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ok with Sec II

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Pierre-Francois Loos 2023-02-20 10:41:07 +01:00
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Manuscript/SRGGW.tex
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@ -205,13 +205,13 @@ with
B_{ia,jb} & = \eri{ij}{ab},
\end{align}
\end{subequations}
and
and where
\begin{equation}
\braket{pq}{rs} = \iint \frac{\SO{p}(\bx_1) \SO{q}(\bx_2)\SO{r}(\bx_1) \SO{s}(\bx_2) }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2
\end{equation}
are bare two-electron integrals in the spin-orbital basis.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problem defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$).
As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
@ -231,7 +231,6 @@ These solutions can be characterized by their spectral weight given by the renor
\end{equation}
The solution with the largest weight is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions).
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined.
%These additional solutions with large weights are the previously mentioned intruder states.
One obvious drawback of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
@ -263,7 +262,7 @@ One of the main results of the present manuscript is the derivation, from first
Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
If it is not the case, the self-consistent qs$GW$ scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
The satellites causing convergence problems are the above-mentioned intruder states. \cite{Monino_2022}
One can deal with them by introducing \textit{ad hoc} regularizers.