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Emmanuel Fromager 2020-05-07 09:39:58 +02:00
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Revised_Manuscript/eDFT.tex
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@ -475,14 +475,14 @@ as follows:
\eeq \eeq
while the ensemble density matrix while the ensemble density matrix
and the ensemble density read and the ensemble density read
\beq \beq\label{eq:ens1RDM}
\bGam{\bw} \bGam{\bw}
= \sum_{K\geq 0} \ew{K} \bGam{(K)} = \sum_{K\geq 0} \ew{K} \bGam{(K)}
\equiv \eGam{\mu\nu}{\bw} \equiv \eGam{\mu\nu}{\bw}
= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)}, = \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)},
\eeq \eeq
and and
\beq \beq\label{eq:ens_dens_from_ens_1RDM}
\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}), \n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
\eeq \eeq
respectively. respectively.
@ -1055,15 +1055,20 @@ as $\ew{2}$ increases. The variations in the ensemble
weights are essentially linear or quadratic. weights are essentially linear or quadratic.
\manurev{This can be rationalized as follows. As readily seen from \manurev{This can be rationalized as follows. As readily seen from
Eqs.~(\ref{eq:EI-eLDA}) and (\ref{eq:ind_HF-like_ener}), the individual Eqs.~(\ref{eq:EI-eLDA}) and (\ref{eq:ind_HF-like_ener}), the individual
HF-like exchange does not depend explicitly on the weights, which means HF-like energies do not depend explicitly on the weights, which means
that the above-mentioned variations originate from the eLDA correlation that the above-mentioned variations originate from the eLDA correlation
functional [second and third terms on the right-hand side of functional [second and third terms on the right-hand side of
Eq.~(\ref{eq:EI-eLDA})]. If, for analysis purposes, we consider the Eq.~(\ref{eq:EI-eLDA})]. If, for analysis purposes, we consider the
Taylor expansions around the uniform density regime in Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and (\ref{eq:Taylor_exp_DDisc_term}) Taylor expansions around the uniform density regime in
}They are induced by the Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and
eLDA correlation functional, as readily seen from (\ref{eq:Taylor_exp_DDisc_term}), contributions with an explicit weight
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and dependence still remain after summation. As both the ensemble density and
\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first the ensemble correlation energy per particle vary linearly with the
weights $\bw$ [see Eqs.~(\ref{eq:ens1RDM}),
(\ref{eq:ens_dens_from_ens_1RDM}), and
(\ref{eq:decomp_ens_correner_per_part})], the latter contributions will contain both linear and quadratic terms in
$\bw$, as readily seen from Eq.~(\ref{eq:Taylor_exp_DDisc_term}) [see the second term on the right-hand
side].} In the biensemble, the weight dependence of the first
excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
systematically enhances the weight dependence, due to the lowering of the systematically enhances the weight dependence, due to the lowering of the
ground-state energy, as $\ew{2}$ increases. ground-state energy, as $\ew{2}$ increases.