diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex
index eb437a7..1ce3a62 100644
--- a/Manuscript/sfBSE.tex
+++ b/Manuscript/sfBSE.tex
@@ -504,13 +504,13 @@ The purpose of the underlying $GW$ calculation is to provide quasiparticle energ
 \label{sec:BSE}
 %================================
 
-The Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)$ is
+Within the so-called static approximation of BSE, the Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6)$ is \cite{ReiningBook,Bruneval_2016a}
 \begin{multline}
 	L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) 
 	= L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
 	\\
 	+ \int L_{0}^{\sig\sigp}(\br_1,\br_4;\br_1',\br_3;\omega) 
-	\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)
+	\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6)
 	\\
 	\times L^{\sig\sigp}(\br_6,\br_2;\br_5,\br_2';\omega)
 	d\br_3 d\br_4 d\br_5 d\br_6
@@ -523,15 +523,18 @@ where
 \end{multline}
 is the non-interacting analog of the two-particle correlation function $L$.
 
-Within the $GW$ approximation, the BSE kernel is
+Within the $GW$ approximation, the static BSE kernel is
 \begin{multline}
-	i \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega) 
+	\label{eq:kernel}
+	i \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6) 
 	= \frac{\delta(\br_3 - \br_4) \delta(\br_5 - \br_6) }{\abs{\br_3-\br_6}} 
 	\\
-	- \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) 
+	- \delta_{\sig\sigp} W(\br_3,\br_4;\omega = 0) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) 
 \end{multline}
 where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980,Strinati_1982,Strinati_1984,Strinati_1988}
-Within the static approximation which consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential, the spin-conserved and spin-flip BSE optical excitations are obtained by solving the usual Casida-like linear response (eigen)problem:
+
+As readily seen in Eq.~\eqref{eq:kernel}, the static approximation consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential.
+In this case, the spin-conserved and spin-flip BSE optical excitations are obtained by solving the usual Casida-like linear response (eigen)problem:
 \begin{equation}
 \label{eq:LR-BSE}
 	\begin{pmatrix}
@@ -1030,7 +1033,7 @@ We hope to these new encouraging results will stimulate new developments around
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
-We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions. 
+We would like to thank Pina Romaniello, Xavier Blase, and Denis Jacquemin for insightful discussions. 
 This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
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