saving work

Manuscript/BSE_JPCL.tex

@@ -176,7 +176,7 @@
 
 \let\oldmaketitle\maketitle
 \let\maketitle\relax
-	\title{A Chemist Guide to the Bethe-Salpeter Equation Formalism}
+	\title{A Chemist Guide to the Bethe-Salpeter Equation}
 \date{\today}
 
 \begin{tocentry}
@@ -200,9 +200,9 @@
 
 \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, allowing 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 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, such as \titou{bla bla bla}.
+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}
 
@@ -226,69 +226,20 @@ Future directions of developments and improvements are also discussed.
 
 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 bold 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 the scientist to develop \textit{``approximate practical methods''}. This is where the methodology community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems. 
+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. 
+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{History}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-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, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to the $N$-electron ground-state energy (see Fig.~\ref{fig:gaps}).
-
-\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 definitions of the other quantities	
-	\label{fig:gaps}}
-\end{figure*} 
-
-Standard $G_0W_0$ calculations starting with Kohn-Sham (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 $\EgOpt < \EgFun$.
-\titou{T2: I will include a figure here.}
-
-Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation (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{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 DFT, Green's function MBPT takes the one-body  Green's function as the central quantity. The (time-ordered) one-body Green's function reads
+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}
@@ -303,23 +254,26 @@ A central property of the one-body Green's function is that its spectral represe
 \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 $\varepsilon_s < \mu$. 
+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.
 \titou{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 [more ??].}
-Unlike 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. 
+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),
+    \qty[  \pdv{}{t_1} - h(\br_1) ] G(1,2) - \int d3 \, \Sigma(1,3) G(3,2)
-    = \delta(1,2)
+    = \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 xc self-energy operator.
+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)
+	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.  [INTRODUCE QUASIPARTICLES and OTHER solutions ??] \\
+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}
@@ -328,7 +282,7 @@ The knowledge of $\Sigma$ allows thus to obtain the true addition/removal energi
 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. 
+\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),
@@ -341,8 +295,8 @@ The neglect of the vertex leads to the so-called $GW$ approximation for $\Sigma$
 	\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 DFT 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. 
+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 DFT xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained as follows:
+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}}.
@@ -353,6 +307,16 @@ In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983}
 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}
 %===================================
@@ -444,6 +408,70 @@ As compared to TD-DFT,
 \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, it is required to perform, first, several calculations (see Fig.~\ref{fig:pentagon}).
+In a first time, a Kohn-Sham DFT calculation has to be performed in order to get orbitals and their corresponding energies .
+Then, a $GW$ calculation has to be performed in order to correct these quantities.
+In the case of a $G_0W_0$ calculation, a single, perturbative correction is applied to the 
+\\
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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} 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -465,6 +493,12 @@ The success of the BSE formalism to treat CT excitations has been demonstrated i
 \titou{T2: introduce discussion about coupling between BSE and solvent models.}
 We now leave the description of successes to discuss difficulties and Perspectives.\\
 
+%==========================================
+\subsection{Unphysical discontinuities} 
+%==========================================
+\titou{T2: talking about multiple solution issues.}
+
+
 %==========================================
 \subsection{The computational challenge} 
 %==========================================

