From 4a8f768229ce51037808e4b0c106d84f9dc25607 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Wed, 11 Mar 2020 21:44:15 +0100 Subject: [PATCH] Manu: saving work --- Manuscript/eDFT.tex | 24 +++++++++++++----------- 1 file changed, 13 insertions(+), 11 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 6d1fb6a..015e05b 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -121,7 +121,9 @@ \begin{document} -\title{Weight-dependent local density-functional approximations for ensembles} +\title{Weight-dependent local density-functional \manu{approximation to +ensemble correlation energies}} +%\title{Weight-dependent local density-functional approximations for ensembles} \author{Pierre-Fran\c{c}ois Loos} \email{loos@irsamc.ups-tlse.fr} @@ -1272,10 +1274,11 @@ drastically. %correlation is strong. It is not clear to me which integral ($W_{01}?$) %drives the all thing.\\} It is important to note that, even though the GIC removes the explicit -quadratic terms from the ensemble energy, a non-negligible curvature -remains in the GIC-eLDA ensemble energy. \manu{This might be due to +quadratic \manu{Hx} terms from the ensemble energy, a non-negligible curvature +remains in the GIC-eLDA ensemble energy \manu{when the electron +correlation is strong}. \manu{This is due to \textit{(i)} the correlation eLDA -functional, which induces linear or quadratic weight dependencies of the individual +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 \textit{(ii)} the optimization of the ensemble KS orbitals in the presence of ghost-interaction errors {[see @@ -1317,10 +1320,10 @@ the ground and first excited-state increase with respect to the first-excited-state weight $\ew{1}$, thus showing that, in this case, we ``deteriorate'' these states by optimizing the orbitals for the -ensemble, rather than for each state individually. The reverse actually occurs for the ground state in the triensemble +ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble as $\ew{2}$ increases. The variations in the ensemble weights are essentially linear or quadratic. They are induced by the -eLDA functional, as readily seen from +eLDA correlation functional, as readily seen from Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}. 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 @@ -1445,7 +1448,7 @@ correlation ensemble derivative contribution $\DD{c}{(I)}$ to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}]. In our case, both single ($I=1$) and double ($I=2$) excitations are considered. 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 $L$ -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}]. +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}]. %\manu{Manu: there is something I do not understand. If you want to %evaluate the importance of the ensemble correlation derivatives you %should only remove the following contribution from the $K$th KS-eLDA @@ -1457,13 +1460,12 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)} %%rather than $E^{(I)}_{\rm HF}$ %} We first stress that although for $\nEl=3$ both single and double excitation energies are -systematically improved, as the strength of electron correlation -increases, when +systematically improved (as the strength of electron correlation +increases) when taking into account the correlation ensemble derivative, this is not always the case for larger numbers of electrons. -The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime. -For 3-boxium, in the zero-weight limit, its contribution is +For 3-boxium, in the zero-weight limit, the ensemble derivative is significantly larger for the single excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble case.