Manu: added (many comments) about the curvature to sec. V

Manuscript/eDFT.tex

@@ -1065,13 +1065,80 @@ equi\manu{-tri}-ensemble (or \manu{equal-weight} state-averaged) limit where $\b
 
 First, we discuss the linearity of the ensemble energy.
 To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
-The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while fulfilling the restrictions on the ensemble weights to ensure the GOK variational principle [\ie, $0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$].
+The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented
+in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
+fulfilling the restrictions on the ensemble weights to ensure the GOK
+variational principle [\ie, $0 \le \manu{\ew{2}} \le 1/3$ and \manu{$\ew{2} \le \ew{1} \le (1-\ew{2})/2$}].
 To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}].
-As one can see in Fig.~\ref{fig:EvsW}, the GOC-free ensemble energy becomes less and less linear as $L$ gets larger, while the GOC makes the ensemble energy almost perfectly linear.
+\manu{Manu: Just to be sure. What you refer to as the GIC ensemble
+energy is
+\beq
+\E{GIC-eLDA}{\bw}=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)},
+\eeq
+right? (I will move this to the theory section later on). The ensemble
+energy with GIE is the one computed in
+Eq.~\eqref{eq:min_with_HF_ener_fun},
+\beq
+\E{HF-eLDA}{\bw}=\E{GIC-eLDA}{\bw}+\WHF[
+\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}]
+\eeq
+\underline{Some suggestions for Fig. 1}: In order to "see" the curvature
+it might be convenient to
+plot $E^{(w_1,0)}-E^{(0,0)}$ and $E^{(1/3,w_2)}-E^{(1/3,0)}$ rather than $E^\bw$. Adding the exact curves
+would be nice (we could see that the slope is also substantially
+improved when introducing the GIC, at least in the strongly correlated
+regime). Showing the linearly-interpolated energies also helps in
+"seeing" the curvature.\\}
+As one can see in Fig.~\ref{fig:EvsW}, \manu{without GIC}, the
+\trashEF{GOC-free} ensemble energy becomes less and less linear as $L$
+gets larger, while the \manu{GIC} makes the ensemble energy almost
+perfectly linear. \manu{Manu: well, after all, it is not that stricking
+for the bi-ensemble (black curves), as you point out in the following.
+"Perfectly linear" is maybe too strong.}
 In other words, the GIE increases as the correlation gets stronger.
-Because the GIE can be easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
+\manu{Manu: discussing GIE while focusing exclusively on the linearity
-It is important to note that, even though the GIC removes the explicit quadratic terms from the ensemble energy, a weak non-linearity remains in the GIC ensemble energy due to the optimization of the ensemble KS orbitals in the presence of GIE.
+is not completely relevant. The GIE is about interactions between two
-However, this ``density-driven'' type of error is small (in our case at least) as the correlation part of the ensemble KS potential $\delta \E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared to the Hx contribution.
+different states. Individual interaction terms also have
+quadratic-in-weight factors in front, which contribute to the curvature
+of course. Our GIC removes not only the GIE (I guess we should see the
+improvement by looking at the slope) but also the wrong factors in front
+of individual interactions.}
+Because the GIE can be easily computed via Eq.~\eqref{eq:WHF} even for
+real, three-dimensional systems, this provides a cheap way of
+quantifying strong correlation in a given electronic system.\manu{This
+is a strong statement I am not sure about. The nature of the excitation
+should also be invoked I guess (charge transfer or not, etc ...). If we look at the GIE:
+\beq
+\WHF[
+\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}]
+\eeq
+For a bi-ensemble ($w_1=w$) it can be written as
+\beq
+\dfrac{1}{2}\left[(w^2-1)W_0+w(w-2)W_1\right]+w(1-w)W_{01} 
+\eeq
+If, for some reason, $W_0\approx W_1\approx W_{01}=W$, then the error
+reduces to $-W/2$, which is weight-independent (it fits for example with
+what you see in the weakly correlated regime). Such an assumption depends on the nature of the
+excitation, not only on the correlation strength, right? Neverthless,
+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 explicit
+quadratic terms from the ensemble energy, a weak \manu{Manu: is it that weak
+when correlation is strong? Look at the bi-ensemble case} non-linearity
+remains in the GIC ensemble energy due to the optimization of the
+ensemble KS orbitals in the presence of GIE [see Eq.~\eqref{eq:min_with_HF_ener_fun}].
+However, this \manu{orbital-driven} error is small \manu{Manu: again, can we
+really say "small" when looking at the strongly correlated case. It
+seems to me that there is some residual curvature which is a signature
+of the error in the orbitals} (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
+to the Hx contribution}.\manu{Manu: well, I guess that the problem arises
+from the density matrices (or orbitals) that are used to compute
+individual Coulomb-exchange energies (I would not expect the DFT
+correlation part to have such an impact, as you say). The best way to check is to plot the
+ensemble energy without the correlation functional.}
 
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