Manu: saving work

Revised_Manuscript/eDFT.tex

@@ -475,14 +475,14 @@ as follows:
 \eeq 
 while the ensemble density matrix
 and the ensemble density read
-\beq
+\beq\label{eq:ens1RDM}
 	\bGam{\bw} 
 	= \sum_{K\geq 0} \ew{K} \bGam{(K)}
 	\equiv \eGam{\mu\nu}{\bw}
 	= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)},
 \eeq
 and
-\beq
+\beq\label{eq:ens_dens_from_ens_1RDM}
 	\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
 \eeq
 respectively. 
@@ -1055,15 +1055,20 @@ as $\ew{2}$ increases. The variations in the ensemble
 weights are essentially linear or quadratic. 
 \manurev{This can be rationalized as follows. As readily seen from
 Eqs.~(\ref{eq:EI-eLDA}) and (\ref{eq:ind_HF-like_ener}), the individual
-HF-like exchange does not depend explicitly on the weights, which means
+HF-like energies do not depend explicitly on the weights, which means
 that the above-mentioned variations originate from the eLDA correlation
 functional [second and third terms on the right-hand side of
 Eq.~(\ref{eq:EI-eLDA})]. If, for analysis purposes, we consider the
-Taylor expansions around the uniform density regime in Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and (\ref{eq:Taylor_exp_DDisc_term})
-}They are induced by the
-eLDA correlation functional, as readily seen from
-Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
-\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
+Taylor expansions around the uniform density regime in
+Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and
+(\ref{eq:Taylor_exp_DDisc_term}), contributions with an explicit weight
+dependence still remain after summation. As both the ensemble density and
+the ensemble correlation energy per particle vary linearly with the
+weights $\bw$ [see Eqs.~(\ref{eq:ens1RDM}),
+(\ref{eq:ens_dens_from_ens_1RDM}), and
+(\ref{eq:decomp_ens_correner_per_part})], the latter contributions will contain both linear and quadratic terms in
+$\bw$, as readily seen from Eq.~(\ref{eq:Taylor_exp_DDisc_term}) [see the second term on the right-hand
+side].} In the biensemble, the weight dependence of the first
 excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
 systematically enhances the weight dependence, due to the lowering of the
 ground-state energy, as $\ew{2}$ increases.