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Manu: polished II C

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Emmanuel Fromager 2020-02-25 14:43:33 +01:00
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Manuscript/eDFT.tex
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@ -4,6 +4,7 @@
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage{mathrsfs}
\usepackage[
colorlinks=true,
@ -769,7 +770,7 @@ What do you think?}
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within KS-eLDA:
\beq\label{eq:EI-eLDA}
\begin{split}
\E{\titou{eLDA}}{(I)}
\E{{eLDA}}{(I)}
& =
\E{HF}{(I)}
\\
@ -781,17 +782,31 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} d\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{split}
\eeq
\titou{T2: I think we should specify what those terms are physically... Maybe earlier in the manuscript?}
where
\beq
\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
\eeq
is the analog for ground {\it and} excited states of the HF energy.
Let us stress that, to the best of our knowledge, eLDA is the first
density-functional approximation that incorporates weight
is the analog for ground and excited states (within an ensemble) of the HF energy.
If, for analysis purposes, we Taylor expand the density-functional
correlation contributions
around the $I$th KS state density
$\n{\bGam{(I)}}{}(\br{})$, the sum of
the second and third terms on the right-hand side
of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
\beq
\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right),
\eeq
and it can therefore be identified as
an individual-density-functional correlation energy where the density-functional
correlation energy per particle is approximated by the ensemble one for
all the states within the ensemble.
Let us finally stress that, to the best of our knowledge, eLDA is the first
density-functional approximation that incorporates ensemble weight
dependencies explicitly, thus allowing for the description of derivative
discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
comment that follows] {\it via} the last term on the right-hand side