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minor corrections b4 resubmission

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Pierre-Francois Loos 2022-11-17 13:40:34 +01:00
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@ -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}