From 11cacfb9cb182b3ff52a83a435341b7352ffc249 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Thu, 7 May 2020 16:28:15 +0200 Subject: [PATCH] Manu: saving work --- Revised_Manuscript/eDFT.tex | 38 ++++++++++++++++++++++++++----------- 1 file changed, 27 insertions(+), 11 deletions(-) diff --git a/Revised_Manuscript/eDFT.tex b/Revised_Manuscript/eDFT.tex index cab7b2c..658ebed 100644 --- a/Revised_Manuscript/eDFT.tex +++ b/Revised_Manuscript/eDFT.tex @@ -682,9 +682,10 @@ 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. \manurev{This perturbation expansion -may not hold in realistic systems, which are all but uniform. Nevertheless, it -gives more insight into the eLDA approximation and becomes useful when -analyzing its performance, as shown in Sec. \ref{sec:res}.\\} +may not hold in realistic systems, which may deviate significantly from +the uniform density regime. Nevertheless, it +gives more insight into the eLDA approximation and becomes useful when +rationalizing its performance, as illustrated in Sec. \ref{sec:res}.\\} Let us 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 @@ -1021,13 +1022,20 @@ drastically. It is important to note that, even though the GIC removes the explicit quadratic Hx terms from the ensemble energy, a non-negligible curvature remains in the GIC-eLDA ensemble energy when the electron -correlation is strong. This is due to -(i) the correlation eLDA -functional, which contributes linearly (or even quadratically) to the individual -energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and -\eqref{eq:Taylor_exp_DDisc_term}], and (ii) the optimization of the +correlation is strong. \manurev{The latter ensemble energy is computed +as the weighted +sum of the individual KS-eLDA energies [see +Eq.~(\ref{eq:Ew-eLDA})]. Therefore, its +curvature can only originate from the weight dependence of the +individual energies. +Note that such a dependence does not exist in the exact theory. Here, +the individual density-functional eLDA correlation energies exhibit an +explicit linear and quadratic dependence on the weights, as discussed +further in the next paragraph. Note also that the individual KS-eLDA energies +may gain an additional (implicit) dependence on the weights through the optimization of the ensemble KS orbitals in the presence of ghost-interaction errors [see Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}]. +} %%% FIG 2 %%% \begin{figure*} @@ -1069,11 +1077,19 @@ weights $\bw$ [see Eqs.~(\ref{eq:ens1RDM}), (\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 +excitation 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 ground-state energy, as $\ew{2}$ increases. -The reverse is observed for the second excited state. - +The reverse is observed for the second excited state. +\manurev{Finally, we notice that the crossover point of the +first excited-state energies based on +bi- and triensemble calculations, respectively, disappears in the strong correlation +regime [see the right panel of Fig. \ref{fig:EIvsW}], thus illustrating +the importance of (individual and ensemble) densities, in +addition to the +weights, in the evaluation of individual energies within +an ensemble. +} %%% FIG 3 %%% \begin{figure} \includegraphics[width=\linewidth]{EvsL_5}