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

Manuscript/eDFT.tex

@@ -1270,16 +1270,16 @@ drastically.
 %when looking at your curves, this assumption cannot be made when the
 %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 explicitly
+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
-\textit{(i)} the correlation eLDA
+(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 \textit{(ii)} 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}]}.
+\eqref{eq:Taylor_exp_DDisc_term}], and (ii) 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}].
 %However, this orbital-driven error is small (in our case at
 %least) \trashEF{as the correlation part of the ensemble KS potential $\delta
 %\E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared
@@ -1333,13 +1333,13 @@ The reverse is observed for the second excited state.
 	\includegraphics[width=\linewidth]{EvsL_5}
 	\caption{
 	\label{fig:EvsL}
-	Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box length $L$.
+	Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box lengths $L$.
 	Graphs for additional values of $\nEl$ can be found as {\SI}.
 	}
 \end{figure}
 %%% %%% %%%
 
-Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
+Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box lengths in the case of 5-boxium (\ie, $\nEl = 5$).
 Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
 For small $L$, the single and double excitations can be labeled as
 ``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
@@ -1539,15 +1539,12 @@ shown).\\
 
 Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime). 
 The difference between the solid and dashed curves
-undoubtedly show that \trashEF{, even in the strong correlation regime,}
-\manu{Manu: in the light of what we already discussed, we expect the
-derivative to be important in the strongly correlated regime so the
-sentence "even in ..." is useless (that's why I would remove it)} the
+undoubtedly show that the
 correlation ensemble derivative has a rather significant impact on the double
 excitation (around $10\%$) with a slight tendency of worsening the excitation energies
 in the case of equal weights, as the number of electrons
-increases. It has a rather large influence \manu{(which decreases with the
-number of electrons)} on the single
+increases. It has a rather large influence (which decreases with the
+number of electrons) on the single
 excitation energies obtained in the zero-weight limit, showing once
 again that the usage of equal weights has the benefit of significantly reducing the magnitude of the correlation ensemble derivative.
 
@@ -1569,9 +1566,9 @@ progress in this direction.
 Unlike any standard functional, eLDA incorporates derivative
 discontinuities through its weight dependence. The latter originates
 from the finite uniform electron gas on which eLDA is
-(partially) based. The KS-eLDA scheme, where exact \manu{individual
-exchange energies are}
-combined with \manu{the eLDA correlation functional}, delivers accurate excitation energies for both
+(partially) based. The KS-eLDA scheme, where exact individual
+exchange energies are
+combined with the eLDA correlation functional	, delivers accurate excitation energies for both
 single and double excitations, especially when an equiensemble is used.
 In the latter case, the same weights are assigned to each state belonging to the ensemble.
 The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.