new color schemes and corrections

Manuscript/SRGGW.tex

@@ -626,12 +626,13 @@ The convergence properties and the accuracy of both static approximations are qu
 To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}.
 Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
 Is it then possible to rely on the SRG machinery to remove discontinuities?
-Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a \titou{finite} value of $s$ is suitable to avoid intruder states in its static part.
-However, performing a bijective transformation of the form,
+Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation.
+However, as we have seen just above the functional form of the renormalized equation \ant{makes it possible to choose $s$ such that there is no intruder states in its static part.}
+But performing a bijective transformation of the form,
 \begin{align}
   e^{- \Delta s} &= 1-e^{-\Delta t},
-       \end{align}
-on the renormalized quasiparticle equation \eqref{eq:GW_renorm} reverses the situation and makes \titou{finite} values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
+\end{align}
+on the renormalized quasiparticle equation \eqref{eq:GW_renorm} reverses the situation and \ant{makes it possible to choose $t$ such that there is no intruder states in the dynamic part, hence removing discontinuities.}
 Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
 
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@@ -652,7 +653,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$
 The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively.
 In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
 However, in order to perform black-box comparisons, these parameters have been fixed to these default values.
-The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations \titou{where we have not linearized the quasiparticle equation} while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
+The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ \ant{(without linearization of the quasiparticle equation)} calculations while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
 
 The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set.
 The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} with default parameters within the restricted and unrestricted HF formalism for the neutral and charged species, respectively.
@@ -770,7 +771,7 @@ Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in th
 As mentioned previously the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.56}{\eV} and a mean absolute error (MAE) of \SI{0.69}{\eV}.
 Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.29}{\eV} and \SI{0.33}{\eV}, respectively.
 However, there are still outliers with large errors. 
-\titou{For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}.}
+\ant{For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}, a large error which is clearly due to the starting point dependence of $G_0W_0$@HF.}
 Self-consistency can mitigate the error of the outliers as the MAE at the qs$GW$ level is now \SI{0.57}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.31}{\eV} for $G_0W_0$ to \SI{0.18}{\eV}.
 In addition, the MSE and MAE (\SI{0.23}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
 
@@ -869,7 +870,7 @@ Indeed, up to $s=\num{e3}$ SRG-qs$GW$ self-consistency can be attained without a
 For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50.
 This means that some intruder states were included in the static correction for this value of $s$.
 However, this is not a problem as the MAE of the test set is already well converged at $s=\num{e3}$.
-This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and \titou{$\eta=1/2s^2$}.
+This is illustrated by the blue curve of Fig.~\ref{fig:fig6} which shows the evolution of the MAE with respect to $s$ and $s=1/2\eta^2$.
 The convergence plateau of the MAE is reached around $s=50$ while the convergence problem arises for $s>\num{e3}$.
 Therefore, for future studies using the SRG-qs$GW$ method, a default value of the flow parameter equal to $\num{5e2}$ or $\num{e3}$ is recommended.