Here, for the sake of simplicity, we consider a Hartree-Fock (HF) starting point.
Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
\begin{equation}
\label{eq:qp_eq}
\omega = \eps{p}{\HF} + \SigC{p}(\omega)
\end{equation}
where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads
Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by
\begin{equation}
\ERI{pq}{m} = \sum_{ia}\ERI{pq}{ia} X_{ia,m}^\RPA
\end{equation}
where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are respectively the $m$th eigenvalue and eigenvector of the RPA problem in the Tamm-Dancoff approximation, \ie,
As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{}$ and their corresponding weight is given by the value of the so-called renormalization factor
The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
is downfolded on the 1h and 1p sectors via their interaction with the 2h1p and 2p1h sectors:
The size of this eigenvalue problem is $N =1+ N^\text{2h1p}+ N^\text{2p1h}=1+ O^2 V + O V^2$.
Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie,
\begin{equation}
Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
\end{equation}
It is important to understand that diagonalizing $\bH^{(p)}$ in Eq.~\eqref{eq:Hp} is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
The main difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites.
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\section{Introducing regularized $GW$ methods}
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One way to hamper such issues is to resort to regularization of the $GW$ self-energy.
Of course, the way of regularizing the self-energy is not unique but here we consider 3 different ways directly imported from MP2 theory.
This helps greatly convergence for (partially) self-consistent $GW$ methods.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
