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add the comments discussed during the meeting to no forget them

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Antoine Marie 2023-02-01 12:01:39 +01:00
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Manuscript/SRGGW.tex
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@ -116,6 +116,7 @@ Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of
For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
We refer the reader to the recent review by Golze and co-workers (see Ref.~\onlinecite{Golze_2019}) for an extensive list of current challenges in many-body perturbation theory.
\ANT{Put emphasis on intruder states rather than discontinuities.}
Recently, it has been shown that a variety of physical quantities, such as charged and neutral excitations energies as well as correlation and total energies, computed within many-body perturbation theory exhibit unphysical discontinuities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
Even more worrying, these discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
These discontinuities have been traced back to a transfer of spectral weight between two solutions of the quasi-particle equation, \cite{Monino_2022} and is another occurrence of the infamous intruder-state problem.\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001}
@ -521,6 +522,7 @@ Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.
Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
\ANT{Change notation here and use multline}
Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
\begin{align}
\label{eq:sym_qsGW}
@ -559,6 +561,7 @@ In fact, the dynamic part after the change of variable is closely related to the
\label{sec:comp_det}
%=================================================================%
\ANT{Frozen-core}
The two qs$GW$ variants considered in this work have been implemented in an in-house program.
The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
The geometries have been optimized at the CC3 level in the aug-cc-pVTZ basis set without frozen core using the CFOUR program.
@ -612,7 +615,7 @@ The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
Add sentence about $GW$ better than GF2 when the results will be here.
The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
The GF(2) IP admits the following perturbation expansion...
The GF(2) IP admits the following perturbation expansion... \ANT{Remove GF2 and try matrix perturbation theory on $GW$, cf Evangelista's talk.}
Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.