From 696b66d7ca45fe702f7462db2a38eef405cede2e Mon Sep 17 00:00:00 2001
From: Antoine MARIE <amarie@irsamc.ups-tlse.fr>
Date: Thu, 3 Nov 2022 14:15:42 +0100
Subject: [PATCH] Saving work in cleaned notes

---
 Notes/MBPTSecondQuant.tex                     | 301 -------------
 Notes/{PerturbativeAnalysis.tex => Notes.tex} | 424 ++++++++++++------
 2 files changed, 291 insertions(+), 434 deletions(-)
 delete mode 100644 Notes/MBPTSecondQuant.tex
 rename Notes/{PerturbativeAnalysis.tex => Notes.tex} (60%)

diff --git a/Notes/MBPTSecondQuant.tex b/Notes/MBPTSecondQuant.tex
deleted file mode 100644
index 18a1a2a..0000000
--- a/Notes/MBPTSecondQuant.tex
+++ /dev/null
@@ -1,301 +0,0 @@
-\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}
\ No newline at end of file
diff --git a/Notes/PerturbativeAnalysis.tex b/Notes/Notes.tex
similarity index 60%
rename from Notes/PerturbativeAnalysis.tex
rename to Notes/Notes.tex
index 5f711d1..6aa4a6b 100644
--- a/Notes/PerturbativeAnalysis.tex
+++ b/Notes/Notes.tex
@@ -15,9 +15,11 @@
 
 \newcommand{\ie}{\textit{i.e.}}
 \newcommand{\eg}{\textit{e.g.}}
+\newcommand{\etal}{\textit{et al.}}
 \newcommand{\alert}[1]{\textcolor{red}{#1}}
 \usepackage[normalem]{ulem}
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
+\newcommand{\ant}[1]{\textcolor{green}{#1}}
 \newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
 \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
 
@@ -37,7 +39,8 @@
 \newcommand{\dbx}{d\bx}
 
 % methods
-\newcommand{\GW}{\text{$GW$}}	
+\newcommand{\GW}{\text{$GW$}}
+\newcommand{\GT}{\text{$GT$}}	
 \newcommand{\evGW}{ev$GW$}	
 \newcommand{\qsGW}{qs$GW$}	
 \newcommand{\GOWO}{$G_0W_0$}	
@@ -91,6 +94,7 @@
 \newcommand{\bO}{\boldsymbol{0}}
 \newcommand{\bI}{\boldsymbol{1}}
 \newcommand{\bH}{\boldsymbol{H}}
+\newcommand{\bSig}{\boldsymbol{\Sigma}}
 \newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
 \newcommand{\be}{\boldsymbol{\epsilon}}
 \newcommand{\bOm}{\boldsymbol{\Omega}}
@@ -98,14 +102,14 @@
 \newcommand{\bB}{\boldsymbol{B}}
 \newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
 \newcommand{\bD}{\boldsymbol{D}}
-\newcommand{\bF}{\boldsymbol{F}}
-\newcommand{\bU}{\boldsymbol{U}}
+\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}}
 \newcommand{\bR}{\boldsymbol{R}}
+\newcommand{\bU}{\boldsymbol{U}}
 \newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
 \newcommand{\bW}{\boldsymbol{W}}
-\newcommand{\bX}{\boldsymbol{X}}
-\newcommand{\bY}{\boldsymbol{Y}}
-\newcommand{\bZ}{\boldsymbol{Z}}
+\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
@@ -135,7 +139,7 @@
 
 \begin{document}	
 
-\title{Notes on the project: Perturbative Analysis of the Similarity Renormalization Group formalism applied to the electronic Hamiltonian and Green's function theory}
+\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}
@@ -158,16 +162,19 @@
 
 %=================================================================%
 \section{Introduction}
+\label{sec:intro}
 %=================================================================%
 
-The aim of this document is two-fold.
-First, we want to re-derive (in details) the perturbative analysis of the similarity renormalisation group (SRG) formalism applied to the non-relativistic electronic Hamiltonian.
-In a second time, we want to apply the same formalism to the unfolded GW Hamiltonian.
-To do so, we first need to find a second quantization effective Hamiltonian for Green function theory.
-Before jumping into these analysis, we do a brief presentation of the SRG formalism.
+This document is a compilation of various notes related to the similarity renormalisation group (SRG) and its application to many-body perturbation theory (MBPT).
+Before tackling its application to MBPT, we give the main SRG equations in Sec.~\ref{sec:intro}.
+Then, we derive in details the perturbative expressions obtained by Evangelista by applying the SRG to the electronic Hamiltonian (see Sec.~\ref{sec:srg}).
+Now turning to MBPT, its various flavors are presented in Sec.~\ref{sec:folded} and the corresponding unfolded equations are given in Sec.~\ref{sec:unfolded}.
+Then, a SRG perturbative analysis is performed on general matrices in Sec.~\ref{sec:matrix_srg}.
+Finally, in Sec.~\ref{sec:second_quant_mbpt} we investigate the possibility to find a second quantized effective Hamiltonian corresponding to the unfolded MBPT equations.
 
 %=================================================================%
 \section{The similarity renormalisation group}
+\label{sec:srg}
 %=================================================================%
 
 The similarity renormalization group aims at continuously transforming an Hamiltonian to a diagonal form, or more often to a block-diagonal form.
@@ -178,7 +185,7 @@ Therefore, the transformed Hamiltonian
 depends on a flow parameter $s$.
 The resulting Hamiltonian possess up to $N$-body operators with $N$ the number of particle.
 \begin{equation}
-  \bH(s) = E_0(s) + \bF(s) + \titou{\bV(s)} + \bW(s) + \dots
+  \bH(s) = E_0(s) + \bF(s) + \bV{}{}(s) + \bW(s) + \dots
 \end{equation}
 In the following, we will truncate every contribution superior to two-body operators.
 We can easily derive an evolution equation for this Hamiltonian by taking the derivative of $\bH(s)$. This gives
@@ -210,7 +217,8 @@ One of the flaws of this generator is that it generates a \titou{stiff} set of O
 However, here we consider analytical perturbative expressions so we will not be affected by this problem.
 
