From 1585ea1e9619c0ac0170c5d7b51af4e1a5f83f96 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sun, 16 Jan 2022 15:41:14 +0100 Subject: [PATCH] tex --- Manuscript/ufGW.tex | 222 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 222 insertions(+) create mode 100644 Manuscript/ufGW.tex diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex new file mode 100644 index 0000000..293baad --- /dev/null +++ b/Manuscript/ufGW.tex @@ -0,0 +1,222 @@ +\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} + +\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} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\SI}{\textcolor{blue}{supporting information}} +\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{\ppRPA}{\text{pp-RPA}} +\newcommand{\BSE}{\text{BSE}} +\newcommand{\dBSE}{\text{dBSE}} +\newcommand{\stat}{\text{stat}} +\newcommand{\dyn}{\text{dyn}} +\newcommand{\TDA}{\text{TDA}} + +% +\newcommand{\Norb}{N} +\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{\Om}[2]{\Omega_{#1}^{#2}} + +% Matrix elements +\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} +\newcommand{\SigC}[1]{\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)} + +%% bold in Table +\newcommand{\bb}[1]{\textbf{#1}} +\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} +\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} + +% Matrices +\newcommand{\bO}{\boldsymbol{0}} +\newcommand{\bI}{\boldsymbol{1}} +\newcommand{\bH}{\boldsymbol{H}} +\newcommand{\bvc}{\boldsymbol{v}} +\newcommand{\bSig}[1]{\boldsymbol{\Sigma}^{#1}} +\newcommand{\be}{\boldsymbol{\epsilon}} +\newcommand{\bOm}[1]{\boldsymbol{\Omega}^{#1}} +\newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}} +\newcommand{\bB}[2]{\boldsymbol{B}_{#1}^{#2}} +\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}} +\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}} +\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}} +\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}} +\newcommand{\bZ}[2]{\boldsymbol{Z}_{#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{\sig}{\sigma} +\newcommand{\bsig}{{\Bar{\sigma}}} +\newcommand{\sigp}{{\sigma'}} +\newcommand{\bsigp}{{\Bar{\sigma}'}} +\newcommand{\taup}{{\tau'}} + +\newcommand{\up}{\uparrow} +\newcommand{\dw}{\downarrow} +\newcommand{\upup}{\uparrow\uparrow} +\newcommand{\updw}{\uparrow\downarrow} +\newcommand{\dwup}{\downarrow\uparrow} +\newcommand{\dwdw}{\downarrow\downarrow} + +% addresses +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} + +\begin{document} + +\title{The $GW$ conundrum} + +\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 + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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) + = \sum_{im} \frac{\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta} + + \sum_{am} \frac{\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta} +\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} +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 +\begin{equation} + 0 \le Z_{p,\ell} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,\ell}{\GW}} ]^{-1} \le 1 +\end{equation} +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 $Z_{\ell} \equiv Z_{p,\ell=0}$ +Note that we have the following important conservation rules +\begin{align} + \sum_{\ell} Z_{p,\ell} & = 1 + & + \sum_{\ell} Z_{p,\ell} \eps{p,\ell}{\GW} & = \eps{p}{\HF} +\end{align} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Upfolding: the linear $GW$ problem} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +The non-linear quasiparticle equation \eqref{eq:qp_ep} can be transformed into a larger linear problem via an upfolding process: +\begin{equation} + 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} +\end{equation} +\begin{align} + V^\text{2h1p}_{p,kld} & = \ERI{pk}{ld} + & + V^\text{2h1p}_{p,cld} & = \ERI{pc}{ld} +\end{align} + +\begin{equation} + \bH = + \begin{pmatrix} + \Tilde{\bA{}{}} & \bV{}{(1)} & \bV{}{(2)} + \\ + \T{(\bV{}{(1)})} & \bC{}{} & \bO + \\ + \T{(\bV{}{(2)})} & \bO & \bC{}{} + \end{pmatrix} +\end{equation} + +%%%%%%%%%%%%%%%%%%%%%%%% +\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}