\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<d} \aeri{pq}{cd} \bX{cd,m}{N+2} \\
  \eri{pa}{\chi^{N-2}_m} &= \sum_{k<l} \aeri{pq}{kl} \bX{kl,m}{N-2}
\end{align}
The GW and GT 
\begin{align}
  \label{eq:selfenergiesGWGT}
  \Sig{pq}{\GW}(\omega) & = \sum_{klc} \frac{\eri{pk}{cl}\eri{qk}{cl}}{\omega - (\e{k}{} + \e{l}{} - \e{c}{}) - \ii \eta}\\
                        & + \sum_{kcd} \frac{\eri{pd}{kc}\eri{qd}{kc}}{\omega - (\e{c}{} + \e{d}{} - \e{k}{}) + \ii \eta} \notag  \\
  \Sig{pq}{\GT}(\omega) & = \sum_{klc} \frac{\aeri{pc}{kl}\aeri{qc}{kl}}{\omega - (\e{k}{} + \e{l}{} - \e{c}{}) - \ii \eta} \\
                          &+ \sum_{kcd} \frac{\aeri{pk}{cd}\aeri{qk}{cd}}{\omega - (\e{c}{} + \e{d}{} - \e{k}{}) + \ii \eta} \notag
\end{align}

The quasi-particle equations involving these self energies can be unfolded into larger linear problems
\begin{equation}
  \label{eqGF(2)lin}
  H_{MBPT} =
	\begin{pmatrix}
          \bF{}{}				                &	\bV{}{\text{2h1p}}		&	\bV{}{\text{2p1h}}	\\
          \T{(\bV{}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}		&	\bO				\\
          \T{(\bV{}{\text{2p1h}})}	&	\bO					&	\bC{}{\text{2p1h}}	\\
	\end{pmatrix}
\end{equation}
In the GF(2) case, the coupling blocks are
\begin{align}
	V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
	&
	V^\text{2p1h}_{p,kcd} & = \aeri{pk}{dc}
\end{align}
and the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
	C^\text{2h1p}_{ija,klc} & = \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} \delta_{ik} 
	\\
	C^\text{2p1h}_{iab,kcd} & = \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} \delta_{bd} 
\end{align}
\end{subequations}
The GW coupling blocks are
\begin{align}
	V^\text{2h1p}_{p,klc} & = \eri{pc}{kl}
	&
	V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}
\end{align}
and the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
	C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} \\
	C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd} 
\end{align}
\end{subequations}
The GT coupling blocks are
\begin{align}
	V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
	&
	V^\text{2p1h}_{p,kcd} & = \aeri{pk}{dc}
\end{align}
and the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
	C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ik} \\
	C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{bd} 
\end{align}
\end{subequations}

\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.}}

%=================================================================%
\section{The IP/EA CI}
%=================================================================%

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.
The natural first idea is to put the electronic Hamiltonian into this IP/EA basis. This gives
\begin{equation}
  \label{eq:H_IPEA}
  H_{CI} =
	\begin{pmatrix}
          \be_i			                     &  \bO		                          &	\bV{1h}{\text{2h1p}}   & \bO		\\
          \bO                                           & \be_a                                       & \bO                                     &\bV{1p}{\text{2p1h}} \\
          \bV{1h}{\text{2h1p},\dagger} & \bO	                                  &  C^\text{2h1p}_{ija,klc}   &	\bO					\\
          \bO                                           & \bV{1p}{\text{2p1h},\dagger} &	\bO				   & C^\text{2p1h}_{iab,kcd}	\\
	\end{pmatrix}
 \end{equation}

 \begin{align}
   V_{i,klc} &= \aeri{kl}{ci} \\
   V_{a,kcd} &= \aeri{ka}{dc} \\
   C^\text{2h1p}_{ija,klc} &= \\
   C^\text{2p1h}_{iab,kcd} &= (-\delta_{ac}\delta_{bd} + \delta_{ad}\delta_{bc}) f_{ki} \notag \\
                           &+ \delta_{ik}(\delta_{ac}f_{bd} + \delta_{ad}f_{bc}  - \delta_{bc}f_{ad}  + \delta_{bd}f_{ac}) \notag \\
                           &+ \delta_{ac}\aeri{kb}{di} - \delta_{ad}\aeri{kb}{ci} - \delta_{bc}\aeri{ka}{di} \notag \\
                           & + \delta_{bd}\aeri{ka}{ci} + \delta_{ik}\aeri{ba}{dc} \notag \\
                           &= \delta_{ac}\delta_{bd}\delta_{ik} (-\eps_i + \eps_a + \eps_b) \notag \\
                           &- \delta_{ad}\delta_{bc}\delta_{ik}(\eps_i - \eps_a - \eps_b) \notag \\
                           &+ \delta_{ac}\aeri{kb}{di} - \delta_{ad}\aeri{kb}{ci} - \delta_{bc}\aeri{ka}{di} \notag \\
                           & + \delta_{bd}\aeri{ka}{ci} + \delta_{ik}\aeri{ba}{dc} \notag
 \end{align}

%=================================================================%
\section{Can we second quantized MBPT?}
% =================================================================%



\appendix

%=================================================================%
\section{Appendix A}
%=================================================================%


\end{document}