 %=================================================================%
-\section{The electronic Hamiltonian}
+\section{The SRG electronic Hamiltonian}
+\label{sec:electronic_ham}
 %=================================================================%
 
 In this part, we derive the perturbative expression for the SRG applied to the non-relativistic electronic Hamiltonian
@@ -228,19 +236,19 @@ In this case, we want to decouple the reference determinant from every singly an
 Hence, we define the off-diagonal Hamiltonian as
 \begin{equation}
   \label{eq:hamiltonianOffDiagonal}
-  \hH^{\text{od}}(s) = \sum_{ia} f_i^a(s)\no{a}{i} + \frac{1}{4} \sum_{ijab}\titou{v(s)_{ij}^{ab}}\no{ab}{ij}.
+  \hH^{\text{od}}(s) = \sum_{ia} f_i^a(s)\no{a}{i} + \frac{1}{4} \sum_{ijab}v_{ij}^{ab}(s)\no{ab}{ij}.
 \end{equation}
 Note that each coefficients depend on $s$.
 
 The perturbative parameter $\la$ is such that
 \begin{equation}
-  \bH(0) = E_0(0) + F(0) + \la V(0)
+  \bH(0) = E_0(0) + \bF{}{}(0) + \la \bV{}{}(0)
 \end{equation}
 In addition, we know the following initial conditions.
-We use the HF basis set of the reference such that $F^{\text{od}}(0) = 0$ and \titou{$F^{\mathrm{d}}(0)=\delta_{pq}\epsilon_p$}
+We use the HF basis set of the reference such that $\bF{}{\text{od}}(0) = 0$ and $\bF{}{\mathrm{d}}(0)_{pq}=\delta_{pq}\epsilon_p$.
 Therefore, we have
 \begin{align}
-  \bH^\text{d}(0)&=E_0(0) + F^{\mathrm{d}}(0) + \la V^{\mathrm{d}}(0) & \bH^\text{od}(0)&= \la V^{\mathrm{od}}(0)
+  \bH^\text{d}(0)&=E_0(0) + \bF{}{\mathrm{d}}(0) + \la \bV{}{\mathrm{d}}(0) & \bH^\text{od}(0)&= \la V^{\mathrm{od}}(0)
 \end{align}
 Now, we want to compute the terms at each order of the following development
 \begin{equation}
@@ -333,22 +341,6 @@ These three differential equations can be integrated to obtain the analytical fo
 \subsection{Second order Hamiltonian}
 %%%%%%%%%%%%%%%%%%%%%%
 
-To compute the second order contribution to the Hamiltonian coefficients, we first need to compute the second order contribution to $\boldsymbol{\eta}(s)$.
-\begin{equation}
-  \boldsymbol{\eta}^{(2)}(s) = \comm{\bH^{\text{d},(0)}(0)}{\bH^{\text{od},(2)}(s)} + \comm{\bH^{\text{d},(1)}(s)}{\bH^{\text{od},(1)}(s)}
-\end{equation}
-The expressions for the first commutator are computed analogously to the one of the previous subsection.
-We focus on deriving expressions for the second term.
-The one-body part of $\bH^{\text{od},(1)}(s)$ is equal to zero so two of the four terms contributing to the one-body part of  $\comm{\bH^{\text{d},(1)}(s)}{\bH^{\text{od},(1)}(s)}$ are zero.
-In addition, the term $\comm{A_1}{B_2}$ is equal to zero as well because the coefficients \titou{$A_{1,i}^a$} are zero (see expression in Appendix).
-So we have
-\begin{align}
-  \eta_a^{i,(2)}(s) &=  \comm{\bH_1^{\text{d},(0)}(0)}{\bH_1^{\text{od},(2)}(s)}_a^i + \comm{\bH_2^{\text{d},(1)}(s)}{\bH_2^{\text{od},(1)}(s)}_a^i \\
-  &= (\epsilon_a - \epsilon_i)f_a^{i,(2)}(s) + \dots
-\end{align}
-Need to continue this derivation but this not needed for EPT2.
-
-
 Now turning to the differential equations, we start by computing the scalar part of Eq.~\eqref{eq:flowEquation}, \ie the differential equation for the second order energy.
 \begin{align}
   \dv{E_0^{(2)}(s)}{s} & = \mel{\phi}{\comm{\boldsymbol{\eta}_1^{(2)}(s)}{\bH_1^{(0)}(s)}}{\phi} + \mel{\phi}{\comm{\boldsymbol{\eta}_2^{(2)}(s)}{\bH_2^{(0)}(s)}}{\phi} \\
@@ -366,13 +358,133 @@ In addition, the one-body hamiltonian has no first order contribution so
 \end{align}
 After integration, using the initial condition $E_0^{(2)}(0)=0$, we obtain
 \begin{equation}
-  E_0^{(2)}(s) = \frac{1}{4} \sum_{i j} \sum_{a b} \titou{\frac{\Delta_{ab}^{ij}}{\aeri{ij}{ab}}}\left(1-e^{-2s (\Delta_{ab}^{ij})^2}\right)
+  E_0^{(2)}(s) = \frac{1}{4} \sum_{i j} \sum_{a b} \frac{\aeri{ij}{ab}^2}{\Delta_{ab}^{ij}}\left(1-e^{-2s (\Delta_{ab}^{ij})^2}\right)
 \end{equation}
 
