minimal corrections

Manuscript/FarDFT.tex

@@ -244,7 +244,7 @@ Equation \eqref{eq:dEdw} is our working equation for computing excitation energi
 \label{sec:func}
 The present work deals with the explicit construction of the (reduced) LDA xc functional $\e{\xc}{\bw}[\n{}{}]$ defined in Eq.~\eqref{eq:exc_def}.
 Here, we restrict our study to the case of a two-state ensemble (\ie, $\Nens = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered.
-The generalisation to a larger number of states is trivial and left for future work.
+The generalisation to a larger number of states (in particular the inclusion of the first singly-excited state) is trivial and left for future work.
 
 We adopt the usual decomposition, and write down the weight-dependent xc functional as
 \begin{equation}
@@ -259,11 +259,12 @@ Extension to spin-polarised systems will be reported in future work.
 \subsection{Weight-dependent exchange functional}
 \label{sec:Ex}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We consider the ground- and doubly-excited states of the two-electron glomium system in its singlet ground state.
+\titou{T2: More details required to understand what is glomium.}
+We consider the ground- and doubly-excited states of the two-electron glomium system in its singlet ground state. \cite{Loos_2009a,Loos_2009c,Loos_2010e}
 These two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome onto which the electrons are confined.
 We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
 
-The reduced (\ie, per electron) Hartree-Fock (HF) energy for these two states is
+The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states are
 \begin{subequations}
 \begin{align}
 	\e{\HF}{(0)}(\n{}{}) & = \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
@@ -292,11 +293,11 @@ with
 \end{align}
 \end{subequations}
 
-Knowing that the exchange functional has the following form
+In analogy with the conventional Dirac exchange functional, \cite{Dirac_1930} we write down the exchange functional of each individual state as
 \begin{equation}
 	\e{\ex}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3},
 \end{equation}
-we obtain
+and we then obtain
 \begin{align}
 	 \Cx{(0)} & = - \frac{4}{3} \qty( \frac{1}{\pi} )^{1/3},
 	 &
@@ -316,7 +317,7 @@ with
 \begin{equation}
 	\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
 \end{equation}
-Quite remarkably, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient, which is expected from a theoretical point of view but also a nice property from a more practical aspect.
+Quite remarkably, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient, which is expected from a theoretical point of view, yet a nice property from a more practical aspect.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -329,7 +330,7 @@ Based on highly-accurate calculations, \cite{Loos_2009a,Loos_2009c,Loos_2010e} o
 \label{eq:ec}
 	\e{\co}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
 \end{equation}
-where the $a_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
+where $a_2^{(I)}$ and $a_3^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
 The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2011b}
 Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
 
@@ -343,7 +344,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
 \begin{figure}
 	\includegraphics[width=\linewidth]{fig1}
 	\caption{
-	Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
+	Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron FUEG.
 	The data gathered in Table \ref{tab:Ref} are also reported. 
 	}
 	\label{fig:Ec}
@@ -354,7 +355,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
 \begin{table}
 	\caption{
 	\label{tab:Ref}
-	$-\e{\co}{(I)}$ as a function of the radius of the glome $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
+	$-\e{\co}{(I)}$ as a function of the radius of the glome $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron FUEG.
 	}
 	\begin{ruledtabular}
 	\begin{tabular}{lcc}
@@ -381,7 +382,8 @@ Combining these, we build a two-state weight-dependent correlation functional:
 \begin{table}
 	\caption{
 	\label{tab:OG_func}
-	Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
+	Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}.
+	The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.}
 	\begin{ruledtabular}
 		\begin{tabular}{lcc}
 					&	\tabc{Ground state}	&	\tabc{Doubly-excited state}	\\