From 74b42641df4954819116ff5204a3cfca7163d2ac Mon Sep 17 00:00:00 2001 From: Antoine MARIE Date: Fri, 12 May 2023 13:23:10 +0200 Subject: [PATCH] put back equations --- Manuscript/SRGGW.tex | 16 +++++++++++++++- 1 file changed, 15 insertions(+), 1 deletion(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 96a6869..f7995a6 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -363,7 +363,21 @@ Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of % \end{pmatrix} % \boldsymbol{\epsilon}, \end{equation} -%where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, +% where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, +where the 2h1p and 2p1h matrix elements are +\begin{subequations} + \begin{align} + C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu}, + \\ + C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu}, + \end{align} +\end{subequations} +and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})] +\begin{align} + W^\text{2h1p}_{p,i\nu} & = W_{pi}^{\nu}, + & + W^\text{2p1h}_{p,a\nu} & = W_{pa}^{\nu}. +\end{align} The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021} \begin{equation}