modifs in IVD

Manuscript/SRGGW.tex

@@ -579,50 +579,48 @@ As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually sta
 
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 \subsection{Alternative form of the static self-energy}
-% ///////////////////////////%
+%///////////////////////////%
 
 Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the self-energy reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{eq:sym_qsgw}.
 Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible.
 However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm}.
 This yields a $s$-dependent static self-energy which matrix elements read
-\begin{equation}
+\begin{multline}
   \label{eq:SRG_qsGW}
-	\begin{split}
    \Sigma_{pq}^{\SRGqsGW}(s) 
-   & = F_{pq}^{(2)}(s) 
+   = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}  
    \\
-   & = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}  
-   \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
-	\end{split}
-\end{equation}
-Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by brute-force symmetrization.
-Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.
-Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
+   \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ].
+\end{multline}
+Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally Hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
+Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator.
+Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms.
 
-It is well-known that in traditional qs$GW$ calculations, increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.
+It is well-known that in traditional qs$GW$ calculations, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
 Similarly, in SRG-qs$GW$, one might need to decrease the value of $s$ to ensure convergence.
-The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
-Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
+Indeed, the fact that SRG-qs$GW$ calculations do not always converge in the large-$s$ limit is expected as, in this limit, potential intruder states have been included.
+Therefore, one should use a value of $s$ large enough to include as many states as possible but small enough to avoid intruder states.
 
-It is instructive to plot the functional form of both regularizing functions, this is done in Fig.~\ref{fig:plot}.
-The surfaces are plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first order Taylor expansion around $(0,0)$ of both functional forms is equal.
-One can observe that the SRG surface is much smoother than its qs counterpart.
-This is due to the fact that the SRG functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities. 
+It is instructive to plot the functional form of both regularizing functions (see Fig.~\ref{fig:plot}).
+These have been plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first-order Taylor expansion around $(x,y) = (0,0)$ of both functional forms is equal.
+One can observe that the SRG-qs$GW$ surface is much smoother than its qs$GW$ counterpart.
+This is due to the fact that the SRG-qs$GW$ functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities. 
 In fact, there is a single singularity at $x=y=0$.
 On the other hand, the function $f^{\qsGW}(x,y;0)$ is singular on the two entire axes, $x=0$ and $y=0$.
-The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$.
-The convergence properties and the accuracy of both static approximations will be quantitatively gauged in Sec.~\ref{sec:results}.
+We believe that the smoothness of the SRG-qs$GW$ surface is the key feature that explains the smoother convergence of SRG-qs$GW$ compared to qs$GW$.
+The convergence properties and the accuracy of both static approximations are quantitatively gauged in Sec.~\ref{sec:results}.
 
-To conclude this section, the case of discontinuities will be briefly discussed.
-Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
-So is it possible to use the SRG machinery developed above to remove discontinuities?
-Not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
-However, doing a change of variable such that
-\begin{align}
-  e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t} 
-\end{align}
-will reverse the situation and now a finite value of $t$ will be well-designed to avoid discontinuities in the renormalized dynamic part.
-The dynamic part after the change of variable is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
+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 finite $s$ values are suitable to avoid intruder states in its static part.
+However, performing the following bijective transformation
+\titou{\begin{align}
+  e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},
+%  s = t/2 - \ln 2 - \ln[\sinh(t/2)]
+\end{align}}
+reverse the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
+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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 \section{Computational details}