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Pierre-Francois Loos 2023-02-14 18:06:36 -05:00
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Manuscript/SRGGW.tex
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@ -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}.
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\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}