301 lines
12 KiB
TeX
301 lines
12 KiB
TeX
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\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,mleftright}
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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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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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\newcommand{\dRPA}{\text{dRPA}}
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% coordinates
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\newcommand{\br}{\boldsymbol{r}}
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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{\GT}{\text{$GT$}}
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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{\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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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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\newcommand{\sERI}[2]{(#1|#2)}
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\newcommand{\e}[2]{\epsilon_{#1}^{#2}}
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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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\newcommand{\ani}[1]{\hat{a}_{#1}}
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\newcommand{\cre}[1]{\hat{a}_{#1}^\dagger}
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\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} }
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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}{\epsilon}
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\newcommand{\reps}{\Tilde{\epsilon}}
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% Matrix elements
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\newcommand{\SigC}{\Sigma^\text{c}}
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\newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}}
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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]{\braket{#1}{#2}}
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\newcommand{\aeri}[2]{\mel{#1}{}{#2}}
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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{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bA}{\boldsymbol{A}}
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\newcommand{\bB}{\boldsymbol{B}}
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\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
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\newcommand{\bD}{\boldsymbol{D}}
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\newcommand{\bF}{\boldsymbol{F}}
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\newcommand{\bU}{\boldsymbol{U}}
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\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
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\newcommand{\bW}{\boldsymbol{W}}
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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}{\boldsymbol{c}}
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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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% shortcuts for greek letters
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\newcommand{\si}{\sigma}
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\newcommand{\la}{\lambda}
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\newcommand{\RHH}{R_{\ce{H-H}}}
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\newcommand{\ii}{\mathrm{i}}
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\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)}
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\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}}
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\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}}
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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{Notes on the project: Similarity Renormalization Group formalism applied to Green's function theory}
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\author{Antoine \surname{Marie}}
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\email{amarie@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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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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The many-body perturbation theory formalism and its various approximations are naturally derived using time-dependent Feynman diagrams.
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These derivation are quite different from wave function methods based on one-body orbitals and second quantization.
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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.
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However, that would be valuable to extend this connection by expressing the MBPT approximations in the second quantization.
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This is the aim of these notes.
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%=================================================================%
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\section{The unfolded Green's function}
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%=================================================================%
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In order to use MBPT in practice, one needs to rely on approximations of the self-energy.
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In the following, we will focus on the GF(2), GW and GT approximations.
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The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of MP perturbation theory.
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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.
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Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA
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The corresponding self-energies read as
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\begin{align}
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\label{eq:selfenergies}
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\Sig{pq}{GF(2)}(\omega) & = \sum_{klc} \frac{\aeri{pc}{kl}\aeri{qc}{kl}}{\omega + \eps _c -\eps_k -\eps_l - \ii \eta} \\
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& + \sum_{kcd} \frac{\aeri{pk}{cd}\aeri{qk}{cd}}{\omega + \eps _k -\eps_c -\eps_d + \ii \eta} \notag \\
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\Sig{pq}{\GW}(\omega) & = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{} + \Om{m}{\dRPA} - \ii \eta}\\
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& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{} - \Om{m}{\dRPA} + \ii \eta} \notag \displaybreak \\
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\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} \\
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&+ \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
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\end{align}
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\begin{align}
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\label{eq:sERI}
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\sERI{pq}{m} &= \sum_{ia} \ERI{pi}{qa} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia} \\
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\eri{pi}{\chi^{N+2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX{cd,m}{N+2} \\
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\eri{pa}{\chi^{N-2}_m} &= \sum_{k<l} \aeri{pq}{kl} \bX{kl,m}{N-2}
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\end{align}
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The GW and GT
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\begin{align}
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\label{eq:selfenergiesGWGT}
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\Sig{pq}{\GW}(\omega) & = \sum_{klc} \frac{\eri{pk}{cl}\eri{qk}{cl}}{\omega - (\e{k}{} + \e{l}{} - \e{c}{}) - \ii \eta}\\
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& + \sum_{kcd} \frac{\eri{pd}{kc}\eri{qd}{kc}}{\omega - (\e{c}{} + \e{d}{} - \e{k}{}) + \ii \eta} \notag \\
