Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/eDFT_FUEG

Manuscript/eDFT.tex

@@ -137,13 +137,13 @@ ensemble correlation energies}}
 We report a local, weight-dependent correlation density-functional approximation that incorporates information about both ground and excited states in the context of density-functional theory for ensembles (eDFT). 
 This density-functional approximation for ensembles is specially
 designed for the computation of single and double excitations within
-Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for \manu{neutral
-excitations} \trashEF{excited states}), and can be seen as a natural extension of the ubiquitous local-density approximation in the context of ensembles.
-The resulting density-functional approximation \trashEF{for ensembles}, based on both finite and infinite uniform electron gas models, automatically incorporates the infamous derivative discontinuity contributions to the excitation energies through its explicit ensemble weight dependence. 
+Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for neutral
+excitations), and can be seen as a natural extension of the ubiquitous local-density approximation in the context of ensembles.
+The resulting density-functional approximation, based on both finite and infinite uniform electron gas models, automatically incorporates the infamous derivative discontinuity contributions to the excitation energies through its explicit ensemble weight dependence. 
 Its accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
 Although the present weight-dependent functional has been specifically
 designed for one-dimensional systems, the methodology proposed here is
-\manu{general}, \ie, directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
+general, \ie, directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
 \end{abstract}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -819,7 +819,7 @@ Note that, within the approximation of Eq.~\eqref{eq:min_with_HF_ener_fun}, the
 optimized with a non-local exchange potential rather than a
 density-functional local one, as expected from
 Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
-applicable to not-necessarily spin polarized and real (higher-dimensional) systems. 
+applicable to not-necessarily spin-polarized and real (higher-dimensional) systems. 
 As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
 ensemble density matrix into the HF interaction energy functional
 introduces unphysical \textit{ghost-interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
@@ -834,9 +834,8 @@ as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
 \eeq
 The ensemble energy is of course expected to vary linearly with the ensemble
 weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}].
-\manu{
 The explicit linear weight dependence of the ensemble Hx energy is actually restored when evaluating the individual energy
-levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.}  
+levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
 
 Turning to the density-functional ensemble correlation energy, the
 following ensemble local-density approximation (eLDA) will be employed  
@@ -1031,9 +1030,9 @@ gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b
 However, an obvious drawback of using finite uniform electron gases is
 that the resulting density-functional approximation for ensembles
 will inexorably depend on the number of electrons in the finite uniform electron gas (see below).
-Here, we propose to construct a weight-dependent eLDA for the
+Here, we propose to construct a weight-dependent LDA functional for the
 calculation of excited states in 1D systems by combining finite uniform electron gases with the
-usual infinite uniform electron gas. 
+usual infinite uniform electron gas paradigm.
 
 As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
 The most appealing feature of ringium regarding the development of
@@ -1209,9 +1208,9 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
 \end{equation} 
 with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
 The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw}
-\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] is set
-to $10^{-5}$. For comparison, regular HF and KS-DFT calculations 
-are performed with the same threshold.
+\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] of the KS-DFT self-consistent calculation is set
+to $10^{-5}$. 
+%For comparison, regular HF and KS-DFT calculations are performed with the same threshold.
 In order to compute the various density-functional
 integrals that cannot be performed in closed form, 
 a 51-point Gauss-Legendre quadrature is employed.