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added paragraph in Theory

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Pierre-Francois Loos 2020-02-11 21:12:20 +01:00
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commit eda91e765c
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Manuscript/FarDFT.tex
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@ -190,9 +190,9 @@ As mentioned above, eDFT for excited states is based on the GOK variational prin
\E{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \E{}{(I)},
\end{equation}
built from an ensemble of $\Nens$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\Nens-1)}$, and (normalized) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\Nens-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\Nens-1}$.
Multiplet degeneracies can be easily handled by assigning the same ensemble weight to the degenerate states.
Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states.
One of the key feature of eDFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
One of the key feature of GOK-DFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
\begin{equation}
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)},
\end{equation}
@ -204,7 +204,7 @@ In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}
\E{}{\bw} = \min_{\n{}{}} \qty{ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} },
\end{equation}
where $\vext(\br{})$ is the external potential.
In the KS formulation of eDFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
In the KS formulation of GOK-DFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
\begin{equation}
\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}],
\end{equation}
@ -231,8 +231,10 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
\end{equation}
where
\begin{align}
\label{eq:nw}
\n{}{\bw}(\br{}) & = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{}),
&
\\
\label{eq:nI}
\n{}{(I)}(\br{}) & = \sum_{p}^{\Norb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
\end{align}
are the ensemble and individual one-electron densities, respectively,
@ -240,7 +242,7 @@ are the ensemble and individual one-electron densities, respectively,
\label{eq:KS-energy}
\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}
\end{equation}
is the weight-dependent KS energy, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ ($\ON{p}{(I)}$ being its occupancy for the state $I$) given by the ensemble KS equation
is the weight-dependent KS energy, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$] given by the ensemble KS equation
\begin{equation}
\label{eq:eKS}
\qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
@ -257,6 +259,7 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vext(\br{})$, and
\end{equation}
is the Hxc potential.
Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
Note that, although we have dropped the weight-dependency in the individual densities $\n{}{(I)}(\br{})$ defined in Eq.~\eqref{eq:nI}, these do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density.
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