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Antoine Marie 2023-02-03 11:06:52 +01:00
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Manuscript/SRGGW.rty
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@ -116,6 +116,7 @@
\newcommand{\hhp}{\text{2h1p}} \newcommand{\hhp}{\text{2h1p}}
\newcommand{\pph}{\text{2p1h}} \newcommand{\pph}{\text{2p1h}}
\newcommand{\dRPA}{\text{dRPA}} \newcommand{\dRPA}{\text{dRPA}}
\newcommand{\TDA}{\text{TDA}}
\newcommand{\RPAx}{\text{RPAx}} \newcommand{\RPAx}{\text{RPAx}}
\newcommand{\QP}{\textsc{quantum package}} \newcommand{\QP}{\textsc{quantum package}}
\newcommand{\Hxc}{\text{Hxc}} \newcommand{\Hxc}{\text{Hxc}}

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Manuscript/SRGGW.tex
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@ -590,22 +590,24 @@ The TDA IPs are now underestimated unlike their RPA counterparts.
For both static self-energies, the TDA leads to a slight increase of the absolute error. For both static self-energies, the TDA leads to a slight increase of the absolute error.
This trend will be investigated in more details in the next subsection. This trend will be investigated in more details in the next subsection.
Before going to the statistical study, the behavior of three particular molecules is investigated. Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems.
The Lithium dimer \ce{Li2} will be considered as a case where HF actually underestimate the IP. The left panel of Fig.~\ref{fig:fig2} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated .
The Lithium hydrid will also be investigated because in this case the usual qs$GW$ IP is worst than the HF one. On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are overestimated
Finally, the Beryllium oxyde will be studied as a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021} Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig1}.
Both TDA results are worse than their RPA counterparts but in this case the SRG-qs$GW_\TDA$ is more accurate than the qs$GW_\TDA$.
Now turning to the Lithium hydrid heterodimer, see middle panel of Fig.~\ref{fig:fig2}.
In this case the qs$GW$ IP is actually worse than the HF one which is already pretty accurate.
However, the SRG-qs$GW$ can improve slightly the accuracy with respect to HF.
Finally, the Beryllium oxyde is considered as a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
The SRG-qs$GW$ could be converged without any problem even for large values of $s$.
Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes. Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes.
However, as we will see in the next subsection these are just particular molecular systems and in average the RPA polarizability performs better than the TDA one. However, as we will see in the next subsection these are just particular molecular systems and in average the RPA polarizability performs better than the TDA one.
Also the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around.
% \ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.} Therefore, it seems that the effect of the TDA can not be systematically predicted.
% 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... \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.
%%% FIG 2 %%% %%% FIG 2 %%%
\begin{figure*} \begin{figure*}