very minor modifs in Sec III which is awesome

Manuscript/eDFT.tex

@@ -927,14 +927,14 @@ Here, we will consider the most simple ringium system featuring electronic corre
 
 The present weight-dependent eDFA is specifically designed for the
 calculation of excited-state energies within GOK-DFT.
-In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including: 
+To take into account both single and double excitations simultaneously, we consider a three-state ensemble including: 
 (i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
-All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring where the electrons are confined.
+\titou{To ensure the GOK variational principle, \cite{Gross_1988a} the
+triensemble weights must fulfil the following conditions: \cite{Deur_2019}
+$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.}
+All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring on which the electrons are confined.
 We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
 Generalization to a larger number of states is straightforward and is left for future work.
-To ensure the GOK variational principle, \cite{Gross_1988a} the
-triensemble weights must fulfil the following conditions: \cite{Deur_2019}
-{$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$}. 
 %The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
 %to consider an equi-bi-ensemble
 %for which $\ew{1}=1/2$. This possibility is excluded with your
@@ -992,7 +992,7 @@ Combining these, one can build the following three-state weight-dependent correl
 \begin{equation}
 \label{eq:ecw}
 	%\e{c}{\bw}(\n{}{})
- \tilde{\epsilon}_{\rm c}^\bw(n)=  (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
+ \Tilde{\epsilon}_{\rm c}^\bw(n)=  (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
 \end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1005,7 +1005,7 @@ The weight-dependence of the correlation functional is then carried exclusively
 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}
-	\tilde{\epsilon}_{\rm c}^\bw(n)\rightarrow{\e{c}{\bw}(\n{}{})} =  (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
+	\Tilde{\epsilon}_{\rm c}^\bw(n)\rightarrow{\e{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}
@@ -1092,7 +1092,7 @@ In order to test the present eLDA functional we perform various sets of calculat
 To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
 For the single excitations, we also perform time-dependent LDA (TDLDA)
 calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}]. 
-\titou{Its Tamm-Dancoff approximation (TDA) version is also considered.} \cite{Dreuw_2005}
+\titou{Its Tamm-Dancoff approximation version  (TDA-TDLDA) is also considered.} \cite{Dreuw_2005}
 
 Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
 (ground-state) limit where $\bw = (0,0)$ and the