update fig 5 and 7

Manuscript/SRGGW.tex

@@ -910,22 +910,25 @@ Yet, one can still compare the $GW$ values with their CCSD(T) counterparts in th
 \label{sec:conclusion}
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-In this work, we have investigated the application of the similarity renormalization group to many-body perturbation theory in its $GW$ form.
+In this work, we have applied the similarity renormalization group 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 central equation of the SRG formalism, the flow equation, can be solved analytically for low perturbation order.
-These analytical expressions for the Fock matrices elements and two-electrons screened integrals lead to a renormalized $GW$ quasiparticle equation.
-Isolating the static part of the equation yields an alternative Hermitian static and intruder-state-free self-energy that can be used for qs$GW$ calculation.
-In addition, to this new static form we also explained how to use the SRG formalism to cure discontinuity problems.
+The central equation of the SRG formalism is the 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).
 This gave a first-principle rationale for the SRG-inspired regularizer introduced in Ref.~\onlinecite{Monino_2022}.
 
-The flow parameter dependence of SRG-qs$GW$ has been studied for a few test cases.
-In particular it has been shown that the flow parameter gradually introduce correction in the static self-energy and therefore the IP gradually evolves from the HF one to a plateau value for $s\to\infty$.
+The flow parameter dependence of SRG-qs$GW$ principal ionization potentials has been studied for a few test cases.
+It has been shown that the IP gradually evolves from the HF one to a plateau value for $s\to\infty$ that is much closer to the reference than the starting point.
 For small values of the flow parameter the SRG-qs$GW$ IPs are actually worst than their starting point so one should always use a value of $s$ as large as possible.
 
 As a second stage to this study, the SRG-qs$GW$ performance has been statistically gauged for a test set of 50 atoms and molecules (referred to as $GW$50).
-It has been shown that in averaged SRG-qs$GW$ is slightly better than its traditional qs$GW$ counterpart for principal ionization energies.
+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.