clean up results and add a bit of conclusion
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@ -739,25 +739,6 @@ Generalization to a larger number of states is straightforward and is left for f
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To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions:
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$0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$ [or $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\label{sec:Ec}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (\ie, per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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\begin{equation}
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\label{eq:ec}
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\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}},
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\end{equation}
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where the $a_k^{(I)}$'s are state-specific fitting parameters provided in Table \ref{tab:OG_func}.
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The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
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Combining these, one can build a three-state weight-dependent correlation eDFA:
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\begin{equation}
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\label{eq:ecw}
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\e{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
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\end{equation}
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%%% TABLE 1 %%%
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\begin{table*}
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\caption{
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@ -777,6 +758,25 @@ Combining these, one can build a three-state weight-dependent correlation eDFA:
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\end{table*}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\label{sec:Ec}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (\ie, per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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\begin{equation}
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\label{eq:ec}
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\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}},
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\end{equation}
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where the $a_k^{(I)}$'s are state-specific fitting parameters provided in Table \ref{tab:OG_func}.
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The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
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Combining these, one can build a three-state weight-dependent correlation eDFA:
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\begin{equation}
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\label{eq:ecw}
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\e{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{LDA-centered functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -862,46 +862,25 @@ Concerning the KS-eDFT and eHF calculations, two sets of weight have been tested
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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First, we discuss the linearity of the ensemble energy.
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To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
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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 fulfilling the restrictions on the ensemble weights to ensure the GOK variational principle [\ie, $0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$].
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To illustrate the magnitude of the ghost interaction error (GIE), we have reported the 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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As one can see in Fig.~\ref{fig:EvsW}, the linearity of the GOC-free ensemble energy deteriorates when $L$ gets larger, while the GOC makes the ensemble energy almost perfectly linear.
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In other words, the GIE increases as the correlation gets stronger.
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Because the GIE can be easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
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It is important to note that, even though the GIC removes the explicit quadratic terms from the ensemble energy, a weak non-linearity remains in the GIC ensemble energy due to the optimization of the ensemble KS orbitals in presence of GIE.
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However, this ``density-driven''-type error is extremely small (in our case at least) as the correlation part of the ensemble KS potential $\delta \E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared to the Hx contribution.
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%%% FIG 1 %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{EvsW_n5}
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\caption{
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\label{fig:EvsW}
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Weight dependence of the KS-eLDA ensemble energy $\E{}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
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Weight dependence of the KS-eLDA ensemble energy $\E{\titou{eLDA}}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
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}
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\end{figure*}
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%%% %%% %%%
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In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
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Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
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For small $L$, the single and double excitations can be labeled as ``pure''.
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In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
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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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This is clearly evidenced if one looks at the weights of the different configurations in the FCI wave function.
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Therefore, it is paramount to construct a two-weight functional (as we have done here) which allows the mixing of single and double 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, evidences a quick 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 this?}
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In the weakly correlated regime (\ie, small $L$), all methods provide accurate estimates of the excitation energies.
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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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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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This is especially true for the single excitation which is significantly improved by using state-averaged weights.
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The effect on the double excitation is less pronounced.
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Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.
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This conclusion is verified for smaller and larger number of electrons (see {\SI}).
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First, we discuss the linearity of the ensemble energy.
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To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
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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 fulfilling the restrictions on the ensemble weights to ensure the GOK variational principle [\ie, $0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$].
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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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As one can see in Fig.~\ref{fig:EvsW}, the GOC-free ensemble energy becomes less and less linear as $L$ gets larger, while the GOC makes the ensemble energy almost perfectly linear.
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In other words, the GIE increases as the correlation gets stronger.
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Because the GIE can be easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
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It is important to note that, even though the GIC removes the explicit quadratic terms from the ensemble energy, a weak non-linearity remains in the GIC ensemble energy due to the optimization of the ensemble KS orbitals in the presence of GIE.
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However, this ``density-driven'' type of error is small (in our case at least) as the correlation part of the ensemble KS potential $\delta \E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared to the Hx contribution.
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%%% FIG 2 %%%
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\begin{figure}
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@ -914,68 +893,90 @@ This conclusion is verified for smaller and larger number of electrons (see {\SI
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\end{figure}
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%%% %%% %%%
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Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\nEl$ for three values of $L$ ($L=\pi/8$, $\pi$, and $8\pi$).
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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.
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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 $L=\pi$.
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Finally, 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-TDLA excitation energies which can deviate by up to $60 \%$.
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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.
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These observations nicely illustrate the robustness of the present state-averaged GOK-DFT scheme in any correlation regime for both single and double excitations.
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Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
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Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
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For small $L$, the single and double excitations can be labeled as ``pure''.
