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Manuscript/FarDFT.tex

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 	\affiliation{\LCPQ}
 
 \begin{abstract}
-Density-functional theory for ensembles (eDFT) is a time-independent formalism which allows to compute individual excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
-Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within eDFT.
+Gross-Oliveira-Kohn (GOK) ensemble density-functional theory (GOK-DFT) is a time-independent formalism which allows to compute individual excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
+Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within GOK-DFT.
 However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous derivative discontinuity contribution to the excitation energies.
-In the present article, we report a first-rung (\ie, local), weight-dependent exchange-correlation density-functional approximation for atoms and molecules specifically designed for the computation of double excitations within eDFT.
-This density-functional approximation for ensembles, based on finite and infinite uniform electron gas models, incorporate information about both ground and excited states. 
+In the present article, we report a first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximation for atoms and molecules specifically designed for the computation of double excitations within GOK-DFT.
+This density-functional approximation for ensembles, based on finite and infinite uniform electron gases, incorporate information about both ground and excited states. 
 Its accuracy is illustrated by computing the double excitation in the prototypical H$_2$ molecule.
 \end{abstract}
 
@@ -138,7 +138,7 @@ Time-dependent density-functional theory (TD-DFT) has been the dominant force in
 At a relatively low computational cost (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
 Importantly, setting up a TD-DFT calculation for a given system is an almost pain-free process from a user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.
 
-Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundation relies on the Runge-Gross theorem. \cite{Runge_1984}
+Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations relie on the Runge-Gross theorem. \cite{Runge_1984}
 The Kohn-Sham (KS) formalism of TD-DFT transfers the complexity of the many-body problem to the xc functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analog which have both the exact same one-electron density.
 
 However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made for the xc functional.
@@ -154,13 +154,13 @@ Although these double excitations are usually experimentally dark (which means t
 One possible solution to access double excitations within TD-DFT is provided by spin-flip TD-DFT which describes double excitations as single excitations from the lowest triplet state. \cite{Huix-Rotllant_2010,Krylov_2001,Shao_2003,Wang_2004,Wang_2006,Minezawa_2009}
 However, spin contamination might be an issue. \cite{Huix-Rotllant_2010}
 In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has been proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} (see also Refs.~\onlinecite{Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012}).
-In this approach the xc kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.
+In this approach the xc kernel is made frequency dependent, which allows to treat doubly-excited states. \cite{Romaniello_2009a,Sangalli_2011,Loos_2019}
 
 Maybe surprisingly, another possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
 DFT for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988a,Gross_1988b,Oliveira_1988} is a viable alternative following such a strategy currently under active development. \cite{Gidopoulos_2002,Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018,Gould_2019,Sagredo_2018,Ayers_2018,Deur_2018,Deur_2019,Kraisler_2013,Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Smith_2016,Senjean_2018}
 In the assumption of monotonically decreasing weights, eDFT for excited states has the undeniable advantage to be based on a rigorous variational principle for ground and excited states,  the so-called Gross-Oliveria-Kohn (GOK) variational principle.  \cite{Gross_1988a} 
 In short, GOK-DFT (\ie, eDFT for excited states) is the density-based analog of state-averaged wave function methods, and excitation energies can then be easily extracted from the total ensemble energy. \cite{Deur_2019}
-Although the formal foundation of GOK-DFT has been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
+Although the formal foundations of GOK-DFT have been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
 We believe that it is partly due to the lack of accurate approximations for GOK-DFT.
 In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules.
 The present contribution is a first step towards this goal.