little piece of introduction

BSEdyn.tex

@@ -52,7 +52,8 @@
 \newcommand{\co}{\text{x}}
 
 % 
-\newcommand{\Norb}{N}
+\newcommand{\Nel}{N}
+\newcommand{\Norb}{N_\text{orb}}
 \newcommand{\Nocc}{O}
 \newcommand{\Nvir}{V}
 \newcommand{\IS}{\lambda}
@@ -61,8 +62,11 @@
 \newcommand{\hH}{\Hat{H}}
 
 % methods
+\newcommand{\KS}{\text{KS}}
+\newcommand{\HF}{\text{HF}}
 \newcommand{\RPA}{\text{RPA}}
 \newcommand{\BSE}{\text{BSE}}
+\newcommand{\GW}{GW}
 
 % energies
 \newcommand{\Enuc}{E^\text{nuc}}
@@ -70,11 +74,7 @@
 \newcommand{\EHF}{E^\text{HF}}
 \newcommand{\EBSE}{E^\text{BSE}}
 \newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
-\newcommand{\EcRPAx}{E_\text{c}^\text{RPAx}}
 \newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
-\newcommand{\IP}{\text{IP}}
-\newcommand{\EA}{\text{EA}}
-\newcommand{\Req}{R_\text{eq}}
 
 % orbital energies
 \newcommand{\e}[1]{\epsilon_{#1}}
@@ -160,6 +160,16 @@
 \DeclareMathOperator*{\argmax}{argmax}
 \DeclareMathOperator*{\argmin}{argmin}
 
+% orbitals, gaps, etc
+\newcommand{\eps}{\varepsilon}
+\newcommand{\IP}{I}
+\newcommand{\EA}{A}
+\newcommand{\HOMO}{\text{HOMO}}
+\newcommand{\LUMO}{\text{LUMO}}
+\newcommand{\Eg}{E_\text{g}}
+\newcommand{\EgFun}{\Eg^\text{fund}}
+\newcommand{\EgOpt}{\Eg^\text{opt}}
+\newcommand{\EB}{E_B}
 
 \newcommand\vari{{\varepsilon}_i}
 \newcommand\vara{{\varepsilon}_a}
@@ -200,6 +210,25 @@ This is the abstract
 \section{Introduction}
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%
+The Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is to the $GW$ approximation \cite{Hedin_1965,Golze_2019} of many-body perturbation theory (MBPT) \cite{Martin_2016} what time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995} is to Kohn-Sham density-functional theory (KS-DFT), \cite{Hohenberg_1964,Kohn_1965} an affordable way of computing the neutral excitations of a given electronic system. 
+In recent years, it has shown to be useful for molecular systems with a large number of systematic benchmark studies appearing in the scientific literature \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} (see Ref.~ \onlinecite{Blase_2018} for a recent review).
+
+Taking the optical gap (\ie, the lowest optical excitation energy) as an example, BSE builds on top of a $GW$ calculation by adding up excitonic effects $\EB$ to the $GW$ HOMO-LUMO gap
+\begin{equation}
+	\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW},
+\end{equation}
+which is itself a corrected version of the Kohn-Sham (KS) gap
+\begin{equation}
+	\Eg^{\KS} = \eps_{\LUMO}^{\KS} - \varepsilon_{\HOMO}^{\KS} \ll \Eg^{\GW} < \EgFun,
+\end{equation}
+in order to approximate the optical gap
+\begin{equation}
+	\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
+\end{equation}
+where $\EgFun = I^\Nel - A^\Nel$ is the the fundamental gap, \cite{Bredas_2014} $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ being the ionization potential and the electron affinity of the $\Nel$-electron system.
+Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
+Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$.
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}