loos/SRGGW
2
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

saving work in conclusion, fix titou's comment, remove TDA from fig3 and from the text

Browse Source
This commit is contained in:
Antoine Marie 2023-03-05 22:29:29 +01:00
commit e9906bd3ad
2 changed files with 41 additions and 38 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

79
Manuscript/SRGGW.tex
View File

@ -681,11 +681,11 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
\begin{figure}
\includegraphics[width=\linewidth]{fig3.pdf}
\caption{
Principal IP of the water molecule 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.
Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
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 the SRG-qs$GW$ method with and without TDA.
The HF and qs$GW$ values are reported as dashed lines.
\PFL{Should we have a similar figure for EAs? (maybe not water though)}
\ANT{I did the plot, let's discuss it at the next meeting}
\label{fig:fig2}}
\label{fig:fig3}}
\end{figure}
%%% %%% %%% %%%
@ -693,9 +693,9 @@ 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{
Principal IP of the \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.
Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
\label{fig:fig3}}
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.
\label{fig:fig4}}
\end{figure*}
%%% %%% %%% %%%
@ -705,11 +705,11 @@ 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:fig2} 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 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.
Figure \ref{fig:fig2} also displays the IP at the SRG-qs$GW$ level as a function of the flow parameter (blue curve).
Figure \ref{fig:fig3} also displays the IP at the SRG-qs$GW$ level as a function of the flow parameter (blue curve).
At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
As $s$ grows, the IP reaches a plateau at an error that is significantly smaller than the HF starting point.
Furthermore, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart.
@ -728,21 +728,21 @@ The denominators of the 2p1h term are positive while the denominators associated
As $s$ increases, the first states that decouple from the HOMO are the 2p1h configurations because their energy difference with respect to the HOMO is larger than the ones associated with the 2h1p block.
Therefore, for small $s$, only the last term of Eq.~\eqref{eq:2nd_order_IP} is partially included, resulting in a positive correction to the IP.
As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreasing and eventually goes below the $s=0$ initial value as observed in Fig.~\ref{fig:fig2}.
As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreasing and eventually goes below the $s=0$ initial value as observed in Fig.~\ref{fig:fig3}.
\ANT{I don't know if we should remove this paragraph and the TDA curves in Fig 3 and 4 or not...}
In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening (dubbed qs$GW^\TDA$ and SRG-qs$GW^\TDA$) are also considered in Fig.~\ref{fig:fig2}.
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.
%\ANT{I don't know if we should remove this paragraph and the TDA curves in Fig 3 and 4 or not...}
%In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening (dubbed qs$GW^\TDA$ and SRG-qs$GW^\TDA$) are also considered in Fig.~\ref{fig:fig3}.
%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.
The left panel of Fig.~\ref{fig:fig3} 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.
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.
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:fig2}.
Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
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:fig2}).
We now turn 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.
@ -750,11 +750,11 @@ Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example
The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values.
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.
However, as we shall see in Sec.~\ref{sec:SRG_vs_Sym}, these are special cases, and, on average, the RPA polarizability performs better than its TDA version.
Also, SRG-qs$GW^\TDA$ is better than qs$GW^\TDA$ in the three cases of Fig.~\ref{fig:fig3}.
However, it is not a general rule.
Therefore, it seems that the effect of the TDA cannot be systematically predicted.
%Note that, for \ce{LiH} and \ce{BeO}, the TDA actually improves the accuracy compared to the RPA-based qs$GW$ schemes.
%However, as we shall see in Sec.~\ref{sec:SRG_vs_Sym}, these are special cases, and, on average, the RPA polarizability performs better than its TDA version.
%Also, SRG-qs$GW^\TDA$ is better than qs$GW^\TDA$ in the three cases of Fig.~\ref{fig:fig4}.
%However, it is not a general rule.
%Therefore, it seems that the effect of the TDA cannot be systematically predicted.
%%%%%%%%%%%%%%%%%%%%%%
@ -766,8 +766,9 @@ Therefore, it seems that the effect of the TDA cannot be systematically predicte
\begin{figure*}
\includegraphics[width=\linewidth]{fig5.pdf}
\caption{
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
\label{fig:fig4}}
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.
\label{fig:fig5}}
\end{figure*}
%%% %%% %%% %%%
@ -788,12 +789,16 @@ The evolution of the statistical descriptors with respect to the various methods
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*}
\caption{First ionization potential (left) and first electron attachment (right) in eV calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. The statistical descriptors are computed for the errors with respect to the reference.}
\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.
All calculations are performed with the aug-cc-pVTZ basis.}
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{l|ddddd|ddddd}
Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} \\
\begin{tabular}{ldddddddddd}
& \mc{5}{c}{Principal IP} & \mc{5}{c}{Principal EA} \\
\cline{2-6} \cline{7-11}
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
Mol. & \mcc{(Ref.)} & & \mcc{($\eta=\num{e-3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{e3}$)} & \mcc{(Ref.)} & & \mcc{($\eta=\num{e-3}$)} & \mcc{($\eta=\num{e-1}$)} & \mcc{($s=\num{e3}$)} \\
\hline
\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.55 & -2.66 & -2.70 & -2.66 & -2.66 & -2.66 \\
\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 & -5.09 & -5.47 & -5.25 & -5.19 & -5.19 \\
@ -845,10 +850,7 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
\ce{OCS} & 11.23 & 11.44 & 11.52 & 11.37 & 11.32 & -1.43 & -1.27 & -1.03 & -0.97 & -0.98 \\
\ce{SO2} & 10.48 & 11.47 & 11.38 & 10.85 & 10.82 & 2.24 & 1.84 & 2.82 & 2.74 & 2.68 \\
\ce{C2H3Cl} & 10.17 & 10.13 & 10.39 & 10.27 & 10.24 & -0.61 & -0.79 & -0.66 & -0.65 & -0.65 \\
\hline
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} \\
\hline
\hline
MSE & & 0.56 & 0.29 & 0.23 & 0.17 & & -0.25 & 0.02 & 0.04 & 0.04 \\
MAE & & 0.69 & 0.33 & 0.25 & 0.19 & & 0.31 & 0.16 & 0.13 & 0.12 \\
SDE & & 0.68 & 0.31 & 0.18 & 0.16 & & 0.43 & 0.29 & 0.23 & 0.22 \\
@ -863,8 +865,8 @@ The decrease of the MSE and SDE correspond to a shift of the maximum toward zero
\centering
\includegraphics[width=\linewidth]{fig6.pdf}
\caption{
SRG-qs$GW$ and qs$GW$ MAE of the IPs for the GW50 test set. The bottom and top axes are equivalent and related by $\eta=1/2s^2$. A different marker has been used for qs$GW$ at $\eta=0.05$ because the MAE includes only 48 molecules.
\label{fig:fig5}}
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}
%%% %%% %%% %%%
@ -872,7 +874,7 @@ In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme has been
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 $s$.
This is illustrated by the blue curve of Fig.~\ref{fig:fig5} 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 $\eta=1/2s^2$.
The plateau of the MAE is reached far before the convergence problem arrives.
On the other hand, the qs$GW$ convergence behaviour seems to be more erratic.
@ -888,13 +890,14 @@ On the other hand, the imaginary shift regularizer acts equivalently on intruder
\begin{figure*}
\includegraphics[width=\linewidth]{fig7.pdf}
\caption{
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
\label{fig:fig6}}
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*}
%%% %%% %%% %%%
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.
The raw results are given in Tab.~\ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
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.
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt}.

BIN
Manuscript/fig3.pdf
View File

Binary file not shown.