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Manu: done (so far) with the theory section

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Emmanuel Fromager 2020-04-23 12:44:55 +02:00
parent 6bba47ef90
commit d4e307bc73
1 changed files with 15 additions and 11 deletions
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26
Manuscript/FarDFT.tex
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@ -380,17 +380,16 @@ where
\label{eq:KS-energy}
\Eps{I}{\bw} = \sum_{p}^{\nOrb} \ON{p}{(I)} \eps{p}{\bw}
\end{equation}
is the energy of the $I$th KS state.
}%%%%%% end manuf
is the energy of the $I$th KS state.\\
Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
Note that the individual KS densities
$\n{\Det{I}{\bw}\left[n^{\bw}\right]}{}(\br{})=\sum_{p}^{\nOrb}
\ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2$ do
not necessarily match the \textit{exact} (interacting) 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.
Nevertheless, these densities can still be extracted in principle exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
\manuf{
not necessarily match the \textit{exact} (interacting) individual-state
densities $n_{\Psi_I}(\br)$ 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.
Nevertheless, these densities can still be extracted in principle
exactly from the KS ensemble as shown by Fromager.
\cite{Fromager_2020}.\\
In the following, we will work at the (weight-dependent) LDA
level of approximation, \ie
\beq
@ -402,13 +401,18 @@ level of approximation, \ie
&\overset{\rm LDA}{\approx}&
\left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
\eeq
In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as
We will also adopt the usual decomposition, and write down the weight-dependent xc functional as
\begin{equation}
\e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}),
\e{\xc}{\bw{}}(\n{}{}) = \e{\ex}{\bw{}}(\n{}{}) + \e{\co}{\bw{}}(\n{}{}),
\end{equation}
where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively.
}
where $\e{\ex}{\bw{}}(\n{}{})$ and $\e{\co}{\bw{}}(\n{}{})$ are the
weight-dependent density-functional exchange and correlation energies
per particle, respectively.
}%%%%%% end manuf
\manu{Maybe we should say a little bit more about how we will design
such approximations, or just say the design of these functionals will be
presented in the following...}
%%%%%%%%%%%%%%%%
%%%%%%% Manu: stuff that I removed from the first version %%%%%
\iffalse%%%%