add the comments discussed during the meeting to no forget them
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@ -116,6 +116,7 @@ Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of
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For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
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For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
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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.
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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.
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\ANT{Put emphasis on intruder states rather than discontinuities.}
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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}
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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}
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Even more worrying, these discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
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Even more worrying, these discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
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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}
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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}
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@ -521,6 +522,7 @@ Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.
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Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
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Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
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This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
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This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
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\ANT{Change notation here and use multline}
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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
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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
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\begin{align}
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\begin{align}
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\label{eq:sym_qsGW}
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\label{eq:sym_qsGW}
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@ -559,6 +561,7 @@ In fact, the dynamic part after the change of variable is closely related to the
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\label{sec:comp_det}
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\label{sec:comp_det}
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%=================================================================%
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%=================================================================%
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\ANT{Frozen-core}
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The two qs$GW$ variants considered in this work have been implemented in an in-house program.
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The two qs$GW$ variants considered in this work have been implemented in an in-house program.
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The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
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The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
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The geometries have been optimized at the CC3 level in the aug-cc-pVTZ basis set without frozen core using the CFOUR program.
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The geometries have been optimized at the CC3 level in the aug-cc-pVTZ basis set without frozen core using the CFOUR program.
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@ -612,7 +615,7 @@ The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
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Add sentence about $GW$ better than GF2 when the results will be here.
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Add sentence about $GW$ better than GF2 when the results will be here.
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The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
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The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
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We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
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We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
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The GF(2) IP admits the following perturbation expansion...
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The GF(2) IP admits the following perturbation expansion... \ANT{Remove GF2 and try matrix perturbation theory on $GW$, cf Evangelista's talk.}
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Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
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Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
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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.
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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.
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