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Manuscript/SRGGW.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2023-02-13 18:31:47 -0500
%% Created for Pierre-Francois Loos at 2023-02-15 10:20:34 -0500
%% Saved with string encoding Unicode (UTF-8)

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Manuscript/SRGGW.rty
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%%% You can import these commands by adding the following line in you .text
%%% \input{commands}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
%%% Latin %%%
\newcommand{\ie}{\textit{i.e.}\xspace}

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Manuscript/SRGGW.tex
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@ -675,53 +675,53 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
\end{figure*}
%%% %%% %%% %%%
This section starts by considering a prototypical molecular system, \ie 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:fig2} 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 overestimated; 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.
Figure \ref{fig:fig2} 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 $s\to\infty$, the IP reaches a plateau at an error that is significantly smaller than their $s=0$ starting point.
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.
However, the SRG-qs$GW$ error does not decrease smoothly between the initial HF value and the $s\to\infty$ 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.
This behavior as a function of $s$ can be understood by applying matrix perturbation theory on Eq.~\eqref{eq:GWlin}.
This behavior as a function of $s$ can be understood by applying matrix perturbation theory on Eq.~\eqref{eq:GWlin}. \cite{Schirmer_2018}
Through second order in the coupling block, the principal IP is
\begin{equation}
\label{eq:2nd_order_IP}
I_k = - \epsilon_k - \sum_{i\nu} \frac{W_{k,i\nu}^2}{\epsilon_k - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{k,a\nu}^2}{\epsilon_k - \epsilon_a - \Omega_\nu} + \order{3}
\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}
where $k$ is the index of the highest molecular MO (HOMO).
The first term is the zeroth order IP and the two following terms come from the 2h1p and 2p1h coupling, respectively.
The denominator of the last term is always positive while the 2h1p term is negative.
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 denominators of the 2p1h term are positive while the denominators associated with the 2h1p term are negative.
When $s$ is increased, the first states that will be decoupled from the HOMO will be the 2p1h ones because their energy difference with the HOMO is larger than the ones of the 2h1p block.
Therefore, for small $s$ only the last term of Eq.~\eqref{eq:2nd_order_IP} will be partially included resulting in a positive correction to the IP.
As soon as $s$ is large enough to decouple the 2h1p block as well the IP will start to decrease and eventually go below the $s=0$ initial value as observed in Fig.~\ref{fig:fig2}.
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}.
In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening are also considered in Fig.~\ref{fig:fig2}.
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 IPs are now underestimated, unlike their RPA counterparts.
For both static self-energies, the TDA leads to a slight increase in the absolute error.
This trend will be investigated in more detail in the next subsection.
This trend is investigated in more detail in the next subsection.
Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems.
The left panel of Fig.~\ref{fig:fig3} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated.
On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are overestimated
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 IP 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 counterparts but in this case the SRG-qs$GW_\TDA$ is more accurate than the qs$GW_\TDA$.
Both TDA results are worse than their RPA counterparts but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
Now turning to the lithium hydride heterodimer, see the middle panel of Fig.~\ref{fig:fig2}.
In this case, the qs$GW$ IP is actually worse than the HF one which is already pretty accurate.
However, the SRG-qs$GW$ can improve slightly the accuracy with respect to HF.
Finally, the beryllium oxide is considered a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
We now turn to the lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2}).
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.
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 because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
The SRG-qs$GW$ could be converged without any problem even for large values of $s$.
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 RPA-based qs$GW$ schemes.
However, as we will see in the next subsection these are just particular molecular systems and on average the RPA polarizability performs better than the TDA one.
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 the TDA one.
Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig3} but this is the other way around.
Therefore, it seems that the effect of the TDA can not be systematically predicted.
