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\author{Xavier Blase}
	\email{xavier.blase@neel.cnrs.fr}
	\affiliation[NEEL, Grenoble]{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
\author{Ivan Duchemin}
	\affiliation[CEA, Grenoble]{Universit\'e Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France}
\author{Denis Jacquemin}
	\affiliation[CEISAM, Nantes]{Universit\'e de Nantes, CNRS,  CEISAM UMR 6230, F-44000 Nantes, France}
\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}

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	\title{A Chemist Guide to the Bethe-Salpeter Equation}
\date{\today}

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%%%%%%%%%%%%%%%%
%%% ABSTRACT %%%
%%%%%%%%%%%%%%%%

\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.
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. 
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.
\end{abstract}


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\noindent

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In its press release announcing the attribution of the 2013 Nobel prize in Chemistry to  Karplus, Levitt, and Warshel, the Royal Swedish Academy of Sciences concluded by stating \textit{``Today the computer is just as important a tool for chemists as the test tube. 
Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003} 
Martin Karplus' Nobel lecture moderated this statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are \textit{``much too complicated to be soluble''}, urging scientists to develop \textit{``approximate practical methods''}. This is where the electronic structure community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems. 

The study of neutral electronic excitations in condensed-matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the  prediction of the colour of precious metals and stones for jewellery, to the understanding, \eg, of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. 
% T2: shall we add a few references?
The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Dreuw_2005} including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}

%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%

The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of  Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} 
While the density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} Green's function MBPT takes the one-body Green's function as the central quantity. 

The (time-ordered) one-body Green's function reads
\begin{equation}
    G(\bx t,\bx't') = -i \mel{N}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{N},
\end{equation}
where $\ket{N}$ is the $N$-electron ground-state wave function. 
The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while \titou{$T$ is the time-ordering operator}. 
For ($t > t'$) the one-body Green's function provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for ($t < t'$) the propagation of a hole is monitored.\\

%===================================
\subsection{Charged excitations}
%===================================
A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation energies of the system
\begin{equation}\label{eq:spectralG}
    G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \, \text{sgn}(\varepsilon_s - \mu ) },  
\end{equation}
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{N+1} - E_0^{N}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{N} - E_s^{N-1}$ for $\varepsilon_s < \mu$, 
Here, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy.
The $f_s$'s  are the so-called Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinant many-body wave functions (see below).
Unlike Kohn-Sham (KS) eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper \titou{charging} energies of the $N$-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbitals. 

Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
\begin{equation}\label{eq:Gmotion}
    \qty[  \pdv{}{t_1} - h(\br_1) ] G(1,2) - \int d3 \, \Sigma(1,3) G(3,2)
    = \delta(1,2),
\end{equation}
where we introduce the usual composite index, \eg, $1 \equiv (\bx_1,t_1)$. 
Here, $h$ is the \titou{one-body Hartree Hamiltonian} and $\Sigma$ the so-called exchange-correlation (xc) self-energy operator.
Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one obtains the familiar eigenvalue equation, \ie,
\begin{equation}
	h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
\end{equation}
which resembles formally the KS equation with the difference that the self-energy $\Sigma$ is non-local, energy dependent and non-hermitian.  
The knowledge of $\Sigma$ allows thus to obtain the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.  
\titou{[INTRODUCE QUASIPARTICLES and OTHER solutions ??]}
\\

%===================================
\subsection{The $GW$ self-energy}
%===================================

While the equations reported above are formally exact, it remains to provide an expression for the xc self-energy operator $\Sigma$.  
This is where Green's function practical theories differ. 
Developed by Lars Hedin in 1965 with application to the interacting homogeneous electron gas, \cite{Hedin_1965} the $GW$ approximation
\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Golze_2019rev} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}). 
The obtained equation, when compared with the equation for the time-evolution of $G$ [Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
\begin{equation}
    \Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
\end{equation}
where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$. 
The neglect of the vertex leads to the so-called $GW$ approximation for $\Sigma$ that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with
\begin{align}
	W(1,2) & = v(1,2) + \int d34 \, v(1,2)   \chi_0(3,4) W(4,2),
	\\
	\chi_0(1,2) & = -i \int d34 \,  G(2,3) G(4,2),
\end{align}
where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential. 
In practice, the input $G$ and $\chi_0$ needed to build $\Sigma$ are taken to be the ``best'' Green's function and susceptibility that can be easily calculated, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates. 
Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained as follows:
\begin{equation}
    \varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
    \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\GW}) - V^{\XC}}{\phi_p^{\KS}}.
\end{equation}
Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique. 
\titou{T2: Shall we introduce the renormalization factor?}
This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988} 
surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV  [REFS] with an $\mathcal{O}(N^4)$ computational scaling (see below). 
Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999} 
However, remaining a low order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995} \\

