From d89470b442b1eab3dbf308c43b49d3908508d761 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Mon, 3 Feb 2020 16:21:06 +0100
Subject: [PATCH] Updates from Overleaf

---
 BSE-PES.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/BSE-PES.tex b/BSE-PES.tex
index 360c7e2..98224ed 100644
--- a/BSE-PES.tex
+++ b/BSE-PES.tex
@@ -333,19 +333,19 @@ In the standard BSE approach, the screened Coulomb potential $W^{\lambda}$ is bu
 with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$ and $\chi_{0}$ the non-interacting polarizability.  In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $W^{\lambda}$ can be written as follows in the case of real molecular orbitals: \cite{complexw}
 \begin{multline}
 \label{eq:W}
-	\W{ij,ab}{\IS}(\omega) = \textcolor{red}{\sout{2}} \ERI{ij}{ab} +  \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
+	\W{ij,ab}{\IS}(\omega) =  \ERI{ij}{ab} +  \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
 	\\
 	\times  \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} \textcolor{red}{-} \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta})
 \end{multline}
-where the \xavier{ \sout{screened two-electron integrals}  spectral weights} $\sERI{pq}{m}$  at coupling strength $\lambda$ read:
+where the  spectral weights $\sERI{pq}{m}$  at coupling strength $\lambda$ read:
 \begin{equation}
 	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}
 \end{equation}
 
 In Eq.~\eqref{eq:W},  $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
 \begin{subequations}
+\label{eq:LR_RPA}
 \begin{align}
-	\label{eq:LR_RPA}
 	\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) +   \IS \ERI{ia}{jb},
 	\\ 
 	\BRPA{ia,jb}{\IS} & =   \IS \ERI{ia}{bj},
@@ -357,8 +357,8 @@ The relation between the BSE formalism and the well-known RPAx approach can be o
 %namely setting  $\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ 
 so that   $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit,  the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\ref{eq:LR_BSE} to the RPAx equations: 
 \begin{subequations}
+\label{eq:LR_RPAx}
 \begin{align}
-	\label{eq:LR_RPAx}
 	\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \left[ \ERI{ia}{jb} -    \ERI{ij}{ab} \right],
 	\\ 
 	\BRPAx{ia,jb}{\IS} & =  \IS \left[ \ERI{ia}{bj} -   \ERI{ib}{aj} \right].