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Manu: minor changes in II A

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Emmanuel Fromager 2020-03-11 15:46:19 +01:00
parent f0874a96ad
commit cb8eb27818
1 changed files with 5 additions and 4 deletions
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Manuscript/eDFT.tex
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@ -353,7 +353,8 @@ auxiliary double-weight ensemble density reads
\beq \beq
\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}). \n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
\eeq \eeq
Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble
densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are obtained from the \textit{same} KS potential (which is unique up to a constant), it comes
from the exact expression in Eq.~\eqref{eq:exact_ens_Hx} that from the exact expression in Eq.~\eqref{eq:exact_ens_Hx} that
\beq \beq
\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}, \E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
@ -402,7 +403,7 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
Note that, when $\bw=0$, the ensemble correlation functional reduces to the Note that, when $\bw=0$, the ensemble correlation functional reduces to the
conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a
result, the regular KS-DFT expression is recovered from result, the regular KS-DFT expression is recovered from
Eq.~\eqref{eq:exact_ener_level_dets} for the ground-state energy Eq.~\eqref{eq:exact_ener_level_dets} for the ground-state energy:
\beq \beq
\E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} + \E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} +
\E{c}{}[\n{\Det{(0)}}{}], \E{c}{}[\n{\Det{(0)}}{}],
@ -445,8 +446,8 @@ potential leaves the density-functional Hamiltonian $\hat{H}[n]$ (and
therefore the individual energy levels) unchanged. As a result, in therefore the individual energy levels) unchanged. As a result, in
this context, this context,
the correlation derivative discontinuities induced by the the correlation derivative discontinuities induced by the
excitation process~\cite{Levy_1995} will be fully described by the ensemble excitation process~\cite{Levy_1995} will be fully described by the
correlation derivatives [second term on the right-hand side of correlation ensemble derivatives [second term on the right-hand side of
Eq.~\eqref{eq:excited_ener_level_gs_lim}]. Eq.~\eqref{eq:excited_ener_level_gs_lim}].
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