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Manu: reformulated the discussion on derivative discontinuities

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Emmanuel Fromager 2020-02-25 10:20:51 +01:00
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Manuscript/eDFT.tex
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@ -346,50 +346,43 @@ Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy:
\eeq
or, equivalently,
\beq\label{eq:gs_ener_level_gs_lim}
\begin{split}
\E{}{(0)}&=\mel*{\Det{(0)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(0)}}
\\
&
+
\E{c}{}[\n{\Det{(0)}}{}]
-\int \fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{})}\n{\Det{(0)}}{}(\br{})
\E{}{(0)}=\mel*{\Det{(0)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(0)}}
,
\end{split}
\eeq
where the ground-state-density-functional Hamiltonian reads
\beq
\hat{H}[\n{\Det{(0)}}{}]=\hH+
\sum^N_{i=1}\fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{i})}.
where the density-functional Hamiltonian reads
\beq\label{eq:dens_func_Hamilt}
\hat{H}[n]=\hH+
\sum^N_{i=1}\left(\fdv{\E{c}{}[n]}{\n{}{}(\br{i})}
+C_{\rm c}[n]
\right),
\eeq
Note that, when divided by the total number $N$ of electrons, the sum of
the last two terms on the right-hand side of
Eq.~(\ref{eq:gs_ener_level_gs_lim}) match the correlation component of
Levy--Zahariev's shift in the KS potential~\cite{Levy_2014}.
The excited-state ($I>0$) energy level expressions
can also be simplified, thus leading to
and
\beq\label{eq:corr_LZ_shift}
C_{\rm c}[n]=\dfrac{\E{c}{}[n]
-\int
\fdv{\E{c}{}[n]}{\n{}{}(\br{})}n(\br{})d\br{}}{\int n(\br{})d\br{}}
\eeq
is the correlation component of
Levy--Zahariev's constant shift in potential~\cite{Levy_2014}.
Similarly, the excited-state ($I>0$) energy level expressions
can be recast as follows:
\beq\label{eq:excited_ener_level_gs_lim}
\begin{split}
\E{}{(I)}
& = \mel*{\Det{(I)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(I)}}
= \mel*{\Det{(I)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(I)}}
+
\left.
\pdv{\E{c}{\bw}[\n{\Det{(0)}}{}]}{\ew{I}}
\right|_{\bw=0}.
\\
&
+
\E{c}{}[\n{\Det{(0)}}{}]
-\int \fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{})}\n{\Det{(0)}}{}(\br{})
\end{split}
\eeq
As readily seen from Eqs.~(\ref{eq:gs_ener_level_gs_lim}) and
(\ref{eq:excited_ener_level_gs_lim}), any constant shift $\delta
As readily seen from Eqs.~(\ref{eq:dens_func_Hamilt}) and
(\ref{eq:corr_LZ_shift}), introducing any constant shift $\delta
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})\rightarrow \delta
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ in the correlation
potential leaves the individual energy levels unchanged. As a result, in
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ into the correlation
potential leaves the density-functional Hamiltonian $\hat{H}[n]$ (and
therefore the individual energy levels) unchanged. As a result, in
this context,
the correlation derivative discontinuities that is expected to appear when the
excitation process occurs~\cite{Levy_1995} are fully described by the ensemble
the correlation derivative discontinuities induced by the
excitation process~\cite{Levy_1995} will be fully described by the ensemble
correlation derivatives [second term on the right-hand side of
Eq.~(\ref{eq:excited_ener_level_gs_lim})].