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Notes/MBPTSecondQuant.tex
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\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}

422
Notes/PerturbativeAnalysis.tex → Notes/Notes.tex
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@ -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)}}
@ -38,6 +40,7 @@
% methods
\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}