saving work in appendix

Manuscript/SRGGW.tex

@@ -186,6 +186,7 @@ where $\eta$ is a positive infinitesimal and the screened two-electron integrals
 \end{equation}
 where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as 
 \begin{equation}
+  \label{eq:full_dRPA}
   \mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) =  \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
 \end{equation}
 with
@@ -669,16 +670,11 @@ The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq
   \label{eq:GWnonTDA_sERI}
   W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
 \end{equation}
-where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem defined as
+where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem obtained by setting $\bB= \bO$ in Eq.~\eqref{eq:full_dRPA}, \ie
 \begin{equation}
-  \label{eq:full_dRPA}
+  \label{eq:TDA_dRPA}
-  \bA \bX = \bX \boldsymbol{\Omega}
+  \bA \bX = \bX \boldsymbol{\Omega}.
 \end{equation}
-with
-\begin{align}
-  A_{ia,jb} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj} \\
-\end{align}
-$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues.
 
 Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
 However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
@@ -705,23 +701,7 @@ However, because we will eventually downfold again the upfolded matrix, we can u
   \boldsymbol{\epsilon},
 \end{equation}
 which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
-
+Within the TDA the renormalized matrix elements have the same $s$ dependence as in the RPA case but the $s=0$ screened integrals $W_{p,q\nu}$ and eigenvalues $\Omega_\nu$ are replaced by the ones of Eq.~\eqref{eq:GWnonTDA_sERI} and \eqref{eq:TDA_dRPA}, respectively.
-
-where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
-\begin{subequations}
-  \begin{align}
-    D^\text{2h1p}_{ija,klc} & = \delta_{ik}\delta_{jl} \delta_{ac} \qty[\epsilon_i - \Omega_{ja}] ,
-    \\
-    D^\text{2p1h}_{iab,kcd} & = \delta_{ik}\delta_{ac} \delta_{bd} \qty[\epsilon_a + \Omega_{ib}] ,
-  \end{align}
-\end{subequations}
-and the corresponding coupling blocks read
-\begin{align}
-  W^\text{2h1p}_{p,klc} & = \sum_{ia}\eri{pi}{ka} \qty( \bX_{lc} + \bY_{lc})_{ia} \\
-    W^\text{2p1h}_{p,kcd} & = \sum_{ia}\eri{pi}{ca} \qty( \bX_{kd} + \bY_{kd})_{ia}
-\end{align}
-
-Using the SRG on this matrix instead of Eq.~\eqref{eq:GWlin} gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
 
 % %%%%%%%%%%%%%%%%%%%%%%
 % \section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}