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Pierre-Francois Loos 2023-03-09 14:28:10 +01:00
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Manuscript/SRGGW.tex
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@ -696,7 +696,7 @@ The numerical data associated with this study are reported in the {\SupInf}.
This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set. 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 in the principal IP [with respect to the $\Delta$CCSD(T) reference value] as a function of the flow parameter in SRG-qs$GW$ (blue curve). Figure \ref{fig:fig3} shows the error in the principal IP [with respect to the $\Delta$CCSD(T) reference value] as a function of the flow parameter in SRG-qs$GW$ (blue curve).
The corresponding HF and qs$GW$ (computed with $\eta = 0.05$) values are also reported for the sake of comparison. The corresponding HF and qs$GW$ (computed with $\eta = 0.05$) values are also reported for the sake of comparison.
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 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 in 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. 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.
At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}. At $s=0$, the SRG-qs$GW$ IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
@ -705,14 +705,14 @@ Furthermore, the value associated with this plateau is slightly more accurate th
However, the SRG-qs$GW$ error does not decrease smoothly between the initial HF value and the large-$s$ limit. However, the SRG-qs$GW$ error does not decrease smoothly between the initial HF value and the large-$s$ limit.
For small $s$, it is actually worse than the HF starting point. For small $s$, it is actually worse than the HF starting point.
This behavior as a function of $s$ can be understood by applying matrix perturbation theory on Eq.~\eqref{eq:GWlin}. \cite{Schirmer_2018} This behavior as a function of $s$ can be understood by applying matrix perturbation theory to Eq.~\eqref{eq:GWlin}. \cite{Schirmer_2018}
Through second order in the coupling block, the principal IP is Through second order in the coupling block, the principal IP is
\begin{equation} \begin{equation}
\label{eq:2nd_order_IP} \label{eq:2nd_order_IP}
\text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{(W_{\text{h}}^{i\nu})^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{(W_{\text{h}}^{a\nu})^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu} \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{(W_{\text{h}}^{i\nu})^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{(W_{\text{h}}^{a\nu})^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu}
\end{equation} \end{equation}
where $\text{h}$ is the index of the highest occupied molecular orbital (HOMO). where $\text{h}$ is the index of the highest occupied molecular orbital (HOMO).
The first term of the right-hand side of Eq.~\eqref{eq:2nd_order_IP} is the zeroth-order IP and the two following terms originate from the 2h1p and 2p1h coupling, respectively. The first term of the right-hand side of Eq.~\eqref{eq:2nd_order_IP} is the zeroth-order IP and the following two terms originate from the 2h1p and 2p1h coupling, respectively.
The denominators of the 2p1h term are positive while the denominators associated with the 2h1p term are negative. The denominators of the 2p1h term are positive while the denominators associated with the 2h1p term are negative.
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. 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.
@ -724,9 +724,9 @@ 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. %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. %For both static self-energies, the TDA leads to a slight increase in the absolute error.
Next, the flow parameter dependence of SRG-qs$GW$ is investigated for the principal IPs of two additional molecular systems as well as the principal EA of the \ce{F2} molecule. Next, the flow parameter dependence of SRG-qs$GW$ is investigated for the principal IP of two additional molecular systems as well as the principal EA of \ce{F2}.
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. 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.
Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimated as in \ce{H2O}. Yet, the qs$GW$ and SRG-qs$GW$ IPs are still overestimating the reference value as in \ce{H2O}.
Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water. 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}. 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$GW^\TDA$. %Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
@ -734,7 +734,7 @@ In addition, the plateau is reached for larger values of $s$ in comparison to Fi
Now turning to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4}), we see that the qs$GW$ IP is actually worse than the fairly accurate HF value. Now turning to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig4}), we see that the qs$GW$ IP is actually worse than the fairly accurate HF value.
However, SRG-qs$GW$ does not suffer from the same problem and 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, 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} displayed in the right panel of 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 (absolute) error is monotonically decreasing from the HF value at $s=0$ to an error close to the qs$GW$ one at $s\to\infty$. 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 $s\to\infty$.