put back equations

Manuscript/SRGGW.tex

@@ -363,7 +363,21 @@ Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of
 %  \end{pmatrix}
 %  \boldsymbol{\epsilon},
 \end{equation}
-%where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, 
+% where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies,
+where the 2h1p and 2p1h matrix elements are
+\begin{subequations}
+  \begin{align}
+    C^\text{2h1p}_{i\nu,j\mu} & =  \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
+    \\
+    C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu},
+  \end{align}
+\end{subequations}
+and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
+\begin{align}
+  W^\text{2h1p}_{p,i\nu} & = W_{pi}^{\nu},
+  &
+    W^\text{2p1h}_{p,a\nu} & = W_{pa}^{\nu}.
+\end{align}
 
 The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021} 
 \begin{equation}