ufGW/Manuscript/ufGW.tex

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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem}
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\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage[
colorlinks=true,
citecolor=blue,
breaklinks=true
]{hyperref}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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% coordinates
\newcommand{\br}{\boldsymbol{r}}
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% methods
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%
\newcommand{\Norb}{N}
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% operators
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% energies
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% orbital energies
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% Matrix elements
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% Matrices
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% orbitals, gaps, etc
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% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\begin{document}
\title{The $GW$ conundrum}
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\author{Enzo \surname{Monino}}
\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract}
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}
\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We consider {\GOWO} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Downfold: The non-linear $GW$ problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here, for the sake of simplicity, we consider a Hartree-Fock (HF) starting point.
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
\begin{equation}
\label{eq:qp_eq}
\omega = \eps{p}{\HF} + \SigC{p}(\omega)
\end{equation}
where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads
\begin{equation}
\SigC{p}(\omega)
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta}
+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta}
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\end{equation}
Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by
\begin{equation}
\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
\end{equation}
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,
\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}
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As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{}$ 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,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{}} ]^{-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) $\eps{p}{} \equiv \eps{p,s=0}{}$ has a large weight $Z_{} \equiv Z_{p,=0}$
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Note that we have the following important conservation rules
\begin{align}
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\sum_{s} Z_{p,s} & = 1
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&
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\sum_{s} Z_{p,s} \eps{p,s}{} & = \eps{p}{\HF}
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\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Upfolding: the linear $GW$ problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
is downfolded on the 1h and 1p sectors via their interaction with the 2h1p and 2p1h sectors:
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\begin{equation}
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\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{} \bc{}{(p,s)}
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\end{equation}
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with
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\begin{equation}
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\label{eq:Hp}
\bH^{(p)} =
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\begin{pmatrix}
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\eps{p}{\HF} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\\
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\T{(\bV{p}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO
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\\
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\T{(\bV{p}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}}
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\end{pmatrix}
\end{equation}
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where
\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}
and the corresponding coupling blocks read
\begin{align}
V^\text{2h1p}_{p,kld} & = \sqrt{2} \ERI{pk}{cl}
&
V^\text{2p1h}_{p,cld} & = \sqrt{2} \ERI{pd}{kc}
\end{align}
The size of this eigenvalue problem is $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$.
Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie,
\begin{equation}
Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
\end{equation}
It is important to understand that diagonalizing $\bH^{(p)}$ in Eq.~\eqref{eq:Hp} is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
The main difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introducing regularized $GW$ methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One way to hamper such issues is to resort to regularization of the $GW$ self-energy.
Of course, the way of regularizing the self-energy is not unique but here we consider 3 different ways directly imported from MP2 theory.
This helps greatly convergence for (partially) self-consistent $GW$ methods.
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%%%%%%%%%%%%%%%%%%%%%%%%
\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).}
%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}