Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/eDFT_FUEG
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@ -121,7 +121,9 @@
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\begin{document}
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\title{Weight-dependent local density-functional approximations for ensembles}
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\title{Weight-dependent local density-functional \manu{approximation to
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ensemble correlation energies}}
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%\title{Weight-dependent local density-functional approximations for ensembles}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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@ -1272,10 +1274,11 @@ drastically.
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%correlation is strong. It is not clear to me which integral ($W_{01}?$)
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%drives the all thing.\\}
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It is important to note that, even though the GIC removes the explicit
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quadratic terms from the ensemble energy, a non-negligible curvature
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remains in the GIC-eLDA ensemble energy. \manu{This might be due to
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quadratic \manu{Hx} terms from the ensemble energy, a non-negligible curvature
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remains in the GIC-eLDA ensemble energy \manu{when the electron
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correlation is strong}. \manu{This is due to
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\textit{(i)} the correlation eLDA
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functional, which induces linear or quadratic weight dependencies of the individual
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functional, which contributes linearly (or even quadratically) to the individual
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energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
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\eqref{eq:Taylor_exp_DDisc_term}], and \textit{(ii)} the optimization of the
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ensemble KS orbitals in the presence of ghost-interaction errors {[see
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@ -1317,10 +1320,10 @@ the ground and first excited-state increase with respect to the
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first-excited-state weight $\ew{1}$, thus showing that, in this
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case, we
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``deteriorate'' these states by optimizing the orbitals for the
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ensemble, rather than for each state individually. The reverse actually occurs for the ground state in the triensemble
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ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble
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as $\ew{2}$ increases. The variations in the ensemble
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weights are essentially linear or quadratic. They are induced by the
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eLDA functional, as readily seen from
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eLDA correlation functional, as readily seen from
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Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
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\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
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excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
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@ -1445,7 +1448,7 @@ correlation ensemble derivative contribution $\DD{c}{(I)}$
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to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
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In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
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To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
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on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
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on the excitation energies obtained at the KS-eLDA level with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
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%\manu{Manu: there is something I do not understand. If you want to
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%evaluate the importance of the ensemble correlation derivatives you
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%should only remove the following contribution from the $K$th KS-eLDA
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@ -1457,13 +1460,12 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}
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%%rather than $E^{(I)}_{\rm HF}$
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%}
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We first stress that although for $\nEl=3$ both single and double excitation energies are
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systematically improved, as the strength of electron correlation
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increases, when
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systematically improved (as the strength of electron correlation
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increases) when
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taking into account
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the correlation ensemble derivative, this is not
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always the case for larger numbers of electrons.
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The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
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For 3-boxium, in the zero-weight limit, its contribution is
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For 3-boxium, in the zero-weight limit, the ensemble derivative is
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significantly larger for the single
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excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
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case.
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