diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index d4df3dd..f0afcf2 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -217,11 +217,7 @@ In the KS formulation of eDFT, the universal ensemble functional (the weight-dep
 \begin{equation}
 	\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}],
 \end{equation}
-where 
-\begin{equation}
-	\Ts{\bw}[\n{}{}] = 
-\end{equation}
-and
+where $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional and 
 \begin{equation}
 \label{eq:exc_def}
 \begin{split}
@@ -232,7 +228,7 @@ and
 	+ \int \e{\xc}{\bw}[\n{}{}(\br{})] \n{}{}(\br{}) d\br{}.
 \end{split}
 \end{equation}
-are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively.
+is the ensemble Hartree-exchange-correlation (Hxc) functional.
 Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$.
 
 From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
@@ -243,10 +239,12 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
 	= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})},
 \end{equation}
 where 
-\begin{equation}
-	\n{}{\bw}(\br{}) = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{})
-\end{equation}
-is the ensemble density, 
+\begin{align}
+	\n{}{\bw}(\br{}) & = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{}),
+	&
+	\n{}{(I)}(\br{}) & = \sum_{p}^{\Norb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
+\end{align}
+are the ensemble and individual one-electron densities, respectively, 
 \begin{equation}
 \label{eq:KS-energy}
 	\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}
@@ -715,13 +713,13 @@ This would be, for example, the case with the exact xc functional.
 
 Extracting excitation energies from Eqs.~\eqref{eq:bEwHF}, \eqref{eq:bEwLDA} and \eqref{eq:bEweLDA} is more tricky.
 To do so, we will employ Eq.~\eqref{eq:dEdw}.
-The two first terms are
+The two first terms are simply
 \begin{align}
-	\Eps{0}{\ew{}} & = 2(1-\ew{}) \eps{1}{\ew{}},
+	\Eps{0}{\ew{}} & = 2 \eps{1}{\ew{}},
 	&
-	\Eps{1}{\ew{}} & = 2 \ew{} \eps{2}{\ew{}},
+	\Eps{1}{\ew{}} & = 2 \eps{2}{\ew{}},
 \end{align}
-where the HF, LDA and eLDA weight-dependent orbital energies are
+and the HF, LDA and eLDA weight-dependent orbital energies are
 \begin{subequations}
 \begin{align}
 	\eps{1}{\ew{},\HF} 
@@ -738,14 +736,14 @@ where the HF, LDA and eLDA weight-dependent orbital energies are
 	\eps{1}{\ew{},\LDA} 
 	& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
 	\\
-	& + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(0)}(\br{}) d\br{},
+	& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
 \end{split}
 	\\
 \begin{split}
 	\eps{2}{\ew{},\LDA} 
 	& = \eHc{2} +  2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
 	\\
-	& + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(1)}(\br{}) d\br{},
+	& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
 \end{split}
 \end{align}
 \end{subequations}
@@ -756,13 +754,13 @@ where the HF, LDA and eLDA weight-dependent orbital energies are
 	\eps{1}{\ew{},\eLDA} 
 	& = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12}) 
 	\\
-	& + \frac{1}{2} \int \left. \fdv{\bE{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(0)}(\br{}) d\br{},
+	& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
 \end{split}
 	\\
 \begin{split}
 	\eps{2}{\ew{},\eLDA} & = \eHc{2} +  (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22}) 
 	\\
-	& + \frac{1}{2} \int \left. \fdv{\bE{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(1)}(\br{}) d\br{},
+	& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
 \end{split}
 \end{align}
 \end{subequations}