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Revised_Manuscript/eDFT.tex
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@ -663,9 +663,9 @@ expressions. It is in fact difficult to readily distinguish from
Eqs.~(\ref{eq:Xic}) and (\ref{eq:Upsic}) purely (uncoupled) individual Eqs.~(\ref{eq:Xic}) and (\ref{eq:Upsic}) purely (uncoupled) individual
contributions from mixed ones. For that purpose, we may contributions from mixed ones. For that purpose, we may
consider a density regime which has a weak deviation from the uniform consider a density regime which has a weak deviation from the uniform
one. In such a regime, for which eLDA is a reasonable approximation, the one. In such a regime, where eLDA is a reasonable approximation, the
deviation of the individual densities from the ensemble one will be deviation of the individual densities from the ensemble one will be
weak. As a result, small. As a result,
we can} Taylor expand the density-functional we can} Taylor expand the density-functional
correlation contributions correlation contributions
around the $I$th KS state density around the $I$th KS state density
@ -682,10 +682,11 @@ Therefore, it can be identified as
an individual-density-functional correlation energy where the density-functional an individual-density-functional correlation energy where the density-functional
correlation energy per particle is approximated by the ensemble one for correlation energy per particle is approximated by the ensemble one for
all the states within the ensemble. \manurev{This perturbation expansion all the states within the ensemble. \manurev{This perturbation expansion
may not hold in realistic systems, which may deviate significantly from is of course less relevant for the (more realistic) systems that exhibit significant
the uniform density regime. Nevertheless, it deviations from the uniform
gives more insight into the eLDA approximation and becomes useful when density regime. Nevertheless, it
rationalizing its performance, as illustrated in Sec. \ref{sec:res}.\\} gives more insight into the eLDA approximation and it becomes useful when
it comes to rationalize its performance, as illustrated in Sec. \ref{sec:res}.\\}
Let us stress that, to the best of our knowledge, eLDA is the first Let us stress that, to the best of our knowledge, eLDA is the first
density-functional approximation that incorporates ensemble weight density-functional approximation that incorporates ensemble weight
dependencies explicitly, thus allowing for the description of derivative dependencies explicitly, thus allowing for the description of derivative
@ -987,7 +988,9 @@ Its Tamm-Dancoff approximation version (TDA-TDLDA) is also considered. \cite{Dr
Concerning the ensemble calculations, two sets of weight are tested: the zero-weight Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
(ground-state) limit where $\bw = (0,0)$ and the (ground-state) limit where $\bw = (0,0)$ and the
equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$. equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
\titou{Note that a zero-weight calculation does correspond to a conventional ground-state KS calculation with exact exchange and LDA correlation.} \titou{Note that a zero-weight calculation does correspond to a
\trashEF{conventional} ground-state KS calculation with \manu{$100\%$} exact exchange and LDA correlation.}
\manu{Manu: OK?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1014,6 +1017,12 @@ linearly-interpolated ensemble energy) is represented
in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
fulfilling the restrictions on the ensemble weights to ensure the GOK fulfilling the restrictions on the ensemble weights to ensure the GOK
variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$]. variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
\manurev{More precisely, we follow a continuous path that connects
ground-state [$\bw=(0,0)$] and equiensemble [$\bw=(1/3,1/3)$]
calculations. For convenience, we use two connected paths. The first
one, for which $\ew{2}=0$ and $0\leq \ew{1}\leq 1/3$, relies on the
biensemble while the second one is defined as follows:
$\ew{1}=1/3$ and $0\leq \ew{2}\leq 1/3$.}
To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without GIC as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}. To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without GIC as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
As one can see in Fig.~\ref{fig:EvsW}, without GIC, the As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
ensemble energy becomes less and less linear as $L$ ensemble energy becomes less and less linear as $L$
@ -1082,7 +1091,7 @@ systematically enhances the weight dependence, due to the lowering of the
ground-state energy, as $\ew{2}$ increases. ground-state energy, as $\ew{2}$ increases.
The reverse is observed for the second excited state. The reverse is observed for the second excited state.
\manurev{Finally, we notice that the crossover point of the \manurev{Finally, we notice that the crossover point of the
first excited-state energies based on first-excited-state energies based on
bi- and triensemble calculations, respectively, disappears in the strong correlation bi- and triensemble calculations, respectively, disappears in the strong correlation
regime [see the right panel of Fig. \ref{fig:EIvsW}], thus illustrating regime [see the right panel of Fig. \ref{fig:EIvsW}], thus illustrating
the importance of (individual and ensemble) densities, in the importance of (individual and ensemble) densities, in
@ -1123,7 +1132,7 @@ Overall, one clearly sees that, with
equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations. equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
This conclusion is verified for smaller and larger numbers of electrons This conclusion is verified for smaller and larger numbers of electrons
(see {\SI}). (see {\SI}).
\titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}).} \titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}.}
%%% FIG 4 %%% %%% FIG 4 %%%
\begin{figure*} \begin{figure*}