fixing up biblio and start of intro

Manuscript/GW-srDFT.tex

@@ -189,7 +189,47 @@ Here, we propose a density-based basis set correction based on short-range corre
 \section{Introduction}
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%
+The so-called GW approximation has a long and successful history in the calculation of the electronic structure of solids \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}.
+The GW approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965} 
+\begin{subequations}
+\begin{align}
+	\label{eq:G}
+	& G(12) = G_\text{0}(12) + \int G_0(13) \Sigma(34) G(42) d(34),
+	\\
+	\label{eq:Gamma}
+	& \Gamma(123) = \delta(12) \delta(13) 
+	\notag 
+	\\
+	& \qquad \qquad		+ \int \fdv{\Sigma(12)}{G(45)} G(46) G(75) \Gamma(673) d(4567),
+	\\
+	\label{eq:P}
+	& P(12) = - i \int G(13) \Gamma(324) G(41) d(34),
+	\\
+	\label{eq:W}
+	& W(12) = v(12) + \int v(13) P(34) W(42) d(34),
+	\\
+	\label{eq:Sig}
+	& \Sigma(12) = i \int G(13) W(14) \Gamma(324) d(34),
+\end{align}
+\end{subequations}
+which connects the Green function $G$, its non-interacting version $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$ 
+and the self-energy $\Sigma$, where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$
+Within the GW approximation, one bypasses the calculation of the vertex corrections by setting \cite{Aryasetiawan_1998, Onida_2002, Reining_2017, Blase_2018}
+\begin{equation}
+	\label{eq:GW}
+	\Gamma(123) = \delta(12) \delta(13).
+\end{equation}
 
+Depending on the degree of self-consistency one is willing to do, there exists several types of GW calculations.
+The simplest and most popular variant is perturbative GW, or {\GOWO}, \cite{Hybertsen_1985a, Hybertsen_1986} which has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
+For finite systems such as atoms and molecules, partially or fully self-consistent GW methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
+
+The paper is organised as follows. 
+%In Sec.~\ref{sec:paradigm}, we briefly review the GW equations for spherium. 
+%Section~\ref{sec:green} provides details about our perturbative and self-consistent GW implementations, and give the expression of the self-energy for GF2 and GW+SOSEX. 
+Results are reported and discussed in Sec.~\ref{sec:results}. 
+Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}. 
+Atomic units are used throughout.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}
@@ -577,6 +617,6 @@ This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the
 \end{acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-\bibliography{GW-srDFT,GW-srDFT-control,biblio}
+\bibliography{GW,GW-srDFT,GW-srDFT-control,biblio}
 
 \end{document}