%%% FIG 1 %%%
\begin{figure}
	\includegraphics[width=0.7\linewidth]{fig1/fig1}
	\caption{
	Hedin's pentagon: its connects the Green's function $G$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
	The path made of back arrow shows the $GW$ process which bypasses the computation of $\Gamma$.
	\label{fig:pentagon}}
\end{figure} 
%%% %%% %%%

%===================================
\subsection{Neutral excitations}
%===================================

While TD-DFT   starts with the  variation of the charge density $\rho$ with respect to an external local perturbation, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation: 
\begin{equation}
    \chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
   \quad  \rightarrow \quad
    L(12,34) \stackrel{\BSE}{=} -i \pdv{G(1,2)}{U(3,4)}.
\end{equation}
%with the relation $\chi(1,2) = L(11,22)$ since $\rho(1) = -iG(1,1^{+})$, as a first bridge between the TD-DFT and BSE worlds. 
The equation of motion for $G$ [Eq.~\ref{eq:Gmotion}] can be reformulated in the form of a Dyson equation
\begin{equation}
	G = G_0 + G_0 \Sigma G,
\end{equation}
that relates the full (interacting) Green's function, $G$, to its Hartree version, $G_0$, obtained by replacing the $\lbrace \varepsilon_s, f_s \rbrace$ by the Hartree eigenvalues and eigenfunctions. 
The derivative with respect to $U$ of the Dyson equation yields
\begin{multline}\label{eq:DysonL}
    L(12,34) = L_0(12,34) 
    \\
     \int d5678 \, L_0(12,34) \Xi^{\BSE}(5,6,7,8) L(78,34),
\end{multline}
where $L_0 = \partial G_0 / \partial U$ is the Hartree 4-point susceptibility and
\begin{equation}
	\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78)  + \pdv{\Sigma(5,6)}{G(7,8)}
\end{equation}  
is the so-called BSE kernel.

This equation can be compared to its TD-DFT analog
\begin{equation}
    \chi(1,2) = \chi_0(1,2) + \int d34 \, \chi_0(1,3) \Xi^{\DFT}(3,4) \chi(4,2),
\end{equation}
where
\begin{equation}
    \Xi^{\DFT}(3,4) = v(3,4) + \pdv{ V^{\XC}(3)}{\rho(4)}
\end{equation} 
is the TD-DFT kernel.
Plugging now the $GW$ self-energy, in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
\begin{equation}\label{eq:BSEkernel}
	\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78)  -W(5,6) \delta(57) \delta(68 ),
\end{equation}
where it is  traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. 
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces thus the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida's equations in TD-DFT:
\begin{equation}
    \begin{pmatrix}
		R & C  
		\\
		-C^*  & R^{*}
	\end{pmatrix}
    \begin{pmatrix}
		X^m
		\\
		Y^m
	\end{pmatrix}
	=
	\Omega_m
    \begin{pmatrix}
		X^m  
		\\
		Y^m
	\end{pmatrix},
\end{equation}
with electron-hole (e-h) eigenstates written as
\begin{equation}
    \psi_{m}^{eh}(\br_e,\br_h) 
    = \sum_{ia} \qty[ X_{ia}^{m} \phi_i(\br_h) \phi_a(\br_e)
	+ Y_{ia}^{m} \phi_i(\br_e) \phi_a(\br_h) ],
\end{equation}
where $\lambda$ index the electronic excitations. 
The $\lbrace \phi_{i/a} \rbrace$ are the input (KS) eigenstates used to build the $GW$ self-energy. 
The resonant and anti-resonant parts of the BSE Hamiltonian read
\begin{gather}
    R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \eta (ai|bj) - W_{ai,bj},
    \\
    C_{ai,bj} = \eta (ai|jb) - W_{ai,jb},
\end{gather}
with $\eta=2,0$ for singlets/triplets and
\begin{equation}\label{eq:Wmatel}
	W_{ai,bj} = \iint d\br d\br' 
	\phi_i(\br) \phi_j(\br) W(\br,\br'; \omega=0)
	\phi_a(\br') \phi_b(\br'),
\end{equation}
where we notice that the two occupied (virtual) eigenstates are taken at the same position of space, in contrast with the 
$(ai|bj)$ bare Coulomb term.  
As compared to TD-DFT,
\begin{itemize}
    \item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues
    \item the non-local screened Coulomb matrix elements replaces the DFT xc kernel.
\end{itemize}    
We emphasise that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations. This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, emphasizing its pros and cons.  \\

