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Manu: continued commenting on the discussion. Reached the "derivative discontinuity" part (not reviewed yet)

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Emmanuel Fromager 2020-02-27 15:36:46 +01:00
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Manuscript/eDFT.tex
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@ -879,7 +879,7 @@ All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
Generalization to a larger number of states is straightforward and is left for future work.
To ensure the GOK variational principle, \cite{Gross_1988a} the
tri-ensemble weights must fulfil the following conditions: \cite{Deur_2019}
triensemble weights must fulfil the following conditions: \cite{Deur_2019}
\titou{$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$}.
%The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
%to consider an equi-bi-ensemble
@ -1138,7 +1138,14 @@ 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.}
ensemble energy without the correlation functional.}\\
\\
\manu{Manu: another idea. As far as I can see we do
not show any individual energies (excitation energies are plotted in the
following). Plotting individual energies (to be compared with the FCI
ones) would immediately show if there is some curvature (in the ensemble
energy). The latter would
be induced by any deviation from the expected horizontal straight lines.}
%%% FIG 2 %%%
\begin{figure}
@ -1158,19 +1165,54 @@ In other words, each excitation is dominated by a sole, well-defined reference S
However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
Therefore, it is paramount to construct a two-weight functional (as we have done here) which allows the mixing of single and double configurations.
Therefore, it is paramount to construct a two-weight \manu{correlation} functional
(\manu{\ie, a triensemble functional}, as we have done here) which
allows the mixing of \trashEF{single and double} \manu{singly- and doubly-excited} configurations.
Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
\titou{Shall we add results for $\ew{2} = 0$ to illustrate this?}
\titou{Shall we add results for $\ew{2} = 0$ to illustrate
this?}\manu{Well, neglecting the second excited state is not the same as
considering the $w_2=0$ limit. I thought you were referring to an
approximation where the triensemble calculation is performed with
the biensemble functional. This is not the same as taking $w_2=0$
because, in this limit, you may still have a derivative discontinuity
correction. The latter is absent if you truly neglect the second excited
state in your ensemble functional. This should be clarified.}\\
\manu{Are the results in the supp mat? We could just add "[not
shown]" if not. This is fine as long as you checked that, indeed, the
results deteriorate ;-)}
\manu{Should we add that, in the bi-ensemble case, the ensemble
correlation derivative $\partial \epsilon^\bw_{\rm c}(n)/\partial w_2$
is neglected (if this is really what you mean (?)). I guess that this is the reason why
the second excitation energy would not be well described (?)}
As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
When the box gets larger, they start to deviate.
For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
TDA-TDLDA slightly corrects this trend thanks to error compensation.
Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
This is especially true for the single excitation which is significantly improved by using state-averaged weights.
The effect on the double excitation is less pronounced.
Overall, one clearly sees that, with state-averaged weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
This conclusion is verified for smaller and larger numbers of electrons (see {\SI}).
This is especially true for the single excitation\manu{Manu: in the
light of your comments about the mixed singly-excited/doubly-excited
character of the first and second excited states when correlation is
strong, I would refer to the
"first excitation" rather than the "single excitation" (to be corrected
everywhere in the discussion if adopted)} which is significantly
improved by using state-averaged weights\manu{Manu: you mean equal-weight?
State-averaged does not mean equal-weight, don't you think? In the state-averaged CASSCF
you do not have to use equal weights, even though most people do}.
The effect on the \trashEF{double} \manu{second?} excitation is less pronounced.
Overall, one clearly sees that, with \trashEF{state-averaged}
\manu{equal} weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
This conclusion is verified for smaller and larger numbers of electrons
(see {\SI}).\\
\manu{Manu: now comes the question that is, I believe, central in this
work. How important are the
ensemble correlation derivatives $\partial \epsilon^\bw_{\rm
c}(n)/\partial w_I$ that, unlike any functional
in the literature, the eLDA functional contains. We have to discuss this
point... I now see, after reading what follows that this question is
addressed later on. We should say something here and then refer to the
end of the section, or something like that ...}
%%% FIG 3 %%%
\begin{figure*}
@ -1183,12 +1225,34 @@ This conclusion is verified for smaller and larger numbers of electrons (see {\S
%%% %%% %%%
For the same set of methods, Fig.~\ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI) as a function of $\nEl$ for three values of $L$ ($\pi/8$, $\pi$, and $8\pi$).
We draw similar conclusions as above: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations, with a very significant improvement brought by the state-averaged KS-eLDA orbitals as compared to their zero-weight analogs.
As a rule of thumb, in the weak and intermediate correlation regimes, we see that KS-eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for these two box lengths.
Moreover, we note that, in the strong correlation regime (left graph of Fig.~\ref{fig:EvsN}), the single excitation energies obtained at the state-averaged KS-eLDA level remain in good agreement with FCI and are much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
This also applies to double excitations, the discrepancy between FCI and KS-eLDA remaining of the order of a few percents in the strong correlation regime.
These observations nicely illustrate the robustness of the present state-averaged GOK-DFT scheme in any correlation regime for both single and double excitations.
This is definitely a very pleasing outcome, which additionally shows that, even though we have designed the eLDA functional based on a two-electron model system, the present methodology is applicable to any 1D electronic system.
We draw similar conclusions as above: irrespectively of the number of
electrons, the eLDA functional with \trashEF{state-averaged} equal
weights is able to accurately model single and double excitations, with
a very significant improvement brought by the \trashEF{state-averaged}
\manu{equiensemble} KS-eLDA orbitals as compared to their zero-weight
\manu{(\ie, conventional ground-state)} analogs.
\manu{As a rule of thumb, in the weak and intermediate correlation regimes, we
see that the \trashEF{single} \manu{first
excitation} obtained from \manu{equiensemble} KS-eLDA is of
the same quality as the one obtained in the linear response formalism
(such as TDLDA). On the other hand, the \trashEF{double} second
excitation energy only deviates
from the FCI value by a few tenth of percent} \trashEF{for these two box
lengths}.
Moreover, we note that, in the strong correlation regime (left graph of
Fig.~\ref{fig:EvsN}), the \trashEF{single} \manu{first} excitation
energy obtained at the equiensemble KS-eLDA level remains in good
agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
This also applies to \trashEF{double} \manu{the second} excitation
\manu{(which has a strong doubly-excited character)}, the discrepancy
between FCI and \manu{equiensemble} KS-eLDA remaining of the order of a few percents in the strong correlation regime.
These observations nicely illustrate the robustness of the
\trashEF{present state-averaged} GOK-DFT scheme in any correlation regime for both single and double excitations.
This is definitely a very pleasing outcome, which additionally shows
that, even though we have designed the eLDA functional based on a
two-electron model system, the present methodology is applicable to any
1D electronic system, \manu{\ie, a system that has more than two
electrons}.
%%% FIG 4 %%%
\begin{figure}
@ -1201,7 +1265,8 @@ This is definitely a very pleasing outcome, which additionally shows that, even
\end{figure}
%%% %%% %%%
\titou{T2: there is a micmac with the derivative discontinuity as it is only defined at zero weight. We should clean up this.}
\titou{T2: there is a micmac with the derivative discontinuity as it is
only defined at zero weight. We should clean up this.}\manu{I will!}
It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage (with respect to FCI) on the excitation energies obtained at the KS-eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium.
The influence of the derivative discontinuity is clearly more important in the strong correlation regime.