working on discussion for He and stretched H2

Manuscript/FarDFT.tex

@@ -390,7 +390,7 @@ and
 \end{align}
 \end{subequations}
 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}).
-As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw}, 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.
+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.
 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} 
 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.
 Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
@@ -622,7 +622,8 @@ They can then be obtained via GOK-DFT ensemble calculations by performing a line
 
 The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
 The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.
-Finally, note that the GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
+It is also important to mention that the GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
+Finally, note that, 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 the Slater exchange functional) between $\ew{} = 0$ and $1$.
 
 %%% TABLE I %%%
 \begin{table}
@@ -689,10 +690,12 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 \label{sec:H2st}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-To investigate the weight dependence of the xc functional in the strongly correlated regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
+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).
 For this particular geometry, the doubly-excited state becomes the lowest excited state.
-We then follow the same protocol as in Sec.~\ref{sec:H2}, and designed a GIC-S functional for this system and the aug-cc-pVTZ basis set.
-The weight-dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw}. One clearly sees that the correction brought by GIC-S is much more subtile 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 and the curvature more gentle.
+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.
+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}.
+The weight-dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve). 
+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.
 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}.
 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}
 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. 
@@ -740,11 +743,12 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 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.
-In \ce{He}, the lowest doubly-excited state is extremely high in energy and lies in the continuum. \cite{Burges_1995}
+In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state which is extremely high in energy known to lie in the continuum. \cite{Burges_1995}
 In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimates an excitation energy of $2.126$ hartree.
 Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
 This is why we have considered for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
 The excitation energies associated with this double excitation computed with various methods and combinations of xc functions are gathered in Table \ref{tab:BigTab_He}.
+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 {H2} (see Fig.~\ref{fig:Cxw}).
 
 %%% TABLE I %%%
 \begin{table}