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Pierre-Francois Loos 2023-03-09 14:08:42 +01:00
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Manuscript/SRGGW.tex
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@ -652,7 +652,7 @@ 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 $G_0W_0$ 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 \titou{where we have not linearized 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.
@ -737,7 +737,7 @@ However, SRG-qs$GW$ does not suffer from the same problem and improves slightly
Finally, we also consider the evolution with respect to $s$ of the principal EA of \ce{F2}, as displayed in Fig.~\ref{fig:fig4}. Finally, we also consider the evolution with respect to $s$ of the principal EA of \ce{F2}, as displayed in Fig.~\ref{fig:fig4}.
The HF value is largely underestimating the $\Delta$CCSD(T) reference. The HF value is largely underestimating the $\Delta$CCSD(T) reference.
Performing a qs$GW$ calculation on top of it brings a quantitative improvement by reducing the error from \SI{-2.03}{\eV} to \SI{-0.24}{\eV}. Performing a qs$GW$ calculation on top of it brings a quantitative improvement by reducing the error from \SI{-2.03}{\eV} to \SI{-0.24}{\eV}.
The SRG-qs$GW$ EA error is monotonically increasing from the HF value at $s=0$ to an error really close to the qs$GW$ one at $s\to\infty$. The SRG-qs$GW$ EA (absolute) error is monotonically decreasing from the HF value at $s=0$ to an error close to the qs$GW$ one at $s\to\infty$.
% Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations due to the presence of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021} % Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations due to the presence of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
% The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values. % The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values.
@ -770,17 +770,17 @@ 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.
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}. \titou{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 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$GW$. 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.
Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative self-consistent scheme. Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative self-consistent scheme.
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=\num{e3}$. 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$GW$. 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 implemented really easily by changing the form of the static approximation. 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 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 \ant{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 toward zero and a shrink of the distribution width, respectively.
\begin{table*} \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$GW$. \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$GW$.
@ -869,11 +869,11 @@ Indeed, up to $s=\num{e3}$ SRG-qs$GW$ self-consistency can be attained without a
For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50. 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 $s$. 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 $s=\num{e3}$. 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 $\eta=1/2s^2$. This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and \titou{$\eta=1/2s^2$}.
The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arises for $s>\num{e3}$. The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arises for $s>\num{e3}$.
Therefore, for future studies using the SRG-qs$GW$ method, a default value of the flow parameter equal to $\num{5e2}$ or $\num{e3}$ seems reasonable. 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 seems to be more erratic. 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 $\eta=\num{5e-2}$ ($s=200$). 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}). 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}. 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}.
@ -893,7 +893,7 @@ 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. 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 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 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.
$G_0W_0$@HF mitigates the average error (MAE equal to \SI{0.16}{\eV}) but the minimum and maximum error values are not yet satisfactory. $G_0W_0$@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}{\electronvolt}.