From fbbfb98d77be7a8fdb1b008ced0071d19bbaf7d6 Mon Sep 17 00:00:00 2001 From: pfloos Date: Tue, 11 Oct 2022 20:21:27 +0200 Subject: [PATCH] remove one comment --- CCvsMBPT.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/CCvsMBPT.tex b/CCvsMBPT.tex index 66888de..b41b480 100644 --- a/CCvsMBPT.tex +++ b/CCvsMBPT.tex @@ -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