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few corrections in results

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Antoine Marie 2023-03-08 09:30:42 +01:00
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Manuscript/SRGGW.tex
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@ -777,7 +777,7 @@ As mentioned previously the HF approximation overestimates the IPs with a mean s
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.
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}.
Self-consistency can mitigate the error of the outliers as the maximum error 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} for qs$GW$.
Self-consistency can mitigate the error of the outliers as the maximum absolute error 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} 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.
Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative self-consistent scheme.
@ -786,7 +786,7 @@ The statistical descriptors corresponding to this alternative static self-energy
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 implemented really easily 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 \ant{shift of the maximum} toward zero and a shrink of the distribution width, respectively.
\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$GW$.
@ -871,15 +871,16 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
%%% %%% %%% %%%
In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
Indeed, up to $s=\num{e3}$ self-consistency can be attained without any problems for the 50 compounds.
For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50, meaning 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 converged for this value of $s$.
Indeed, up to $s=\num{e3}$ SRG-qs$GW$ self-consistency can be attained without any problems 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 $s=\num{e3}$.
This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and $\eta=1/2s^2$.
The plateau of the MAE is reached far before the convergence problem arrives.
The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arise for $s>\num{e3}$.
On the other hand, the qs$GW$ convergence behaviour seems to be 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 this convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau when these problem arise.
But this 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 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 behaviour is due to the energy (in)dependence of the regularizers.
@ -896,16 +897,17 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
\end{figure*}
%%% %%% %%% %%%
Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal electron attachement (EA) energies.
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 Tab.~\ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig7}.
The HF EAs are understimated 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.
$G_0W_0$@HF mitigates the average error (\SI{0.16}{\eV} MAE) but the minimum and maximum values are not yet satisfactory.
$G_0W_0$@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 two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavours 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 the EAs.
On the other hand, a positive EA means that this is a resonance state and the methods considered in this study, even the reference, are not able to describe the physics of resonance states.
Yet, one can still compare the $GW$ values with their CCSD(T) counterparts in these cases.
On the other hand, a negative EA means that this is a resonance state.
The methods considered in this study, even the CCSD(T) reference, are not able to describe the physics of resonance states, therefore one should not try to give a physical interpretation to these values.
Yet, one can still compare the $GW$ values with their CCSD(T) counterparts within a given basis set in these cases.
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\section{Conclusion}