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Gross-Oliveira-Kohn (GOK) ensemble density-functional theory (GOK-DFT) is a time-independent formalism which allows to compute excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
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Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within GOK-DFT.
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However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous derivative discontinuity contribution to the excitation energies.
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In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) specifically designed for the computation of double excitations within GOK-DFT.
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In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) \bruno{but it can be applied to larger systems as well right ? thanks to your shift} specifically designed for the computation of double excitations within GOK-DFT.
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\end{abstract}
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\maketitle
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@ -208,8 +208,8 @@ where $\hH = \hT + \hWee + \hVne$ contains the kinetic, electron-electron and nu
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\begin{eqnarray}
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\hGam{\bw} = \sum_{I=0}^{\nEns - 1} \ew{I} \dyad*{\overline{\Psi}^{(I)}},
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\end{eqnarray}
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where $\lbrace \overline{\Psi}^{(I)} \rbrace_{1 \le I \le \nEns}$ is a set of $\nEns$ orthonormal trial wave functions.
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The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{1 \le I \le \nEns} = \lbrace \Psi^{(I)} \rbrace_{1 \le I \le \nEns}$.
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where $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1}$ is a set of $\nEns$ orthonormal trial wave functions.
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The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1} = \lbrace \Psi^{(I)} \rbrace_{0 \le I \le \nEns-1}$.
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Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states.
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One of the key feature of the GOK ensemble is that individual excitation energies are extracted from the ensemble energy via differentiation with respect to individual weights:
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\begin{equation}
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@ -234,7 +234,7 @@ $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional,
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\begin{equation}
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\hgam{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}}
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\end{equation}
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is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{1 \le I \le \nEns}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le \nOrb}$, and
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is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{0 \le I \le \nEns-1}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le \nOrb}$, and
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\begin{equation}
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\label{eq:exc_def}
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\begin{split}
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@ -258,10 +258,10 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
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where
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\begin{align}
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\label{eq:nw}
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\n{}{\bw}(\br{}) & = \sum_{I=0}^{\nEns-1} \ew{I} \n{}{(I)}(\br{}),
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\n{}{\bw}(\br{}) & = \sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}}{}(\br{}),
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\\
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\label{eq:nI}
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\n{}{(I)}(\br{}) & = \sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
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\n{\Det{I}{\bw}}{}(\br{}) & = \sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
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\end{align}
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are the ensemble and individual one-electron densities, respectively,
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\begin{equation}
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@ -272,7 +272,7 @@ is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orb
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The latters are determined by solving the ensemble KS equation
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\begin{equation}
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\label{eq:eKS}
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\qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
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\qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
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\end{equation}
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where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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\begin{equation}
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@ -286,7 +286,10 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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\end{equation}
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is the Hxc potential.
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Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
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Note that, although we have dropped the weight-dependency in the individual densities $\n{}{(I)}(\br{})$ defined in Eq.~\eqref{eq:nI}, these do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density.
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Note that the individual densities $\n{\Det{I}{\bw}}{}(\br{})$ defined in Eq.~\eqref{eq:nI} do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density.
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Nevertheless,
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these densities can still be extracted in principle exactly
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from the KS ensemble as shown by Fromager~\cite{Fromager_2020}.
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In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as
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\begin{equation}
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@ -305,9 +308,9 @@ For more details about the self-consistent implementation of GOK-DFT, we refer t
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For all calculations, we use the aug-cc-pVXZ (X = D, T, and Q) Dunning's family of atomic basis sets.
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Numerical quadratures are performed with the \texttt{numgrid} library using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,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 two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered.
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Although we will sometimes ``violate'' this variational constraint, we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle.
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However, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained wit the maximum overlap method developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered.
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Although we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
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Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -315,7 +318,7 @@ Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for wh
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-pVTZ basis set and the weight-independent Slater local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by
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First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-pVTZ basis set and the weight-independent Slater local exchange functional, \cite{Dirac_1930, Slater_1951} \bruno{notée S51 sur les figures?} which is explicitly given by
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\begin{align}
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\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
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&
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@ -323,7 +326,7 @@ First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-p
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\end{align}
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The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$.
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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, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies greatly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
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As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
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Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and the same value of the excitation energy independently of $\ew{}$.
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\begin{figure}
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@ -360,7 +363,7 @@ and
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&
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\gamma & = - 0.367\,189,
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\end{align}
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makes the ensemble much more linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2})
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makes the ensemble much more linear (see Fig.~\ref{fig:Ew_H2})\bruno{C'est celle notée ``GIC'' sur la figure ? Pourquoi pas MSFL ? A clarifier pour le lecteur}, and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2})
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As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
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Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
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We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limit at $\ew{} = 0$ and $1$.
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@ -540,6 +543,7 @@ Hence, we employ a simple embedding scheme where the two-electron FUEG (the impu
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The weight-dependence of the xc functional is then carried exclusively by the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
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Consistently with such a strategy, Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} are ``centred'' on their corresponding jellium reference
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\bruno{you commented the exchange part, why ?}
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\begin{equation}
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\label{eq:becw}
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\be{\xc}{\ew{}}(\n{}{}) = (1-\ew{}) \be{\xc}{(0)}(\n{}{}) + \ew{} \be{\xc}{(1)}(\n{}{})
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@ -596,19 +600,12 @@ This embedding procedure can be theoretically justified by the generalised adiab
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(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014}
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Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional.
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In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir
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$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ? Can it be :
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\begin{equation}
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\label{eq:GACE2}
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\E{\xc}{\bw}[\n{}{}]
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= \E{\xc}{}[\n{}{}]
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+ \sum_{I=0}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I+1},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi
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\end{equation}
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?}
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$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
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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) of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets.
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Excitation energies (in eV) of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets. \bruno{Why you don't report results from the eLDA functional which is not system-specific like MSFL ? What is MSFL for the correlation part ? Is it what you referred to eLDA in the text ?}
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\label{tab:Energies}
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}
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\begin{ruledtabular}
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