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Manuscript/FarDFT.tex
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@ -238,12 +238,14 @@ Unless otherwise stated, atomic units are used throughout.
\section{Theory}
\label{sec:theo}
Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
Let us consider a GOK ensemble of $\nEns$ electronic states with
individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and
(normalised) monotonically decreasing weights $\bw \equiv (\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
The corresponding ensemble energy
\begin{equation}
\E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)}
\end{equation}
can be obtained from the variational principle
can be obtained from the GOK variational principle
as follows\cite{Gross_1988a}
\begin{eqnarray}\label{eq:ens_energy}
\E{}{\bw} = \min_{\hGam{\bw}} \Tr[\hGam{\bw} \hH],
@ -259,20 +261,21 @@ where $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1}$ is a set of
The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1} = \lbrace \Psi^{(I)} \rbrace_{0 \le I \le \nEns-1}$.
Multiplet degeneracies can be easily handled by assigning the same
weight to the degenerate states. \cite{Gross_1988b}
One of the key feature of the GOK ensemble is that individual excitation
energies can be extracted from the ensemble energy via differentiation with respect to individual weights:
One of the key feature of the GOK ensemble is that \trashEF{individual} excitation
energies can be extracted from the ensemble energy via differentiation
with respect to the individual \manu{excited-state} weights \manu{$\ew{I}$ ($I>0$)}:
\begin{equation}\label{eq:diff_Ew}
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)}.
\end{equation}
Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles allows to rewrite the exact variational expression for the ensemble energy as\cite{Gross_1988a}
Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles allows to rewrite the exact variational expression for the ensemble energy as\cite{Gross_1988b}
\begin{equation}
\label{eq:Ew-GOK}
\E{}{\bw} = \min_{\n{}{}} \qty{ \F{}{\bw}[\n{}{}] + \int \vne(\br{}) \n{}{}(\br{}) d\br{} },
\end{equation}
where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional
(the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles).
In the KS formulation, this functional can be decomposed as
In the KS formulation, this functional is decomposed as
\begin{equation}\label{eq:FGOK_decomp}
\F{}{\bw}[\n{}{}]
= \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}],