micro modif in conclusion and start clean up results
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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 $
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The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented
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The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented
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in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
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in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
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fulfilling the restrictions on the ensemble weights to ensure the GOK
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fulfilling the restrictions on the ensemble weights to ensure the GOK
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variational principle [\ie, $0 \le \manu{\ew{2}} \le 1/3$ and \manu{$\ew{2} \le \ew{1} \le (1-\ew{2})/2$}].
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variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
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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}].
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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}].
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\manu{Manu: Just to be sure. What you refer to as the GIC ensemble
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\manu{Manu: Just to be sure. What you refer to as the GIC ensemble
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energy is
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energy is
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@ -1212,18 +1212,18 @@ In other words, each excitation is dominated by a sole, well-defined reference S
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However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
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However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
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In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
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In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
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This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
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This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
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Therefore, it is paramount to construct a two-weight \manu{correlation} functional
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Therefore, it is paramount to construct a two-weight correlation functional
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(\manu{\ie, a triensemble functional}, as we have done here) which
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(\ie, a triensemble functional, as we have done here) which
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allows the mixing of \trashEF{single and double} \manu{singly- and doubly-excited} configurations.
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allows the mixing of singly- and doubly-excited configurations.
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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.
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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.
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\titou{Shall we add results for $\ew{2} = 0$ to illustrate
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\titou{Shall we add results for the biensemble to illustrate this?}
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this?}\manu{Well, neglecting the second excited state is not the same as
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%\manu{Well, neglecting the second excited state is not the same as
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considering the $w_2=0$ limit. I thought you were referring to an
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%considering the $w_2=0$ limit. I thought you were referring to an
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approximation where the triensemble calculation is performed with
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%approximation where the triensemble calculation is performed with
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the biensemble functional. This is not the same as taking $w_2=0$
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%the biensemble functional. This is not the same as taking $w_2=0$
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because, in this limit, you may still have a derivative discontinuity
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%because, in this limit, you may still have a derivative discontinuity
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correction. The latter is absent if you truly neglect the second excited
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%correction. The latter is absent if you truly neglect the second excited
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state in your ensemble functional. This should be clarified.}\\
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%state in your ensemble functional. This should be clarified.}\\
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\manu{Are the results in the supp mat? We could just add "[not
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\manu{Are the results in the supp mat? We could just add "[not
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shown]" if not. This is fine as long as you checked that, indeed, the
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shown]" if not. This is fine as long as you checked that, indeed, the
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results deteriorate ;-)}
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results deteriorate ;-)}
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@ -1237,18 +1237,11 @@ When the box gets larger, they start to deviate.
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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.
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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.
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TDA-TDLDA slightly corrects this trend thanks to error compensation.
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TDA-TDLDA slightly corrects this trend thanks to error compensation.
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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, $\bw = (0,0)$].
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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, $\bw = (0,0)$].
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This is especially true for the single excitation\manu{Manu: in the
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This is especially true for the single excitation
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light of your comments about the mixed singly-excited/doubly-excited
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which is significantly improved by using equal weights.
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character of the first and second excited states when correlation is
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The effect on the double excitation is less pronounced.
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strong, I would refer to the
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Overall, one clearly sees that, with
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"first excitation" rather than the "single excitation" (to be corrected
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equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
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everywhere in the discussion if adopted)} which is significantly
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improved by using state-averaged weights\manu{Manu: you mean equal-weight?
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State-averaged does not mean equal-weight, don't you think? In the state-averaged CASSCF
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you do not have to use equal weights, even though most people do}.
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The effect on the \trashEF{double} \manu{second?} excitation is less pronounced.
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Overall, one clearly sees that, with \trashEF{state-averaged}
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\manu{equal} weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
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This conclusion is verified for smaller and larger numbers of electrons
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This conclusion is verified for smaller and larger numbers of electrons
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(see {\SI}).\\
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(see {\SI}).\\
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\manu{Manu: now comes the question that is, I believe, central in this
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\manu{Manu: now comes the question that is, I believe, central in this
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@ -1273,33 +1266,31 @@ end of the section, or something like that ...}
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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 $8\pi$).
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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 $8\pi$).
