working on results

Data/DD_n2.dat

@@ -14,4 +14,3 @@
 -0.0074	-0.0067	-0.0055	-0.0036	-0.0010	0.0019	
 -0.0010	-0.0011	-0.0014	-0.0017	-0.0021	-0.0022	
 0.0084	0.0079	0.0069	0.0053	0.0031	0.0003	
-

Manuscript/eDFT.tex

@@ -1105,94 +1105,67 @@ equi-tri-ensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$
 	\includegraphics[width=\linewidth]{EvsW_n5}
 	\caption{
 	\label{fig:EvsW}
-	Weight dependence of the KS-eLDA ensemble energy $\E{\titou{eLDA}}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
+	Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
 	}
 \end{figure*}
 %%% %%% %%%
 
 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
+The deviation from linearity of the three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the hypothetical linear ensemble energy obtained by linear interpolation) 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, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
 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}].
-\manu{Manu: Just to be sure. What you refer to as the GIC ensemble
-energy is
+\titou{Manu will move this to the theory section later on
 \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[
+\E{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.
-\manu{Manu: discussing GIE while focusing exclusively on the linearity
-is not completely relevant. The GIE is about interactions between two
-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.\\} 
+\eeq}
+As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
+ensemble energy becomes less and less linear as $L$
+gets larger, while the GIC makes the ensemble energy almost
+linear. 
+%\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
+quadratic terms from the ensemble energy, a non-negligible curvature
+remains in the GIC-eLDA 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.}\\
-\\
-\manu{Manu: another idea. As far as I can see we do
-not show any individual energies (excitation energies are plotted in the
-following). Plotting individual energies (to be compared with the FCI
-ones) would immediately show if there is some curvature (in the ensemble
-energy). The latter would
-be induced by any deviation from the expected horizontal straight lines.}
+%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
+%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.}\\
+%\\
+%\manu{Manu: another idea. As far as I can see we do
+%not show any individual energies (excitation energies are plotted in the
+%following). Plotting individual energies (to be compared with the FCI
+%ones) would immediately show if there is some curvature (in the ensemble
+%energy). The latter would
+%be induced by any deviation from the expected horizontal straight lines.}
 
 %%% FIG 2 %%%
 \begin{figure}