@ -730,8 +730,8 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
\label{tab:tab1}
\begin{ruledtabular}
\begin{tabular}{l|ddddd|ddddd}
Mol. & \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} & \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} \\
& \multicolumn{1}{c}{(Reference)} & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} & \multicolumn{1}{c}{(Reference)} & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
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{$s=\num{e2}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
\hline
\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.54 \\
\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 \\
@ -784,8 +784,8 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
\ce{SO2} & & & & & & & & & & \\
\ce{C2H3Cl} & & & & & & & & & & \\
\hline
& \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} & \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} \\
& \multicolumn{1}{c}{(Reference)} & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} & \multicolumn{1}{c}{(Reference)} & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
& \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{$s=\num{e2}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
\hline
MSE & & 0.64 & 0.26 & 0.24 & 0.17 \\
MAE & & 0.74 & 0.32 & 0.25 & 0.19 \\
@ -810,10 +810,10 @@ Therefore, it seems that the effect of the TDA can not be systematically predict
\end{figure*}
%%% %%% %%% %%%
The test set considered in this study is based on the GW20 set introduced by Lewis and Berkelbach, \cite{Lewis_2019} which is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
This set has been augmented with the MgO and O3 molecules (which are part of GW100 as well) because they are known to possess intruder states. \cite{vanSetten_2015,Forster_2021}
The test set considered in this study is made of the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
%This set has been augmented with the MgO and O3 molecules (which are part of GW100 as well) because they are known to possess intruder states. \cite{vanSetten_2015,Forster_2021}
Table~\ref{tab:tab1} shows the principal IP of the 22 molecules considered in this work computed at various level of theories.
Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in this work computed at various levels of theory.
As mentioned previously the HF IPs are overestimated with a mean signed error (MSE) of \SI{0.64}{\electronvolt} and a mean absolute error (MAE) of \SI{0.74}{\electronvolt}.
Performing a one-shot $G_0W_0$ calculation on top these mean-field results allows to divided by more than two the MSE and MAE, \SI{0.26}{\electronvolt} and \SI{0.32}{\electronvolt}, respectively.
However, there are still outliers with quite large errors, for example the IP of the dinitrogen is overestimated by \SI{1.56}{\electronvolt}.
@ -856,8 +856,8 @@ The values of the IP that could be converged for $\eta=0.01$ can vary between $1
% \label{tab:tab1}
% \begin{ruledtabular}
% \begin{tabular}{lddd}
% Mol. & \multicolumn{1}{c}{$G_0W_0^{\text{TDA}}$@HF} & \multicolumn{1}{c}{qs$GW^{\text{TDA}}$} & \multicolumn{1}{c}{SRG-qs$GW^{\text{TDA}}$} \\
% & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{5e-2}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
% Mol. & \mcc{$G_0W_0^{\text{TDA}}$@HF} & \mcc{qs$GW^{\text{TDA}}$} & \mcc{SRG-qs$GW^{\text{TDA}}$} \\
% & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{5e-2}$} & \mcc{$s=\num{e2}$} \\
% \hline
% \ce{He} & 24.45 & 24.48 & 24.39 \\
% \ce{Ne} & 20.85 & 21.23 & 20.92 \\
@ -896,8 +896,8 @@ The values of the IP that could be converged for $\eta=0.01$ can vary between $1
\label{tab:tab2}
\begin{ruledtabular}
\begin{tabular}{lddddd}
Mol. & \multicolumn{1}{c}{$\Delta\text{CCSD(T)}$} & \multicolumn{1}{c}{HF} & \multicolumn{1}{c}{$G_0W_0$@HF} & \multicolumn{1}{c}{qs$GW$} & \multicolumn{1}{c}{SRg-qs$GW$} \\
& & & \multicolumn{1}{c}{$\eta=\num{e-3}$} & \multicolumn{1}{c}{$\eta=\num{e-1}$} & \multicolumn{1}{c}{$s=\num{e2}$} \\
Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
& & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
\hline
\ce{He} & 2.66 & 2.70 & 2.66 & 2.66 & 2.66 \\
\ce{Ne} & 5.09 & 5.47 & 5.25 & 5.19 & 5.19 \\