338 lines
11 KiB
TeX
338 lines
11 KiB
TeX
\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,siunitx}
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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{\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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%
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\newcommand{\Ne}{N}
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\newcommand{\Norb}{K}
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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{\reps}[2]{\Tilde{\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{\rSigC}[1]{\Tilde{\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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% 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}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
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\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}_{#1}^{\text{c}}}
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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{\bD}[2]{\boldsymbol{D}_{#1}^{#2}}
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\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
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\newcommand{\bW}[2]{\boldsymbol{W}_{#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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\newcommand{\bc}[2]{\boldsymbol{c}_{#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{\RHH}{R_{\ce{H-H}}}
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\newcommand{\ii}{\mathrm{i}}
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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{Undressing $GW$ one determinant at a time}
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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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Here comes the 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{Introduction}
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%%%%%%%%%%%%%%%%%%%%%%
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Here comes the introduction.
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%%%%%%%%%%%%%%%%%
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\section{Theory}
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%%%%%%%%%%%%%%%%%
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In the case of {\GOWO}, the quasiparticle equation reads
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\begin{equation}
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\label{eq:qp_eq}
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\eps{p}{} + \SigC{p}(\omega) - \omega = 0
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\end{equation}
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where $\eps{p}{}$ is the one-electron energy of the HF spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is
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\begin{equation}
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\label{eq:SigC}
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\SigC{p}(\omega)
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{} + \Om{m}{}}
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+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{} - \Om{m}{}}
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\end{equation}
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where
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\begin{equation}
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\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}
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\end{equation}
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are the screened two-electron repulsion integrals where $\Om{m}{}$ and $\bX{m}{}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system within the Tamm-Dancoff approximation:
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\begin{equation}
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\bA{}{\RPA} \cdot \bX{m}{} = \Om{m}{\RPA} \bX{m}{}
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\end{equation}
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with
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\begin{equation}
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A_{ia,jb}^{} = (\eps{a}{} - \eps{i}{}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
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\end{equation}
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and
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\begin{equation}
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\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
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\end{equation}
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The spectral weight of the solution $\eps{p,s}{\GW}$ is given by
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\begin{equation}
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\label{eq:Z}
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0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
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\end{equation}
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with the following sum rules:
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\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}{\GW} & = \eps{p}{}
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\end{align}
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As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
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\begin{equation}
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\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \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}
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\bH^{(p)} =
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\begin{pmatrix}
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\eps{p}{} & \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}
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\end{equation}
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and where the expressions of the 2h1p and 2p1h blocks reads
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\begin{subequations}
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\begin{align}
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\label{eq:C2h1p}
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C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps{I}{} + \eps{J}{} - \eps{A}{}) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK}
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\\
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\label{eq:C2p1h}
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C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps{A}{} + \eps{B}{} - \eps{I}{}) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD}
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\end{align}
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\end{subequations}
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with the following expressions for the coupling blocks:
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\begin{subequations}
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\begin{align}
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\label{eq:V2h1p}
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V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
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\\
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\label{eq:V2p1h}
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V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
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\end{align}
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\end{subequations}
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Here, we use lower case letters for the electronic configurations belonging to the reference model state and upper case letters for the external determinants (\ie, the perturbers).
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By solving the secular equation
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\begin{equation}
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\det[ \bH^{(p)} - \omega \bI ] = 0
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\end{equation}
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we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie,
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\begin{multline}
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\SigC{p}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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\\
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{multline}
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with
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\begin{equation}
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\label{eq:Z_proj}
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Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
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\end{equation}
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In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
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Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$)
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\begin{equation}
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\label{eq:Hp_qia}
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\bH^{(p,qia)} =
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\begin{pmatrix}
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\eps{p}{} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\\
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V_{qia,p} & \eps{qia}{} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}}
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\\
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\T{(\bV{p}{\text{2h1p}})} & \T{(\bC{qia}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO
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\\
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\T{(\bV{p}{\text{2p1h}})} & \T{(\bC{qia}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}}
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\end{pmatrix}
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\end{equation}
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with new blocks defined as
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\begin{gather}
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V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia}
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\\
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\eps{qia}{} \equiv C_{qia,qia} = \text{sgn}(\eps{q}{} - \mu) \qty[ \qty(\eps{q}{} + \eps{a}{} - \eps{i}{} ) + 2 \ERI{ia}{ia} ]
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\\
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C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
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\\
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C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
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\end{gather}
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The expressions of $\bC{p}{\text{2h1p}}$, $\bC{p}{\text{2p1h}}$, $\bV{}{\text{2h1p}}$, and $\bV{}{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has remove the contribution from the 2h1p or 2p1h configuration $qia$.
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While $\eps{p}{\HF}$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps{qia}{} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration.
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Therefore, when $\eps{p}{\HF}$ and $\eps{qia}{}$ becomes of similar mangitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems.
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Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix
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\begin{equation}
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\label{eq:Hp}
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\bSigC{p,qia}(\omega) =
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\begin{pmatrix}
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\eps{p}{} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega)
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\\
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V_{qia,p} + \SigC{qia,p}(\omega) & \eps{qia}{} + \SigC{qia}(\omega)
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\\
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\end{pmatrix}
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\end{equation}
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with
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\begin{subequations}
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\begin{gather}
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\begin{split}
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\SigC{p}(\omega)
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& = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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\\
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& + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{split}
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\\
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\begin{split}
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\SigC{qia}(\omega)
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& = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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\\
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& + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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\end{split}
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\\
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\begin{split}
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\SigC{p,qia}(\omega)
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& = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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\\
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& + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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\end{split}
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\\
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\begin{split}
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\SigC{qia,p}(\omega)
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& = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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\\
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& + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{split}
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\end{gather}
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\end{subequations}
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Of course, the present procedure can be generalized to any number of states.
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%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%%%
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Here comes the conclusion.
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Data availability statement}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The data that supports the findings of this study are available within the article.% and its supplementary material.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{MRGW}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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