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+\documentclass[journal=jctcce,manuscript=article]{achemso}
+
+%%\usepackage{natbib}
+%%\usepackage{notoccite}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{graphicx}% Include figure files:q
+
+\usepackage{dcolumn}% Align table columns on decimal point
+\usepackage{bm}% bold math
+\usepackage{longtable}
+\usepackage[usenames, dvipsnames]{color}
+\usepackage{subfigure}
+\usepackage{multirow}
+\usepackage{dsfont}
+\usepackage{ulem}
+\usepackage{xcolor}
+
+\DeclareMathOperator*{\argmax}{argmax}
+\DeclareMathOperator*{\argmin}{argmin}
+
+
+\newcommand\vari{{\varepsilon}_i}
+\newcommand\vara{{\varepsilon}_a}
+\newcommand\varj{{\varepsilon}_j}
+\newcommand\varb{{\varepsilon}_b}
+\newcommand\varn{{\varepsilon}_n}
+\newcommand\varm{{\varepsilon}_m}
+\newcommand\Oms{{\Omega}_s}
+\newcommand\hOms{\frac{{\Omega}_s}{2}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\newcommand{\correction}[1]{\textcolor{blue}{#1}}
+%\definecolor{myblue}{rgb}{0.0, 0.0, 1.0}
+
+\author{Pierre-Fran\c{c}ois Loos}
+	\email{loos@irsamc.ups-tlse.fr}
+	\affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
+	
+\author{Xavier Blase}
+	\email{xavier.blase@neel.cnrs.fr}
+	\affiliation[NEEL, Grenoble]{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
+
+
+
+\title{  Dynamical BSE  }
+
+\date{\today}
+\begin{document}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{abstract}
+
+ \end{abstract}
+ 
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+ \section{Theory}
+
+The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) =  G(1, 3)G(4, 1')$   reads, dropping the (space/spin)-variables:  
+\begin{align*}
+[iL_0]( \omega_1 )  = \frac{ 1 }{  2\pi } \int d \omega   \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2}  )   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 1-body Green's function Lehman representation, e.g.
+$$
+G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) }
+$$
+and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component  
+\begin{align*}
+\int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'})  L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times    \\
+     &       \frac{ \phi_a^*(x_3) \phi_i(x_4)  } {   \Oms - (  \vara - \vari )  + i \eta }   
+       \Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau }   
+     +   \theta( - \tau  )   e^{i (\vara -  \hOms \tau }  \Big)  
+\end{align*}
+with $\tau = \tau_{34}$.
+We further obtain the spectral representation of
+$\langle N  | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$
+expanding the field operators  over a complete orbital basis creation/destruction operators:
+\begin{align*}
+\langle N  | T   {\hat \psi}(3)     {\hat \psi}^{\dagger}(4) & | N,s \rangle = - \Big( e^{  -i \Omega_s t^{34} }  \Big)    \sum_{mn} \phi_m(x_3) \phi_n^*(x_4)  \langle N | {\hat a}_n^{\dagger} {\hat a}_m  | N,s \rangle    \times \nonumber \\
+\times & \Big( \theta( \tau )   e^{- i   ( \varepsilon_m  -  \hOms ) \tau  }  
+ + \theta( -\tau )   e^{ - i   ( \varepsilon_n  + \hOms)  \tau  }      \Big)
+\end{align*}
+with $\tau = \tau_{34}$ and where the $ \lbrace \varepsilon_{n/m} \rbrace$ are proper addition/removal energies such that e.g.
+$$
+e^{  i H \tau  } {\hat a}_m^{\dagger}   | N  \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau  }  {\hat a} _m^{\dagger}   | N  \rangle
+$$
+Selecting (n,m)=(j,b) yields the largest components
+$A_{jb}^{s} = \langle N | {\hat a}_j^{\dagger} {\hat a}_b  | N,s \rangle $, while  (n,m)=(b,j) yields much weaker  
+$B_{jb}^{s} = \langle N | {\hat a}_b^{\dagger} {\hat a}_j  | N,s \rangle $ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the   $B_{jb}^{s}$ leads to the Tamm Dancoff approximation (TDA). Obtaining similarly the spectral representation of $ \langle N  | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle$ ($t_{1'} = t_1^{+}$) projected onto $\phi_a^*(x_1) \phi_i(x_{1'})$, 
+one obtains after a few tedious manipulations (see Supplemental Information) the dynamical  Bethe-Salpeter equation (DBSE) :
+ \begin{align}
+      ( \varepsilon_a - \varepsilon_i    - \Omega_s    )       A_{ia}^{s}         
+       &+ \sum_{jb}  \Big(  v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms)    \Big)  A_{jb}^{s}  \\ 
+       &+  \sum_{bj}  \Big(  v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms)    \Big)  B_{jb}^{s} 
+       = 0
+\end{align}
+ with an effective dynamically  screened Coulomb potential (see Pina  eq. 24): 
+ \begin{align}
+ \widetilde{W}_{ij,ab}(\Oms)  &= { i \over 2  \pi}  \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
+    \hskip 1cm &\times \left[ \frac{1}{ (\Oms - \omega) - ( \varb - \vari ) +i \eta } + \frac{1}{ (\Oms + \omega) - ( \vara - \varj ) + i\eta } \right] \nonumber
+\end{align}
+In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
+\begin{align*}
+W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
+   & \times \Big(  \frac{1}{ \omega-\Omega_m^{RPA} + i\eta }  -   \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) 
+\end{align*}
+so that
+\begin{align}
+ \widetilde{W}_{ij,ab}( \Oms ) &=   (ij|ab) + 2 \sum_m^{OV}    [ij|m] [ab|m] \times \\ 
+    & \times     \left[ \frac{  1 }{    \Omega_{ib}^{s} - \Omega_m^{RPA}    + i\eta   }   +   \frac{ 1}{   \Omega_{ja}^{s}   - \Omega_m^{RPA}   + i\eta }
+     \right]   \nonumber
+\end{align} 
+with e.g.  $  \Omega_{ib}^{s} = \Oms - ( \varepsilon_b -  \varepsilon_i) $.   \textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation  energy, so that 
+e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and cannot diverge. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap   low lying BSE excitations, such that
+$$
+\left[ \frac{  1 }{    \Omega_{ib}^{s} - \Omega_m^{RPA}    + i\eta   }   +   \frac{ 1}{   \Omega_{ja}^{s}   - \Omega_m^{RPA}   + i\eta }
+     \right] 
+     < 
+  \Big(  \frac{1}{ \omega-\Omega_m^{RPA} + i\eta }  -   \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) < 0   
+$$
+in the limit $(\omega \rightarrow 0)$ of the standard adiabatic BSE . WELL, do we know the sign of 
+$[ij|m] [ab|m]$ ?? }
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{document}