diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 0058166..6552ada 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -901,13 +901,16 @@ Eqs.~\eqref{eq:MOM1} and \eqref{eq:MOM2}. The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional. The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI. +Interestingly, the CC-S functional +leads to a substantial improvement of the LIM +excitation energy, getting close to the reference value +(with an error of up to 0.24 eV) when no correlation +functional is used. When correlation functionals are +added (\ie VWN5 or eVWN5), LIM tends to overestimate +the excitation energy by about 1 eV but still performs +better than when no correction of the curvature is considered. It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes in this limit (\textit{vide supra}). Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits. -\manuf{There is a quite detailed discussion about LIM but nothing about -its performance. One should say something. LIM is a ``poor man (or -woman)'' approach when weight-dependent functionals are not available. I -would at least compare regular LDA LIM with results obtained (by -differentiations) with the more advanced CC-S+eVWN5 approach.} %%% TABLE III %%% \begin{table} @@ -1003,6 +1006,9 @@ For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the exci %work. I guess the latter option is what you did. We need to explain more %what we do!!!} As expected from the linearity of the ensemble energy, the CC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits. +As a direct consequence of this linearity, LIM and MOM +do not provide any noticeable improvement on the excitation +energy. Nonetheless, the excitation energy is still off by $3$ eV. The fundamental theoretical reason of such a poor agreement is not clear. The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error. @@ -1074,7 +1080,13 @@ The CC-S exchange functional attenuates significantly this dependence, and when As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion was made in Ref.~\onlinecite{Loos_2020}. -This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy. +This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy. Here again, the LIM excitation energy +when the CC-S functional is used is very accurate with +only 22 millihartree error compared to the reference value, +while adding the correlation contribution to the functional +tends to overestimate the excitation energy. +Hence, in the light of the results obtained in this paper, it seems that the weight-dependent curvature correction to the exchange functional makes the biggest impact in providing +accurate excitation energies. As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation. %%% TABLE V %%%