+We can continue the derivation further than Evangelista's paper.
+To compute the second order contribution to the Hamiltonian coefficients, we first need to compute the second order contribution to $\boldsymbol{\eta}(s)$.
+\begin{equation}
+  \boldsymbol{\eta}^{(2)}(s) = \comm{\bH^{\text{d},(0)}(0)}{\bH^{\text{od},(2)}(s)} + \comm{\bH^{\text{d},(1)}(s)}{\bH^{\text{od},(1)}(s)}
+\end{equation}
+The expressions for the first commutator are computed analogously to the one of the previous subsection.
+We focus on deriving expressions for the second term.
+The one-body part of $\bH^{\text{od},(1)}(s)$ is equal to zero so two of the four terms contributing to the one-body part of  $\comm{\bH^{\text{d},(1)}(s)}{\bH^{\text{od},(1)}(s)}$ are zero.
+In addition, the term $\comm{A_1}{B_2}$ is equal to zero as well because the coefficients $(A_1)_{i}^a$ are zero (see expression in Appendix).
+So we have
+\begin{align}
+  \eta_a^{i,(2)}(s) &=  \comm{\bH_1^{\text{d},(0)}(0)}{\bH_1^{\text{od},(2)}(s)}_a^i + \comm{\bH_2^{\text{d},(1)}(s)}{\bH_2^{\text{od},(1)}(s)}_a^i \\
+  &= (\epsilon_a - \epsilon_i)f_a^{i,(2)}(s) + \dots
+\end{align}
+Need to continue this derivation but this not needed for EPT2.
+
 %=================================================================%
-\section{The unfolded Green's function}
+\section{The various flavors of MBPT}
+\label{sec:mbpt}
 % =================================================================%
 
+The central equation of MBPT in practice is the following
+\begin{equation}
+  \label{eq:quasipart_eq}
+  \bF{}{} + \bSig(\omega) = \omega \mathbb{1}.
+\end{equation}
+However, in order to use it we need to rely on approximations of the self-energy $\bSig(\omega)$.
+                        
+%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Self-energies and quasiparticle equations}
+\label{sec:folded}
+%%%%%%%%%%%%%%%%%%%%%%
+
+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 M\"oller-Plesset perturbation theory.
+\begin{align}
+  \label{eq:GF2_selfenergy}
+  \Sig{pq}{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \eps _a -\eps_i -\eps_j - \ii \eta} \notag \\
+  &+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \eps _i -\eps_a -\eps_b + \ii \eta} \notag  
+\end{align}
+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.
+\begin{equation}
+  \label{eq:GW_selfenergy}
+  \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
+\end{equation}
+with
+\begin{equation}
+  \label{eq:GW_sERI}
+    \sERI{pq}{m} = \sum_{ia} \eri{pq}{ia} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia} \notag
+\end{equation}
+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{equation}
+  \label{eq:GT_selfenergy}
+  \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{equation}
+with
+\begin{align}
+	\label{eq:GT_sERI}
+  \eri{pq}{\chi^{N+2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX{cd,m}{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY{kl,m}{N+2} \notag \\
+  \eri{pq}{\chi^{N-2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX{cd,m}{N-2}  + \sum_{k<l} \aeri{pq}{kl} \bY{kl,m}{N-2} \notag
+\end{align}
+The two RPA problems giving the eigenvectors needed to build the GW and GT self-energies are given in Appendix~\ref{sec:rpa}.
+
+%%%%%%%%%%%%%%%%%%%%%%
+\subsection{The unfolded equations}
+\label{sec:unfolded}
+%%%%%%%%%%%%%%%%%%%%%%
+
+Following Schirmer for the GF(2) case or Bintrim \etal, the non-linear quasi-particle equations for each approximations can be unfolded in larger linear problems
+\begin{equation}
+  \label{eq:unfolded_equation}
+  \bH  \bc_{(s)} = \epsilon_s \bc_{(s)}
+\end{equation}
+where $\bH$ depends on the approximation chosen for the self-energy.
+For the three approximations considered here, the three matrices $\bH$ share the general form
+\begin{equation}
+  \label{eq:unfolded_matrice}
+  \bH =
+  \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}
+The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}$ in the different cases is given below.
+\begin{itemize}
+\item \textbf{GF(2)}
+  \begin{align}
+    V^\text{2h1p}_{p,klc} & = \frac{1}{\sqrt{2}}\aeri{pc}{kl}
+    &
+      V^\text{2p1h}_{p,kcd} & = \frac{1}{\sqrt{2}}\aeri{pk}{dc} \\
+    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} \notag
+  \end{align}
+\item \textbf{GW}
+  \begin{align}
+    V^\text{2h1p}_{p,klc} & = \eri{pc}{kl}
+    &
+      V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc} \notag \\
+    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} \notag & &
+  \end{align}
+\item \textbf{GT}
+  \begin{align}
+    V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
+    &
+      V^\text{2p1h}_{p,kcd}&= \aeri{pk}{cd} \notag \\
+    C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac} & & \\
+    C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik} & & \notag
+  \end{align}
+\end{itemize}
+
+\textbf{\textcolor{red}{That would be nice to add electron-hole T matrix to see if it also correspond to one term that can be found in the CI below.}}
+
+\textbf{\textcolor{red}{That would be nice to add the self-energies of the various post-RPA correction.}}
+
+% =================================================================%
+\section{A first quantization approach to SRG for MBPT}
+\label{sec:matrix_srg}
+%=================================================================%
+
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Initial conditions}
 %%%%%%%%%%%%%%%%%%%%%%
@@ -383,7 +495,7 @@ A general upfolded MBPT matrix can be written as
   \label{eq:H_MBPT}
   H =
          \begin{pmatrix}
-         \bF & \bV{}{} \\
+         \bF{}{} & \bV{}{} \\
          \bV{}{\dagger} & \bC{}{} 
          \end{pmatrix}
 \end{equation}
@@ -392,7 +504,7 @@ Using SRG language, we define the diagonal and off-diagonal parts as
   \label{eq:H_MBPT_partitioning}
   H(0) =
   \begin{pmatrix}
-    \bF & \bO \\
+    \bF{}{} & \bO \\
     \bO & \bC{\text{d}}{} 
   \end{pmatrix}
   + \lambda
@@ -404,14 +516,16 @@ Using SRG language, we define the diagonal and off-diagonal parts as
 which gives the following conditions
 \begin{align}
   \bHd{0}(0) &=   \begin{pmatrix}
-    \bF & \bO \\
+    \bF{}{} & \bO \\
     \bO & \bC{\text{d}}{} 
-  \end{pmatrix}  & \bHod{0}(0) &= \bO \\
+  \end{pmatrix}  & \bHod{0}(0) &= \bO \notag \\
   \bHd{1}(0) &= \bO & \bHod{1}(0) &=   \begin{pmatrix}
     \bO & \bV{}{} \\
-    \bV{}{\dagger} & \bC{\text{od}}{}
+    \bV{}{\dagger} & \bC{\text{od}}{} \notag
   \end{pmatrix}  
 \end{align}
+At this point, we aren't sure if the off-diagonal part of $\bC{}{}$ should be included or not in the off-diagonal part of the Hamiltonian $\bH_\text{od}$.
+In the following derivation, we choose to do so because the other case can be obtained simply by taking $\bC{\text{od}}{} = \boldsymbol{0}$ and $\bC{\text{d}}{} = \bC{}{}$.
 
