put back equations

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Antoine Marie 2023-05-12 13:23:10 +02:00
parent aac34cff9a
commit 74b42641df

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@ -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}