From fffe24314495efb464cc068ec06b623433c4c0b0 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Thu, 27 Feb 2020 13:06:02 +0100 Subject: [PATCH] Manu: added (many comments) about the curvature to sec. V --- Manuscript/eDFT.tex | 77 ++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 72 insertions(+), 5 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index ca7fad7..363c54f 100644 --- a/Manuscript/eDFT.tex +++ b/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. -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. -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. +\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.\\} +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.} %%% FIG 2 %%% \begin{figure}