Manu: done (so far) with the computational details

Manuscript/FarDFT.tex

@@ -342,11 +342,21 @@ equals the exact ensemble one
 n^{\bw}(\br)=\sum_{I=0}^{\nEns-1}
 \ew{I}n_{\Psi_I}(\br).
 \eeq
- The minimizing density-functional KS
+In practice, the minimizing KS density matrix operator
-wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq
+$\hgam{\bw}\left[\n{}{\bw}\right]$ 
-I\leq M-1}$ are constructed from (weight-dependent) KS orbitals
+can be determined from the following KS reformulation of the
+GOK variational principle, \cite{Gross_1988b,Senjean_2015} 
+\beq\label{eq:min_KS_DM}
+\E{}{\bw}  = \min_{\hGam{\bw}} \left\{\Tr[\hGam{\bw}
+\left(\hT+\hVne\right)]+\E{\Ha}{}[\n{\hGam{\bw}}{}]+\E{\xc}{\bw}[\n{\hGam{\bw}}{}]\right\},
+\eeq 
+where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1}
+\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is the trial ensemble density. As a
+result, the orbitals
 $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
-\nOrb}$. The latters are determined by solving the ensemble KS equation
+\nOrb}$ from which the KS
+wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq
+I\leq M-1}$ are constructed can be obtained by solving the following ensemble KS equation
 \begin{equation}
 \label{eq:eKS}
 	\qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
@@ -491,8 +501,19 @@ This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\
 Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. 
 Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
 Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
-\manu{Maybe we should be more clear about what we mean with $\ew{} = 1$.
+\manu{I think we need to be a bit more clear about the $\ew{} = 1$ case, as
-In the range $1/2\leq \ew{}\leq 1$, }
+it stands a little bit beyond the theory discussed previously. What you
+are looking at in the range $1/2\leq \ew{}\leq 1$ are, indeed, other
+stationary points (than the minimizing ones) of the density matrix
+operator functional in Eq.~(\ref{eq:min_KS_DM}). I would say that we
+look at these solutions for analysis purposes. I personally never looked
+(formally) at these solutions and their physical meaning. One should clearly
+mention that applying GOK-DFT in this range of weights would simply
+consists in switching ground and first excited states if a true
+minimization of the ensemble energy were performed. From this point of
+view we do not violate anything. 
+} 
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}
 \label{sec:res}