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remove TDA from fig4 and some more corrections

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Antoine Marie 2023-03-08 10:21:43 +01:00
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2 changed files with 13 additions and 11 deletions
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Manuscript/SRGGW.tex
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@ -668,8 +668,8 @@ In practice, one may achieve convergence, in some cases, by adjusting these para
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 (SRG-)qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
These many-body perturbation theory values are compared with a CCSD(T) reference.
The CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively.
These many-body perturbation theory values are compared with a $\Delta$CCSD(T) reference.
The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively.
%=================================================================%
\section{Results}
@ -696,8 +696,8 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
\begin{figure*}
\includegraphics[width=\linewidth]{fig4.pdf}
\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 with and without TDA.
The HF and qs$GW$ (with and without TDA) values are reported as dashed lines.
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.
\label{fig:fig4}}
\end{figure*}
%%% %%% %%% %%%
@ -708,7 +708,7 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
%%%%%%%%%%%%%%%%%%%%%%
This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set.
Figure \ref{fig:fig3} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
Figure \ref{fig:fig3} shows the error of various methods for the principal IP with respect to the $\Delta$CCSD(T) reference value.
The IP at the HF level (dashed black line) is too large; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019}
The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\eV} of the reference value.
@ -738,19 +738,20 @@ As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreas
%The TDA values are now underestimating the IP, unlike their RPA counterparts.
%For both static self-energies, the TDA leads to a slight increase in the absolute error.
Next, we investigate the flow parameter dependence of SRG-qs$GW$ for three more challenging molecular systems.
Next, the flow parameter dependence of SRG-qs$GW$ is investigated for three more challenging molecular systems.
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.
On the other hand, the qs$GW$ and SRG-qs$GW$ IPs are too large.
Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimated as in the water molecule case.
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}.
%Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
We now turn to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4}).
Now turning to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4}).
In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value.
However, SRG-qs$GW$ improves slightly the accuracy as compared to HF.
However, SRG-qs$GW$ does not suffer from the same problem and improves slightly the accuracy as compared to HF.
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 convergence properties of both schemes will be systematically investigated in the next subsection.
Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
%Note that, for \ce{LiH} and \ce{BeO}, the TDA actually improves the accuracy compared to the RPA-based qs$GW$ schemes.
@ -880,6 +881,7 @@ This means that some intruder states were included in the static correction for
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 convergence plateau of the MAE is reached around $s=50$ while the convergence problem arise 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.
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 $\eta=\num{5e-2}$ ($s=200$).
@ -909,8 +911,8 @@ The two partially self-consistent methods reduce as well the minimum value but i
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.
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.
The methods considered in this study, even the $\Delta$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 $\Delta$CCSD(T) counterparts within a given basis set in these cases.
%=================================================================%
\section{Conclusion}

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Manuscript/fig4.pdf
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