saving work
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@ -94,7 +94,6 @@
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% elements
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\newcommand{\ew}[1]{w_{#1}}
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\newcommand{\eW}{\xi}
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\newcommand{\eHc}[1]{h_{#1}}
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\newcommand{\eJ}[1]{J_{#1}}
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\newcommand{\eK}[1]{K_{#1}}
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@ -559,37 +558,38 @@ is the Hxc potential, with
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The self-consistent GOK-DFT calculations [see Eqs.~\eqref{eq:eKS} and \eqref{eq:ens_KS_dens}] have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented.
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For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
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For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994}
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Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001}
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Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-7}$. \cite{Becke_1988b,Lindh_2001}
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This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
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Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), the first singly-excited state ($I=1$ with weight $\ew{1}$), as well as the first doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
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To ensure the GOK variational principle, one should then have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$.
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Assuming that the singly-excited state is lower in energy than the doubly-excited state (which is not always the case as one would notice later), one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
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%Taking a generic two-electron system as an example, the individual one-electron densities read
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%\begin{subequations}
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%\begin{align}
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% \n{}{(0)} & = 2 \HOMO{2},
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% \\
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% \n{}{(1)} & = \HOMO{2} + \LUMO{2},
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% \\
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% \n{}{(2)} & = 2 \LUMO{2},
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%\end{align}
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%\end{subequations}
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%and they can be combined to produce the ensemble density
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%\begin{equation}
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% \label{eq:nw1w2}
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% \n{}{(\ew{1},\ew{2})}
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% = (1 - \ew{1} - \ew{2}) \n{}{(0)}
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% + \ew{1} \n{}{(1)} + \ew{2} \n{}{(2)}.
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%\end{equation}
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%For analysis purposes, Eq.~\eqref{eq:nw1w2} can be conveniently recast as a single-weight quantity
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%\begin{equation}
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% \n{}{\eW} = (1 - \eW) \n{}{(0)} + \eW \n{}{(2)},
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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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Taking a generic two-electron system as an example, the individual one-electron densities read
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\begin{subequations}
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\begin{align}
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\n{}{(0)} & = 2 \HOMO{2},
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\\
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\n{}{(1)} & = \HOMO{2} + \LUMO{2},
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\\
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\n{}{(2)} & = 2 \LUMO{2},
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\end{align}
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\end{subequations}
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and they can be combined to produce the ensemble density
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\begin{equation}
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\label{eq:nw1w2}
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\n{}{(\ew{1},\ew{2})}
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= (1 - \ew{1} - \ew{2}) \n{}{(0)}
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+ \ew{1} \n{}{(1)} + \ew{2} \n{}{(2)}.
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\end{equation}
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For analysis purposes, Eq.~\eqref{eq:nw1w2} can be conveniently recast as a single-weight quantity
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\begin{equation}
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\n{}{\eW} = (1 - \eW) \n{}{(0)} + \eW \n{}{(2)},
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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, $\ew{1} = \ew{2}$), and we consider the zero-weight limit $\eW = 0$ (\ie, $\ew{1} = \ew{2} = 0$), and the equiweight ensemble $\eW = 1/2$ (\ie, $\ew{1} = \ew{2} = 1/3$).
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Nonetheless, we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals.
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Indeed, the pure-state limits (\ie, $\ew{1} = 1 \land \ew{2} = 0$ or $\ew{1} = 0 \land \ew{2} = 1$) are of particular interest as they are genuine saddle points of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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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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The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is nonetheless of particular interest as it is a genuine saddle point of the restricted KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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%Although the range $1/2 < \ew{} \leq 1$ stands a little bit beyond the theory discussed previously, we look at these solutions for analysis purposes mainly.
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%These solutions of the density matrix operator functional in Eq.~\eqref{eq:min_KS_DM} correspond to stationary points rather than minimising ones.
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%Applying GOK-DFT in this range of weights would simply consists in switching the ground and excited states if true minimisations of the ensemble energy were performed.
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@ -629,16 +629,16 @@ 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 the composite weight $0 \le \eW \le 1/2$.
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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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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 $1/2$.
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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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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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\includegraphics[width=\linewidth]{Ew_H2}
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\caption{
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\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\eW}$ (in hartree) as a function of the composite weight $\eW = \ew{1}/2 + \ew{2}$ and $\ew{1} = \ew{2}$ for various functionals and the aug-cc-pVTZ basis set.
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\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
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See main text for the definition of the various functional's acronyms.