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\Sig{pq}{\GT}(\omega) & = \sum_{klc} \frac{\aeri{pc}{kl}\aeri{qc}{kl}}{\omega - (\e{k}{} + \e{l}{} - \e{c}{}) - \ii \eta} \\
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&+ \sum_{kcd} \frac{\aeri{pk}{cd}\aeri{qk}{cd}}{\omega - (\e{c}{} + \e{d}{} - \e{k}{}) + \ii \eta} \notag
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\end{align}
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The quasi-particle equations involving these self energies can be unfolded into larger linear problems
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\begin{equation}
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\label{eqGF(2)lin}
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H_{MBPT} =
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\begin{pmatrix}
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\bF{}{} & \bV{}{\text{2h1p}} & \bV{}{\text{2p1h}} \\
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\T{(\bV{}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\
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\T{(\bV{}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \\
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\end{pmatrix}
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\end{equation}
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In the GF(2) case, the coupling blocks are
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\begin{align}
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V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
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&
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V^\text{2p1h}_{p,kcd} & = \aeri{pk}{dc}
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\end{align}
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and the 2h1p and 2p1h matrix elements are
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\begin{subequations}
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\begin{align}
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C^\text{2h1p}_{ija,klc} & = \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} \delta_{ik}
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\\
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C^\text{2p1h}_{iab,kcd} & = \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} \delta_{bd}
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\end{align}
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\end{subequations}
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The GW coupling blocks are
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\begin{align}
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V^\text{2h1p}_{p,klc} & = \eri{pc}{kl}
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&
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}
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\end{align}
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and the 2h1p and 2p1h matrix elements are
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\begin{subequations}
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\begin{align}
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C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} \\
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C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd}
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\end{align}
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\end{subequations}
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The GT coupling blocks are
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\begin{align}
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V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
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&
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V^\text{2p1h}_{p,kcd} & = \aeri{pk}{dc}
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\end{align}
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and the 2h1p and 2p1h matrix elements are
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\begin{subequations}
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\begin{align}
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C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ik} \\
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C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{bd}
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\end{align}
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\end{subequations}
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\textbf{\textcolor{red}{That would be nice to add electron-hole T matrix to see it also correspond to one term that can be found in the CI below.}}
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%=================================================================%
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\section{The IP/EA CI}
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%=================================================================%
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We would like to find second quantized effective hamiltonians for each of these MBPT approximate methods such that if these hamiltonians are put in the basis $\{\ket{\Psi_i},\ket{\Psi^a},\ket{\Psi_{ij}^a},\ket{\Psi_i^{ab}} \}$ we get back the matrices above.
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The natural first idea is to put the electronic Hamiltonian into this IP/EA basis. This gives
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\begin{equation}
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\label{eq:H_IPEA}
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H_{CI} =
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\begin{pmatrix}
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\be_i & \bO & \bV{1h}{\text{2h1p}} & \bO \\
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\bO & \be_a & \bO &\bV{1p}{\text{2p1h}} \\
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\bV{1h}{\text{2h1p},\dagger} & \bO & C^\text{2h1p}_{ija,klc} & \bO \\
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\bO & \bV{1p}{\text{2p1h},\dagger} & \bO & C^\text{2p1h}_{iab,kcd} \\
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\end{pmatrix}
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\end{equation}
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\begin{align}
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V_{i,klc} &= \aeri{kl}{ci} \\
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V_{a,kcd} &= \aeri{ka}{dc} \\
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C^\text{2h1p}_{ija,klc} &= \\
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C^\text{2p1h}_{iab,kcd} &= (-\delta_{ac}\delta_{bd} + \delta_{ad}\delta_{bc}) f_{ki} \notag \\
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&+ \delta_{ik}(\delta_{ac}f_{bd} + \delta_{ad}f_{bc} - \delta_{bc}f_{ad} + \delta_{bd}f_{ac}) \notag \\
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&+ \delta_{ac}\aeri{kb}{di} - \delta_{ad}\aeri{kb}{ci} - \delta_{bc}\aeri{ka}{di} \notag \\
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& + \delta_{bd}\aeri{ka}{ci} + \delta_{ik}\aeri{ba}{dc} \notag \\
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&= \delta_{ac}\delta_{bd}\delta_{ik} (-\eps_i + \eps_a + \eps_b) \notag \\
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&- \delta_{ad}\delta_{bc}\delta_{ik}(\eps_i - \eps_a - \eps_b) \notag \\
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&+ \delta_{ac}\aeri{kb}{di} - \delta_{ad}\aeri{kb}{ci} - \delta_{bc}\aeri{ka}{di} \notag \\
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& + \delta_{bd}\aeri{ka}{ci} + \delta_{ik}\aeri{ba}{dc} \notag
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\end{align}
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%=================================================================%
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\section{Can we second quantized MBPT?}
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% =================================================================%
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\appendix
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%=================================================================%
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\section{Appendix A}
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%=================================================================%
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\end{document}
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