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Manu: started polishing the discussion and the theory. Saving work.

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Emmanuel Fromager 2020-03-09 14:43:53 +01:00
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Manuscript/eDFT.tex
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@ -838,6 +838,16 @@ example, from a finite uniform electron gas model.
%\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
%What do you think?}
The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
reads
\beq\label{eq:Ew-GIC-eLDA}
\E{eLDA}{\bw}=\Tr[\bGam{\bw}\bh] + \WHF[\bGam{\bw}] +\int
\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{\bw}}{}(\br{}) d\br{}.
\eeq
%Manu, would it be useful to add this equation and the corresponding text?
%I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within KS-eLDA:
\beq\label{eq:EI-eLDA}
\begin{split}
@ -875,7 +885,19 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
Therefore, it can be identified as
an individual-density-functional correlation energy where the density-functional
correlation energy per particle is approximated by the ensemble one for
all the states within the ensemble.
all the states within the ensemble. Note that the weighted sum of the
individual KS-eLDA energy levels delivers a \manu{\it ghost-interaction-corrected (GIC)} version of
the KS-eLDA ensemble energy:
\beq\label{eq:Ew-eLDA}
\begin{split}
\E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
\\
&=
\E{eLDA}{\bw}
-\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}].
\end{split}
\eeq
Let us finally stress that, to the best of our knowledge, eLDA is the first
density-functional approximation that incorporates ensemble weight
dependencies explicitly, thus allowing for the description of derivative
@ -883,18 +905,7 @@ discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
comment that follows] {\it via} the last term on the right-hand side
of Eq.~\eqref{eq:EI-eLDA}.
\titou{The GIC KS-eLDA ensemble energy is thus given by
\beq\label{eq:Ew-eLDA}
\E{GIC-eLDA}{\bw}=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)},
\eeq
while the uncorrected KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun} can be recast as
\beq\label{eq:Ew-GIC-eLDA}
\E{eLDA}{\bw}=\E{GIC-eLDA}{\bw}+\WHF[
\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}].
\eeq
%Manu, would it be useful to add this equation and the corresponding text?
%I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
\titou{
The corresponding excitation energies are
\beq\label{eq:Om-eLDA}
\Ex{eLDA}{(I)}
@ -1139,10 +1150,15 @@ equi-tri-ensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$
\end{figure*}
%%% %%% %%%
First, we discuss the linearity of the ensemble energy.
First, we discuss the linearity of the \manu{computed} (approximate)
ensemble \manu{energies}.
To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
The deviation from linearity of the three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the hypothetical linear ensemble energy obtained by linear interpolation) is represented
in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
The deviation from linearity of the three-state ensemble energy
$\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the
\trashEF{hypothetical} \manu{linearly-interpolated} ensemble energy
\trashEF{obtained by linear interpolation}) is represented
in Fig.~\ref{fig:EvsW} as a function of \trashEF{both} $\ew{1}$
\trashEF{and} \manu{or} $\ew{2}$ while
fulfilling the restrictions on the ensemble weights to ensure the GOK
variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above \titou{[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.