Manuscript/fig1/fig1.tex

@@ -0,0 +1,53 @@
+\documentclass{standalone}
+
+\usepackage{graphicx,bm,microtype,hyperref,algpseudocode,subfigure,algorithm,algorithmicx,multirow,footnote,xcolor,physics,lipsum,wasysym,physics}
+
+
+\usepackage{tikz}
+\usetikzlibrary{arrows,positioning,shapes.geometric}
+\usetikzlibrary{decorations.pathmorphing}
+
+\tikzset{snake it/.style={
+decoration={snake, 
+    amplitude = .4mm,
+    segment length = 2mm},decorate}
+}
+
+%\usepackage{tgchorus}
+%\usepackage[T1]{fontenc}
+
+\begin{document}
+
+\begin{tikzpicture}
+	\begin{scope}[very thick,
+		node distance=2cm,on grid,>=stealth',
+		Op1/.style={circle,draw,fill=yellow!40},
+		Op2/.style={circle,draw,fill=orange!40},
+		Op3/.style={circle,draw,fill=red!40},
+		Op4/.style={circle,draw,fill=violet!40},
+		DeadOp/.style={circle,draw,fill=gray!40},
+		Input/.style={fill=white!40},
+		Output/.style={fill=white!40}]
+		\node [Op1, align=center]		(G)		at (3*0.587785, 3*0.809017)		{$G$};
+		\node [DeadOp, align=center] 		(Gamma)	at (3*0.951057, -3*0.309017)		{$\Gamma$};
+		\node [Op2, align=center]		(P)		at (3*0, -3*1.00000)				{$P$};
+		\node [Op3, align=center]		(W)		at (-3*0.951057, -3*0.309017)		{$W$};
+		\node [Op4, align=center] 		(Sigma)	at (-3*0.587785, 3*0.809017)		{$\Sigma$};
+		\node [Input, align=center] 		(In)		[above=of G]					{};
+		\node [Output, align=center] 		(Out)		[above=of Sigma]				{};
+		\node [Input, align=center] 		(In)		[above=of G]					{KS-DFT};
+		\node [Output, align=center] 		(Out)		[above=of Sigma]				{BSE};
+		\path 
+		(G) 			edge [->,color=gray!50] 		node [above,sloped,black] {$\Gamma = 1 + \fdv{\Sigma}{G} GG \Gamma$}	(Gamma)
+		(Gamma) 		edge [->,color=gray!50] 		node [below,sloped,black] {$P = - i GG \Gamma$}								(P)
+		(P) 			edge [->,color=black] 			node [above,sloped,black] {$W = v + vPW$}								(W)
+		(W) 			edge [->,color=black] 			node [above,sloped,black] {$\Sigma = i GW\Gamma$}						(Sigma)
+		(Sigma) 		edge [->,color=black] 			node [above,sloped,black] {$G = G_\text{0} + G_\text{0} \Sigma G$}					(G)
+		(G) 			edge [->,color=black] 			node [above,sloped,black] {$P = - i GG \quad (\Gamma = 1)$}								(P)
+		(In) 			edge [->,color=black] 			node [above,sloped,black] {$\varepsilon^\text{KS}$}						(G)
+		(Sigma) 		edge [->,color=black] 			node [above,sloped,black] {$\varepsilon^\text{GW}$}						(Out)
+		;
+	\end{scope}
+\end{tikzpicture}
+
+\end{document}

Manuscript/xbbib.bib

@@ -1,13 +1,48 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-05-14 22:49:58 +0200 
+%% Created for Pierre-Francois Loos at 2020-05-15 09:00:18 +0200 
 
 
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+@book{ParrBook,
+	Address = {Clarendon Press},
+	Author = {R. G. Parr and W. Yang},
+	Date-Added = {2020-05-15 08:59:25 +0200},
+	Date-Modified = {2020-05-15 08:59:25 +0200},
+	Keywords = {dft; qmech},
+	Publisher = {Oxford},
+	Title = {Density-functional theory of atoms and molecules},
+	Year = {1989}}
+
+@article{Hohenberg_1964,
+	Annote = {Hohenberg-Kohn theorem},
+	Author = {P. Hohenberg and W. Kohn},
+	Date-Added = {2020-05-15 08:59:18 +0200},
+	Date-Modified = {2020-05-15 08:59:18 +0200},
+	Doi = {10.1103/PhysRev.136.B864},
+	Journal = {Phys. Rev.},
+	Pages = {B864--B871},
+	Title = {Inhomogeneous electron gas},
+	Volume = {136},
+	Year = {1964},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.136.B864}}
+
+@article{Kohn_1965,
+	Author = {W. Kohn and L. J. Sham},
+	Date-Added = {2020-05-15 08:58:59 +0200},
+	Date-Modified = {2020-05-15 08:58:59 +0200},
+	Doi = {10.1103/PhysRev.140.A1133},
+	Journal = {Phys. Rev.},
+	Pages = {A1133--A1138},
+	Title = {Self-consistent equations including exchange and correlation effects},
+	Volume = {140},
+	Year = {1965},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.140.A1133}}
+
 @article{Lettmann_2019,
 	Author = {Tobias Lettmann and Michael Rohlfing},
 	Date-Added = {2020-05-14 22:27:01 +0200},