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We draw similar conclusions as above: irrespectively of the number of
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We draw similar conclusions as above: irrespectively of the number of
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electrons, the eLDA functional with \trashEF{state-averaged} equal
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electrons, the eLDA functional with equal
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weights is able to accurately model single and double excitations, with
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weights is able to accurately model single and double excitations, with
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a very significant improvement brought by the \trashEF{state-averaged}
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a very significant improvement brought by the
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\manu{equiensemble} KS-eLDA orbitals as compared to their zero-weight
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equiensemble KS-eLDA orbitals as compared to their zero-weight
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\manu{(\ie, conventional ground-state)} analogs.
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(\ie, conventional ground-state) analogs.
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\manu{As a rule of thumb, in the weak and intermediate correlation regimes, we
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As a rule of thumb, in the weak and intermediate correlation regimes, we
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see that the \trashEF{single} \manu{first
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see that the single
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excitation} obtained from \manu{equiensemble} KS-eLDA is of
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excitation obtained from equiensemble KS-eLDA is of
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the same quality as the one obtained in the linear response formalism
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the same quality as the one obtained in the linear response formalism
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(such as TDLDA). On the other hand, the \trashEF{double} second
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(such as TDLDA). On the other hand, the double
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excitation energy only deviates
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excitation energy only deviates
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from the FCI value by a few tenth of percent} \trashEF{for these two box
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from the FCI value by a few tenth of percent.
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lengths}.
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Moreover, we note that, in the strong correlation regime (left graph of
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Moreover, we note that, in the strong correlation regime (left graph of
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Fig.~\ref{fig:EvsN}), the \trashEF{single} \manu{first} excitation
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Fig.~\ref{fig:EvsN}), the single excitation
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energy obtained at the equiensemble KS-eLDA level remains in good
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energy obtained at the equiensemble KS-eLDA level remains in good
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agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
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agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
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This also applies to \trashEF{double} \manu{the second} excitation
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This also applies to the double excitation, the discrepancy
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\manu{(which has a strong doubly-excited character)}, the discrepancy
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between FCI and equiensemble KS-eLDA remaining of the order of a few percents in the strong correlation regime.
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between FCI and \manu{equiensemble} KS-eLDA remaining of the order of a few percents in the strong correlation regime.
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These observations nicely illustrate the robustness of the
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These observations nicely illustrate the robustness of the
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\trashEF{present state-averaged} GOK-DFT scheme in any correlation regime for both single and double excitations.
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GOK-DFT scheme in any correlation regime for both single and double excitations.
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This is definitely a very pleasing outcome, which additionally shows
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This is definitely a very pleasing outcome, which additionally shows
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that, even though we have designed the eLDA functional based on a
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that, even though we have designed the eLDA functional based on a
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two-electron model system, the present methodology is applicable to any
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two-electron model system, the present methodology is applicable to any
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1D electronic system, \manu{\ie, a system that has more than two
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1D electronic system, \ie, a system that has more than two
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electrons}.
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electrons.
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%%% FIG 4 %%%
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%%% FIG 4 %%%
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\begin{figure}
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\begin{figure}
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@ -1395,23 +1386,14 @@ calculations.}\manu{to be updated ...}
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Let us finally stress that the present methodology can be extended
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Let us finally stress that the present methodology can be extended
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straightforwardly to other types of ensembles like, for example, the
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straightforwardly to other types of ensembles like, for example, the
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$N$-centered ones\cite{Senjean_2018,Senjean_2020}, thus allowing for the design an LDA-type functional for the
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$\nEl$-centered ones, \cite{Senjean_2018,Senjean_2020} thus allowing for the design of a LDA-type functional for the
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calculation of ionization potentials, electron affinities, and
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calculation of ionization potentials, electron affinities, and
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fundamental gaps.
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fundamental gaps.
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Like in the present
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Like in the present
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eLDA, such a functional would incorporate the infamous derivative
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eLDA, such a functional would incorporate the infamous derivative
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discontinuity contribution to the gap through its explicit weight
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discontinuity contribution to the fundamental gap through its explicit weight
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dependence. We hope to report on this in the near future.
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dependence. We hope to report on this in the near future.
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\trashEF{This can be done by constructing a functional for the one- and
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three-electron ground-state systems, and combining them with the
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two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and
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\eqref{eq:ecw}.}\manu{I find the sentence too technical for a
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conclusion.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supplementary material}
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\section*{Supplementary material}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -1419,14 +1401,11 @@ See {\SI} for the additional details about the construction of the functionals,
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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\begin{acknowledgements}
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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.
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The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
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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''.}
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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''.}
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\end{acknowledgements}
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{eDFT}
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\bibliography{eDFT}
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\end{document}
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\end{document}
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