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Pierre-Francois Loos 2022-10-13 10:54:56 +02:00
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CCvsMBPT.tex
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@ -454,7 +454,7 @@ Within the perturbative $GW$ scheme (commonly known as $G_0W_0$), the quasiparti
\cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007}
Partial self-consistency can be attained via the \textit{``eigenvalue''} self-consistent $GW$ (ev$GW$) \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018} or the quasiparticle self-consistent $GW$ (qs$GW$) \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} schemes.
In the most general setting, the quasiparticle energies and their corresponding orbitals are obtained by diagonalizing the so-called non-linear and frequency-dependent quasiparticle equation
In the most general setting, the quasiparticle energies and their corresponding orbitals are obtained self-consistently by diagonalizing the so-called non-linear and frequency-dependent quasiparticle equation
\begin{equation}
\label{eq:GW}
\qty[ \be{}{} + \bSig{}{\GW}\qty(\omega = \e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW}
@ -470,7 +470,7 @@ where $\be{}{}$ is a diagonal matrix gathering the HF orbital energies and the e
\end{equation}
Because both the left- and right-hand sides of Eq.~\eqref{eq:GW} depend on $\e{p}{\GW}$, this equation has to be solved iteratively via a self-consistent procedure.
As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the frequency-dependent $GW$ equations can be recast as a larger set of frequency-independent equations, which reads in the Tamm-Dancoff approximation
As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the quasiparticle equation \eqref{eq:GW} can be recast as a larger set of linear and frequency-independent equations (that still needs to be solved self-consistently), which reads in the Tamm-Dancoff approximation
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}