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corrections in the conclusion

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Antoine Marie 2023-03-08 09:47:32 +01:00
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Manuscript/SRGGW.tex
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@ -904,7 +904,7 @@ $G_0W_0$@HF mitigates the average error (MAE equal to \SI{0.16}{\eV}) but the mi
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt}.
The two partially self-consistent methods reduce as well the minimum value but interestingly, the three flavours of many-body perturbation theory considered here can not decrease the maximum error with respect to their HF starting point.
Note that a positive EA means that the anion state is bound and therefore the methods considered here are well-suited to describe the EAs.
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.
@ -914,34 +914,35 @@ Yet, one can still compare the $GW$ values with their CCSD(T) counterparts withi
\label{sec:conclusion}
%=================================================================%
In this work, we have applied the similarity renormalization group to many-body perturbation theory in its $GW$ form.
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 perturbatin theory are multiple but here the focus was on convergence problems caused by such states.
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 central equation of the SRG formalism is the the flow equation and needs to be solved numerically in the general case.
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 matrices elements and two-electrons screened integrals.
These renormalized quantities lead to a renormalized $GW$ quasiparticle equation after downfolding.
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 referred to as SRG-qs$GW$.
In addition, to this new static form we also explained how to use the SRG formalism to cure discontinuity problems (which are also due to intruder states).
Doing so in the (upfolded) $GW$ context yields analytical expressions for second-order renormalized Fock matrices 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}.
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 evolves from the HF starting point at $s=0$ to a plateau value for $s\to\infty$ that is much closer to the reference than the HF initial value.
It has been shown that the IPs gradually evolves 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 need to use the smallest possible $\eta$.
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$50).
It has been shown that in average SRG-qs$GW$ is slightly better than its traditional qs$GW$ counterpart for principal ionization energies.
Note that while the accuracy improvement is quite small, it comes with no additional computational cost and its really fast to implement as one just need to change the static self-energy expression.
In addition, the SRG-qs$GW$ can be converged in a much more black-box fashion than traditional qs$GW$ thanks to its intruder-state free nature.
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.
Finally, the EA have been investigated as well.
It has been found that the performance of the two qs$GW$ flavours for the $GW$50 set are quite similar.
However, there is a caveat because most of the anions of the $GW$50 set are actually resonance states and their associated physcis can not be accurately described.
A test set of molecules with bound anions and a benchmark of accurate reference values would certainly valuable to many-body perturbation theory community.
Finally, the EAs have been investigated as well.
It has been found that the performance of the two qs$GW$ flavours for the EAs of the $GW$50 set are quite similar.
However, there is a caveat because most of the anions of the $GW$50 set are actually resonance states and their associated physcis 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.
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\acknowledgements{