From fdfdd9319287e6533b521cde71b077451c67631e Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@dhcp-35-94.u-strasbg.fr>
Date: Fri, 28 Feb 2020 09:59:41 +0100
Subject: [PATCH] Manu: polished the conclusion

---
 Manuscript/eDFT.tex | 49 +++++++++++++++++++++++++++++++++++++--------
 1 file changed, 41 insertions(+), 8 deletions(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index 30f66dc..3705a45 100644
--- a/Manuscript/eDFT.tex
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@@ -1322,15 +1322,48 @@ again that the usage of equal weights has the benefit of significantly reducing
 \section{Concluding remarks}
 \label{sec:conclusion}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-In the present article, we have constructed a local, weight-dependent three-state DFA in the context of ensemble DFT.
-The KS-eLDA scheme delivers accurate excitation energies for both single and double excitations, especially within its state-averaged version where the same weights are assigned to each state belonging to the ensemble.
-Generalization to a larger number of states is straightforward and will be investigated in future work.
-We have observed that, although the derivative discontinuity has a non-negligible effect on the excitation energies (especially for the single excitations), its magnitude can be significantly reduced by performing state-averaged calculations instead of zero-weight calculations.
 
-Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.
-Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap. \cite{Senjean_2018}
-This can be done by constructing DFAs for the one- and three-electron ground state systems, and combining them with the two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and \eqref{eq:ecw}.
-We hope to report on this in the near future.
+A local and ensemble-weight-dependent correlation density-functional approximation
+(eLDA) has been constructed in the context of GOK-DFT for spin-polarized
+triensembles in
+1D. The approach is actually general and can be extended to real
+(three-dimensional)
+systems~\cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} 
+and larger ensembles in order to  
+model excited states in molecules and solids. Work is currently in
+progress in this direction.
+
+Unlike any standard functional, eLDA incorporates derivative
+discontinuities through its weight dependence. The latter originates
+from the finite uniform electron gas eLDA is
+(partially) based on. The KS-eLDA scheme, where exact exchange is
+combined with eLDA, delivers accurate excitation energies for both
+single and double excitations, especially when an equiensemble is used.
+In the latter case, the same weights are assigned to each state belonging to the ensemble.
+{\it We have observed that, although the derivative discontinuity has a
+non-negligible effect on the excitation energies (especially for the
+single excitations), its magnitude can be significantly reduced by
+performing state-averaged calculations instead of zero-weight
+calculations.}\manu{to be updated ...}
+
+Let us finally stress that the present methodology can be extended
+straightforwardly to other types of ensembles like, for example, the
+$N$-centered ones, thus allowing for the design an LDA-type functional for the
+calculation of ionization potentials, electron affinities, and
+fundamental gaps. \cite{Senjean_2018,Senjean_2020}. 
+Like in the present
+eLDA, such a functional would incorporate the infamous derivative
+discontinuity contribution to the gap through its explicit weight
+dependence. We hope to report on this in the near future.
+
+
+\trashEF{This can be done by constructing a functional for the one- and
+three-electron ground-state systems, and combining them with the
+two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and
+\eqref{eq:ecw}.}\manu{I find the sentence too technical for a
+conclusion.} 
+
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Supplementary material}