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Pierre-Francois Loos 2023-03-09 16:09:27 +01:00
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@ -627,12 +627,12 @@ To conclude this section, we briefly discussed the case of discontinuities menti
Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023} Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
Is it then possible to rely on the SRG machinery to remove discontinuities? Is it then possible to rely on the SRG machinery to remove discontinuities?
Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation. Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation.
However, as we have seen just above the functional form of the renormalized equation \ant{makes it possible to choose $s$ such that there is no intruder states in its static part.} However, as we have seen just above the functional form of the renormalized equation makes it possible to choose $s$ such that there is no intruder states in its static part.
But performing a bijective transformation of the form, Performing a bijective transformation of the form,
\begin{align} \begin{align}
e^{- \Delta s} &= 1-e^{-\Delta t}, e^{- \Delta s} &= 1-e^{-\Delta t},
\end{align} \end{align}
on the renormalized quasiparticle equation \eqref{eq:GW_renorm} reverses the situation and \ant{makes it possible to choose $t$ such that there is no intruder states in the dynamic part, hence removing discontinuities.} on the renormalized quasiparticle equation \eqref{eq:GW_renorm} reverses the situation and makes it possible to choose $t$ such that there is no intruder states in the dynamic part, hence removing discontinuities.
Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}. Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
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@ -653,7 +653,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$
The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively. The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively.
In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme. In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
However, in order to perform black-box comparisons, these parameters have been fixed to these default values. However, in order to perform black-box comparisons, these parameters have been fixed to these default values.
The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ \ant{(without linearization of the quasiparticle equation)} calculations while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set. The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations (where we eschew linearizing the quasiparticle equation) while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set. The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set.
The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} with default parameters within the restricted and unrestricted HF formalism for the neutral and charged species, respectively. The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} with default parameters within the restricted and unrestricted HF formalism for the neutral and charged species, respectively.
@ -673,8 +673,8 @@ The numerical data associated with this study are reported in the {\SupInf}.
\begin{figure} \begin{figure}
\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$GW$. 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 and qs$GW$ values are reported as dashed lines. The HF (cyan curve) and qs$GW$ (blue curve) values are reported as dashed lines.
\label{fig:fig3}} \label{fig:fig3}}
\end{figure} \end{figure}
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@ -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. 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 and qs$GW$ values are reported as dashed lines. The HF (cyan curves) and qs$GW$ (blue curves) values are reported as dashed lines.
\label{fig:fig4}} \label{fig:fig4}}
\end{figure*} \end{figure*}
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@ -771,7 +771,7 @@ Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in th
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 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.
However, there are still outliers with large errors. However, there are still outliers with large errors.
\ant{For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}, a large error which is clearly due to the starting point dependence of $G_0W_0$@HF.} 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.
Self-consistency can mitigate 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 $G_0W_0$ to \SI{0.18}{\eV}. Self-consistency can mitigate 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 $G_0W_0$ to \SI{0.18}{\eV}.
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.
@ -861,7 +861,7 @@ 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$GW$ and qs$GW$ 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. 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.
\label{fig:fig6}} \label{fig:fig6}}
\end{figure} \end{figure}
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