From 39b874f305543c4e84dffaf1847f1a6746d3e683 Mon Sep 17 00:00:00 2001 From: pfloos Date: Tue, 14 Feb 2023 18:06:36 -0500 Subject: [PATCH] saving work in results --- Manuscript/SRGGW.tex | 21 +++++++++++---------- 1 file changed, 11 insertions(+), 10 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 45e3336..9e12004 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -647,7 +647,7 @@ However, in order to perform a black-box comparison, these parameters have been The results section is divided into two parts. The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases. -Then the accuracy of the IP yielded by the traditional and SRG schemes will be statistically gauged over a test set of molecules. +Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-qs$GW$ schemes are statistically gauged over the test set of molecules described in Sec.~\ref{sec:comp_det}. %%%%%%%%%%%%%%%%%%%%%% \subsection{Flow parameter dependence of the SRG-qs$GW$ scheme} @@ -675,18 +675,19 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s \end{figure*} %%% %%% %%% %%% -This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set. +This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ basis set. Figure \ref{fig:fig2} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value. -The HF IP (dashed black line) is overestimated; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019} -The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference. +The IP at the HF level (dashed black line) is overestimated; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019} +The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference. -Figure~\ref{fig:fig2} also displays the SRG-qs$GW$ IP as a function of the flow parameter (blue curve). -At $s=0$, the IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}. -For $s\to\infty$, the IP reaches a plateau at an error that is significantly smaller than their $s=0$ starting point. -Even more, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart. -However, the SRG-qs$GW$ error do not decrease smoothly between the initial HF value and the $s\to\infty$ limit as for small $s$ values it is actually worst than the HF starting point. +Figure \ref{fig:fig2} also displays the IP at the SRG-qs$GW$ level as a function of the flow parameter (blue curve). +At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}. +As $s\to\infty$, the IP reaches a plateau at an error that is significantly smaller than their $s=0$ starting point. +Furthermore, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart. +However, the SRG-qs$GW$ error does not decrease smoothly between the initial HF value and the $s\to\infty$ limit. +For small $s$, it is actually worse than the HF starting point. -This behavior as a function of $s$ can be approximately rationalized by applying matrix perturbation theory on Eq.~(\ref{eq:GWlin}). +This behavior as a function of $s$ can be understood by applying matrix perturbation theory on Eq.~\eqref{eq:GWlin}. Through second order in the coupling block, the principal IP is \begin{equation} \label{eq:2nd_order_IP}