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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} \includegraphics[width=\linewidth]{fig3}
\caption{ \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). 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}} \label{fig:fig3}}
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -683,8 +683,8 @@ The numerical data associated with this study are reported in the {\SupInf}.
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{fig4} \includegraphics[width=\linewidth]{fig4}
\caption{ \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). 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. The HF (cyan lines) and qs$GW$ (blue lines) values are also reported.
\label{fig:fig4}} \label{fig:fig4}}
\end{figure*} \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. %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}. 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}. 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. 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}. 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. 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}. 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. 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$GW$. 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. 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 \centering
\includegraphics[width=\linewidth]{fig6} \includegraphics[width=\linewidth]{fig6}
\caption{ \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}} \label{fig:fig6}}
\end{figure} \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}. 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. 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. The SRG regularizer first incorporates the terms with a large denominator and subsequently adds the intruder states.
On the other hand, the imaginary shift regularizer acts equivalently on all terms. Conversely, the imaginary shift regularizer treats all terms equivalently.
%%% FIG 7 %%% %%% FIG 7 %%%
\begin{figure*} \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. 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. 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. As such, it is not advisable to assign 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. 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} \section{Conclusion}