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Antoine Marie 2023-03-09 15:34:21 +01:00
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Manuscript/SRGGW.tex
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@ -583,7 +583,7 @@ For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \
%%% FIG 2 %%%
\begin{figure*}
\includegraphics[width=0.8\linewidth]{fig2.pdf}
\includegraphics[width=0.8\linewidth]{fig2}
\caption{
Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.}
\label{fig:plot}
@ -671,7 +671,7 @@ The numerical data associated with this study are reported in the {\SupInf}.
%%% FIG 3 %%%
\begin{figure}
\includegraphics[width=\linewidth]{fig3.pdf}
\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$GW$.
The HF and qs$GW$ values are reported as dashed lines.
@ -681,7 +681,7 @@ The numerical data associated with this study are reported in the {\SupInf}.
%%% FIG 4 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig4.pdf}
\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.
The HF and qs$GW$ values are reported as dashed lines.
@ -759,7 +759,7 @@ The SRG-qs$GW$ EA (absolute) error is monotonically decreasing from the HF value
%%% FIG 5 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig5.pdf}
\includegraphics[width=\linewidth]{fig5}
\caption{
Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal IP of the $GW$50 test set calculated using HF, $G_0W_0$@HF, qs$GW$, and SRG-qs$GW$.
All calculations are performed with the aug-cc-pVTZ basis.
@ -775,14 +775,15 @@ However, there are still outliers with large errors.
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.
Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative self-consistent scheme.
Let us now turn to the new method of this manuscript, the SRG-qs$GW$ self-consistent scheme.
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=\num{e3}$.
The statistical descriptors corresponding to this alternative static self-energy are all improved with respect to qs$GW$.
In particular, the MSE and MAE are decreased by \SI{0.06}{\eV}
In particular, the MSE and MAE are decreased by \SI{0.06}{\eV}.
Of course, these are slight improvements but this is done with no additional computational cost and can be easily implemented by changing the form of the static approximation.
The evolution of the statistical descriptors with respect to the various methods considered in Table \ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
The decrease of the MSE and SDE correspond to a shift of the maximum toward zero and a shrink of the distribution width, respectively.
The decrease of the MSE and SDE correspond to a shift of the maximum of the distribution toward zero and a contraction of the distribution width, respectively.
%%% TABLE I %%%
\begin{table*}
\caption{Principal IP and EA (in eV) of the $GW$50 test set calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$, and SRG-qs$GW$.
The statistical descriptors associated with the errors with respect to the reference values are also reported.
@ -858,15 +859,15 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
%%% FIG 6 %%%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig6.pdf}
\includegraphics[width=\linewidth]{fig6}
\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.
\label{fig:fig6}}
\end{figure}
%%% %%% %%% %%%
In addition to this improvement of accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=\num{e3}$ SRG-qs$GW$ self-consistency can be attained without any problems for the 50 compounds.
In addition to this improvement in accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=\num{e3}$ SRG-qs$GW$, it is straightforward to reach self-consistency for the 50 compounds.
For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50.
This means that some intruder states were included in the static correction for this value of $s$.
However, this is not a problem as the MAE of the test set is already well converged at $s=\num{e3}$.
@ -875,19 +876,19 @@ The convergence plateau of the MAE is reached around $s=50$ while the convergenc
Therefore, for future studies using the SRG-qs$GW$ method, a default value of the flow parameter equal to $\num{5e2}$ or $\num{e3}$ is recommended.
On the other hand, the qs$GW$ convergence behavior is more erratic.
At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be achieved for 13 molecules while 2 molecules were already problematic at $\eta=\num{5e-2}$ ($s=200$).
But these convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau before these problem arise (see the orange curve in Fig.~\ref{fig:fig6}).
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}.
At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be reached for 13 molecules while 2 systems were already problematic at $\eta=\num{5e-2}$ ($s=200$).
These convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE has not reached a convergence plateau before these problems arise (see the orange curve in Fig.~\ref{fig:fig6}).
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 of 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 important for the energy and finally adds the intruder states.
On the other hand, the imaginary shift regularizer acts equivalently on intruder states and terms that contribute to the energy.
%%% FIG 7 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig7.pdf}
\includegraphics[width=\linewidth]{fig7}
\caption{
Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal EA of the $GW$50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
Histogram of the errors [with respect to $\Delta$CCSD(T)] for the principal EA of the $GW$50 test set calculated using HF, $G_0W_0$@HF, qs$GW$, and SRG-qs$GW$.
All calculations are performed with the aug-cc-pVTZ basis.
\label{fig:fig7}}
\end{figure*}
@ -895,9 +896,9 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal EA energies.
The raw results are given in Table \ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig7}.
The HF EAs are underestimated in average with a MAE of \SI{0.31}{\eV} and some large outliers, for \SI{-2.03}{\eV} for \ce{F2} and \SI{1.04}{\eV} for \ce{CH2O} for example.
The HF EAs are, on average, underestimated with a MAE of \SI{0.31}{\eV} and some clear outliers, for \SI{-2.03}{\eV} for \ce{F2} and \SI{1.04}{\eV} for \ce{CH2O} for example.
$G_0W_0$@HF mitigates the average error (MAE equal to \SI{0.16}{\eV}) but the minimum and maximum error values are not yet satisfactory.
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt}.
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\eV}.
The two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavors of many-body perturbation theory considered here can not decrease the maximum error with respect to their HF starting point.
Note that a positive EA means that the anion state is bound and therefore the methods considered here are well-suited to describe these EAs.