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RSDFT

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Pierre-Francois Loos 2019-04-12 14:12:02 +02:00
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Manuscript/G2-srDFT.tex
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@ -310,16 +310,29 @@ Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals to approx
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05} As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline} \begin{multline}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}]
= \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
\\ \\
- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}, - \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
\end{multline} \end{multline}
where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
\begin{equation} \begin{equation}
\label{eq:argmin} \label{eq:argmin}
\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation} \end{equation}
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$. with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
%\begin{multline}
% \label{eq:ec_md_mu}
% \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
% \\
% - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
%\end{multline}
%where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
%\begin{equation}
%\label{eq:argmin}
% \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
%\end{equation}
%with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
%and %and
%\begin{equation} %\begin{equation}
%\label{eq:erf} %\label{eq:erf}