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Pierre-Francois Loos 2022-10-11 20:21:27 +02:00
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CCvsMBPT.tex
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@ -168,7 +168,7 @@ In particular, it may provide a path for the computation of ground- and excited-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The random-phase approximation (RPA), introduced by Bohm and Pines \cite{Bohm_1951,Pines_1952,Bohm_1953} in the context of the uniform electron gas, \cite{Loos_2016} is a quasibosonic approximation where one treats fermion products as bosons.
In the particle-hole (ph) channel, which is quite popular in the electronic structure community, \cite{Ren_2012,Chen_2017} particle-hole fermionic excitations and deexcitations are assumed to be bosons.
Because \ant{ph-}RPA corresponds to a resummation of all ring diagrams, it is adequate in the high-density (or weakly correlated) regime and catch effectively long-range correlation effects (such as dispersion). \cite{Gell-Mann_1957,Nozieres_1958} \ant{Maybe explain more this last sentence, I feel like it's a bit fast.}
Because ph-RPA corresponds to a resummation of all ring diagrams, it is adequate in the high-density (or weakly correlated) regime and catch effectively long-range correlation effects (such as dispersion). \cite{Gell-Mann_1957,Nozieres_1958} \ant{Maybe explain more this last sentence, I feel like it's a bit fast.}
Roughly speaking, the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} of many-body perturbation theory \cite{Martin_2016} can be seen as a cheap and efficient way of introducing correlation in order to go \textit{beyond} RPA physics.
In the ph channel, BSE is commonly performed on top of a $GW$ calculation \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} from which one extracts the quasiparticle energies as well as the dynamically-screened Coulomb potential $W$.
@ -400,7 +400,7 @@ with
B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
\end{align}
\end{subequations}
\ant{We maybe we can remove the above equations and just say that it's Eq.~\eqref{eq:RPA}\eqref{eq:A_RPAx}\eqref{eq:B_RPAx} without double bars. If we keep the equations, I think the GW superscript on the energies is wrong.}
%\ant{We maybe we can remove the above equations and just say that it's Eq.~\eqref{eq:RPA}\eqref{eq:A_RPAx}\eqref{eq:B_RPAx} without double bars. If we keep the equations, I think the GW superscript on the energies is wrong.}
As readily seen in Eqs.~\eqref{eq:A_RPAx}, \eqref{eq:B_RPAx}, \eqref{eq:A_BSE} and \eqref{eq:B_BSE}, the only difference between RPAx and BSE lies in the definition of the matrix elements, where one includes, via the presence of the $GW$ quasiparticle energies in the one-body terms and the screening of the electron-electron interaction [see Eq.~\eqref{eq:W}] in the two-body terms, correlation effects at the BSE level.
Therefore, following the derivation detailed in Sec.~\ref{sec:RPAx}, one can show that the BSE correlation energy obtained using the trace formula