%===================================
\subsection{Practical considerations}
%===================================

From a practical point of view, it is important to understand that, to compute the BSE neutral excitations, one must perform, first, several calculations.
First, a KS-DFT (or HF) calculation has to be performed in order to get orbitals and their corresponding energies.
Then, these are used as input variables for the $GW$ calculation, whose main purpose is to correct these quantities.
Depending on the level of self-consistency, only the eigenvalues or both the eigenvalues and the orbitals are updated.
In the case of a $G_0W_0$ calculation, a single, perturbative correction is applied to the orbital energies only.
The partially self-consistent ev$GW$ scheme update

The simplest and most popular variant is perturbative GW, or $G_0W_0$, \cite{Hybertsen_1985, 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}
Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature, and violates some important conservation laws, such as the conservation of energy, momentum and particle number. \cite{Martin_1959, Baym_1961, Baym_1962}
Improvements may be obtained via partial self-consistency while the conservation laws are satisfied at full self-consistency.
However, things are not that simple, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in GW. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.
A famous example has been provided by the calculations performed on the uniform electron gas, \cite{Holm_1998, Holm_1999,Holm_2000, Garcia-Gonzalez_2001} 
a paradigm central to many areas of physics and chemistry. \cite{Loos_2016}
This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017}
However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006}
These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
For finite systems such as atoms and molecules, the situation is less controversial, and 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}


There exist two main types of partially self-consistent GW methods: 
i) \textit{``eigenvalue-only quasiparticle''} GW (ev$GW$), \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} 
where the quasiparticle (QP) energies are updated at each iteration, and 
ii) \textit{``quasiparticle self-consistent''} GW (qs$GW$), \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011, Kaplan_2016} 
where one updates both the QP energies and the corresponding orbitals.
Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Historical overview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} the BSE formalism has emerged in condensed-matter physics around the 1960's at the semi-empirical tight-binding level with the study of the optical properties of simple semiconductors.  \cite{Sham_1966,Strinati_1984,Delerue_2000}
Three decades later, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999} paved the way to the popularization in the solid-state physics community of the BSE formalism. 

Following early applications to periodic polymers and molecules, [REFS] BSE gained much momentum in the quantum chemistry community with, in particular, several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same parameters (geometries, basis sets, etc) than the available higher-level reference calculations, \cite{Schreiber_2008} such as CC3. \cite{Christiansen_1995}
Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by well-documented correlation-consistent Gaussian basis sets, \cite{Dunning_1989} together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques were used. [REFS] 

An important conclusion drawn from these calculations was that the quality of the BSE excitation energies is strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap 
\begin{equation}
	\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW},
\end{equation}
with the experimental (photoemission) fundamental gap \cite{Bredas_2014}
\begin{equation}
	\EgFun = I^N - A^N,
\end{equation}
where $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ are the ionization potential and the electron affinity of the $N$-electron system (see Fig.~\ref{fig:gaps}).

%%% FIG 2 %%%
\begin{figure*}
	\includegraphics[width=0.7\linewidth]{gaps}
	\caption{
	Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
	$\EB$ is the excitonic effect, while $I^N$ and $A^N$ are the ionization potential and the electron affinity of the $N$-electron system.
	$\Eg^{\KS}$ and $\Eg^{\GW}$ are the KS and $GW$ HOMO-LUMO gaps.
	See main text for the definition of the other quantities	
	\label{fig:gaps}}
\end{figure*} 
%%% %%% %%%

Standard $G_0W_0$ calculations starting with KS eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input KS gap
\begin{equation}
	\Eg^{\KS} = \eps_{\LUMO}^{\KS} - \varepsilon_{\HOMO}^{\KS},
\end{equation}
but still too small as compared to the experimental value, \ie,
\begin{equation}
	\Eg^{\KS} \ll \Eg^{G_0W_0} < \EgFun.
\end{equation}
Such an underestimation of the fundamental gap leads to a similar underestimation of the optical gap $\EgOpt$, \ie, the lowest optical excitation energy.
\begin{equation}
	\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
\end{equation}
where $\EB$ is the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind.
Because of this, we have $\EgOpt < \EgFun$.

