diff --git a/BSEdyn.tex b/BSEdyn.tex
index 1db90b4..f68370d 100644
--- a/BSEdyn.tex
+++ b/BSEdyn.tex
@@ -192,7 +192,7 @@
 
 \begin{document}	
 
-\title{Dynamical Correction to the Bethe-Salpeter Equation}
+\title{ \textcolor{red}{Assessing} Dynamical Corrections to the Bethe-Salpeter Equation}
 
 \author{Pierre-Fran\c{c}ois \surname{Loos}}
 	\email{loos@irsamc.ups-tlse.fr}
@@ -329,15 +329,14 @@ The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space
 \begin{align} \label{eq:iL0}
 	[iL_0]( \omega_1 ) 
 	= \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} )
-	e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} },
+	e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
    \end{align}
 with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. 
-Plugging now the Lehman representation of the one-body Green's function
+We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation : 
 \begin{equation}
-	G(x_1,x_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \text{sgn} \times (\e{p} - \mu) }
+	G(x_1,x_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn}  (\e{p} - \mu) }
 \end{equation}
-\titou{T2: did you introduce a new basis here?}
-(where $\eta$ is a positive infinitesimal) into Eq.~\eqref{eq:iL0} and projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component  
+where the $\lbrace \varepsilon_p \rbrace$ are quasiparticle energies and  $\lbrace \phi_p \rbrace$  the associated one-body molecular orbitals, namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component  
 \begin{multline}
 	\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'})  L_0(\bx_1,3;\bx_{1'},4; \Oms) 
 	\\
@@ -367,20 +366,20 @@ We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hps
 \end{multline}
 with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that
 \begin{equation}
-	e^{i H \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
+	e^{i {\hat H} \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
 \end{equation}
-\titou{T2: what is $H$?}
+ with ${\hat H}$ the exact many-body Hamiltonian. 
 The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies. 
 Selecting $(p,q)=(j,b)$ yields the largest components
-$A_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker  
-$B_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. 
-Neglecting the $B_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). 
+$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker  
+$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. 
+Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). 
 Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE) :
 \begin{equation}
 \begin{split}
-	( \e{a} - \e{i} - \Oms ) A_{ia}^{s}         
-	& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] A_{jb}^{s}  \\ 
-	& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] B_{jb}^{s} 
+	( \e{a} - \e{i} - \Oms ) X_{ia}^{s}         
+	& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s}  \\ 
+	& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s} 
 	= 0
 \end{split}
 \end{equation}
@@ -391,7 +390,17 @@ with an effective dynamically  screened Coulomb potential (see Pina  eq. 24):
 	\\
 	\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ]
 \end{multline}
-where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.  
+where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.  The Coulomb matrix elements are defined following the Mulliken notations : 
+\begin{align*}
+    v_{ai,bj} &= \int d{\bf r} d{\bf r}' \; \phi_a({\bf r}) \phi_i^*({\bf r}) v({\bf r} -{\bf r}')
+         \phi_b^*({\bf r}') \phi_j({\bf r}') \\
+            W_{ij,ab}({\omega}) &= \int d{\bf r} d{\bf r}' \; \phi_i({\bf r}) \phi_j^*({\bf r}) W({\bf r} ,{\bf r}'; \omega)
+         \phi_a^*({\bf r}') \phi_b({\bf r}')
+\end{align*}
+where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with MO with complex conjugation.
+
+\xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto  the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : }
+
 In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
 \begin{multline}
 	W_{ij,ab}(\omega)