Manu: saving work in the introduction

Manuscript/eDFT.tex

@@ -147,26 +147,48 @@ Their accuracy is illustrated by computing single and double excitations in one-
 \section{Introduction}
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964,Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials. \cite{ParrBook}
-The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function. 
-The complexity of the many-body problem is then transferred to the xc functional. 
+Over the last two decades, density-functional theory (DFT)
+\cite{Hohenberg_1964,Kohn_1965} has become the method of choice for
+modeling the electronic structure of large molecular systems and
+materials. \cite{ParrBook}\manu{why this ref?}
+The main reason is that, within DFT, the quantum contributions to the
+electronic repulsion energy --- the so-called exchange-correlation (xc)
+energy --- is rewritten as a functional of the electron density $n\equiv\n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function. 
+The complexity of the many-body problem is then transferred to the xc
+density functional. 
 Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003,Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tapavicza_2008,Levine_2006}
-The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge. \cite{Gori-Giorgi_2010,Gagliardi_2017}
+The description of strongly multiconfigurational ground states (often
+referred to as ``strong correlation problem'') still remains a
+challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017}
 Another issue, which is partly connected to the previous one, is the description of electronically-excited states. 
 
-The standard approach for modeling excited states in DFT is linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida}
+The standard approach for modeling excited states in DFT is
+linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012}
 In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong. 
-Moreover, in exact TDDFT, the xc functional is time dependent. 
-The simplest and most widespread approximation in state-of-the-art electronic structure programs where TDDFT is implemented consists in neglecting memory effects. \cite{Casida}
-In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time. 
+Moreover, in exact TDDFT, the xc energy is in fact an xc action \cite{Vignale_2008} which is a
+functional of the time-dependent density $n\equiv n(\br,t)$ and, as
+such, has memory. Standard implementations of TDDFT rely on 
+the adiabatic approximation where memory effects are neglected. In other
+words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
 As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
 
-When affordable (\ie, for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above. 
-The basic idea is to describe a finite ensemble of states (ground and excited) altogether, \ie, with the same set of orbitals. 
-Interestingly, a similar approach exists in DFT. 
-Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988a, Oliveira_1988, Gross_1988b} and is a generalization of Theophilou's variational principle for equiensembles. \cite{Theophilou_1979}
-In GOK-DFT (\ie, eDFT for excited states), the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest. 
-This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies. 
+When affordable (\ie, for relatively small molecules), time-independent
+state-averaged wave function methods
+\cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002,Helgakerbook} can be employed to fix the various issues mentioned above. 
+The basic idea is to describe a finite (canonical) ensemble of ground
+and excited states altogether, \ie, with the same set of orbitals. 
+Interestingly, a similar approach exists in DFT. Referred to as
+Gross--Oliveira--Kohn (GOK) DFT\cite{Gross_1988a,Gross_1988b,Oliveira_1988}, it was proposed at the end of the 80's as a generalization
+of Theophilou's DFT for equiensembles. \cite{Theophilou_1979}
+In exact GOK-DFT, the ensemble xc energy not only a functional of the
+density but also a  
+function of the ensemble weights. Note that, unlike in conventional
+Boltzmann ensembles~\cite{Pastorczak_2013}, the weights (each state in the ensemble
+is assigned a given and fixed weight) are allowed to vary
+independently in a GOK ensemble.   
+The weight dependence of the xc functional plays a crucial role in the
+calculation of excitation energies.
+\cite{Gross_1988b,Yang_2014,Deur_2017,Deur_2019,Senjean_2018,Senjean_2020} 
 It actually accounts for the infamous derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
 %\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}