Corrected my comments.

Manuscript/FarDFT.tex

@@ -193,36 +193,51 @@ Unless otherwise stated, atomic units are used throughout.
 %%%%%%%%%%%%%%%%%%%%
 \section{Theory}
 \label{sec:theo}
-As mentioned above, eDFT for excited states is based on the GOK variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy\bruno{I would write the variational principle equation here}
+
+Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
+The corresponding ensemble energy
 \begin{equation}
 	\E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)}
 \end{equation}
-built from an ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
+fulfils the variational principle
+as follows\cite{Gross_1988a}
+\begin{eqnarray}\label{eq:ens_energy}
+\E{}{\bw}  = \min_{\hat{\Gamma}^w} {\rm Tr}\left[ \hat{\Gamma}^w \hat{H} \right],
+\end{eqnarray}
+where $\hat{H} = \hat{T} + \hat{W}_{\rm ee} + \hat{V}_{\rm ne}$ contains the kinetic, electron-electron and nuclei-electron interaction potential operators, respectively,
+Tr denotes the trace and $\hat{\Gamma}^w$
+is a trial density matrix of the form
+\begin{eqnarray}
+\hat{\Gamma}^w = \sum_{I=0}^{\nEns - 1}
+\ew{I} \dyad{\overline{\Psi}^{(I)}},
+\end{eqnarray}
+where $\lbrace \overline{\Psi}^{(I)} \rbrace$ is a set of $M$ orthonormal trial wavefunctions. The lower bound of Eq.~(\ref{eq:ens_energy}) is reached when the set of wavefunctions correspond to the exact eigenstates of $\hat{H}$, \ie, $\lbrace \Psi^{(I)} \rbrace$.
 Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states.
-
-One of the key feature of GOK-DFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
+One of the key feature of the GOK ensemble is that individual excitation energies are extracted from the ensemble energy via differentiation with respect to individual weights:
 \begin{equation}
 	\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)}.
 \end{equation}
 
-\bruno{Turning to the DFT formulation... $\rightarrow$ I think you should not mention GOK-DFT prior to this part, as you simply described an ensemble without any DFT contributions.}
-In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}{\bw}[\n{}{}]$ such that 
+Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles allows to rewrite the exact variational expression for the ensemble energy as\cite{Gross_1988a}
 \begin{equation}
 \label{eq:Ew-GOK}
 	\E{}{\bw} = \min_{\n{}{}} \qty{ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} },
 \end{equation}
-where $\vext(\br{})$ is the external potential.
-In the KS formulation of GOK-DFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
+where $\vext(\br{})$ is the external potential
+and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional
+(the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles).
+In the KS formulation, this functional is decomposed as
 \begin{equation}
 	\F{}{\bw}[\n{}{}] 
 	= \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
-	= \Tr[ \hGam{\bw} \hT ] + \Tr[ \hGam{\bw} \hWee ],
+	= \Tr[ \hat{\gamma}^w \hT ] + \Tr[ \hat{\gamma}^w \hWee ],
 \end{equation}
-where $\hT$ and $\hWee$ are the kinetic and electron-electron interaction potential operators, respectively, $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional,
+where
+$\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional,
 \begin{equation}
-	\hGam{\bw} = \sum_{I=0}^{M-1} \ew{I} \dyad{\Det{I}{\bw}}
+	\hat{\gamma}^w = \sum_{I=0}^{M-1} \ew{I} \dyad{\Det{I}{\bw}}
 \end{equation}
-is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\MO{p}{\bw}(\br{})$, and 
+is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{0 \leq I \leq \nEns - 1}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace$, and 
 \begin{equation}
 \label{eq:exc_def}
 \begin{split}
@@ -230,7 +245,7 @@ is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave funct
 	& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
 	\\
 	& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' 
-	+ \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}.
+	+ \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}
 \end{split}
 \end{equation}
 is the ensemble Hartree-exchange-correlation (Hxc) functional.
@@ -256,7 +271,8 @@ are the ensemble and individual one-electron densities, respectively,
 \label{eq:KS-energy}
 	\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}
 \end{equation}
-is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$]. The latters are determined by solving the ensemble KS equation
+is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$]. 
+The latters are determined by solving the ensemble KS equation
 \begin{equation}
 \label{eq:eKS}
 	\qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
@@ -484,11 +500,11 @@ Equation \eqref{eq:becw} can be recast
 which nicely highlights the centrality of the LDA in the present eDFA. 
 In particular, $\be{\xc}{(0)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{})$.
 Consequently, in the following, we name this weight-dependent xc functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
-Also, we note that, by construction,\bruno{no need to specify $n = n^w$ for now right ?}
+Also, we note that, by construction,
 \begin{equation}
 \label{eq:dexcdw}
-	\left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} 
-	= \be{\xc}{(1)}(\n{}{\ew{}}(\br)) - \be{\xc}{(0)}(\n{}{\ew{}}(\br)).
+	\pdv{\be{\xc}{\ew{}}(\n{}{})}{\ew{}}
+	= \be{\xc}{(1)}(n(\br)) - \be{\xc}{(0)}(n(\br)).
 \end{equation}
 
 This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
@@ -501,7 +517,14 @@ This embedding procedure can be theoretically justified by the generalised adiab
 (where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} 
 Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional. 
 In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir 
-$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
+$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ? Can it be :
+\begin{equation}
+\label{eq:GACE2}
+	\E{\xc}{\bw}[\n{}{}] 
+	= \E{\xc}{}[\n{}{}] 
+	+ \sum_{I=0}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I+1},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi
+\end{equation}
+?}
 
 %%% TABLE I %%%
 \begin{table*}