From baf12cbbc0cc0d9f8c31f680f4d817b61d07df02 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 7 Feb 2023 14:48:30 +0100 Subject: [PATCH] commit b4 tricky part --- Manuscript/SRGGW.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 2c9988e..f8e1389 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -336,8 +336,8 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation. However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward. -\ant{A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms. -Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} } +A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms. +Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} \begin{equation} \label{eq:GWlin} \begin{pmatrix} @@ -431,7 +431,7 @@ where the supermatrices \end{align} \end{subequations} collect the 2h1p and 2p1h channels. -Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma}\trashant{\eqref{eq:GWlin} before applying the downfolding process to obtain} to define a renormalized version of the quasiparticle equation. +Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma} to define a renormalized version of the quasiparticle equation. In particular, we focus here on the second-order renormalized quasiparticle equation. %///////////////////////////% @@ -488,13 +488,13 @@ Equation \eqref{eq:F0_C0} implies \begin{align} \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, \end{align} -and, thanks to the \ant{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 +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_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s} \end{equation} At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero. Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals. -\ant{It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} for the usual electronic Hamiltonian in the context of wave-function theory within a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).} +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}). %///////////////////////////% \subsection{Second-order matrix elements} @@ -532,7 +532,7 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends 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}. \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. -\ant{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.} +\titou{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 %%%