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\label{fig:Ew_H2}
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}
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@ -647,7 +647,7 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
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\begin{figure}
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\includegraphics[width=\linewidth]{Om_H2}
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\caption{
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\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) of the doubly-excited state $\Ex{}{(2)}$ as a function of the composite weight $\eW = \ew{1}/2 + \ew{2}$ and $\ew{1} = \ew{2}$ for various functionals and the aug-cc-pVTZ basis set.
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\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) of the doubly-excited state $\Ex{}{(2)}$ as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
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\label{fig:Om_H2}
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}
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\end{figure}
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@ -659,7 +659,7 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
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Second, in order to remove 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 $\E{}{\eW}$ as linear as possible for $0 \le \eW \le 1/2$.
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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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%\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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@ -675,36 +675,36 @@ one can easily reverse-engineer (for this particular system, geometry, and basis
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%two-step procedure.}
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Doing so, we have found that the present weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
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\begin{equation}
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\e{\ex}{\eW,\text{CC-S}}(\n{}{}) = \Cx{\eW} \n{}{1/3},
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\e{\ex}{\ew{},\text{CC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
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\end{equation}
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with
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\begin{equation}
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\label{eq:Cxw}
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\frac{\Cx{\eW}}{\Cx{}} = 1 - \eW (1 - \eW)\qty[ \alpha + \beta (\eW - 1/2) + \gamma (\eW - 1/2)^2 ],
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\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (\ew{} - 1/2)^2 ],
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\end{equation}
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and
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\begin{subequations}
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\begin{align}
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\alpha & = + 0.573\,919,
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\alpha & = + 0.575\,178,
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&
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\beta & = - 0.000\,347,
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\beta & = - 0.021\,108,
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&
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\gamma & = - 0.267\,634,
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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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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 $\eW = 0$ and $\eW = 1/2$ by steps of $0.025$.
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Note that this procedure makes, by construction, the ensemble energy also linear with respect to $\ew{1}$ and $\ew{2}$.
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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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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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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 $\ew{1} = \ew{2} = 0$, $\ew{1} = 1 \land \ew{2} = 0$, and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\eW}$ reduces to $\Cx{}$ in these three 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 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.
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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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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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@ -861,7 +861,7 @@ showing that the weight correction is purely linear in eVWN5 and entirely depend
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As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
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For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
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In particular, we report the excitation energies obtained with GOK-DFT
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in the zero-weight limit (\ie, $\eW = 0$) and for equi-weights, \ie, $\ew{1} = \ew{2} = 1/3$ (or $\eW = 1/2$).
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in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$).
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These excitation energies are computed using Eq.~\eqref{eq:dEdw}.
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For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016}
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@ -886,10 +886,10 @@ which also require three separate calculations at a different set of ensemble we
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%They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
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The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
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The CC-SeVWN5 excitation energies at equi-weights (\ie, $\eW = 1/2$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
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It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\eW = 1$ (\textit{vide supra}).
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\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\eW = 1/4$ and $\eW = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
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Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\eW = 0$ and $\eW = 1/2$.
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The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
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It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
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\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
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Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\ew{} = 0$ and $\ew{} = 1/2$.
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\titou{This two-step procedure can be related to optimally-tuned range-separated hybrid functionals. T2: more to come.}
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@ -903,7 +903,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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\begin{tabular}{llccccc}
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\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
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\cline{1-2} \cline{4-5}
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\tabc{x} & \tabc{c} & Basis & $\eW = 0$ & $\eW = 1/2$ & LIM & MOM \\
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\tabc{x} & \tabc{c} & Basis & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\
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\hline
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HF & & aug-cc-pVDZ & 38.52 & 30.86 & 34.00 & 28.65 \\
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& & aug-cc-pVTZ & 38.58 & 35.82 & 35.80 & 28.65 \\
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@ -959,12 +959,13 @@ 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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\titou{For this particular geometry, the doubly-excited state becomes the
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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.}
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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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%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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The weight dependence of $\Cx{\eW}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
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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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%In other words, the \titou{curvature ``hole''} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
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@ -1001,7 +1002,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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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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\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\
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\hline
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HF & & 19.09 & 6.59 & 12.92 & 6.52 \\
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HF & VWN5 & 19.40 & 6.54 & 13.02 & 6.49 \\
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@ -1045,7 +1046,7 @@ The excitation energies associated with this double excitation computed with var
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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 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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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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As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
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This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
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As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
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@ -1060,7 +1061,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
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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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\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\
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\hline
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HF & & 1.874 & 2.212 & 2.080 & 2.142 \\
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HF & VWN5 & 1.988 & 2.260 & 2.153 & 2.193 \\
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