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Manu: II

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Emmanuel Fromager 2020-05-09 10:48:47 +02:00
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17
Manuscript/FarDFT.tex
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@ -275,7 +275,7 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al
\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 is decomposed as
In the KS formulation\cite{Gross_1988b}, this functional can be decomposed as
\begin{equation}\label{eq:FGOK_decomp}
\F{}{\bw}[\n{}{}]
= \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}],
@ -285,9 +285,9 @@ $\Tr{ \hgamdens{\bw} \hT } =\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kin
\begin{equation}
\hgam{\bw}[n] = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}[n]}
\end{equation}
is the KS density-functional density matrix operator, and $\lbrace
is the density-functional KS density matrix operator, and $\lbrace
\Det{I}{\bw}[n] \rbrace_{0 \le I \le \nEns-1}$ are single-determinant
wave functions (or configuration state functions).
wave functions (or configuration state functions\cite{Gould_2017}).
Their dependence on the density is determined from the ensemble density constraint
\begin{equation}
\sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}[n]}{}(\br) = \n{}{}(\br).
@ -324,7 +324,7 @@ GOK variational principle, \cite{Gross_1988b,Senjean_2015}
\left(\hT+\hVne\right)]+\E{\Ha}{}[\n{\hGam{\bw}}{}]+\E{\xc}{\bw}[\n{\hGam{\bw}}{}]\right\},
\eeq
where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1}
\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is the trial ensemble density. As a
\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is a trial ensemble density. As a
result, the orbitals
$\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
\nOrb}$ from which the KS
@ -351,7 +351,7 @@ where $\ON{p}{(I)}$ denotes the occupation of $\MO{p}{\bw}(\br{})$ in
the $I$th KS wave function $\Det{I}{\bw}\left[n^{\bw}\right]$. Turning
to the excitation energies, they can be extracted from the
density-functional ensemble as follows [see Eqs.~\eqref{eq:diff_Ew}
and \eqref{eq:Ew-GOK} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]:
and \eqref{eq:min_KS_DM} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]:
\beq
\label{eq:dEdw}
\Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
@ -372,15 +372,16 @@ densities $n_{\Psi_I}(\br)$ as the non-interacting KS ensemble is expected to re
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
In the following, we will work at the (weight-dependent) \manu{ensemble}
LDA \manu{(eLDA)}
level of approximation, \ie
\beq
\E{\xc}{\bw}[\n{}{}]
&\overset{\rm LDA}{\approx}&
&\overset{\rm \manu{e}LDA}{\approx}&
\int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{},
\\
\fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
&\overset{\rm LDA}{\approx}&
&\overset{\rm \manu{e}LDA}{\approx}&
\left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
\eeq
We will also adopt the usual decomposition, and write down the weight-dependent xc functional as