From 7822bc83cb8bd72ab8d3f2904b46b5ea7167ad07 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Wed, 27 May 2020 19:51:13 +0200
Subject: [PATCH] fix

---
 BSEdyn.tex | 18 +++++++++++-------
 1 file changed, 11 insertions(+), 7 deletions(-)

diff --git a/BSEdyn.tex b/BSEdyn.tex
index 1c5f48f..319b866 100644
--- a/BSEdyn.tex
+++ b/BSEdyn.tex
@@ -442,7 +442,7 @@ In Eq.~\eqref{eq:BSE-final},
 	\ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'),
 \end{equation}
 are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
-\begin{multline}
+\begin{multline} \label{eq:wtilde}
 	\widetilde{W}_{ij,ab}(\Om{s}{})
 	= \frac{ i }{ 2  \pi}  \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) 
 	\\
@@ -507,8 +507,8 @@ with
 where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$.
 The RPA matrices $\bA{\RPA}$ and $\bB{\RPA}$ in Eq.~\eqref{eq:LR-RPA} are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}$, and $\bY{m}{}$ are (eigen)vectors of length $\Nocc \Nvir$. 
 
-Because $\Om{m}{\RPA} > 0$, we have
-\begin{multline}
+The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
+\begin{multline} 
 	\widetilde{W}_{ij,ab}( \Om{s}{} ) 
 	= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m}  
 	\\ 
@@ -527,8 +527,10 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
 \begin{equation}
 \label{eq:LR-dyn}
 	\begin{pmatrix}
-		\bA{}(\Om{s}{})		&	\bB{}(\Om{s}{})	\\
-		-\bB{}(\titou{-}\Om{s}{})	&	-\bA{}(\titou{-}\Om{s}{})	\\
+		\bA{}(\Om{s}{})				&	\bB{}(\Om{s}{})	
+		\\
+		-\bB{}(\titou{-}\Om{s}{})	&	-\bA{}(\titou{-}\Om{s}{})	
+		\\
 	\end{pmatrix}
 	\cdot
 	\begin{pmatrix}
@@ -577,8 +579,10 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
 	\\
 	=
 	\begin{pmatrix}
-		\bA{(0)}	&	\bB{(0)}	\\
-		-\bB{(0)}	&	-\bA{(0)}	\\
+		\bA{(0)}	&	\bB{(0)}	
+		\\
+		-\bB{(0)}	&	-\bA{(0)}	
+		\\
 	\end{pmatrix}
 	+
 	\begin{pmatrix}