still unhappy with IVC and IVD

Manuscript/SRGGW.tex

@@ -520,19 +520,26 @@ Collecting every second-order term in the flow equation and performing the block
   \label{eq:diffeqF2}
   \dv{\bF^{(2)}}{s}  = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger},
 \end{multline}
-which can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
+which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to give
 \begin{multline}
   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^{\dagger}_{r\nu,q}  \\
   \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
 \end{multline}
 
-At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit
+At $s=0$, the second-order correction vanishes, hence giving
+\begin{equation}
+	\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)},
+\end{equation}
+while, for $s\to\infty$, it tends towards the following static limit
 \begin{equation}
   \label{eq:static_F2}
-  F_{pq}^{(2)}(\infty) =  \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}.
+  \lim_{s\to\infty} 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}.
 \end{equation}
-Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
-\titou{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
+Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
+\begin{equation}
+	\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
+\end{equation}
+Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
 %%% FIG 1 %%% 
@@ -541,7 +548,7 @@ This transformation is done gradually starting from the states that have the lar
   \includegraphics[width=\linewidth]{fig1.pdf}
   \caption{
     Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2)$.
-    \label{fig:fig1}}
+    \label{fig:plot}}
 \end{figure*}
 %%% %%% %%% %%%
 
@@ -550,14 +557,14 @@ This transformation is done gradually starting from the states that have the lar
 % ///////////////////////////%
 
 Because the $s\to\infty$ limit of Eq.~(\ref{eq:GW_renorm}) is purely static, it can be seen as a qs$GW$ calculation with an alternative static approximation than the usual one of Eq.~(\ref{eq:sym_qsgw}).
-Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, in the $s\to\infty$ limit self-consistently solving the renormalized quasi-particle equation is once again quite difficult, if not impossible.
+Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, in the $s\to\infty$ limit, self-consistently solving the renormalized quasi-particle equation is once again quite difficult, 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.~(\ref{eq:GW_renorm}).
 This yields a $s$-dependent static self-energy which matrix elements read
 \begin{multline}
   \label{eq:SRG_qsGW}
-  \Sigma_{pq}^{\text{SRG}}(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}  \\ \times \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
+  \Sigma_{pq}^{\text{SRG}}(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}  \\ \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
 \end{multline}
-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 symmetrization.
+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 state.
 Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
 
@@ -566,13 +573,13 @@ Similarly, in SRG-qs$GW$ one might need to decrease the value of $s$ to ensure c
 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.
 
-It is instructive to plot both regularizing functions, this is done in Fig.~\ant{???}.
+It is instructive to plot both regularizing functions, this is done in Fig.~\ref{fig:plot}.
 The surfaces correspond to a value of the regularizing parameter value of 1.
 The SRG surface is much smoother than its qs counterpart.
 In fact the SRG regularization has less work to do because for $\eta=0$ there is a single singularity at $x=y=0$.
 On the other hand the function $f_{\text{qs}}(x,y;0)$ is singular on two entire axis, $x=0$ 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 the results section.
+The convergence properties and the accuracy of both static approximations will be 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.