diff --git a/Notes/MBPTSecondQuant.tex b/Notes/MBPTSecondQuant.tex new file mode 100644 index 0000000..18a1a2a --- /dev/null +++ b/Notes/MBPTSecondQuant.tex @@ -0,0 +1,301 @@ +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,mleftright} +\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}} +\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} +\newcommand{\dRPA}{\text{dRPA}} + +% coordinates +\newcommand{\br}{\boldsymbol{r}} +\newcommand{\bx}{\boldsymbol{x}} +\newcommand{\dbr}{d\br} +\newcommand{\dbx}{d\bx} + +% methods +\newcommand{\GW}{\text{$GW$}} +\newcommand{\GT}{\text{$GT$}} +\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{\Om}[2]{\Omega_{#1}^{#2}} +\newcommand{\sERI}[2]{(#1|#2)} +\newcommand{\e}[2]{\epsilon_{#1}^{#2}} + +% +\newcommand{\Ne}{N} +\newcommand{\Norb}{K} +\newcommand{\Nocc}{O} +\newcommand{\Nvir}{V} + +% operators +\newcommand{\hH}{\Hat{H}} +\newcommand{\hS}{\Hat{S}} +\newcommand{\ani}[1]{\hat{a}_{#1}} +\newcommand{\cre}[1]{\hat{a}_{#1}^\dagger} +\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} } + +% energies +\newcommand{\Enuc}{E^\text{nuc}} +\newcommand{\Ec}[1]{E_\text{c}^{#1}} +\newcommand{\EHF}{E^\text{HF}} + +% orbital energies +\newcommand{\eps}{\epsilon} +\newcommand{\reps}{\Tilde{\epsilon}} + +% Matrix elements +\newcommand{\SigC}{\Sigma^\text{c}} +\newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}} +\newcommand{\MO}[1]{\phi_{#1}} +\newcommand{\SO}[1]{\psi_{#1}} +\newcommand{\eri}[2]{\braket{#1}{#2}} +\newcommand{\aeri}[2]{\mel{#1}{}{#2}} +\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{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}} +\newcommand{\be}{\boldsymbol{\epsilon}} +\newcommand{\bOm}{\boldsymbol{\Omega}} +\newcommand{\bA}{\boldsymbol{A}} +\newcommand{\bB}{\boldsymbol{B}} +\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}} +\newcommand{\bD}{\boldsymbol{D}} +\newcommand{\bF}{\boldsymbol{F}} +\newcommand{\bU}{\boldsymbol{U}} +\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}} +\newcommand{\bW}{\boldsymbol{W}} +\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}} +\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}} +\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}} +\newcommand{\bc}{\boldsymbol{c}} + +% 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} + +% shortcuts for greek letters +\newcommand{\si}{\sigma} +\newcommand{\la}{\lambda} + + +\newcommand{\RHH}{R_{\ce{H-H}}} +\newcommand{\ii}{\mathrm{i}} + +\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)} +\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}} +\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}} + +% addresses +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} + +\begin{document} + +\title{Notes on the project: Similarity Renormalization Group formalism applied to Green's function theory} + +\author{Antoine \surname{Marie}} + \email{amarie@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} + +\author{Pierre-Fran\c{c}ois \surname{Loos}} + \email{loos@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} + +%\begin{abstract} +%Here comes the abstract. +%\bigskip +%\begin{center} +% \boxed{\includegraphics[width=0.5\linewidth]{TOC}} +%\end{center} +%\bigskip +%\end{abstract} + +\maketitle + +%=================================================================% +\section{Introduction} +%=================================================================% + +The many-body perturbation theory formalism and its various approximations are naturally derived using time-dependent Feynman diagrams. +These derivation are quite different from wave function methods based on one-body orbitals and second quantization. +One can study the link between these formalisms by expanding the MBPT Feynman diagrams into time-independent Goldstone diagrams and then compare them to the ones that appear in WFT. +However, that would be valuable to extend this connection by expressing the MBPT approximations in the second quantization. +This is the aim of these notes. + +%=================================================================% +\section{The unfolded Green's function} +%=================================================================% + +In order to use MBPT in practice, one needs to rely on approximations of the self-energy. +In the following, we will focus on the GF(2), GW and GT approximations. +The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of MP perturbation theory. +On the other hand, the GW self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy. +Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA +The corresponding self-energies read as +\begin{align} + \label{eq:selfenergies} +\Sig{pq}{GF(2)}(\omega) & = \sum_{klc} \frac{\aeri{pc}{kl}\aeri{qc}{kl}}{\omega + \eps _c -\eps_k -\eps_l - \ii \eta} \\ + & + \sum_{kcd} \frac{\aeri{pk}{cd}\aeri{qk}{cd}}{\omega + \eps _k -\eps_c -\eps_d + \ii \eta} \notag \\ + \Sig{pq}{\GW}(\omega) & = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{} + \Om{m}{\dRPA} - \ii \eta}\\ + & + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{} - \Om{m}{\dRPA} + \ii \eta} \notag \displaybreak \\ + \Sig{pq}{\GT}(\omega) & = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \e{i}{} - \Om{m}{N+2} - \ii \eta} \\ + &+ \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \e{a}{} - \Om{m}{N-2} + \ii \eta} \notag +\end{align} + +\begin{align} + \label{eq:sERI} + \sERI{pq}{m} &= \sum_{ia} \ERI{pi}{qa} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia} \\ + \eri{pi}{\chi^{N+2}_m} &= \sum_{c