Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW
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commit
1af1b6ea33
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@ -122,12 +122,12 @@
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\newcommand{\xc}{\text{xc}}
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\newcommand{\x}{\text{x}}
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\newcommand{\GW}{\text{GW}}
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\newcommand{\GW}{GW}
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\newcommand{\GF}{\text{GF(2)}}
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\newcommand{\GT}{\text{$GT$}}
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\GT}{GT}
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\newcommand{\evGW}{\text{ev}$GW$}
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\newcommand{\qsGW}{\text{qs}GW}
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\newcommand{\GOWO}{G_0W_0}
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%%% Notations %%%
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@ -19,6 +19,9 @@
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\ant}[1]{\textcolor{green}{#1}}
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% addresses
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@ -274,55 +277,61 @@ The central equation of MBPT in practice is the following
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\label{eq:quasipart_eq}
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\bF{}{} + \bSig(\omega) = \omega \mathbb{1}.
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\end{equation}
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However, in order to use it we need to rely on approximations of the self-energy $\bSig(\omega)$.
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\PFL{Not quite. You're missing the eigenvectors to make it a non-linear eigenvalue problem.}
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However, in order to use it we need to rely on approximations of the dynamical self-energy $\bSig(\omega)$.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Self-energies and quasiparticle equations}
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\label{sec:folded}
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%%%%%%%%%%%%%%%%%%%%%%
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In the following, we will focus on the GF(2), GW and GT approximations.
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In the following, we will focus on the GF(2), $GW$ and $GT$ approximations.
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The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory.
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\begin{align}
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\begin{equation}
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\label{eq:GF2_selfenergy}
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\Sigma_{pq}^{GF(2)}(\omega) &= \sum_{ija} \frac{W_{pa,ij}W_{qa,ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\
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&+ \sum_{iab} \frac{W_{pi,ab}W_{qi,ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag
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\end{align}
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\Sigma_{pq}^{\text{GF(2)}}(\omega)
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= \sum_{ija} \frac{W_{pa,ij}^{\text{GF(2)}}W_{qa,ij}^{\text{GF(2)}}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta}
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+ \sum_{iab} \frac{W_{pi,ab}^{\text{GF(2)}}W_{qi,ab}^{\text{GF(2)}}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta}
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\end{equation}
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with
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\begin{equation}
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\label{eq:GF2_sERI}
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W^{\GF}_{pq,rs}= \frac{1}{\sqrt{2}}\aeri{pq}{rs}
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\end{equation}
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On the other hand, the GW self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy.
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On the other hand, the $GW$ self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy.
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\begin{equation}
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\label{eq:GW_selfenergy}
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\Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{W_{pi,v} W_{qi,v}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{av} \frac{W_{pa,v}W_{qa,v}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag
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\Sigma_{pq}^{\GW}(\omega)
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= \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta}
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+ \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag
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\end{equation}
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with
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\begin{equation}
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\label{eq:GW_sERI}
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W_{pq,v}^\GW = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v}^{\dRPA} + \bY_{v}^{\dRPA} )_{ia}
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W_{pq,v}^{\GW} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v}^{\dRPA} + \bY_{v}^{\dRPA} )_{ia}
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\end{equation}
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Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA
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Finally, the $GT$ approximation corresponds to another approximation to the polarizability than in $GW$, namely the one coming from pp-hh-RPA
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The corresponding self-energies read as
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\begin{equation}
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\label{eq:GT_selfenergy}
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\Sigma_{pq}^{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \epsilon_i - \Omega_{m}^{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \epsilon_a - \Omega_{m}^{N-2} - \ii \eta} \notag
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\Sigma_{pq}^{\GT}(\omega)
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= \sum_{iv} \frac{W_{pi,v}^{N+2} W_{qi,v}^{N+2}}{\omega + \epsilon_i - \Omega_{v}^{N+2} + \ii \eta}
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+ \sum_{av} \frac{W_{pa,v}^{N-2} W_{qa,v}^{N-2}}{\omega + \epsilon_a - \Omega_{v}^{N-2} - \ii \eta} \notag
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\end{equation}
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with
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\begin{align}
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\label{eq:GT_sERI}
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\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 \\
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\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
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W_{pq,v}^{N+2} & = \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N+2} \notag \\
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W_{pq,v}^{N-2} & = \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N-2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N-2} \notag
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\end{align}
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The two RPA problems giving the eigenvectors needed to build the GW and GT self-energies are given in Appendix~\ref{sec:rpa}.
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The two RPA problems giving the eigenvectors needed to build the $GW$ and $GT$ self-energies are given in Appendix \ref{sec:rpa}.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The unfolded equations}
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\label{sec:unfolded}
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%%%%%%%%%%%%%%%%%%%%%%
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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
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Following Schirmer for the GF(2) case or Bintrim \etal, the non-linear quasi-particle equations for each approximation can be unfolded in larger linear problems
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\begin{equation}
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\label{eq:unfolded_equation}
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\bH \bc_{(s)} = \epsilon_s \bc_{(s)}
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@ -344,33 +353,37 @@ The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}
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\begin{align}
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\label{eq:GF2_unfolded}
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V^\text{2h1p}_{p,klc} & = \frac{1}{\sqrt{2}}\aeri{pc}{kl}
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&
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V^\text{2p1h}_{p,kcd} & = \frac{1}{\sqrt{2}}\aeri{pk}{dc} \\
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\\
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V^\text{2p1h}_{p,kcd} & = \frac{1}{\sqrt{2}}\aeri{pk}{dc}
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\\
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C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
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&
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C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd} \notag
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\\
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C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd}
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\end{align}
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\item \textbf{GW}
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\begin{align}
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\label{eq:GW_unfolded}
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V^\text{2h1p}_{p,klc} & = \eri{pc}{kl}
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&
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc} \notag \\
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C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} & &
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\\
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C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd} \notag & &
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}
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\\
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C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik}
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\\
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C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd}
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\end{align}
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\item \textbf{GT}
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\begin{align}
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\label{eq:GT_unfolded}
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V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
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&
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V^\text{2p1h}_{p,kcd}&= \aeri{pk}{cd} \notag \\
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C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac} & & \\
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C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik} & & \notag
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\\
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V^\text{2p1h}_{p,kcd} & = \aeri{pk}{cd}
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\\
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C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac}
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\\
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C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik}
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\end{align}
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\end{itemize}
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The downfolding procedure to obtain the GW self-energy is derived in details in Appendix~\ref{sec:downfolding}.
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The downfolding procedure to obtain the $GW$ self-energy is derived in details in Appendix~\ref{sec:downfolding}.
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\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.}}
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