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Manu: done with III (see my comments/corrections)

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Manuscript/eDFT.tex
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@ -933,22 +933,23 @@ One of the main driving force behind the popularity of DFT is its ``universal''
Obviously, the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the FUEG.
However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
Following this simple strategy, which is further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
\begin{equation}
\label{eq:becw}
\be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
\manu{\e{c}{\bw}(\n{}{})\rightarrow}\be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
\end{equation}
where
\begin{equation}
\be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}).
\end{equation}
The local-density approximation (LDA) correlation functional,
In the following, we will use the LDA correlation functional that has been specifically designed for 1D systems in
Ref.~\onlinecite{Loos_2013}:
\begin{equation}
\label{eq:LDA}
\e{c}{\text{LDA}}(\n{}{})
= a_1^\text{LDA} F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}],
\end{equation}
specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
\begin{subequations}
\begin{align}
a_1^\text{LDA} & = - \frac{\pi^2}{360},
@ -958,7 +959,10 @@ specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used
a_3^\text{LDA} & = 2.408779.
\end{align}
\end{subequations}
Equation \eqref{eq:becw} can be recast
\manu{Note that the strategy described in Eq.~(\ref{eq:becw}) is general and
can be applied to real (higher-dimension) systems.} In order to make the
connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may
recast Eq.~\eqref{eq:becw} as
\begin{equation}
\label{eq:eLDA}
\begin{split}
@ -968,14 +972,33 @@ Equation \eqref{eq:becw} can be recast
& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
\end{split}
\end{equation}
which nicely highlights the centrality of the LDA in the present eDFA.
or, equivalently,
\begin{equation}
\label{eq:eLDA_gace}
\begin{split}
\be{c}{\bw}(\n{}{})
& = \e{c}{\text{LDA}}(\n{}{})
\\
& + \sum_{K>0}\int_0^{\ew{K}}
\qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K,
\end{split}
\end{equation}
where the $K$th correlation excitation energy (per electron) is integrated over the
ensemble weight $\xi_K$ at fixed (uniform) density $n$.
Eq.~(\ref{eq:eLDA_gace}) nicely highlights the centrality of the LDA in the present eDFA.
In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
Finally, we note that, by construction,
\manu{
\begin{equation}
\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \be{c}{(J)}(\n{}{}(\br{})) - \be{c}{(0)}(\n{}{}(\br{})).
\end{equation}
Manu: I guess that the "overlines" and the dependence in $\bf r$ of the
densities on the RHS should be removed. The final expression should be
\beq
\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).
\eeq
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:comp_details}