blush

BSEdyn.tex

@@ -274,7 +274,7 @@ Unless otherwise stated, atomic units are used.
 \label{sec:theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-In this Section, following the seminal work of Strinati, \cite{Strinati_1988} we describe, first, the theoretical foundations leading to the dynamical Bethe-Salpeter equation. 
+In this Section, following Strinati's seminal work, \cite{Strinati_1988} we first derive in some details the theoretical foundations leading to the dynamical Bethe-Salpeter equation. 
 We present, in a second step, the perturbative implementation of the dynamical correction \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} as compared to the standard static approximation. 
 More details about this derivation are provided as {\SI}.
 
@@ -335,7 +335,7 @@ with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
 The $\Oms$'s are the neutral excitation energies of interest. 
 \titou{T2: shall we specify the physical meaning of $\chi_s$ and $\tchi_s$?}
 
-Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the modified dynamical Bethe-Salpeter equation:
+Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$ functions, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the modified dynamical Bethe-Salpeter equation:
 \begin{multline} \label{eq:BSE_2}
     \mel{N}{T \hpsi(\bx_1) &\hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
 	\theta ( \tau_{12} )  = \int d3456 \times
@@ -344,7 +344,7 @@ Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$
 	\mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s} 
 	\theta (t^{56}_m - t_2)
 \end{multline}
-with $t^{56}_m = \min(t_5,t_6)$. \titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system}, $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential.
+with $t^{56}_m = \min(t_5,t_6)$. \titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system} due to excitonic effects, $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response 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 $L_0(1,4;1',3)$ reads, dropping the (space/spin) variables
@@ -363,13 +363,13 @@ The set $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \
 After projecting onto \titou{$\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) 
-	\\
-	= e^{i \Oms t^{34} }   
+ = e^{i \Oms t^{34} } \times	\\
+	\times   
 	\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4)  } {   \Oms - (  \e{a} - \e{i} )  + i \eta }   
 	\times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) }  ]  
 \end{multline}
-with $\tau = \tau_{34}$. From then on, 
- $(i,j)$ index  occupied orbitals and $(a,b)$ virtual ones. 
+with $\tau = \tau_{34}$ and where 
+ $(i,j)$/$(a,b)$ index  occupied/virtual orbitals, respectively. 
 Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, 
 \begin{equation}
 	\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),