micro modif in conclusion and start clean up results

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Pierre-Francois Loos 2020-02-28 22:19:49 +01:00
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@ -1115,7 +1115,7 @@ To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $
The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented
in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
fulfilling the restrictions on the ensemble weights to ensure the GOK fulfilling the restrictions on the ensemble weights to ensure the GOK
variational principle [\ie, $0 \le \manu{\ew{2}} \le 1/3$ and \manu{$\ew{2} \le \ew{1} \le (1-\ew{2})/2$}]. variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}]. To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}].
\manu{Manu: Just to be sure. What you refer to as the GIC ensemble \manu{Manu: Just to be sure. What you refer to as the GIC ensemble
energy is energy is
@ -1212,18 +1212,18 @@ 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. 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} 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. 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 \manu{correlation} functional Therefore, it is paramount to construct a two-weight correlation functional
(\manu{\ie, a triensemble functional}, as we have done here) which (\ie, a triensemble functional, as we have done here) which
allows the mixing of \trashEF{single and double} \manu{singly- and doubly-excited} configurations. allows the mixing of 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. 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 \titou{Shall we add results for the biensemble to illustrate this?}
this?}\manu{Well, neglecting the second excited state is not the same as %\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 %considering the $w_2=0$ limit. I thought you were referring to an
approximation where the triensemble calculation is performed with %approximation where the triensemble calculation is performed with
the biensemble functional. This is not the same as taking $w_2=0$ %the biensemble functional. This is not the same as taking $w_2=0$
because, in this limit, you may still have a derivative discontinuity %because, in this limit, you may still have a derivative discontinuity
correction. The latter is absent if you truly neglect the second excited %correction. The latter is absent if you truly neglect the second excited
state in your ensemble functional. This should be clarified.}\\ %state in your ensemble functional. This should be clarified.}\\
\manu{Are the results in the supp mat? We could just add "[not \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 shown]" if not. This is fine as long as you checked that, indeed, the
results deteriorate ;-)} results deteriorate ;-)}
@ -1237,18 +1237,11 @@ 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. 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. 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, $\bw = (0,0)$]. 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\manu{Manu: in the This is especially true for the single excitation
light of your comments about the mixed singly-excited/doubly-excited which is significantly improved by using equal weights.
character of the first and second excited states when correlation is The effect on the double excitation is less pronounced.
strong, I would refer to the Overall, one clearly sees that, with
"first excitation" rather than the "single excitation" (to be corrected equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
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 This conclusion is verified for smaller and larger numbers of electrons
(see {\SI}).\\ (see {\SI}).\\
\manu{Manu: now comes the question that is, I believe, central in this \manu{Manu: now comes the question that is, I believe, central in this
@ -1273,33 +1266,31 @@ end of the section, or something like that ...}
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 $8\pi$). 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 We draw similar conclusions as above: irrespectively of the number of
electrons, the eLDA functional with \trashEF{state-averaged} equal electrons, the eLDA functional with equal
weights is able to accurately model single and double excitations, with weights is able to accurately model single and double excitations, with
a very significant improvement brought by the \trashEF{state-averaged} a very significant improvement brought by the
\manu{equiensemble} KS-eLDA orbitals as compared to their zero-weight equiensemble KS-eLDA orbitals as compared to their zero-weight
\manu{(\ie, conventional ground-state)} analogs. (\ie, conventional ground-state) analogs.
\manu{As a rule of thumb, in the weak and intermediate correlation regimes, we As a rule of thumb, in the weak and intermediate correlation regimes, we
see that the \trashEF{single} \manu{first see that the single
excitation} obtained from \manu{equiensemble} KS-eLDA is of excitation obtained from equiensemble KS-eLDA is of
the same quality as the one obtained in the linear response formalism the same quality as the one obtained in the linear response formalism
(such as TDLDA). On the other hand, the \trashEF{double} second (such as TDLDA). On the other hand, the double
excitation energy only deviates excitation energy only deviates
from the FCI value by a few tenth of percent} \trashEF{for these two box from the FCI value by a few tenth of percent.
lengths}.
Moreover, we note that, in the strong correlation regime (left graph of Moreover, we note that, in the strong correlation regime (left graph of
Fig.~\ref{fig:EvsN}), the \trashEF{single} \manu{first} excitation Fig.~\ref{fig:EvsN}), the single excitation
energy obtained at the equiensemble KS-eLDA level remains in good 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 $60 \%$. 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 This also applies to the double excitation, the discrepancy
\manu{(which has a strong doubly-excited character)}, the discrepancy between FCI and equiensemble KS-eLDA remaining of the order of a few percents in the strong correlation regime.
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 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. GOK-DFT scheme in any correlation regime for both single and double excitations.
This is definitely a very pleasing outcome, which additionally shows This is definitely a very pleasing outcome, which additionally shows
that, even though we have designed the eLDA functional based on a that, even though we have designed the eLDA functional based on a
two-electron model system, the present methodology is applicable to any two-electron model system, the present methodology is applicable to any
1D electronic system, \manu{\ie, a system that has more than two 1D electronic system, \ie, a system that has more than two
electrons}. electrons.
%%% FIG 4 %%% %%% FIG 4 %%%
\begin{figure} \begin{figure}
@ -1395,23 +1386,14 @@ calculations.}\manu{to be updated ...}
Let us finally stress that the present methodology can be extended Let us finally stress that the present methodology can be extended
straightforwardly to other types of ensembles like, for example, the straightforwardly to other types of ensembles like, for example, the
$N$-centered ones\cite{Senjean_2018,Senjean_2020}, thus allowing for the design an LDA-type functional for the $\nEl$-centered ones, \cite{Senjean_2018,Senjean_2020} thus allowing for the design of a LDA-type functional for the
calculation of ionization potentials, electron affinities, and calculation of ionization potentials, electron affinities, and
fundamental gaps. fundamental gaps.
Like in the present Like in the present
eLDA, such a functional would incorporate the infamous derivative eLDA, such a functional would incorporate the infamous derivative
discontinuity contribution to the gap through its explicit weight discontinuity contribution to the fundamental gap through its explicit weight
dependence. We hope to report on this in the near future. dependence. We hope to report on this in the near future.
\trashEF{This can be done by constructing a functional for the one- and
three-electron ground-state systems, and combining them with the
two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and
\eqref{eq:ecw}.}\manu{I find the sentence too technical for a
conclusion.}
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\section*{Supplementary material} \section*{Supplementary material}
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@ -1419,14 +1401,11 @@ See {\SI} for the additional details about the construction of the functionals,
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\begin{acknowledgements} \begin{acknowledgements}
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support. The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
\end{acknowledgements} \end{acknowledgements}
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\bibliography{eDFT} \bibliography{eDFT}
\end{document} \end{document}