This commit is contained in:
Emmanuel Fromager 2020-03-11 23:45:53 +01:00
commit 1e5ec03018
4 changed files with 13 additions and 16 deletions

Binary file not shown.

Binary file not shown.

Binary file not shown.

View File

@ -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.