Manu: some polishing in II B and C

Manuscript/eDFT.tex

@@ -450,8 +450,15 @@ Eq.~(\ref{eq:excited_ener_level_gs_lim})].
 %%%%%%%%%%%%%%%%
 For implementation purposes, we will use in the rest of this work 
 (one-electron reduced) density matrices 
-as basic variables, rather than Slater determinants. If we expand the
-ensemble KS (spin) orbitals [from which the latter determinants are constructed] in an atomic orbital (AO) basis,
+as basic variables, rather than Slater determinants. 
+As the theory is applied later on to {\it spin-polarized}
+systems, we drop spin indices in the density matrices, for convenience.
+If we expand the
+ensemble KS orbitals [from which the determinants are constructed] in an atomic orbital (AO) basis,
+\beq
+	\MO{p}{}(\br{}) =  \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
+\eeq
+\iffalse%%%%%%%%%%%%%%%%%%%%%%%%
 \titou{\beq
 	\SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
 \eeq
@@ -465,16 +472,14 @@ where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatia
 		\beta(\omega),	&	\text{for spin-down electrons,}	
 	\end{cases}
 \eeq
-}then the density matrix of the   
+}
+\fi%%%%%%%%%%%%%%%%%%%%%
+then the density matrix of the   
 determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
 \beq
 	\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
 \eeq
-where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$. 
-\trashPFL{Note that, as the theory is applied later on to spin-polarized
-systems, we drop spin indices in the density matrices, for convenience.}
-\manu{Is the latter sentence ok with you?}
-\titou{I don't think we need it anymore. What do you think?}
+where the summation runs over the orbitals that are occupied in $\Det{(K)}$. 
 The electron density of the $K$th KS determinant can then be evaluated
 as follows:
 \beq
@@ -501,7 +506,7 @@ p}}c^\sigma_{{\nu p}}
 \fi%%%
 %%%% end Manu
 while the ensemble density matrix
-and ensemble density read
+and the ensemble density read
 \beq
 	\bGam{\bw} 
 	= \sum_{K\geq 0} \ew{K} \bGam{(K)}
@@ -880,12 +885,6 @@ dependencies explicitly, thus allowing for the description of derivative
 discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
 comment that follows] {\it via} the last term on the right-hand side
 of Eq.~\eqref{eq:EI-eLDA}.\\
-\titou{In order to test the influence of the derivative discontinuity on the excitation energies, it is useful to perform ensemble HF (labeled as eHF) calculations in which the correlation effects are removed.
-In this case, the individual energies are simply defined as
-\beq\label{eq:EI-eHF}
-	\E{eHF}{(I)} \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}].
-\eeq
-}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}