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Zeroth order Hamiltonian}
@@ -441,43 +555,22 @@ Now turning to the first-order contribution to the MBPT matrix, we start by comp
 \begin{align}
   &\bEta{1} = \comm{\bHd{0}}{\bHod{1}} \\
   &= \begin{pmatrix}
-    \bO & \bF^{(0)}\bV{}{(1)} - \bV{}{(1)}\bC{\text{d}}{(0)}\\
-    \bC{\text{d}}{(0)}\bV{}{(1),\dagger} - \bV{}{(1),\dagger} \bF^{(0)} & \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} - \bC{\text{od}}{(1)} \bC{\text{d}}{(0)}
+    \bO & \bF{}{(0)}\bV{}{(1)} - \bV{}{(1)}\bC{\text{d}}{(0)}\\
+    \bC{\text{d}}{(0)}\bV{}{(1),\dagger} - \bV{}{(1),\dagger} \bF{}{(0)} & \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} - \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag
   \end{pmatrix}  
 \end{align}
 
 \begin{align}
   \dv{\bH^{(1)}}{s} &= \comm{\bEta{1}}{\bHd{0}} =  \begin{pmatrix}
-        \dv{\bF^{(1)}}{s} & \dv{\bV{}{(1)}}{s} \\
+        \dv{\bF{}{(1)}}{s} & \dv{\bV{}{(1)}}{s} \\
         \dv{\bV{}{(1),\dagger}}{s} & \dv{\bC{}{(1)}}{s}
       \end{pmatrix}  \\
-  \dv{\bF^{(1)}}{s}  &= \bO  \Longleftrightarrow \color{red}{\boxed{\color{black}{\bF^{(1)}= \bO}}} \\
-  \dv{\bV{}{(1)}}{s} &= 2 \bF^{(0)}\bV{}{(1)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{\text{d}}{(0)})^2 \\
-  \dv{\bV{}{(1),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(1),\dagger}\bF^{(0)} - \bV{}{(1),\dagger}(\bF^{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(1),\dagger} \\
+  \dv{\bF{}{(1)}}{s}  &= \bO  \\
+  \dv{\bV{}{(1)}}{s} &= 2 \bF{}{(0)}\bV{}{(1)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{\text{d}}{(0)})^2 \\
+  \dv{\bV{}{(1),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(1),\dagger}\bF{}{(0)} - \bV{}{(1),\dagger}(\bF{}{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(1),\dagger} \\
   \dv{\bC{}{(1)}}{s}  &= 2 \bC{\text{d}}{(0)}\bC{\text{od}}{(1)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(1)} - \bC{\text{od}}{(1)}(\bC{\text{d}}{(0)})^2
 \end{align}
-The last two equations can be solved differently depending on the form of $\bF$ and $\bC{}{}$.
-\subsubsection*{Diagonal $\bC{}{(0)}$}
-In the following, upper case indices correspond to the 2h1p and 2p1h sectors while lower case indices correspond to the 1h and 1p sectors.  Also the $\Delta\eps_R$ corresponds to the diagonal elements of the 2h1p and 2p1h sectors.
-
-\begin{align}
-  (\dv{\bV{}{(1)}}{s})_{pQ} &= (2 \bF^{(0)}\bV{}{(1)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{\text{d}}{(0)})^2 )_{pQ}\\
-                            &= \sum_{rS} 2 f^{(0)}_{pr} v^{(1)}_{rS}c^{(0)}_{SQ} - \sum_{rs} f^{(0)}_{pr} f^{(0)}_{rs} v^{(1)}_{sQ} - \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS}c^{(0)}_{SQ} \\
-                            &=  \sum_{rS} 2 \epsilon^{(0)}_p\delta_{pr} v^{(1)}_{rS}\Delta\epsilon^{(0)}_Q\delta_{SQ} \\
-                            &- \sum_{rs} \epsilon^{(0)}_p\delta_{pr} \epsilon^{(0)}_r\delta_{rs} v^{(1)}_{sQ} \\
-                            &- \sum_{RS} v^{(1)}_{pR} \Delta\epsilon^{(0)}_R\delta_{RS}  \Delta\epsilon^{(0)}_Q\delta_{SQ} \\
-                            &= (2 \epsilon^{(0)}_p\Delta\epsilon^{(0)}_Q - (\epsilon^{(0)}_p)^2 - (\Delta\epsilon^{(0)}_Q )^2) v^{(1)}_{pQ} \\
-  \dv{v^{(1)}_{pQ}}{s}  &= - (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2 v^{(1)}_{pQ} \\
-  &\color{red}{\boxed{\color{black}{v^{(1)}_{pQ}(s) = v^{(1)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} }}} 
-\end{align}
-Note the close similarity with Evangelista's expressions for the off-diagonal part at first order!
-\begin{align}
-  (\dv{\bC{}{(1)}}{s})_{PQ} &= (2 \bC{\text{d}}{(0)}\bC{\text{od}}{(1)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(1)} - \bC{\text{od}}{(1)}(\bC{\text{d}}{(0)})^2)_{PQ} \\
-                            &= \sum_{RS} 2 c^{(0)}_{PR} c^{(1)}_{RS} c^{(0)}_{SQ} - c^{(0)}_{PR} c^{(0)}_{RS} c^{(1)}_{SQ} - c^{(1)}_{PR} c^{(0)}_{RS} c^{(0)}_{SQ} \\
-                            &= 2 \Delta\epsilon^{(0)}_Pc^{(1)}_{PQ}\Delta\epsilon^{(0)}_Q - (\Delta\epsilon^{(0)}_P)^2  c^{(1)}_{PQ} -  c^{(1)}_{PQ} (\Delta\epsilon^{(0)}_Q)^2 \\
-                            &= - (\Delta\epsilon^{(0)}_P - \Delta\epsilon^{(0)}_Q )^2 c^{(1)}_{PQ} \\
-  &\color{red}{\boxed{\color{black}{c^{(1)}_{PQ}(s) = c^{(1)}_{PQ}(0) e^{-s(\Delta\epsilon^{(0)}_P - \Delta\epsilon^{(0)}_Q )^2} }}} 
-\end{align}
+The last two equations can be solved differently depending on the form of $\bF{}{}$ and $\bC{}{}$.
 