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In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
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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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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 functional (as we have done here) which allows the mixing of single and double 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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\titou{Shall we add results for $\ew{2} = 0$ to illustrate this?}
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As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
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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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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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This is especially true for the single excitation which is significantly improved by using state-averaged weights.
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The effect on the double excitation is less pronounced.
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Overall, one clearly sees that, with state-averaged 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 (see {\SI}).
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%%% FIG 3 %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{EvsN}
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\caption{
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\label{fig:EvsN}
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Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and number of electrons $\nEl$ at $L=\pi$.
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Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and electron numbers $\nEl$ at $L=\pi/8$ (left), $L=\pi$ (center), and $L=8\pi$ (right).
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}
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\end{figure*}
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%%% %%% %%%
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It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
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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.
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The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
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Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influence the double excitation.
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Importantly, one realizes that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).
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This could explain why equiensemble calculations are clearly more accurate as it reduces the influence of the derivative discontinuity: for a given method, state-averaged orbitals partially remove the burden of modeling properly the derivative discontinuity.
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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 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.
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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.
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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 $60 \%$.
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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.
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These observations nicely illustrate the robustness of the present state-averaged GOK-DFT scheme in any correlation regime for both single and double excitations.
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This is definitely a very pleasing outcome.
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{EvsL_5_HF}
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\caption{
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\label{fig:EvsLHF}
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Error with respect to FCI (in \%) associated with the single excitation $\Ex{(1)}$ (bottom) and double excitation $\Ex{(2)}$ (top) as a function of the box length $L$ for 5-boxium at the KS-eLDA (solid lines) and eHF (dashed lines) levels of theory.
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Error with respect to FCI (in \%) associated with the single excitation $\Ex{(1)}$ (bottom) and double excitation $\Ex{(2)}$ (top) as a function of the box length $L$ for 5-boxium at the KS-eLDA (solid lines) and eHF (dashed lines) levels.
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Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and state-averaged (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
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}
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\end{figure}
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%%% %%% %%%
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Finally, in Fig.~\ref{fig:EvsN_HF}, we report the same quantities as a function of the number of electrons for a box of length $8\pi$ (\ie, in the strongly-correlated regime).
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The difference between the eHF and KS-eLDA excitation energies undoubtedly show that, even in the strongly-correlated regime, the derivative discontinuity has i) a small impact on the double excitations and has a slight tendency of worsening the excitation energies in the case of state-averaged weights, and ii) a rather large influence on the single excitation energies obtained in the zero-weight limit, showing once again that the usage of state-averaged weights has the benefit of significantly reducing the magnitude of the derivative discontinuity.
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It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
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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.
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The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
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Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influences the double excitation.
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Importantly, one realizes that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).
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This could explain why equiensemble calculations are clearly more accurate as it reduces the influence of the derivative discontinuity: for a given method, state-averaged orbitals partially remove the burden of modeling properly the derivative discontinuity.
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%%% FIG 5 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{EvsN_HF}
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\caption{
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\label{fig:EvsN_HF}
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Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA (solid lines) and eHF (dashed lines) levels of theory.
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Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA (solid lines) and eHF (dashed lines) levels.
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Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, black and red lines) and state-averaged (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported.
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}
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\end{figure}
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%%% %%% %%%
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Finally, in Fig.~\ref{fig:EvsN_HF}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime).
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The difference between the eHF and KS-eLDA excitation energies undoubtedly show that, even in the strong correlation regime, the derivative discontinuity has a small impact on the double excitations with a slight tendency of worsening the excitation energies in the case of state-averaged weights, and a rather large influence on the single excitation energies obtained in the zero-weight limit, showing once again that the usage of state-averaged weights has the benefit of significantly reducing the magnitude of the derivative discontinuity.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Concluding remarks}
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\label{sec:conclusion}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In the present article, we have constructed a weight-dependent three-state DFA in the context of ensemble DFT.
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This eDFA delivers accurate excitation energies for both single and double excitations.
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In the present article, we have constructed a local, weight-dependent three-state DFA in the context of ensemble DFT.
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The KS-eLDA scheme delivers accurate excitation energies for both single and double excitations, especially within its state-averaged version where the same weights are assigned to each state belonging to the ensemble.
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Generalization to a larger number of states is straightforward and will be investigated in future work.
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Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.
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We have observed that, although the derivative discontinuity has a non-negligible effect on the excitation energies (especially for the single excitations), its magnitude can be significantly reduced by performing state-averaged calculations instead of zero-weight calculations.
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Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap.\cite{Senjean_2018}
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Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.
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Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap. \cite{Senjean_2018}
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This can be done by constructing DFAs 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}.
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We hope to report on this in the near future.
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%However, as shown by Senjean and Fromager, \cite{Senjean_2018} one must modify the weights accordingly in order to maintain a constant density.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supplementary material}
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