From b5063a4d887c819a79ccb301f6121ece9751fba2 Mon Sep 17 00:00:00 2001 From: Antoine MARIE Date: Fri, 3 Feb 2023 11:06:52 +0100 Subject: [PATCH] saving work --- Manuscript/SRGGW.rty | 1 + Manuscript/SRGGW.tex | 28 +++++++++++++++------------- 2 files changed, 16 insertions(+), 13 deletions(-) diff --git a/Manuscript/SRGGW.rty b/Manuscript/SRGGW.rty index c6de653..f17fec8 100755 --- a/Manuscript/SRGGW.rty +++ b/Manuscript/SRGGW.rty @@ -116,6 +116,7 @@ \newcommand{\hhp}{\text{2h1p}} \newcommand{\pph}{\text{2p1h}} \newcommand{\dRPA}{\text{dRPA}} +\newcommand{\TDA}{\text{TDA}} \newcommand{\RPAx}{\text{RPAx}} \newcommand{\QP}{\textsc{quantum package}} \newcommand{\Hxc}{\text{Hxc}} diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 926a221..bba8db3 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -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. 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. -The Lithium dimer \ce{Li2} will be considered as a case where HF actually underestimate the IP. -The Lithium hydrid will also be investigated because in this case the usual qs$GW$ IP is worst than the HF one. -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} +Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems. +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 . +On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are overestimated +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. 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. - -% \ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.} -% 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. +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. +Therefore, it seems that the effect of the TDA can not be systematically predicted. %%% FIG 2 %%% \begin{figure*}