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@ -586,6 +586,7 @@ Assuming that the singly-excited state is lower in energy than the doubly-excite
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%\end{equation}
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%with $\eW = \ew{1}/2 + \ew{2}$ and $0 \le \eW \le 1/2$.
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Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$), and we consider the zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$).
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In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
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Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals.
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However, let us stress that we will not compute excitation energies with these ensembles inconsistent with GOK theory.
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@ -618,7 +619,7 @@ First, we compute the ensemble energy of the \ce{H2} molecule at equilibrium bon
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&
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\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
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\end{align}
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In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state, the lowest singly-excited state $1\sigma_g 1\sigma_u$ and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which has an autoionising resonance nature \cite{Bottcher_1974}).
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In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state, the lowest singly-excited state $1\sigma_g 1\sigma_u$ of the same symmetry as the ground state (\ie, $\Sigma_g^+$), and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
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%\manu{At equilibrium, I expect the singly-excited configuration
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%$1\sigma_g2\sigma_g$ to be lower in energy. From the point of view of
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%GOK-DFT I do not see how we can reach the doubly-excited state while
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@ -629,10 +630,11 @@ In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground sta
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%ensemble? In one way or another
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%we have to look at this, even within the simplest weight-independent
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%approximation.}
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The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$.
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The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
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Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
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As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see Fig.~\ref{fig:Om_H2}).
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Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/3$.
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As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
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Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $7$ eV from $\ew{} = 0$ to $1/3$.
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Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of the ensemble weights.
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\begin{figure}
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@ -657,9 +659,9 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
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\subsubsection{Weight-dependent exchange functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Second, in order to remove this spurious curvature of the ensemble
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Second, in order to remove some of this spurious curvature of the ensemble
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energy (which is mostly due to the ghost-interaction error, \cite{Loos_2020} but not only),
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one can easily reverse-engineer (for this particular system, geometry, and basis set) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0 \le \ew{2} \le 1$ assuming a perfect linearity between the pure-state limits $ \ew{1} = \ew{2} = 0$ (ground state) and $\ew{1} = 0 \land \ew{2} = 1$ (doubly-excited state).
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one can easily reverse-engineer (for this particular system, geometry, basis set, and excitation) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0 \le \ew{2} \le 1$ assuming a perfect linearity between the pure-state limits $ \ew{1} = \ew{2} = 0$ (ground state) and $\ew{1} = 0 \land \ew{2} = 1$ (doubly-excited state).
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%\manu{Something that seems important to me: you may require linearity in
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%the range $0\leq \ew{}\leq 1/2$. The excitation energy you would obtain
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%is simply the one of LIM, right? I suspect that by considering the
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@ -692,19 +694,20 @@ and
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\gamma & = - 0.367\,189,
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\end{align}
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\end{subequations}
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makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable (with respect to $\ew{}$) and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
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makes the ensemble energy $\E{}{(0,\ew{2})}$ almost perfectly linear (by construction), and removes some of the curvature of $\E{}{\ew{}}$ (see yellow curve in Fig.~\ref{fig:Ew_H2}).
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It also makes the excitation energy much more stable (with respect to $\ew{}$), and closer to the FCI reference (see yellow curve in Fig.~\ref{fig:Om_H2}).
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The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have been obtained via a least-square fit of the non-linear component of the ensemble energy computed between $\ew{2} = 0$ and $\ew{2} = 1$ by steps of $0.025$.
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\titou{Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure ...}
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Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure a correct behavior on the whole range of weights in order to obtain accurate excitation energies.
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As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the three ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits thanks to the factor $\ew{} (1 - \ew{})$.
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Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limit, which is a genuine saddle point of the KS equations, as mentioned above.
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Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
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As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
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Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above.
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Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature one needs to catch in order to get accurate excitation energies in the zero-weight limit.
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We shall come back to this point later on.
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\begin{figure}
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\includegraphics[width=\linewidth]{Cxw}
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\caption{
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$\Cx{\ew{}}/\Cx{}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH = 1.4$ bohr (red) and $\RHH = 3.7$ bohr (green).