Such a residual HOMO-LUMO gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016} 
Alternatively, self-consistent schemes, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011} where corrected eigenvalues, and possibly orbitals, \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011} are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016} 
As a result, BSE excitation singlet energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations.
For sake of illustration,  an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering more than hundred representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017} 
This is equivalent to the best TD-DFT results obtained by scanning a large variety of global hybrid functionals with various amounts of exact exchange. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Successes \& Challenges} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%==========================================
\subsection{Charge-transfer excited states} 
%==========================================
A very remarkable success of the BSE formalism lies in the description of the charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT calculations adopting standard functionals. \cite{Dreuw_2004} 
Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes, cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004} 
These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
It can be cured through an exact exchange contribution, a solution that explains in particular the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012} 
The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where screening reduces the long-range electron-hole interactions. 
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\

%==========================================
\subsection{Solvent effects} 
%==========================================
\titou{T2: introduce discussion about coupling between BSE and solvent models.}
We now leave the description of successes to discuss difficulties and Perspectives.\\

%==========================================
\subsection{The computational challenge} 
%==========================================
As emphasized above, the BSE eigenvalue equation in the occupied-to-virtual product space is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
Searching iteratively for the lowest eigenstates presents the same $\order*{N^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT. 
Concerning the construction of the BSE Hamiltonian, it is no more expensive than building the TD-DFT one with hybrid functionals, reducing again to $\order*{N^4}$ operations with standard RI techniques. 
At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{N^3}$ operations using iterative techniques. \cite{Ljungberg_2015} 
With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015}

In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculations that scale as $\order{N^4}$ with system size using plane-wave basis sets or RI techniques, but with a rather large prefactor. 
%%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states. 
%%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019} 
The field of low-scaling $GW$ calculations is however witnessing significant advances. 
While the sparsity of ..., \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid  and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017} 
The separability of occupied and virtual states summations lying at the heart of these approaches are now blooming in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in RPA and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}   These ongoing developments pave the way to applying the $GW$/BSE formalism to systems comprising several hundred atoms on standard laboratory clusters.  \\

%==========================================
\subsection{The triplet instability challenge}
%==========================================
The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or 
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels. 
contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range. \cite{Sears_2011}
While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.

benchmarks \cite{Jacquemin_2017b,Rangel_2017}

a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$.  \cite{Holzer_2018b}\\

%==========================================
\subsection{The challenge of analytic gradients}
%==========================================
An additional issue concerns the formalism taken to calculate the ground-state energy for a given atomic configuration. Since the BSE formalism presented so far concerns the calculation of the electronic excitations, namely the difference of energy between the GS and the ES, gradients of the ES absolute energy require 

This points to another direction for the BSE formlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}\\

%==========================================
\subsection{Unphysical discontinuities} 
%==========================================
\titou{T2: talking about multiple solution issues.}


%==========================================
\subsection{The double excitation challenge}
%==========================================
As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009,Sangalli_2011,Zhang_2013,ReiningBook}
To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
In most cases, this can be done by solving a set of linear equations of the form
\begin{equation}
	\label{eq:lin_sys}
	\bA \bx = \omega \bx,
\end{equation}
where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies.
However, in practice, $K$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that 
\begin{equation}
	\label{eq:lin_sys_split}
	\begin{pmatrix}
		\bA_1	&	\tr{\bb}		\\
		\bb			&	\bA_2		\\
	\end{pmatrix}
	\begin{pmatrix}
		\bx_1	\\
		\bx_2	\\
	\end{pmatrix}
	= \omega
	\begin{pmatrix}
		\bx_1	\\
		\bx_2  	\\
	\end{pmatrix},
\end{equation}
where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 =  K$), can be associated with, for example, the single and double excitations of the system.
Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.