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Second order Hamiltonian}
@@ -486,105 +579,124 @@ Note the close similarity with Evangelista's expressions for the off-diagonal pa
 Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive 
 \begin{align}
   &\bEta{2} = \comm{\bHd{0}}{\bHod{2}} + \comm{\bHd{1}}{\bHod{1}} \\
-  &= \comm{\bHd{0}}{\bHod{2}} \\
+  &= \comm{\bHd{0}}{\bHod{2}} \notag \\
   &= \begin{pmatrix}
-    \bO & \bF^{(0)}\bV{}{(2)} - \bV{}{(2)}\bC{\text{d}}{(0)}\\
-    \bC{\text{d}}{(0)}\bV{}{(2),\dagger} - \bV{}{(2),\dagger}\bF^{(0)} & \bC{\text{d}}{(0)} \bC{\text{od}}{(2)} - \bC{\text{od}}{(2)} \bC{\text{d}}{(0)}
+    \bO & \bF{}{(0)}\bV{}{(2)} - \bV{}{(2)}\bC{\text{d}}{(0)}\\
+    \bC{\text{d}}{(0)}\bV{}{(2),\dagger} - \bV{}{(2),\dagger}\bF{}{(0)} & \bC{\text{d}}{(0)} \bC{\text{od}}{(2)} - \bC{\text{od}}{(2)} \bC{\text{d}}{(0)} \notag
   \end{pmatrix}  
 \end{align}
 
 \begin{align}
-  &\dv{\bH^{(2)}}{s} = \comm{\bEta{2}}{\bHd{0}} + \comm{\bEta{1}}{\bHd{1}}  \\
+  \dv{\bH^{(2)}}{s} &= \comm{\bEta{2}}{\bHd{0}} + \comm{\bEta{1}}{\bHd{1}}  \\
   &=  \begin{pmatrix}
-        \dv{\bF^{(2)}}{s} & \dv{\bV{}{(2)}}{s} \\
+        \dv{\bF{}{(2)}}{s} & \dv{\bV{}{(2)}}{s} \\
         \dv{\bV{}{(2),\dagger}}{s} & \dv{\bC{}{(2)}}{s} 
-      \end{pmatrix} \\
-  \dv{\bF^{(2)}}{s}  &= \bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger}\\
+      \end{pmatrix} \notag \\
+  \dv{\bF{}{(2)}}{s}  &= \bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger}\\
   \dv{\bC{}{(2)}}{s}  &= 2 \bC{\text{d}}{(0)}\bC{\text{od}}{(2)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(2)} - \bC{\text{od}}{(2)}(\bC{\text{d}}{(0)})^2  \\
-  &-2 \bC{\text{d}}{(1)}\bC{\text{od}}{(0)}\bC{\text{d}}{(1)}- (\bC{\text{d}}{(1)})^2\bC{\text{od}}{(0)} - \bC{\text{od}}{(0)}(\bC{\text{d}}{(1)})^2 \notag \\
-  \dv{\bV{}{(2)}}{s} &= 2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
-  &- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag \\
-  \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(2),\dagger}\bF^{(0)} - \bV{}{(2),\dagger}(\bF^{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(2),\dagger} \\
-  &- 2 \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \bV{}{(1),\dagger} +  \bC{\text{od}}{(1)} \bV{}{(1),\dagger}  \bF^{(0)} + \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \notag
-\end{align}
-
-\begin{align}
- &(\dv{\bF^{(2)}}{s})_{pq}  = (\bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger})_{pq}  \notag \\
-                            &= \sum_{rS} f^{(0)}_{pr} v^{(1)}_{rS} v^{(1),\dagger}_{Sq} + \sum_{Rs}   v^{(1)}_{pR} v^{(1),\dagger}_{Rs} f^{(0)}_{sq} - 2\sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS} v^{(1),\dagger}_{Sq} \notag \\
-                            &= \sum_{S} \eps^{(0)}_{p} v^{(1)}_{pS} v^{(1)}_{qS} + \sum_{R} \eps^{(0)}_{q} v^{(1)}_{pR} v^{(1)}_{qR} - 2\sum_{R} \Delta\eps^{(0)}_R v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
-                            &= \sum_R (\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
- &= \sum_R (\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR}(0) v^{(1)}_{qR}(0) e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]} \notag \\
- &f^{(2)}_{pq}(s) = \notag \\
- &\color{red}{\boxed{\color{black}{- \sum_R \frac{\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \notag
-\end{align}
-
-\begin{align}
-  (\dv{\bV{}{(2)}}{s})_{pQ} &=  (2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
-  & - 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)})_{pQ} \notag \\
-  v^{(2)}_{pQ}(s) &= v^{(2)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} + \text{Non-homogeneous solution} \notag \\
-  v^{(2)}_{pQ}(s) &= \text{Non-homogeneous solution}
+  &-2 \bC{\text{d}}{(1)}\bC{\text{od}}{(0)}\bC{\text{d}}{(1)} + (\bC{\text{d}}{(1)})^2\bC{\text{od}}{(0)} + \bC{\text{od}}{(0)}(\bC{\text{d}}{(1)})^2 \notag \\
+  &+ \bC{\text{d}}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bC{\text{d}}{(0)} - 2 \bV{}{(1)}\bF{}{(0)}\bV{}{(1),\dagger} \notag \\
+  \dv{\bV{}{(2)}}{s} &= 2 \bF{}{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
+  &- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF{}{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag \\
+  \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(2),\dagger}\bF{}{(0)} - \bV{}{(2),\dagger}(\bF{}{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(2),\dagger} \\
+  &- 2 \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \bV{}{(1),\dagger} +  \bC{\text{od}}{(1)} \bV{}{(1),\dagger}  \bF{}{(0)} + \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \notag
 \end{align}
 
