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Manu: started my revision. Saving work

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Emmanuel Fromager 2020-05-05 13:40:08 +02:00
parent cbd6902635
commit dc671d9ce1
1 changed files with 22 additions and 6 deletions
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Revised_Manuscript/eDFT.tex
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@ -17,6 +17,7 @@
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\manu}[1]{\textcolor{blue}{#1}}
\newcommand{\manurev}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashEF}[1]{\textcolor{blue}{\sout{#1}}}
@ -589,15 +590,30 @@ following ensemble local-density approximation (eLDA) will be employed
\beq\label{eq:eLDA_corr_fun}
\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
\eeq
where the ensemble correlation energy per particle
where the \manurev{\textit{weight-dependent}} ensemble correlation
energy per particle \manurev{will have the general
expression}
\beq\label{eq:decomp_ens_correner_per_part}
\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{}).
\eeq
is explicitly \textit{weight dependent}.
As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
from a finite uniform electron gas model.
\titou{Note that, here, only the correlation part is treated at the KS level while we rely on exact HF exchange.
\manurev{Note that, at this level of approximation, which is expected to
be exact for any \textit{uniform}
system, the
density-functional correlation components $\be{c}{(K)}(\n{}{})$ are
weight-\textit{independent}, unlike in the exact theory \cite{Fromager_2020}.
As discussed further in Sec.~\ref{sec:eDFA}, these components can be
extracted from a
finite uniform electron gas model for which density-functional correlation excitation
energies can be computed.
}
\titou{Note also that, here, only the correlation part of the ensemble
energy is treated at the
DFT level while we rely on HF exchange.
This is different from the usual context where both exchange and correlation are treated at the LDA level which gives compensation of errors.}
\manu{Manu: I changed a bit your sentence. Is this fine? Maybe we should add
that we are not interested in accurate ensemble energies. Error
cancellations may occur when computing excitation
energies, which are the quantities we are truly interested in.}
The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
reads