1st draft for He and H2st
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@ -335,7 +335,7 @@ Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a g
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\subsubsection{Weight-independent exchange functional}
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\subsubsection{Weight-independent exchange functional}
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First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac 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 (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac (LDA) local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by
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\begin{align}
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\begin{align}
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\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
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\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
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&
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&
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@ -390,6 +390,7 @@ and
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\end{align}
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\end{align}
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\end{subequations}
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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 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 $\ew{} = 1$ by steps of $0.025$.
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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 $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
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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 $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
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Maybe surprisingly, one would have noticed that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is deterred by the GOK variational principle. \cite{Gross_1988a}
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Maybe surprisingly, one would have noticed that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is deterred by the GOK variational principle. \cite{Gross_1988a}
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However, it is important to ensure that the weight-dependent functional does not alter the $\ew{} = 1$ limit, which is a genuine saddle point of the KS equations, as mentioned above.
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However, it is important to ensure that the weight-dependent functional does not alter the $\ew{} = 1$ limit, which is a genuine saddle point of the KS equations, as mentioned above.
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@ -692,13 +693,14 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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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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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 lowest excited state.
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For this particular geometry, the doubly-excited state becomes the lowest excited state.
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We then follow the same protocol as in Sec.~\ref{sec:H2}, and design a GIC-S functional for this system and the aug-cc-pVTZ basis set.
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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 GIC-S functional for this system at $\RHH = 3.7$ bohr.
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For $\RHH = 3.7$ bohr, we have $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ in 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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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 GIC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the Slater-Dirac functional is much more linear at $\RHH = 3.7$ bohr.
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One clearly sees that the correction brought by GIC-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 ghost-interaction ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
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Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers converged results with respect to the size of the basis set), the same set of calculations as in Table \ref{tab:BigTab_H2}.
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Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers converged results with respect to the size of the basis set), the same set of calculations as in Table \ref{tab:BigTab_H2}.
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As a reference value, we have computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
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As a reference value, we have computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
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For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the best match being obtained with HF exchange.
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For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the best match being reached with HF exchange.
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The GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
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The GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
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Nonetheless, the excitation energy is still off by 3 eV.
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Nonetheless, the excitation energy is still off by 3 eV.
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The fundamental theoretical reason of such a poor agreement is not clear.
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The fundamental theoretical reason of such a poor agreement is not clear.
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@ -743,13 +745,14 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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As a final example, we consider the \ce{He} atom which can be seen as the limiting form of the \ce{H2} molecule for very short bond lengths.
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As a final example, we consider the \ce{He} atom which can be seen as the limiting form of the \ce{H2} molecule for very short bond lengths.
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In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lying in the continuum. \cite{Madden_1963}
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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.
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In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate an excitation energy of $2.126$ hartree.
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Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
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Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
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This is why we have considered for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
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This is why we have considered 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 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 GIC-S weight-dependent exchange functional 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} (see Fig.~\ref{fig:Cxw}).
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The parameters of the GIC-S weight-dependent exchange functional 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} (see the 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 Slater-Dirac exchange.
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In other words, the ghost-interaction hole is deeper.
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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 GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree off the reference value.
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The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree 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.
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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.
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As a final comment, let us stress that the present protocole does not rely on high-level calculations as the sole requirement for constructing the GIC-S functional is the linearity of the ensemble energy.
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As a final comment, let us stress that the present protocole does not rely on high-level calculations as the sole requirement for constructing the GIC-S functional is the linearity of the ensemble energy.
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@ -793,7 +796,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
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\section{Conclusion}
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\section{Conclusion}
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\label{sec:ccl}
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\label{sec:ccl}
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In the light of the results obtained in this study on double excitations computed within the GOK-DFT framework, we believe that the development of more universal weight-dependent exchange and correlation functionals has a bright future.
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In the light of the results obtained in this study on double excitations computed within the GOK-DFT framework, we believe that the development of more universal weight-dependent exchange and correlation functionals has a bright future, and we hope to be able to report on this in the near future.
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%%% ACKNOWLEDGEMENTS %%%
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%%% ACKNOWLEDGEMENTS %%%
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