From 5bcfaf48dc4df7bdaff25c89af118e7688164b81 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 17 Nov 2022 13:40:34 +0100 Subject: [PATCH] minor corrections b4 resubmission --- CCvsMBPT.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/CCvsMBPT.tex b/CCvsMBPT.tex index 533bc01..ed990cd 100644 --- a/CCvsMBPT.tex +++ b/CCvsMBPT.tex @@ -421,7 +421,7 @@ can be equivalently obtained via a set of rCCD-like amplitude equations, where o \end{multline} with $\Delta_{ijab}^{\GW} = \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW} - \e{j}{\GW}$. Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these approximate CCD amplitude equations can be solved with $\order*{N^6}$ cost via the definition of appropriate intermediates. -As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE correlation energy do exist, \footnote{\alert{To the best of our knowledge, the trace (or plasmon) formula has been introduced first by Sawada \cite{Sawada_1957a} to calculate the correlation energy of the uniform electron gas as an alternative to the Gell-Mann-Brueckner formulation \cite{Gell-Mann_1957} where one integrates along the adiabatic connection path. +As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE correlation energy do exist, \footnote{\alert{To the best of our knowledge, the trace (or plasmon) formula has been first introduced by Sawada \cite{Sawada_1957a} to calculate the correlation energy of the uniform electron gas as an alternative to the Gell-Mann-Brueckner formulation \cite{Gell-Mann_1957} where one integrates along the adiabatic connection path. More precisely, the trace formula can be justified via the introduction of a quadratic Hamiltonian made of boson transition operators (quasiboson approximation). See Ref.~\onlinecite{Li_2020} for more details}.} either based on the plasmon formula \cite{Li_2020,Li_2021,DiSabatino_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e,Berger_2021,DiSabatino_2021} @@ -622,7 +622,7 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $ \bSig{}{\GW} = \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} \end{equation} \alert{Due to the non-linear nature of these equations, the iterative procedure proposed in Eqs.~\eqref{eq:t_2h1p_update} and \eqref{eq:t_2p1h_update} can potentially converge to satellite solutions. -This is also the case at the CC level when one relies on more elaborated algorithm to converge the amplitude equations to higher-energy solutions. \cite{Piecuch_2000,Mayhall_2010,Lee_2019,Kossoski_2021,Marie_2021b}} +This is also the case at the CC level when one relies on more elaborated algorithms to converge the amplitude equations to higher-energy solutions. \cite{Piecuch_2000,Mayhall_2010,Lee_2019,Kossoski_2021,Marie_2021b}} Again, similarly to the dynamical equations defined in Eq.~\eqref{eq:GW} which requires the diagonalization of the dRPA eigenproblem [see Eq.~\eqref{eq:dRPA}], the CC equations reported in Eqs.~\eqref{eq:r_2h1p} and \eqref{eq:r_2p1h} can be solved with $\order*{N^6}$ cost by defining judicious intermediates. Cholesky decomposition, density fitting, and other related techniques may be employed to further reduce this scaling as it is done in conventional $GW$ calculations. \cite{Bintrim_2021,Forster_2020,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}