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Manu: saving work in the discussion (Fig. 2).

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Emmanuel Fromager 2020-03-10 12:33:10 +01:00
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Manuscript/eDFT.tex
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@ -1290,7 +1290,7 @@ ensemble KS orbitals in the presence of GIE {[see Eqs.~\eqref{eq:min_with_HF_ene
Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights. Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
\manu{ \manu{
We first notice that, unlike in the exact theory, we do not obtain Unlike in the exact theory, we do not obtain
straight horizontal lines when plotting these straight horizontal lines when plotting these
energies, which is in agreement with energies, which is in agreement with
the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
@ -1300,13 +1300,12 @@ We see for example that, within the biensemble [$w_2=0$], the energy of
the ground state increases with the first-excited-state weight $w_1$, thus showing that we the ground state increases with the first-excited-state weight $w_1$, thus showing that we
``deteriorate'' this state a little by optimizing the orbitals also for ``deteriorate'' this state a little by optimizing the orbitals also for
the first excited state. The reverse actually occurs in the triensemble the first excited state. The reverse actually occurs in the triensemble
as $w_2$ increases. In most cases, the variations in the ensemble as $w_2$ increases. The variations in the ensemble
weights are essentially weights are essentially linear or quadratic. They are induced by the
linear, as expected from the Taylor expansion in eLDA functional, as readily seen from
Eq.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and the expression in Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and
Eq.~(\ref{eq:eLDA}) of the ensemble correlation energy per particle in \eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
eLDA. As the correlation increases, the weight dependence of the first excitation energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
excitation energy is reduced. On the other hand, switching from a bi- to a tri-ensemble
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 in this case, as $w_2$ increases. ground-state energy in this case, as $w_2$ increases.
The reverse is observed for the second excitation energy. The reverse is observed for the second excitation energy.