starting writing He and H2st

Manuscript/FarDFT.bib

@@ -1,13 +1,35 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-04-09 21:27:50 +0200 
+%% Created for Pierre-Francois Loos at 2020-04-09 22:26:01 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Becke_1988a,
+	Author = {A. D. Becke},
+	Date-Added = {2020-04-09 22:22:05 +0200},
+	Date-Modified = {2020-04-09 22:24:01 +0200},
+	Doi = {10.1103/PhysRevA.38.3098},
+	Journal = {Phys. Rev. A},
+	Pages = {3098},
+	Title = {Density-functional exchange-energy approximation with correct asymptotic behavior},
+	Volume = {38},
+	Year = {1988}}
+
+@article{Lee_1988,
+	Author = {C. Lee and W. Yang and R. G. Parr},
+	Date-Added = {2020-04-09 22:20:57 +0200},
+	Date-Modified = {2020-04-09 22:21:44 +0200},
+	Doi = {10.1103/PhysRevB.37.785},
+	Journal = {Phys. Rev. B},
+	Pages = {785},
+	Title = {Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density},
+	Volume = {37},
+	Year = {1988}}
+
 @article{Burges_1995,
 	Author = {A. Burgers and D. Wintgen and J.-M. Rost},
 	Date-Added = {2020-04-09 14:56:36 +0200},
@@ -177,10 +199,10 @@
 	Year = {2001},
 	Bdsk-Url-1 = {https://doi.org/10.1007/s002140100263}}
 
-@article{Becke_1988,
+@article{Becke_1988b,
 	Author = {A. D. Becke},
 	Date-Added = {2020-03-30 09:58:02 +0200},
-	Date-Modified = {2020-03-30 09:59:13 +0200},
+	Date-Modified = {2020-04-09 22:24:05 +0200},
 	Doi = {10.1063/1.454033},
 	Journal = {J. Chem. Phys.},
 	Pages = {2547},

Manuscript/FarDFT.tex

@@ -311,15 +311,15 @@ where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-depen
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
 \label{sec:compdet}
+
 The self-consistent GOK-DFT calculations 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. 
 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.
 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}
-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_1988,Lindh_2001}
+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}
 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).
 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. 
 Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
 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 (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
-\titou{Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}
@@ -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:Cx_H2}, 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}, 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.
@@ -400,7 +400,7 @@ We shall come back to this point later on.
 	\includegraphics[width=\linewidth]{Cxw}
 	\caption{
 	$\Cx{\ew{}}/\Cx{\ew{}=0}$ 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).
-	\label{fig:Cx_H2}
+	\label{fig:Cxw}
 	}
 \end{figure}
 
@@ -689,6 +689,18 @@ 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).
+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.
+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. 
+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.
+Nonetheless, the excitation energy is still off by 3 eV.
+The fundamental theoretical reason of such a poor agreement is not clear.
+The fact that HF exchange yields better excitation energy hints at the effect of self-interaction error.
+
 %%% TABLE I %%%
 \begin{table}
 \caption{
@@ -702,7 +714,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 	\tabc{x}	&	\tabc{c}		&	 $\ew{} = 0$	&	$\ew{} = 1/2$	&	LIM		&	MOM		\\
 	\hline
 	HF			&					&	19.09			&	6.59			&	12.92	&	6.52	\\
-	HF			&	VWN5			&	19.40			&	6.54			&	13.02	&	6.49\\
+	HF			&	VWN5			&	19.40			&	6.54			&	13.02	&	6.49	\\
 	HF			&	eVWN5			&	19.59			&	6.72			&	13.11	&			\\
 	S			&					&	5.31			&	5.60			&	5.46	&	5.56	\\
 	S			&	VWN5			&	5.34			&	5.57			&	5.46	&	5.52	\\
@@ -727,6 +739,13 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 \label{sec:He}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+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 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}.
+
 %%% TABLE I %%%
 \begin{table}
 \caption{