Solving separately each row of the system \eqref{eq:lin_sys_split} yields
\begin{subequations}
\begin{gather}
	\label{eq:row1}
	\bA_1 \bx_1  + \tr{\bb} \bx_2 = \omega \bx_1,
	\\
	\label{eq:row2}
	\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1.
\end{gather}
\end{subequations}
Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
\begin{equation}
	\label{eq:non_lin_sys}
	\Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
\end{equation}
with 
\begin{equation}
	\Tilde{\bA}_1(\omega) = \bA_1 +  \tr{\bb} (\omega \bI - \bA_2)^{-1} \bb 
\end{equation}
which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{Romaniello_2009,Sangalli_2011,ReiningBook}

How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension. 
This procedure converting degrees of freedom into frequency or energy dependence is very  general and can be applied in various contexts. \cite{Gatti_2007,Garniron_2018}
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension. 
However, because there is no free lunch, this non-linear system is obviously  harder to solve than its corresponding linear analogue given by Eq.~\eqref{eq:lin_sys}.
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.

One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value. 
For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.\\

Beyond the static approximation \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009,Romaniello_2009,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}

%==========================================
\subsection{Core-level spectroscopy}
%==========================================
XANES, \cite{Olovsson_2009,Vinson_2011}


diabatization and conical intersections \cite{Kaczmarski_2010}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here goes the conclusion.

%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. 
This work has also been supported through the EUR Grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support. 

%%%%%%%%%%%%%%%%%%%%
%%% BIBLIOGRAPHY %%%
%%%%%%%%%%%%%%%%%%%%
\bibliography{xbbib}

\clearpage

\section*{Biographies}

\begin{center}
	\includegraphics[width=3cm]{XBlase}
\end{center}
\noindent{\bfseries Xavier Blase} holds his PhD in Physics from the University of California at Berkeley. 
He is presently CNRS Research Director at Institut N\'eel, Grenoble, France. 
His research focuses on the electronic and optical properties of systems relevant to condensed matter physics, materials sciences and physical chemistry, with much emphasis on methodology. 
He is the co-author of the FIESTA code (Bull-Fourier prize 2014) that implements the $GW$ and Bethe-Salpeter formalisms with Gaussian basis sets. 
He received the 2008 CNRS silver medal.

\begin{center}
	\includegraphics[width=3cm]{IDuchemin}
\end{center}
\noindent{\bfseries Ivan Duchemin} holds his PhD from the CEMES in Toulouse, France. 
He stayed as a postdoctoral fellow at the UC Davis Physics department, California, and the Max Plank Institute for Polymer Research in Mainz, Germany, before joining the French Center for Alternative Energies (CEA), Grenoble, France, as a senior scientist. 
His research interests focus on methodological and code developments in ab initio quantum simulations. 
He is presently the main developer of the FIESTA code (Bull-Fourier prize 2014).

\begin{center}
	\includegraphics[width=4cm]{DJacquemin}
\end{center}
\noindent{\bfseries Denis Jacquemin} received his PhD in Chemistry from the University of Namur in 1998, before moving to the University of Florida for his postdoctoral stay. He is currently full Professor at the University of Nantes (France). 
His research is focused on modeling electronically excited-state processes in organic and inorganic dyes as well as photochromes using a large panel of \emph{ab initio} approaches. His group collaborates with many experimental
and theoretical groups. He is the author of more than 500 scientific papers. He has been ERC grantee (2011--2016), member of Institut Universitaire de France (2012--2017) and received the WATOC's Dirac Medal (2014).

\begin{center}
	\includegraphics[width=3cm]{PFLoos}
\end{center}
\noindent{\bfseries Pierre-Fran\c{c}ois Loos} was born in Nancy, France in 1982. He received his M.S.~in Computational and Theoretical Chemistry from the Universit\'e Henri Poincar\'e (Nancy, France) in 2005 and his Ph.D.~from the same university in 2008. From 2009 to 2013, He was undertaking postdoctoral research with Peter M.W.~Gill at the Australian National University (ANU). From 2013 to 2017, he was a \textit{``Discovery Early Career Researcher Award''} recipient and, then, a senior lecturer at the ANU. Since 2017, he holds a researcher position from the \textit{``Centre National de la Recherche Scientifique (CNRS)} at the \textit{Laboratoire de Chimie et Physique Quantiques} in Toulouse (France), and was awarded, in 2019, an ERC consolidator grant for the development of new excited-state methodologies.

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