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Theory section done

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Pierre-Francois Loos 2019-09-09 11:56:53 +02:00
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Manuscript/eDFT.tex
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@ -235,7 +235,7 @@ In Eq.~\eqref{eq:F}, $\hHc$ is the core Hamiltonian (including kinetic and elect
(\mu\nu|\la\si) = \iint \frac{\AO\mu(\br_1) \AO\nu(\br_1) \AO\la(\br_2) \AO\si(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2 (\mu\nu|\la\si) = \iint \frac{\AO\mu(\br_1) \AO\nu(\br_1) \AO\la(\br_2) \AO\si(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
\end{equation} \end{equation}
are two-electron repulsion integrals, are two-electron repulsion integrals,
$\bE{Hxc}{\bw}[\n{}{}(\br)] = \n{}{}(\br) \be{Hxc}{\bw}[\n{}{}(\br)]$ and $\be{Hxc}{\bw}[\n{}{}(\br)]$ is the weight-dependent correlation functional to be built in the present study. $\bE{Hxc}{\bw}[\n{}{}(\br)] = \n{}{}(\br) \be{Hxc}{\bw}[\n{}{}(\br)]$ and $\be{Hxc}{\bw}[\n{}{}(\br)]$ is the weight-dependent Hartree-exchange-correlation functional to be built in the present study.
The one-electron ensemble density is The one-electron ensemble density is
\begin{equation} \begin{equation}
\n{}{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br), \n{}{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
@ -266,15 +266,15 @@ Following Deur and Fromager, \cite{Deur_2018b} it is possible to extract individ
+ \LZ{Hxc}{} + \DD{Hxc}{(I)}. + \LZ{Hxc}{} + \DD{Hxc}{(I)}.
\end{multline} \end{multline}
Note that a \emph{single} KS-eDFT calculation is required to extract the three individual energies. Note that a \emph{single} KS-eDFT calculation is required to extract the three individual energies.
The correlation part of the (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by \alert{Mention LIM?}
The (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by
\begin{align} \begin{align}
\LZ{Hxc}{} & = - \int \left. \fdv{\be{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br)^2 d\br, \LZ{Hxc}{} & = - \int \left. \fdv{\be{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br)^2 d\br,
\\ \\
\DD{Hxc}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int \left. \pdv{\be{Hxc}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br) d\br. \DD{Hxc}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int \left. \pdv{\be{Hxc}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br) d\br.
\end{align} \end{align}
Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}]. Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\be{Hxc}{\bw}(\n{}{})$. The only remaining piece of information to define at this stage is the weight-dependent Hartree-exchange-correlation functional $\be{Hxc}{\bw}(\n{}{})$.
\alert{Mention LIM?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Density-functional approximations for ensembles} \section{Density-functional approximations for ensembles}