}^{#2}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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% Matrix elements
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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\SO}[1]{\psi_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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\newcommand{\rbra}[1]{(#1|}
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\newcommand{\rket}[1]{|#1)}
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%% bold in Table
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\newcommand{\bb}[1]{\textbf{#1}}
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\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
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\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
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% Matrices
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bvc}{\boldsymbol{v}}
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\newcommand{\bSig}[1]{\boldsymbol{\Sigma}^{#1}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bOm}[1]{\boldsymbol{\Omega}^{#1}}
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\newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}}
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\newcommand{\bB}[2]{\boldsymbol{B}_{#1}^{#2}}
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\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
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\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
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\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}}
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\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}}
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\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
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% orbitals, gaps, etc
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\newcommand{\IP}{I}
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\newcommand{\EA}{A}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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\newcommand{\Eg}{E_\text{g}}
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\newcommand{\EgFun}{\Eg^\text{fund}}
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\newcommand{\EgOpt}{\Eg^\text{opt}}
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\newcommand{\EB}{E_B}
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\newcommand{\sig}{\sigma}
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\newcommand{\bsig}{{\Bar{\sigma}}}
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\newcommand{\sigp}{{\sigma'}}
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\newcommand{\bsigp}{{\Bar{\sigma}'}}
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\newcommand{\taup}{{\tau'}}
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\newcommand{\up}{\uparrow}
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\newcommand{\dw}{\downarrow}
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\newcommand{\upup}{\uparrow\uparrow}
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\newcommand{\updw}{\uparrow\downarrow}
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\newcommand{\dwup}{\downarrow\uparrow}
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\newcommand{\dwdw}{\downarrow\downarrow}
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% addresses
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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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\begin{document}
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\title{The $GW$ conundrum}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Downfold: The non-linear $GW$ problem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, for the sake of simplicity, we consider a Hartree-Fock (HF) starting point.
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Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
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\begin{equation}
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\label{eq:qp_eq}
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\omega = \eps{p}{\HF} + \SigC{p}(\omega)
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\end{equation}
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where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads
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\begin{equation}
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\SigC{p}(\omega)
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= \sum_{im} \frac{\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta}
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+ \sum_{am} \frac{\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta}
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\end{equation}
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Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by
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\begin{equation}
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\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
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\end{equation}
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where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are respectively the $m$th eigenvalue and eigenvector of the RPA problem in the Tamm-Dancoff approximation, \ie,
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\begin{equation}
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\bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA}
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\end{equation}
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with
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\begin{equation}
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A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
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\end{equation}
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As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,\ell}{\GW}$ and their corresponding weight is given by the value of the so-called renormalization factor
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\begin{equation}
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0 \le Z_{p,\ell} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,\ell}{\GW}} ]^{-1} \le 1
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\end{equation}
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In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW} \equiv \eps{p,\ell=0}{\GW}$ has a large weight $Z_{\ell} \equiv Z_{p,\ell=0}$
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Note that we have the following important conservation rules
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\begin{align}
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\sum_{\ell} Z_{p,\ell} & = 1
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&
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\sum_{\ell} Z_{p,\ell} \eps{p,\ell}{\GW} & = \eps{p}{\HF}
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\end{align}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Upfolding: the linear $GW$ problem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The non-linear quasiparticle equation \eqref{eq:qp_ep} can be transformed into a larger linear problem via an upfolding process:
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\begin{equation}
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C^\text{1h1p}_{iajb,kcld} = \qty[ \qty( \eps{b}{\GW} + \eps{a}{\HF} - \eps{i}{\GW} - \eps{j}{\HF} ) \delta_{jl} \delta_{ac} + 2 \ERI{ja}{cl} ] \delta_{ik} \delta_{bd}
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\end{equation}
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\begin{align}
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V^\text{2h1p}_{p,kld} & = \ERI{pk}{ld}
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&
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V^\text{2h1p}_{p,cld} & = \ERI{pc}{ld}
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\end{align}
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\begin{equation}
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\bH =
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\begin{pmatrix}
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\Tilde{\bA{}{}} & \bV{}{(1)} & \bV{}{(2)}
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\\
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\T{(\bV{}{(1)})} & \bC{}{} & \bO
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\\
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\T{(\bV{}{(2)})} & \bO & \bC{}{}
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\end{pmatrix}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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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).}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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