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@ -695,10 +695,10 @@ 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.
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$GW$ (green curve).
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 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 IP at the HF level (cyan 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 (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}.
As $s$ grows, the IP reaches a plateau at an error that is significantly smaller than the HF starting point.
@ -902,45 +902,42 @@ $G_0W_0$@HF mitigates the average error (MAE equals to \SI{0.16}{\eV}) but the m
The performance of the two qs$GW$ schemes are quite similar for EAs with MAEs of the order of \SI{0.1}{\eV}.
These two partially self-consistent methods reduce also the minimum errors but, interestingly, they do not decrease the maximum error compared to HF.
Note that a positive EA means that the anion state is bound and, therefore, the methods that we consider here are well-suited to describe these states.
On the other hand, a negative EA means that we are potentially dealing with a resonance state.
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.
Note that a positive EA indicates a bounded anion state, which can be accurately described by the methods considered in this study.
However, a negative EA suggests a resonance state, which is beyond the scope of the methods used in this study, including the $\Delta$CCSD(T) reference.
Therefore, one should avoid giving a physical interpretation to these values.
Yet, one can still compare, for a given basis set, the $GW$-based and $\Delta$CCSD(T) values in these cases.
%=================================================================%
\section{Conclusion}
\label{sec:conclusion}
%=================================================================%
In this manuscript, the similarity renormalization group has been applied to many-body perturbation theory in its $GW$ form.
The latter one is known to be plagued by intruder states while the first one is designed to avoid them.
The problems caused by intruder states in many-body perturbation theory are multiple but here the focus was on convergence problems caused by such states.
The present manuscript applies the similarity renormalization group (SRG) to the $GW$ approximation of many-body perturbation theory, which is known to be plagued by intruder states.
The problems caused by intruder states in many-body perturbation theory are numerous but here we focus on the convergence issues caused by them.
The central equation of the SRG formalism is the flow equation and needs to be solved numerically in the general case.
Yet, it can still be solved analytically for low perturbation order.
Doing so in the (upfolded) $GW$ context yields analytical expressions for second-order renormalized Fock matrix elements and two-electrons screened integrals.
These renormalized quantities lead to a renormalized $GW$ quasiparticle equation, referred to as SRG-$GW$, which is the main equation of this work.
The usual approximate solving schemes of the quasiparticle equation can be applied to SRG-$GW$ as well.
In particular, isolating the static part of this equation yields an alternative Hermitian static and intruder-state-free self-energy that can be used for qs$GW$ calculation.
This new qs$GW$ approximation is therefore referred to as SRG-qs$GW$.
In addition to this new static form, we also explained how to use the SRG-$GW$ to avoid discontinuity problems that were arising in $GW$ due to intruder states.
This gave a first-principle rationale for the SRG-inspired regularizer introduced in Ref.~\onlinecite{Monino_2022}.
SRG's central equation is the flow equation, which is usually solved numerically but can be solved analytically for low perturbation order.
Applying this approach in the $GW$ context yields analytical renormalized expressions for the Fock matrix elements and the screened two-electron integrals.
These renormalized quantities lead to a renormalized $GW$ quasiparticle equation, referred to as SRG-$GW$, which is the main result of this work.
The flow parameter dependence of the SRG-qs$GW$ IPs has been studied for a few test cases.
It has been shown that the IPs gradually evolve from the HF starting point at $s=0$ to a plateau value for $s\to\infty$ that is much closer to the CCSD(T) reference than the HF initial value.
For small values of the flow parameter, the SRG-qs$GW$ IPs are actually worst than their starting point.
Therefore, in practice, one should always use a value of $s$ as large as possible.
This is similar to qs$GW$ calculations where one needs to use the smallest possible $\eta$.
By isolating the static component of SRG-$GW$, we obtain an alternative Hermitian and intruder-state-free self-energy that can be used in the context of qs$GW$ calculations.
This new variant is called SRG-qs$GW$.
Additionally, we demonstrate how SRG-$GW$ can effectively resolve the discontinuity problems that arise in $GW$ due to intruder states.
This provides a first-principles justification for the SRG-inspired regularizer proposed in Ref.~\onlinecite{Monino_2022}.
The second stage of this study was to statistically gauge the accuracy of the SRG-qs$GW$ IP for a test set of 50 atoms and molecules (referred to as $GW$50).
It has been shown that on average SRG-qs$GW$ is slightly better than its traditional qs$GW$ counterpart for principal ionization energies.
Note that while the accuracy improvements are quite small, it comes with no additional computational cost, and its really fast to implement as one only needs to change the expression of the static self-energy.
In addition, it has been shown that the SRG-qs$GW$ can be converged in a much more black-box fashion than the traditional qs$GW$ thanks to its intruder-state-free nature.
We first study the flow parameter dependence of the SRG-qs$GW$ IPs for a few test cases.
The results show that the IPs gradually evolve from the HF starting point at $s=0$ to a plateau value for $s\to\infty$ that is much closer to the CCSD(T) reference than the HF initial value.
For small values of the flow parameter, the SRG-qs$GW$ IPs are actually worse than their starting point.
Therefore, it is advisable to use the largest possible value of $s$, similar to qs$GW$ calculations where one needs to use the smallest possible $\eta$ value.
Finally, the EAs have been investigated as well.
It has been found that the performances of qs$GW$ and SRG-qs$GW$ are quite similar for the EAs of the $GW$50 set.
However, there is a caveat because most of the anions of the $GW$50 set are actually resonance states and their associated physics can not be accurately described by the methods considered in this study.
A test set of molecules with bound anions with an accompanying benchmark of accurate reference values would certainly be valuable to the many-body perturbation theory community.
Next, we gauge the accuracy of the SRG-qs$GW$ principal IP for a test set of 50 atoms and molecules (referred to as $GW$50).
The results show that, on average, SRG-qs$GW$ is slightly better than its qs$GW$ parent.
Despite the fact that the increase in accuracy is relatively modest, it comes with no additional computational cost and is straightforward to implement, as only the expression of the static self-energy needs to be modified.
Moreover, SRG-qs$GW$ calculations are much easier to converge than their traditional qs$GW$ counterparts thanks to the intruder-state-free nature of SRG-qs$GW$.
Finally, the principal EAs of the $GW$50 set are also investigated.
It is found that the performances of qs$GW$ and SRG-qs$GW$ are quite similar in this case.
However, it should be noted that most of the anions of the $GW$50 set are resonance states, and the associated physics cannot be accurately described by the methods considered in this study.
Therefore, a test set of molecules with bound anions and their accompanying accurate reference values would be valuable to the many-body perturbation theory community.
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\acknowledgements{