Manu: polished II B

Manuscript/eDFT.tex

@@ -15,7 +15,7 @@
 \newcommand{\alert}[1]{\textcolor{red}{#1}}
 \usepackage[normalem]{ulem}
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
-\newcommand{\manu}[1]{\textcolor{blue}{#1}}
+\newcommand{\manu}[1]{\textcolor{blue}{Manu: #1}}
 \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
 \newcommand{\trashEF}[1]{\textcolor{blue}{\sout{#1}}}
 
@@ -392,19 +392,23 @@ 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 spinorbitals [from which the latter are constructed] in an atomic orbital (AO) basis,
+ensemble KS (spin) orbitals [from which the latter determinants are constructed] in an atomic orbital (AO) basis,
 \beq
 	\SO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
 \eeq
-then the density matrix elements obtained from the   
+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 $\Det{(K)}$. 
+where the summation runs over the spinorbitals that are occupied in
-Note that the density of the $K$th KS state reads
+$\Det{(K)}$. 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?}
+The electron density of the $K$th KS determinant can then be evaluated
+as follows:
 \beq
-	\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}).
+	\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}),
 \eeq 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Manu's derivation %%%
@@ -426,8 +430,8 @@ p}}c^\sigma_{{\nu p}}
 }
 \fi%%%
 %%%% end Manu
-We can then construct the ensemble density matrix
+while the ensemble density matrix
-and the ensemble density as follows:
+and ensemble density read
 \beq
 	\bGam{\bw} 
 	= \sum_{K\geq 0} \ew{K} \bGam{(K)}
@@ -439,7 +443,7 @@ and
 	\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
 \eeq
 respectively. 
-The exact expression of the individual energies in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
+The individual energy expression in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
 \beq\label{eq:exact_ind_ener_rdm}
 \begin{split}
 	\E{}{(I)} 
@@ -460,7 +464,7 @@ where
 	\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}	
 \eeq
 denotes the one-electron integrals matrix.
-The individual Hx energy is obtained from the following trace formula
+The exact individual Hx energies are obtained from the following trace formula
 \beq
 	\Tr[\bGam{(K)} \bG \bGam{(L)}]
 	= \sum_{\mu\nu\la\si} \eGam{\mu\nu}{(K)} \eG{\mu\nu\la\si} \eGam{\la\si}{(L)},