LDA centered

Manuscript/FarDFT.tex

@@ -164,7 +164,7 @@ The present contribution is a first step towards this goal.
 
 When one talks about constructing functionals, the local-density approximation (LDA) has always a special place.
 The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
-Although the Hohenberg-Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behavior in a real system. \cite{Kohn_1965}
+Although the Hohenberg-Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
 However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
 Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
 Here, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA functional for ensembles (eLDA).
@@ -237,7 +237,7 @@ 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 generalization to a larger number of states is trivial and left for future work.
+The generalisation to a larger number of states is trivial and left for future work.
 
 We adopt the usual decomposition, and write down the weight-dependent xc functional as
 \begin{equation}
@@ -391,35 +391,36 @@ Combining these, we build a two-state weight-dependent correlation functional:
 \subsection{LDA-centered functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\titou{Here, I shall explain our embedding scheme where we consider that a two-electron system (the impurity) is embedded in a larger system (the bath).
-Here the bath is the IUEG while the impurity is our two-electron systems.
-The weight-dependence only comes from the impurity, while the remaining effect originates from the bath.
-}
-
 Our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
-Therefore, we ... an 
+Hence, we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
-In order to make the two-electron-based eDFA defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
+The weight-dependence of the xc functional solely originates from the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
+ 
+Consistently with such a strategy, Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} are ``centred'' on their corresponding jellium reference
 \begin{equation}
 \label{eq:becw}
-	\be{\xc}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{\xc}{(0)}(\n{}{}) + \ew{} \be{\xc}{(1)}(\n{}{}),
+	\be{\xc}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{\xc}{(0)}(\n{}{}) + \ew{} \be{\xc}{(1)}(\n{}{})
 \end{equation}
-where
+via the following shift:
 \begin{equation}
 	\be{\xc}{(I)}(\n{}{}) = \e{\xc}{(I)}(\n{}{}) + \e{\xc}{\LDA}(\n{}{}) - \e{\xc}{(0)}(\n{}{}).
 \end{equation}
-The local-density approximation (LDA) xc functional is
+The LDA xc functional is similarly decomposed as
 \begin{equation}
-	\e{\xc}{\LDA}(\n{}{}) = \e{\ex}{\LDA}(\n{}{}) + \e{\co}{\LDA}(\n{}{}).
+	\e{\xc}{\LDA}(\n{}{}) = \e{\ex}{\LDA}(\n{}{}) + \e{\co}{\LDA}(\n{}{}),
 \end{equation}
-where we use here the Dirac exchange functional \cite{Dirac_1930} and the VWN5 correlation functional \cite{Vosko_1980}
+where we consider here the Dirac exchange functional \cite{Dirac_1930} 
-\begin{subequations}
+\begin{equation}
-\begin{align}
+	\e{\ex}{\LDA}(\n{}{}) = \Cx{\LDA} \n{}{1/3},
-	\e{\ex}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3},
+\end{equation}
-	\\
+with 
-	\e{\co}{\LDA}(\n{}{}) & \equiv \e{\co}{\text{VWN5}}(\n{}{}).
+\begin{equation}
-\end{align}
+	\Cx{\LDA} = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3},
-\end{subequations}
+\end{equation}
-with $\Cx{\LDA} = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}$.
+and the VWN5 correlation functional \cite{Vosko_1980}
+\begin{equation}
+	\e{\co}{\LDA}(\n{}{}) \equiv \e{\co}{\text{VWN5}}(\n{}{}).
+\end{equation}
+For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
 
 Equation \eqref{eq:becw} can be recast
 \begin{equation}
@@ -436,7 +437,7 @@ Also, we note that, by construction,
 	\left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{I}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{\xc}{(I)}[\n{}{\ew{}}(\br)] - \be{\xc}{(0)}[\n{}{\ew{}}(\br)].
 \end{equation}
 
-This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) 
+This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE) 
 \begin{equation}
 \label{eq:GACE}
 	\E{\xc}{\bw}[\n{}{}]