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Manuscript/BSE_JPCL.tex
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\let\oldmaketitle\maketitle
\let\maketitle\relax
\title{A Chemist Guide to the Bethe-Salpeter Equation \\
\xavier{ Challenges for the Bethe-Salpeter Equation Formalism : from Physics to Chemistry } }
\title{The Bethe-Salpeter Equation Formalism: \\ From Physics to Chemistry}
\date{\today}
\begin{tocentry}
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%%%%%%%%%%%%%%%%
\begin{abstract}
The many-body Green's function Bethe-Salpeter formalism is steadily asserting itself as a new efficient and accurate tool in the armada of computational methods available to chemists in order to predict neutral electronic excitations in molecular systems.
The many-body Green's function Bethe-Salpeter equation (BSE) formalism is steadily asserting itself as a new efficient and accurate tool in the armada of computational methods available to chemists in order to predict neutral electronic excitations in molecular systems.
In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable charged excitations and quasiparticle energies, and the Bethe-Salpeter formalism, able to catch excitonic effects, has shown to provide accurate excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
With a similar computational cost than time-dependent density-functional theory (TD-DFT), BSE@$GW$ is then able to provide an accuracy on par with the most accurate global hybrid functionals without the unsettling choice of the exchange-correlation functional.
In this \textit{Perspective} article, we provide a historical overview of the Bethe-Salpeter formalism, with a particular focus on its condensed-matter roots.
We also propose a critical review of its strengths and weaknesses for different chemical situations.
Future directions of developments and improvements are also discussed.

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