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Pierre-Francois Loos 2022-04-13 13:23:19 +02:00
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\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}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\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}[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{\bD}[2]{\boldsymbol{D}_{#1}^{#2}}
\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
\newcommand{\bW}[2]{\boldsymbol{W}_{#1}^{#2}}
\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}}
\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}
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}
\maketitle
\begin{equation}
\label{eq:qp_eq}
\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
\end{equation}
\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}
\begin{equation}
\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
\end{equation}
\begin{equation}
\bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA}
\end{equation}
\begin{equation}
A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
\end{equation}
\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}
\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}
\begin{align}
\sum_{s} Z_{p,s} & = 1
&
\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
\end{align}
\begin{equation}
\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
\end{equation}
\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}
\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}
\begin{align}
V^\text{2h1p}_{p,klc} & = \sqrt{2} \ERI{pk}{cl}
&
V^\text{2p1h}_{p,kcd} & = \sqrt{2} \ERI{pd}{kc}
\end{align}
\begin{equation}
\det[ \bH^{(p)} - \omega \bI ] = 0
\end{equation}
\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}
\begin{equation}
\label{eq:Z_proj}
Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
\end{equation}
\begin{equation}
\label{eq:Hp}
\bH^{(P,QIA)} =
\begin{pmatrix}
\eps{P}{\HF} & W_{P,QIA} & \bW{P}{\text{2h1p}} & \bW{P}{\text{2p1h}}
\\
W_{QIA,P} & D_{QIA,QIA} & \bD{QIA}{\text{2h1p}} & \bD{QIA}{\text{2p1h}}
\\
\T{(\bW{P}{\text{2h1p}})} & \T{(\bD{QIA}{\text{2h1p}})} & \bD{}{\text{2h1p}} & \bO
\\
\T{(\bW{P}{\text{2p1h}})} & \T{(\bD{QIA}{\text{2p1h}})} & \bO & \bD{}{\text{2p1h}}
\end{pmatrix}
\end{equation}
with
\begin{gather}
W_{P,QIA} = \sqrt{2} \ERI{PQ}{IA}
\\
D_{QIA,QIA} = \text{sgn}(\eps{Q}{\HF} - \mu) \qty[ \qty(\eps{Q}{\HF} + \eps{A}{\HF} - \eps{I}{\HF} ) + 2 \ERI{IA}{IA} ]
\\
D_{QIA,klc}^\text{2h1p} = - 2 \ERI{IA}{cl} \delta_{Qk}
\\
D_{QIA,klc}^\text{2p1h} = + 2 \ERI{IA}{kc} \delta_{Qd}
\end{gather}
\begin{equation}
\label{eq:Hp}
\bH^{(P,QIA)} =
\begin{pmatrix}
\eps{P}{\HF} + \SigC{P}(\omega) & W_{P,QIA} + \SigC{P,QIA}(\omega)
\\
W_{QIA,P} + \SigC{QIA,P}(\omega) & D_{QIA,QIA} + \SigC{QIA}(\omega)
\\
\end{pmatrix}
\end{equation}
with
\begin{align}
\SigC{P}(\omega)
& = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})}
+ \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})}
\\
\SigC{QIA}(\omega)
& = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})}
+ \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})}
\\
\SigC{P,QIA}(\omega)
& = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})}
+ \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})}
\\
\SigC{QIA,P}(\omega)
& = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})}
+ \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%
\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.
%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{MRGW}
%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}