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Manu: saving work.

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Manuscript/FarDFT.tex
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@ -580,10 +580,21 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
Second, in order to remove this spurious curvature of the ensemble
energy (which is mostly due to the ghost-interaction error, but not only
\manu{I would be more explicit. We can also cite Ref. \cite{Loos_2020}}), one can easily reverse-engineer (for this particular system, geometry, and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
\manu{Something that seems important to me: you may require linearity in
the range $0\leq \ew{}\leq 1/2$. The excitation energy you would obtain
is simply the one of LIM, right? I suspect that by considering the
endpoint $\ew{}=1$ you change the excitation energy substantially. How
different are the results? At first sight, it seems like MOM gives you
the excitation energy that drives the
parameterization of the functional. Regarding the excitation energy, the
parameterized functional does not bring any additional information,
right? Maybe I miss something. Of course it gives ideas about how to
construct functionals. Maybe we need to elaborate more on this.}
Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error)
\manu{As mentioned in our previous work, the individual-state Hartree
energies (which have nothing to do with the ghost-interaction) also have a quadratic-in-$\ew{}$ pre-factor. I am not a big fan
of the acronym GIC-S (why S?). Something like ``curvature-corrected'' seems more
of the acronym GIC-S (why S?). Something like ``curvature-corrected'' or
``linearized'' (?) seems more
appropriate to me.}
\begin{equation}
\e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
@ -786,8 +797,8 @@ LIM and MOM can be reduced to a single calculation
at $w = 1/4$ and $w=1/2$, respectively, instead of performing an interpolation between two different calculations.
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 $\ew{} = 0$ and $1$.
\manu{That is a good point. Maybe I was too hard with you when referring
to GIC-S as ``semi-empirical''. Actually, I see here an analogy with the
optimally-tuned range-separated functionals. Maybe we should elaborate
to GIC-S as ``semi-empirical''. Actually, it makes me think about the
optimally-tuned range-separated functionals. Maybe we could elaborate
more on this.}
%%% TABLE III %%%
@ -864,11 +875,21 @@ It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,9
The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
One clearly sees that the correction brought by GIC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH = 3.7$ bohr.
In other words, the ghost-interaction ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
In other words, the ghost-interaction ``hole'' \manu{see my previous
comments on curvature} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-independent xc functionals in Ref.~\onlinecite{Senjean_2015}.
Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}.
As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and eVWN5 correlation at equi-ensemble.
\manu{We did not mention HF exchange neither in the theory section nor
in the computational details. We should be clear about this. Is this an
ad-hoc correction, like in our previous work on ringium? Is HF exchange
used for the full ensemble energy (i.e. the HF interaction energy is
computed with the ensemble density matrix and therefore with
ghost-interaction errors) or for
individual energies (that you state-average then), like in our previous
work. I guess the latter option is what you did. We need to explain more
what we do!!!}
%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}.
As expected from the linearity of the ensemble energy, the GIC-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.
Nonetheless, the excitation energy is still off by $3$ eV.
@ -925,12 +946,18 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
As a final example, we consider the \ce{He} atom which can be seen as the limiting form of the \ce{H2} molecule for very short bond lengths.
In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963}
In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate an excitation energy of $2.126$ hartree for this $1s^2 \rightarrow 2s^2$ transition.
\manu{same comment as for H$_2$ at equilibrium. I would expect the
singly-excited configuration $1s2s$ to be considered in the ensemble
with the doubly-excited one. We need to know if the former has an impact
(I guess it does)
on the computations.}
Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
Consequently, we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
The excitation energies associated with this double excitation computed with various methods and combinations of xc functionals are gathered in Table \ref{tab:BigTab_He}.
The parameters of the GIC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
In other words, the ghost-interaction hole is deeper.
In other words, the ghost-interaction hole \manu{see my previous
comments on curvature} is deeper.
The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.