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Downfolding the SRG-transformed matrix}
 %%%%%%%%%%%%%%%%%%%%%%
 
-Now that we obtained the SRG-transformed Hamiltonian to a given order we can downfold it back to obtain a SRG-renormalized self-energy up to a given order.
+In order to choose what to do with $\bC{\text{od}}{}$ we look at the downfolded SRG quasiparticle equation.
+
 \begin{equation}
   \label{eq:H_SRGMBPT}
   H(s) =
   \begin{pmatrix}
-    \bF^{(0)}(0) + \bF^{(2)}(s) & \bV{}{(1)}(s)  + \bV{}{(2)}(s) \\
+    \bF{}{(0)}(0) + \bF{}{(2)}(s) & \bV{}{(1)}(s)  + \bV{}{(2)}(s) \\
     \bV{}{(1),\dagger}(s)  + \bV{}{(2),\dagger}(s) & \bC{}{(0)}(0) +\bC{}{(2)}(s) 
   \end{pmatrix}
 \end{equation}
 
-\begin{equation}
-\left\{
-  \begin{aligned}
-    (\bF^{(0)}(0) + \bF^{(2)}(s)) \bR^{1h/1p} + \bV{}{(1)}(s) \bR^{2h1p/2p1h} &= \omega \bR^{1h/1p} \\
-    \bV{}{(1),\dagger}(s) \bR^{1h/1p} + (\bC{}{(0)}(0) +\bC{}{(2)}(s) ) \bR^{2h1p/2p1h}&=   \omega \bR^{2h1p/2p1h}             
-  \end{aligned}
- \right.
-\end{equation}
-
 \begin{widetext}
   \begin{equation}
-    (\bF^{(0)}(0) + \bF^{(2)}(s)) + (\bV{}{(1)}(s) + \bV{}{(2)}(s)) (\omega \mathbb{1} - \bC{\text{d}}{(0)}(0) - \bC{\text{od}}{(1)}(s)  -\bC{}{(2)}(s)  )^{-1}  (\bV{}{(1),\dagger}(s) + \bV{}{(2),\dagger}(s)) \bR^{1h/1p} = \omega \bR^{1h/1p}   
+    \left\{
+      \begin{aligned}
+        (\bF{}{(0)}(0) + \bF{}{(2)}(s)) \bR^{1h/1p} + (\bV{}{(1)}(s) + \bV{}{(2)}(s)) \bR^{2h1p/2p1h} &= \omega \bR^{1h/1p} \\
+        (\bV{}{(1),\dagger}(s) + \bV{}{(2),\dagger}(s))  \bR^{1h/1p} + (\bC{}{(0)}(0) +\bC{}{(2)}(s) ) \bR^{2h1p/2p1h}&=   \omega \bR^{2h1p/2p1h}             
+      \end{aligned}
+    \right.
+  \end{equation}
+  
+  \begin{equation}
+    (\bF{}{(0)}(0) + \bF{}{(2)}(s)) + (\bV{}{(1)}(s) + \bV{}{(2)}(s)) (\omega \mathbb{1} - \bC{\text{d}}{(0)}(0) - \bC{\text{od}}{(1)}(s)  -\bC{}{(2)}(s)  )^{-1}  (\bV{}{(1),\dagger}(s) + \bV{}{(2),\dagger}(s)) \bR^{1h/1p} = \omega \bR^{1h/1p}   
+  \end{equation}
+
+  If we want to truncate the quasiparticle equation to the second order we obtain 
+
+    \begin{equation}
+    (\bF{}{(0)}(0) + \bF{}{(2)}(s)) + \bV{}{(1)}(s)(\omega \mathbb{1} - \bC{\text{d}}{(0)}(0))^{-1} \bV{}{(1),\dagger}(s)  \bR^{1h/1p} = \omega \bR^{1h/1p}   
   \end{equation}
 \end{widetext}
+So if we choose to put the off-diagonal part of $\bC{}{}$ in the off-diagonal $\bH{}{}$ we see that the off diagonal part of $\bC{}{}$ is not present in the second order quasi-particle equation.
+We believe that this is not desirable.
+In the following, we will integrate order by order the differential equations obtained above in the case $\bC{\text{od}}{} = \boldsymbol{0}$ and $\bC{\text{d}}{} = \bC{}{}$.
+The expression in the other case are given in Appendix~\ref{sec:diagC}.
+
+%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Integrating order by order}
+%%%%%%%%%%%%%%%%%%%%%%
+
+In the following, upper case indices correspond to the 2h1p and 2p1h sectors while lower case indices correspond to the 1h and 1p sectors.  Also the $\Delta\eps_R$ corresponds to the diagonal elements of the 2h1p and 2p1h sectors.
+
+\subsubsection{First order}
 
