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Emmanuel Fromager 2020-05-05 17:18:32 +02:00
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Revised_Manuscript/eDFT.tex
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@ -605,15 +605,16 @@ As discussed further in Sec.~\ref{sec:eDFA}, these components can be
extracted from a extracted from a
finite uniform electron gas model for which density-functional correlation excitation finite uniform electron gas model for which density-functional correlation excitation
energies can be computed. energies can be computed.
} }\titou{Note also that, here, only the correlation part of the
\titou{Note also that, here, only the correlation part of the ensemble energy will be treated at the
energy is treated at the DFT level while we rely on HF for the exchange part.
DFT level while we rely on HF exchange. This is different from the usual context where both exchange and
This is different from the usual context where both exchange and correlation are treated at the LDA level which gives compensation of errors.} correlation are treated at the LDA level which gives compensation of
\manu{Manu: I changed a bit your sentence. Is this fine? Maybe we should add errors. Despite the errors
that we are not interested in accurate ensemble energies. Error that might be introduced into the ensemble energy within such a scheme,
cancellations may occur when computing excitation cancellations may actually occur when computing excitation energies,
energies, which are the quantities we are truly interested in.} which are energy {\it differences}.}
\manu{Manu: I changed a bit and complemented your sentence. Is this fine?}
The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun} The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
reads reads
@ -640,6 +641,7 @@ where
is the analog for ground and excited states (within an ensemble) of the HF energy, and is the analog for ground and excited states (within an ensemble) of the HF energy, and
\begin{gather} \begin{gather}
\begin{split} \begin{split}
\label{eq:Xic}
\Xi_\text{c}^{(I)} \Xi_\text{c}^{(I)}
& = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{} & = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
\\ \\
@ -650,15 +652,24 @@ is the analog for ground and excited states (within an ensemble) of the HF energ
\\ \\
\end{split} \end{split}
\\ \\
\label{eq:Upsic}
\Upsilon_\text{c}^{(I)} \Upsilon_\text{c}^{(I)}
= \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{}) = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}. \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
\end{gather} \end{gather}
\manurev{
If, for analysis purposes, we Taylor expand the density-functional One may naturally wonder about the physical content of the above correlation energy
expressions. It is in fact difficult to readily distinguish from
Eqs.~(\ref{eq:Xic}) and (\ref{eq:Upsic}) purely (uncoupled) individual
contributions from mixed ones. For that purpose, we may
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
deviation of the individual densities from the ensemble one will be
weak. As a result,
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
$\n{\bGam{(I)}}{}(\br{})$, the $\n{\bGam{(I)}}{}(\br{})$, \manurev{so that} the
second term on the right-hand side second term on the right-hand side
of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
@ -670,9 +681,10 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
Therefore, it can be identified as 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. all the states within the ensemble. \manurev{This perturbation expansion
may not hold in realistic systems, which are all but uniform. Nevertheless, it
gives more insight into the eLDA approximation and becomes useful when
analyzing its performance, as shown 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