From 327d151f2da7c56ef17c3d272058f069573b17a4 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 20 Feb 2023 11:24:19 +0100 Subject: [PATCH] ok with Sec III --- Manuscript/SRGGW.tex | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 1781b45..c30355f 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -488,9 +488,13 @@ and % \end{align} %\end{subequations} Equation \eqref{eq:F0_C0} implies +\begin{subequations} \begin{align} - \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, + \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & + \\ + \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, \end{align} +\end{subequations} and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields \begin{equation} W_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s} @@ -499,7 +503,7 @@ At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals de \begin{equation} \lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0. \end{equation} -Therefore, $W_{pq}^{\nu(1)}(s)$ are genuine renormalized two-electron screened integrals. +Therefore, $W_{pq}^{\nu(1)}(s)$ is a genuine renormalized two-electron screened integral. It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}). %///////////////////////////% @@ -620,7 +624,7 @@ 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 finite $s$ values are suitable to avoid intruder states in its static part. +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 value of $s$ is 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},