-Therefore we have to solve the following equation
 \begin{align}
-  &(\tilde{\bF} + \tilde{\boldsymbol{\Sigma}}(\omega)) \bX{}{} = \omega \bX \\
-  &\tilde{\bF} =\bF^{(0)}(0) + \bF^{(2)}(s) + \dots \\
-  &\tilde{\boldsymbol{\Sigma}}(\omega) = \tilde{\bV{}{}}(s) (\omega \mathbb{1} - \tilde{\bC{}{}}(s) )^{-1} \tilde{\bV{}{}}^\dagger(s) \\
-  &\tilde{\bC{}{}}(s) = \bC{\text{d}}{(0)}(0) + \bC{\text{od}}{(1)}(s) + \bC{}{(2)}(s)  + \dots \\
-  &\tilde{\bV{}{}}(s) = \bV{}{(1)}(s) + \bV{}{(2)}(s) + \dots
+  \dv{\bF{}{(1)}}{s}  &= \bO  \Longleftrightarrow \bF{}{(1)}(s) = \bF{}{(1)}(0) \Longleftrightarrow \color{red}{\boxed{\color{black}{\bF{}{(1)}(s)= \bO}}} \\
+  \dv{\bV{}{(1)}}{s} &= 2 \bF{}{(0)}\bV{}{(1)}\bC{}{(0)} - (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 \\
+  \dv{\bV{}{(1),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(1),\dagger}\bF{}{(0)} - \bV{}{(1),\dagger}(\bF{}{(0)})^2 - (\bC{}{(0)})^2\bV{}{(1),\dagger} \\
+  \dv{\bC{}{(1)}}{s}  &= \bO  \Longleftrightarrow \bC{}{(1)}(s) = \bC{}{(1)}(0) \Longleftrightarrow \color{red}{\boxed{\color{black}{\bC{}{(1)}(s)= \bO}}} 
+\end{align}
+The differential equation for the coupling blocks can be solved in the GF(2) case because in this case $\bC{}{}$ is diagonal (see Appendix~\ref{sec:diagC}).
+However, in the general case this matrix differential equation is not trivial to solve.
+In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_p$.
+\begin{align}
+  \dv{\bV{}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \\
+  \dv{\bV{}{(1),\dagger}}{s} &=  (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger} \\
+\end{align}
+These matrix differential equations can be solved if we know how to diagonalize $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2- (\bC{}{(0)})^2$.
+We know how to diagonalize $\bC{}{(0)}$ so we know how to diagonalize polynomial of $\bC{}{(0)}$.
+
+\textcolor{red}{\textbf{TODO Give analytical expression for the different cases.}}
+
+\subsubsection{Second order}
+
+\begin{align}
+  \dv{\bF{}{(2)}}{s}  &= \bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger} \\
+  \dv{\bC{}{(2)}}{s}  &= \bC{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bC{}{(0)} - 2 \bV{}{(1)}\bF{}{(0)}\bV{}{(1),\dagger} \\
+  \dv{\bV{}{(2)}}{s} &= 2 \bF{}{(0)}\bV{}{(2)}\bC{}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{}{(0)})^2 \\
+  \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(2),\dagger}\bF{}{(0)} - \bV{}{(2),\dagger}(\bF{}{(0)})^2 - (\bC{}{(0)})^2\bV{}{(2),\dagger}
 \end{align}
 
-Note that the inverse of $\omega\mathbb{1} - \tilde{\bC{}{}}(s)$ does not need to be approximated using a Taylor expansion.
-Indeed, $\tilde{\bC{}{}}(s)$ can be used to define a renormalized RPA problem to diagonalize instead of the usual one.
-We need to find a way to truncate the above quasi-particle equation.
-For the moment, I'm not sure if we should truncate according to the quasi-particle equation or truncate directly the subblocks of the unfolded matrix\dots
+The two first equations can be solved by simple integrations.
+The two last equations admit the same solutions as the first order coupling blocks differential equations with different initial conditions.
 
-I'm not sure if this valid from a perturbation theory point of view but truncating each terms after their first non-zero correction could be handy.
-This would give first order for the coupling terms and second order for the diagonal ones.
+% =================================================================%
+\section{Towards second quantized effective Hamiltonians for MBPT?}
+\label{sec:second_quant_mbpt}
+%=================================================================%
+
+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 like configuration interaction (CI) and coupled-cluster (CC) which are naturally expressed in 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 formalism.
+This is the aim of this section.
 
 %%%%%%%%%%%%%%%%%%%%%%
-\subsection{The SRG(2) quasi-particle equations}
+\subsection{The IP/EA CI}
+\label{sec:ip_ea_ci}
 %%%%%%%%%%%%%%%%%%%%%%
 
-In this section, we report the GF(2), GW and GT quasi-particle equations.
-
-We start with the GF(2) equation because it does not have non-diagonal $\bC{}{}$ contributions, this gives the following renormalized quasi-particle equation 
-\begin{widetext}
-  \begin{align}
-    \label{eq:selfenergies}
-    \tilde{f}_{pq} + \tilde{\Sigma}_{pq}^{GF(2)}(\omega) &= \delta_{pq}\eps_p + - \sum_R \frac{\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]}  \\
-                                                         &+\sum_{klc} \frac{\aeri{pc}{kl}\aeri{qc}{kl}}{\omega + \eps _c -\eps_k -\eps_l - \ii \eta}e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_{ckl} )^2} e^{-s(\epsilon^{(0)}_q - \Delta\epsilon^{(0)}_{ckl} )^2}  + \sum_{kcd} \frac{\aeri{pk}{cd}\aeri{qk}{cd}}{\omega + \eps _k -\eps_c -\eps_d + \ii \eta} e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_{kcd} )^2} e^{-s(\epsilon^{(0)}_q - \Delta\epsilon^{(0)}_{kcd} )^2} \notag
-  \end{align}
-\end{widetext}
+We start by expressing the electronic Hamiltonian in the IP/EA basis to compare its expressions to the matrices of Sec.~\ref{sec:unfolded}.
 
