diff --git a/BSEdyn.tex b/BSEdyn.tex
index 2ce37da..4ec3d15 100644
--- a/BSEdyn.tex
+++ b/BSEdyn.tex
@@ -39,8 +39,8 @@
 \newcommand{\T}[1]{#1^{\intercal}}
 
 % coordinates
-\newcommand{\br}[1]{\mathbf{r}_{#1}}
-\newcommand{\dbr}[1]{d\br{#1}}
+\newcommand{\br}{\mathbf{r}}
+\newcommand{\dbr}{d\br}
 
 % methods
 \newcommand{\evGW}{ev$GW$}	
@@ -336,7 +336,7 @@ Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$
 \titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system}, $L_0(1,2;1',2')$ cannot contribute \titou{to?} since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential.
 \titou{T2: I think we should specify at which level of theory this quasiparticle gap is computed. What do you think?}
 
-The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space/spin) variables:  
+The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space/spin) variables
 \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} )
@@ -347,7 +347,9 @@ We now adopt the Lehman representation of the one-body Green's function in the q
 \begin{equation}
 	G(\bx_1,\bx_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}
-where the $\lbrace \e{p} \rbrace$ 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  
+where \titou{$\mu$ is the chemical potential}. 
+The set $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body \titou{(spin)} orbitals, \titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
+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) 
 	\\
@@ -356,6 +358,7 @@ where the $\lbrace \e{p} \rbrace$ are quasiparticle energies and  $\lbrace \phi_
 	\times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) }  ]  
 \end{multline}
 with $\tau = \tau_{34}$. 
+\titou{We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. (T2: I wouldn't call that chemist notations...)}
 
 Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, 
 \begin{equation}
@@ -366,26 +369,26 @@ leads to the following simplified BSE kernel
 	\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
 \end{equation}
 where $W$ is its dynamically-screened Coulomb operator.
-We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators  over a complete orbital basis creation/destruction operators:
+We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators  over a complete orbital basis creation/destruction operators \titou{(T2: I don't understand why we need this spectral representation)}:
 \begin{multline}
 	\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s} 
 	\\
 	= - \qty( e^{  -i \Omega_s t^{34} }  ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4) 
 	\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
 	\\
-	\times \qty[ \theta( \tau ) e^{- i ( \e{p} - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \e{q}  + \hOms) \tau } ]
+	\times \qty[ \theta( \tau_{34} ) e^{- i ( \e{p} - \hOms ) \tau_{34} } + \theta( -\tau_{34} ) e^{ - i ( \e{q}  + \hOms) \tau_{34} } ]
 \end{multline}
-with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that
+where the $ \lbrace \eps_{p/q} \rbrace$ are proper addition/removal energies \titou{(T2: shall it be mentioned earlier around Eq. (14)?)} such that
 \begin{equation}
-	e^{i {\hat H} \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
+	e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
 \end{equation}
- with ${\hat H}$ the exact many-body Hamiltonian. 
+with $\hH$ 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
 $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, 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. 
+$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. 
 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) :
+Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ \titou{(where do we need these terms?)}, 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 ) X_{ia}^{s}         
@@ -401,14 +404,16 @@ 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 $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.  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.
+where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.  
+Following Mulliken's notations, the Coulomb matrix elements are defined as
+\begin{align}
+	v_{ai,bj} 
+	& = \int d\br d\br' \; \phi_a(\br) \phi_i^*(\br) v(\br -\br') \phi_b^*(\br') \phi_j(\br'),
+	\\
+	W_{ij,ab}({\omega}) 
+	& = \int d\br d\br' \; \phi_i(\br) \phi_j^*(\br) W(\br ,\br'; \omega) \phi_a^*(\br') \phi_b(\br'),
+\end{align}
+where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with orbitals 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 : }
 
@@ -471,7 +476,7 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
 \label{eq:LR-dyn}
 	\begin{pmatrix}
 		\bA{}(\omega)	&	\bB{}(\omega)	\\
-		-\bB{}(\omega)	&	-\bA{}(\omega)	\\
+		-\bB{}(\titou{-}\omega)	&	-\bA{}(\titou{-}\omega)	\\
 	\end{pmatrix}
 	\cdot
 	\begin{pmatrix}
@@ -548,12 +553,13 @@ 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$GW$.
 
 Now, let us decompose, using basic perturbation theory, the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (hence linear) reference and a first-order dynamic (hence non-linear) perturbation, such that
-\begin{equation}
+\begin{multline}
 \label{eq:LR-PT}
 	\begin{pmatrix}
 		\bA{}(\omega)	&	\bB{}(\omega)	\\
-		-\bB{}(\omega)	&	-\bA{}(\omega)	\\
+		-\bB{}(\titou{-}\omega)	&	-\bA{}(\titou{-}\omega)	\\
 	\end{pmatrix}
+	\\
 	=
 	\begin{pmatrix}
 		\bA{(0)}	&	\bB{(0)}	\\
@@ -561,10 +567,10 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
 	\end{pmatrix}
 	+
 	\begin{pmatrix}
-		\bA{(1)}(\omega)	&	\bB{(1)}(\omega)	\\
-		-\bB{(1)}(\omega)	&	-\bA{(1)}(\omega)	\\
+		\bA{(1)}(\omega)			&	\bB{(1)}(\omega)	\\
+		-\bB{(1)}(\titou{-}\omega)	&	-\bA{(1)}(\titou{-}\omega)	\\
 	\end{pmatrix}
-\end{equation}
+\end{multline}
 with
 \begin{subequations}
 \begin{align}
@@ -642,7 +648,7 @@ and, thanks to first-order perturbation theory, the first-order correction to th
 	\cdot
 	\begin{pmatrix}
 		\bA{(1)}(\Om{m}{(0)})	&	\bB{(1)}(\Om{m}{(0)})	\\
-		-\bB{(1)}(\Om{m}{(0)})	&	-\bA{(1)}(\Om{m}{(0)})	\\
+		-\bB{(1)}(\titou{-}\Om{m}{(0)})	&	-\bA{(1)}(\titou{-}\Om{m}{(0)})	\\
 	\end{pmatrix}
 	\cdot
 	\begin{pmatrix}