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$\Cx{\ew{}}/\Cx{}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH = 1.4$ bohr (red), and $\RHH = 3.7$ bohr (green).
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\label{fig:Cxw}
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}
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\end{figure}
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@ -959,12 +962,11 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
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For this particular geometry, the doubly-excited state becomes the
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lowest excited state with the same symmetry as
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the ground state, so we can safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state.
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Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
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Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble as defined in Sec.~\ref{sec:H2}
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%In other words, we set the weight of the single excitation to zero (\ie, $\ew{1} = 0$) and we have thus $\ew = \ew{2}$ for the rest of this example.
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We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
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It yields $\alpha = +0.019\,182$, $\beta = -0.015\,453$, and $\gamma = -0.012\,720$ [see Eq.~\eqref{eq:Cxw}].
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It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
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The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
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One clearly sees that the correction brought by CC-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.
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@ -1040,10 +1042,10 @@ Similar to \ce{H2}, our ensemble contains the ground state of configuration $1s^
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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}
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In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate an excitation energy of $2.126$ hartree for this $1s^2 \rightarrow 2s^2$ transition.
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Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
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Consequently, we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
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Consequently, we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
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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}.
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The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.901\,572$, $\beta = +2.523\,660$, and $\gamma = +1.665\,228$ [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}).
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The parameters of the CC-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}).
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The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
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The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.
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@ -1083,39 +1085,6 @@ Excitation energies (in hartree) associated with the lowest double excitation of
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\fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
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\end{table}
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%%% TABLE I %%%
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%\begin{table}
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%\caption{
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%Excitation energies (in eV) associated with the lowest double excitation of \ce{HNO} obtained with the aug-cc-pVDZ basis set for various methods and combinations of xc functionals.
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%\label{tab:BigTab_H2st}
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%}
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%\begin{ruledtabular}
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%\begin{tabular}{llcccc}
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% \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
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% \cline{1-2} \cline{3-4}
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% \tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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% \hline
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% HF & & & & & \\
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% HF & VWN5 & & & & \\
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% S & & 1.72 & 4.00 & 2.86 & 3.99 \\
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% S & VWN5 & & & & \\
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% CC-S & & 3.99 & 3.99 & 3.99 & 3.99 \\
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% CC-S & VWN5 & 4.05 & 4.03 & 4.04 & 4.03 \\
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% \hline
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% S & PW92 & & & & 4.00\fnm[1] \\
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% PBE & PBE & & & & 4.13\fnm[1] \\
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% SCAN & SCAN & & & & 4.24\fnm[1] \\
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% B97M-V & B97M-V & & & & 4.33\fnm[1] \\
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% PBE0 & PBE0 & & & & 4.24\fnm[1] \\
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% \hline
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% \mc{5}{l}{Theoretical best estimate\fnm[2]} & 4.32 \\
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%\end{tabular}
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%\end{ruledtabular}
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%\fnt[1]{Square gradient minimization (SGM) approach from Ref.~\onlinecite{Hait_2020} obtained with the aug-cc-pVTZ basis set. SGM is theoretically equivalent to MOM.}
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%\fnt[2]{Theoretical best estimate from Ref.~\onlinecite{Loos_2019} obtained at the (extrapolated) FCI/aug-cc-pVQZ level.}
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%\end{table}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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%%%%%%%%%%%%%%%%%%
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@ -1123,7 +1092,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
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\label{sec:ccl}
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In the present article, we have discussed the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron systems (\ce{He} and \ce{H2}) specifically designed for the computation of double excitations within GOK-DFT, a time-\textit{independent} formalism thanks to which one can extract excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
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In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing most of the curvature of the ensemble energy).
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In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing some of the curvature of the ensemble energy), and improves excitation energies.
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Although the weight-dependent correlation functional developed in this paper (eVWN5) performs systematically better than their weight-independent counterpart (VWN5), the improvement remains rather small.
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To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure,
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\ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead
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