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Manu: polished theory and eLDA

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Emmanuel Fromager 2020-02-27 10:05:46 +01:00
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23
Manuscript/eDFT.tex
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@ -457,7 +457,7 @@ and
\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}), \n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
\eeq \eeq
respectively. respectively.
The individual energy expression in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as The exact individual energy expression in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
\beq\label{eq:exact_ind_ener_rdm} \beq\label{eq:exact_ind_ener_rdm}
\begin{split} \begin{split}
\E{}{(I)} \E{}{(I)}
@ -936,7 +936,8 @@ Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a
Combining these, one can build the following three-state weight-dependent correlation eDFA: Combining these, one can build the following three-state weight-dependent correlation eDFA:
\begin{equation} \begin{equation}
\label{eq:ecw} \label{eq:ecw}
\e{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}). %\e{c}{\bw}(\n{}{})
\tilde{\epsilon}_{\rm c}^\bw(n)= (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
\end{equation} \end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -949,7 +950,7 @@ The weight-dependence of the correlation functional is then carried exclusively
Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows: Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
\begin{equation} \begin{equation}
\label{eq:becw} \label{eq:becw}
\titou{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}), \tilde{\epsilon}_{\rm c}^\bw(n)\rightarrow{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
\end{equation} \end{equation}
where where
\begin{equation} \begin{equation}
@ -979,7 +980,7 @@ recast Eq.~\eqref{eq:becw} as
\begin{equation} \begin{equation}
\label{eq:eLDA} \label{eq:eLDA}
\begin{split} \begin{split}
\titou{\e{c}{\bw}(\n{}{})} {\e{c}{\bw}(\n{}{})}
& = \e{c}{\text{LDA}}(\n{}{}) & = \e{c}{\text{LDA}}(\n{}{})
\\ \\
& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})], & + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
@ -988,7 +989,7 @@ recast Eq.~\eqref{eq:becw} as
or, equivalently, or, equivalently,
\begin{equation} \begin{equation}
\label{eq:eLDA_gace} \label{eq:eLDA_gace}
\titou{\e{c}{\bw}(\n{}{})} {\e{c}{\bw}(\n{}{})}
= \e{c}{\text{LDA}}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})
+ \sum_{K>0}\int_0^{\ew{K}} + \sum_{K>0}\int_0^{\ew{K}}
\qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K, \qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K,
@ -996,18 +997,14 @@ or, equivalently,
where the $K$th correlation excitation energy (per electron) is integrated over the where the $K$th correlation excitation energy (per electron) is integrated over the
ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$. ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$.
Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present eDFA. Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present eDFA.
In particular, $\titou{\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$. In particular, ${\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles. Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
Finally, we note that, by construction, Finally, we note that, by construction,
\begin{equation} \begin{equation}
\titou{\pdv{\e{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).} {\pdv{\e{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).}
\end{equation} \end{equation}
%Manu: I guess that the "overlines" and the dependence in $\bf r$ of the
%densities on the RHS should be removed. The final expression should be
%\beq
%\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).
%\eeq
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details} \section{Computational details}
\label{sec:comp_details} \label{sec:comp_details}