saving work 1st part of Xav part

BSEdyn.tex

@@ -273,9 +273,9 @@ Unless otherwise stated, atomic units are used.
 \label{sec:theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-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 as compared to the standard static approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} 
-More details about this derivation are provided as {\SI}.
+In this Section, following Strinati's seminal work, \cite{Strinati_1988} we first derive in some details the theoretical foundations leading to the dynamical BSE. 
+Additional details about this derivation are provided as {\SI}.
+We present, in a second step, the perturbative implementation of the dynamical correction as compared to the standard static approximation.
 
 %================================
 \subsection{General dynamical BSE theory}
@@ -286,15 +286,36 @@ The two-particle correlation function $L(1,2; 1',2')$ --- a central quantity in
 	iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)}
 \end{equation}
 where, \eg, $1 \equiv (\bx_1 t_1)$ is a space-spin plus time composite variable. 
-The relation between $G$ and the charge density $\rho(1) = -i G(1,1^+)$ provides a direct connection with the density-density susceptibility $\chi(1,2) = L(1,2;1^+,2^+)$ at the core of TD-DFT.
-(The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$ where $0^+$ is a small positive infinitesimal.)
-The two-body correlation function $L$ satisfies the self-consistent Bethe-Salpeter equation \cite{Strinati_1988}  
+The relation between $G$ and the one-body charge density $\rho(1) = -i G(1,1^+)$ provides a direct connection with the density-density susceptibility $\chi(1,2) = L(1,2;1^+,2^+)$ at the core of TD-DFT.
+(The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$, where $0^+$ is a small positive infinitesimal.)
+
+The two-body correlation function $L$ satisfies the self-consistent BSE \cite{Strinati_1988}  
 \begin{multline} \label{eq:BSE}
 	L(1,2; 1',2') = L_0(1,2;1',2')
 	\\
 	+ \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'),
 \end{multline}
 where
+\begin{subequations}
+\begin{align}
+	\label{eq:L0}
+	iL_0(1, 4; 1', 3) & = G(1, 3)G(4, 1'),
+	\\
+	\label{eq:L}
+	iL(1,2; 1',2') & = - G_2(1,2;1',2') + G(1,1') G(2,2'),
+\end{align}
+\end{subequations}
+can be expressed as a function of the one- and two-body Green's functions
+\begin{subequations}
+\begin{align}
+	\label{eq:G1}
+	G(1,1') & = - i \mel{N}{T \hpsi(1) \hpsi^{\dagger}(1')}{N},
+	\\
+	\label{eq:G2}
+	G_2(1,2;1',2') & = - \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
+\end{align}
+\end{subequations}
+and
 \begin{equation}
 	\Xi(3,5;4,6) = i \fdv{[v_\text{H}(3) \delta(3,4) + \Sigma_\text{xc}(3,4)]}{G(6,5)}
 \end{equation}
@@ -303,20 +324,9 @@ is the BSE kernel that takes into account the self-consistent variation of the H
 	v_\text{H}(1) = - i \int d2 \, v(1,2) G(2,2^+),
 \end{equation}
 [where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the exchange-correlation self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$.  
-$L$ and $L_0$ can be expressed as a function of the one- and two-body ($G_2$) Green's functions as follows:
-\begin{gather}
-	\label{eq:L0}
-	iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1'),
-	\\
-	\label{eq:L}
-	iL(1,2; 1',2') = - G_2(1,2;1',2') + G(1,1') G(2,2'),
-	\\
-	\label{eq:G2}
-	i^2 G_2(1,2;1',2') = \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
-\end{gather}
-where the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.
+In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.
 
-The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $G_2$ and $L$ over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$). 
+The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$). 
 In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one gets
 \begin{equation}
 \begin{split}