 \appendix
 
@@ -636,4 +748,50 @@ $$
 
 In these equations $P(r s)$ is the antisymmetric permutation operator.
 
+%=================================================================%
+\section{The ph- and pp-RPA problems}
+\label{sec:rpa}
+%=================================================================%
+
+%=================================================================%
+\section{Perturbative matrix coefficients for $C^{(0)}$ diagonal}
+\label{sec:diagC}
+%=================================================================%
+
+\begin{align}
+  (\dv{\bV{}{(1)}}{s})_{pQ} &= (2 \bF{}{(0)}\bV{}{(1)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{\text{d}}{(0)})^2 )_{pQ}\\
+                            &= \sum_{rS} 2 f^{(0)}_{pr} v^{(1)}_{rS}c^{(0)}_{SQ} - \sum_{rs} f^{(0)}_{pr} f^{(0)}_{rs} v^{(1)}_{sQ} - \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS}c^{(0)}_{SQ} \\
+                            &=  \sum_{rS} 2 \epsilon^{(0)}_p\delta_{pr} v^{(1)}_{rS}\Delta\epsilon^{(0)}_Q\delta_{SQ} \\
+                            &- \sum_{rs} \epsilon^{(0)}_p\delta_{pr} \epsilon^{(0)}_r\delta_{rs} v^{(1)}_{sQ} \\
+                            &- \sum_{RS} v^{(1)}_{pR} \Delta\epsilon^{(0)}_R\delta_{RS}  \Delta\epsilon^{(0)}_Q\delta_{SQ} \\
+                            &= (2 \epsilon^{(0)}_p\Delta\epsilon^{(0)}_Q - (\epsilon^{(0)}_p)^2 - (\Delta\epsilon^{(0)}_Q )^2) v^{(1)}_{pQ} \\
+  \dv{v^{(1)}_{pQ}}{s}  &= - (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2 v^{(1)}_{pQ} \\
+  &\color{red}{\boxed{\color{black}{v^{(1)}_{pQ}(s) = v^{(1)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} }}} 
+\end{align}
+Note the close similarity with Evangelista's expressions for the off-diagonal part at first order!
+\begin{align}
+  (\dv{\bC{}{(1)}}{s})_{PQ} &= (2 \bC{\text{d}}{(0)}\bC{\text{od}}{(1)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(1)} - \bC{\text{od}}{(1)}(\bC{\text{d}}{(0)})^2)_{PQ} \\
+                            &= \sum_{RS} 2 c^{(0)}_{PR} c^{(1)}_{RS} c^{(0)}_{SQ} - c^{(0)}_{PR} c^{(0)}_{RS} c^{(1)}_{SQ} - c^{(1)}_{PR} c^{(0)}_{RS} c^{(0)}_{SQ} \\
+                            &= 2 \Delta\epsilon^{(0)}_Pc^{(1)}_{PQ}\Delta\epsilon^{(0)}_Q - (\Delta\epsilon^{(0)}_P)^2  c^{(1)}_{PQ} -  c^{(1)}_{PQ} (\Delta\epsilon^{(0)}_Q)^2 \\
+                            &= - (\Delta\epsilon^{(0)}_P - \Delta\epsilon^{(0)}_Q )^2 c^{(1)}_{PQ} \\
+  &\color{red}{\boxed{\color{black}{c^{(1)}_{PQ}(s) = c^{(1)}_{PQ}(0) e^{-s(\Delta\epsilon^{(0)}_P - \Delta\epsilon^{(0)}_Q )^2} }}} 
+\end{align}
+
+\begin{align}
+ &(\dv{\bF{}{(2)}}{s})_{pq}  = (\bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger})_{pq}  \notag \\
+                            &= \sum_{rS} f^{(0)}_{pr} v^{(1)}_{rS} v^{(1),\dagger}_{Sq} + \sum_{Rs}   v^{(1)}_{pR} v^{(1),\dagger}_{Rs} f^{(0)}_{sq} - 2\sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS} v^{(1),\dagger}_{Sq} \notag \\
+                            &= \sum_{S} \eps^{(0)}_{p} v^{(1)}_{pS} v^{(1)}_{qS} + \sum_{R} \eps^{(0)}_{q} v^{(1)}_{pR} v^{(1)}_{qR} - 2\sum_{R} \Delta\eps^{(0)}_R v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
+                            &= \sum_R (\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
+ &= \sum_R (\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR}(0) v^{(1)}_{qR}(0) e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]} \notag \\
+ &f^{(2)}_{pq}(s) = \notag \\
+ &\color{red}{\boxed{\color{black}{- \sum_R\frac{ v^{(1)}_{pR}(0) v^{(1)}_{qR}(0)  (\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R)}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \notag
+\end{align}
+
+\begin{align}
+  (\dv{\bV{}{(2)}}{s})_{pQ} &=  (2 \bF{}{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
+  & - 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF{}{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)})_{pQ} \notag \\
+  v^{(2)}_{pQ}(s) &= v^{(2)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} + \text{Non-homogeneous solution} \notag \\
+  v^{(2)}_{pQ}(s) &= \text{Non-homogeneous solution}
+\end{align}
+
 \end{document}
\ No newline at end of file