Added more discussion on the results about LIM (or MOM).

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Bruno Senjean 2020-05-10 19:38:16 +02:00
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@ -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 %%%