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Manuscript/FarDFT.tex

@@ -288,6 +288,12 @@ is the Hxc potential.
 Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
 Note that, although we have dropped the weight-dependency in the individual densities $\n{}{(I)}(\br{})$ defined in Eq.~\eqref{eq:nI}, these do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density.
 
+In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as
+\begin{equation}
+	\e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}),
+\end{equation}
+where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively.
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%% COMPUTATIONAL DETAILS %%%
@@ -298,6 +304,46 @@ The self-consistent GOK-DFT calculations have been performed with the \texttt{Qu
 For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
 For all calculations, we use the aug-cc-pVXZ (X = D, T, and Q) Dunning's family of atomic basis sets.
 Numerical quadratures are performed with the \texttt{numgrid} library using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001}
+This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
+Moreover, 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. 
+Although we will sometimes ``violate'' this variational constraint, we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle. 
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Results}
+\label{sec:res}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Weight-independent local exchange-correlation functionals}
+\label{sec:S51}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+First, we compute the ensemble energy of the \ce{H2} molecule using the weight-independent Slater's local exchange functional, \cite{Dirac_1930}
+which is explicitly given by 
+\begin{align}
+	\e{\ex}{\text{S51}}(\n{}{}) & = \Cx{} \n{}{1/3},
+	&
+	\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
+\end{align}
+The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew-H2} as a function of the weight $0 \le \ew{} \le 1$.
+Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
+As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies greatly with the weight (see Fig.~\ref{fig:Om-H2}).
+Note that the exact xc correlation ensemble functional should yield a perfectly linear energy and the same excitation energy independently of $\ew{}$.
+
+
+As a first example, we compute the ensemble energy of the \ce{H2} molecule as a function of the weight $\ew{}$ using the SVWN5 local functional which corresponds to the combination of Slater's local exchange functional \cite{Dirac_1930} and the VNW5 local correlation functional. \cite{Vosko_1980}
+The SVWN5 xc functional is explicitly given by 
+\begin{equation}
+	\e{\xc}{\text{SVWN5}}(\n{}{}) = \e{\ex}{\text{S51}}(\n{}{}) + \e{\co}{\text{VWN5}}(\n{}{}),
+\end{equation}
+with 
+\begin{align}
+	\e{\ex}{\text{S51}}(\n{}{}) & = \Cx{} \n{}{1/3},
+	&
+	\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
+\end{align}
+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}.
+
 
 
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@@ -306,17 +352,9 @@ Numerical quadratures are performed with the \texttt{numgrid} library using 194
 \section{Functional}
 \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. 
-Thus, we have $0 \le \ew{} \le 1/2$.
 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}
-	\e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}),
-\end{equation}
-where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively.
 The construction of these two functionals is described below.
-Here, we restrict our study to spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
 Extension to spin-polarised systems will be reported in future work.
 
 To build our weight-dependent xc functional, we propose to consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEG which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
@@ -585,26 +623,6 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o
 \end{figure}
 %%% %%% %%% %%%
 
-%%%%%%%%%%%%%%%
-%%% RESULTS %%%
-%%%%%%%%%%%%%%%
-\section{Results and discussion}
-\label{sec:resdis}
-Here, we consider as testing ground the \ce{H2} molecule, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr).
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Results}
-\label{sec:res}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Discussion}
-\label{sec:dis}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Numerical results are reported in Table \ref{tab:Energies}.
-
 
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 %%% CONCLUSION %%%