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Pierre-Francois Loos 2023-03-09 22:14:03 +01:00
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Manuscript/SRGGW.tex
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@ -674,7 +674,7 @@ The numerical data associated with this study are reported in the {\SupInf}.
\includegraphics[width=\linewidth]{fig3}
\caption{
Error [with respect to $\Delta$CCSD(T)] in the principal IP of water in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for SRG-qs$GW$ (green curve).
The HF (cyan curve) and qs$GW$ (blue curve) values are reported as dashed lines.
The HF (cyan line) and qs$GW$ (blue line) values are also reported.
\label{fig:fig3}}
\end{figure}
%%% %%% %%% %%%
@ -683,8 +683,8 @@ The numerical data associated with this study are reported in the {\SupInf}.
\begin{figure*}
\includegraphics[width=\linewidth]{fig4}
\caption{
Error [with respect to $\Delta$CCSD(T)] in the principal IP of \ce{Li2}, \ce{LiH} and \ce{BeO} in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method (green curves).
The HF (cyan curves) and qs$GW$ (blue curves) values are reported as dashed lines.
Error [with respect to $\Delta$CCSD(T)] in the principal IP of \ce{Li2}, \ce{LiH}, and \ce{BeO} in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method (green curves).
The HF (cyan lines) and qs$GW$ (blue lines) values are also reported.
\label{fig:fig4}}
\end{figure*}
%%% %%% %%% %%%
@ -726,7 +726,7 @@ As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreas
%For both static self-energies, the TDA leads to a slight increase in the absolute error.
Next, the flow parameter dependence of SRG-qs$GW$ is investigated for the principal IP of two additional molecular systems as well as the principal EA of \ce{F2}.
The left panel of Fig.~\ref{fig:fig4} shows the results for the lithium dimer, \ce{Li2}, which is an interesting case because, unlike in water, the HF approximation underestimates the reference IP.
The left panel of Fig.~\ref{fig:fig4} shows the results for the lithium dimer, \ce{Li2}, which is an interesting case because, unlike in water, HF underestimates the reference IP.
Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimating the reference value as in \ce{H2O}.
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:fig3}.
@ -769,9 +769,9 @@ The SRG-qs$GW$ EA (absolute) error is monotonically decreasing from the HF value
Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in this work computed at various levels of theory.
As mentioned previously, the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.56}{\eV} and a mean absolute error (MAE) of \SI{0.69}{\eV}.
Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.29}{\eV} and \SI{0.33}{\eV}, respectively.
Performing a $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.29}{\eV} and \SI{0.33}{\eV}, respectively.
However, there are still outliers with large errors.
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}, a large error that is due to the HF starting point.
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}, a large discrepancy that is due to the HF starting point.
Self-consistency mitigates the error of the outliers as the MAE at the qs$GW$ level is now \SI{0.57}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.31}{\eV} for $G_0W_0$@HF to \SI{0.18}{\eV} for qs$GW$.
In addition, the MSE and MAE (\SI{0.23}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
@ -862,7 +862,9 @@ The decrease of the MSE and SDE correspond to a shift of the maximum of the dist
\centering
\includegraphics[width=\linewidth]{fig6}
\caption{
SRG-qs$GW$ (green) and qs$GW$ (blue) MAEs for the principal IPs of the $GW$50 test set. The bottom and top axes are equivalent and related by $s=1/(2\eta^2)$. A different marker has been used for qs$GW$ at $\eta=0.05$ because the MAE includes only 48 molecules.
Evolution of the SRG-qs$GW$ (green) and qs$GW$ (blue) MAEs for the principal IPs of the $GW$50 test set as functions of $s$ and $\eta$, respectively.
The bottom and top axes are related by $s=1/(2\eta^2)$.
A different marker has been used for qs$GW$ at $\eta=0.05$ because the MAE includes only 48 molecules.
\label{fig:fig6}}
\end{figure}
%%% %%% %%% %%%
@ -882,8 +884,8 @@ These convergence problems are much more dramatic than for SRG-qs$GW$ because th
For example, out of the 37 molecules that could be converged for $\eta=\num{e-2}$, the variation of the IP with respect to $\eta=\num{5e-2}$ can go up to \SI{0.1}{\eV}.
This difference in behavior is due to the energy (in)dependence of the regularizers.
Indeed, the SRG regularizer first includes the terms that are \titou{contributing to} the energy and finally adds the intruder states.
On the other hand, the imaginary shift regularizer acts equivalently on all terms.
The SRG regularizer first incorporates the terms with a large denominator and subsequently adds the intruder states.
Conversely, the imaginary shift regularizer treats all terms equivalently.
%%% FIG 7 %%%
\begin{figure*}
@ -904,8 +906,8 @@ These two partially self-consistent methods reduce also the minimum errors but,
Note that a positive EA indicates a bounded anion state, which can be accurately described by the methods considered in this study.
However, a negative EA suggests a resonance state, which is beyond the scope of the methods used in this study, including the $\Delta$CCSD(T) reference.
Therefore, one should avoid giving a physical interpretation to these values.
Yet, one can still compare, for a given basis set, the $GW$-based and $\Delta$CCSD(T) values in these cases.
As such, it is not advisable to assign a physical interpretation to these values.
Nonetheless, it is possible to compare $GW$-based and $\Delta$CCSD(T) values in such cases, provided that the comparison is limited to a given basis set.
%=================================================================%
\section{Conclusion}