Manu: first commit :-) Just saving work.

Manuscript/FarDFT.tex

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+\newcommand{\manu}[1]{\textcolor{magenta}{Manu: #1}}
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 	\affiliation{\LCPQ}
 
 \begin{abstract}
-Gross--Oliveira--Kohn (GOK) ensemble density-functional theory (GOK-DFT) is a time-\textit{independent} formalism which allows to compute excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
+Gross--Oliveira--Kohn (GOK) ensemble density-functional theory (GOK-DFT)
+is a time-\textit{independent} formalism \manu{extension of DFT?} which
+allows to compute excitation energies \manu{I would say excited-state
+energies (or energy levels)} via the derivative of the ensemble energy
+with respect to the weight of each excited state \manu{and then, to be
+consistent: {\it ``via the derivatives of the ensemble energy with
+respect to the ensemble weights''}}.
 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 ensemble derivative contribution to the excitation energies.
 In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) specifically designed for the computation of double excitations within GOK-DFT.
@@ -149,13 +156,33 @@ A specific protocol is proposed to obtain accurate energies associated with doub
 %%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
 Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida,Ulrich_2012,Loos_2020a}
-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).
+At a relatively low \manu{Tim suggested me to refer to TD-DFT as a
-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.
+method with a moderate computational cost which I
+think is fair. It is more involved than a regular DFT or GOK-DFT
+calculation} 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
+\manu{I would say ``ground-state functional'' and mention the widely used adiabatic approximation} 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 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.
+The Kohn-Sham (KS) formalism \manu{formulation?} 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
+\manu{exactly the?} same one-electron density.
 
-However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made for the xc functional.
+However, TD-DFT is far from being perfect as, in practice, drastic
+approximations must be made for the xc functional. \manu{At this point I
+would mention the time dependence of the functional which is treated at
+the local approximation level within the standard adiabatic
+approximation. In other words, memory effects are absent from the
+functional (which is an action functional, not an energy functional). I
+guess the choice of functional you discuss in the following refers to
+standard (frequency-independent) ground-state functionals. As you refer
+to exact TD-DFT first, the different levels of approximation should be
+clearly highlighted. You also discuss only the linear response regime
+without referring explicitly to it.}
 One of its issues actually originates directly from the choice of the xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
 Moreover, because it is so popular, it has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
 
@@ -176,7 +203,16 @@ In the assumption of monotonically decreasing weights, eDFT for excited states h
 In short, GOK-DFT (\ie, eDFT for neutral excitations) 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 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.
+In particular, to the best of our knowledge, an explicitly
+weight-dependent density-functional approximation for ensembles (eDFA)
+has never been developed for atoms and molecules \manu{I would add
+``from first principles'' to be on the safe side. I remember this work
+by Nagy [J. Phys. B: At. Mol. Opt. Phys. 29 (1996) 389–394] where she
+did build an exchange functional for a bi-ensemble. Her approach was
+semi-empirical as she used experimental excitation energies. It would be
+fair to cite her paper. There is also this paper [Journal of Molecular
+Structure (Theochem) 571 (2001) 153-161] that I never really understood
+but they tried something}.
 The present contribution is a small step towards this goal.
 
 When one talks about constructing functionals, the local-density approximation (LDA) is never far away.