From 500d21cceb0f9bebc23ba4cc3483b0996c777ebb Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Fri, 10 Apr 2020 17:27:26 +0200
Subject: [PATCH] 1st draft for He and H2st

---
 FarDFT.nb             | 7035 +++++++++++++++++++++++++++++++++++------
 Manuscript/FarDFT.tex |   21 +-
 2 files changed, 6147 insertions(+), 909 deletions(-)

diff --git a/FarDFT.nb b/FarDFT.nb
index a6903ce..b4d82a2 100644
--- a/FarDFT.nb
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diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index 189c566..6e7f09b 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -335,7 +335,7 @@ Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a g
 \subsubsection{Weight-independent exchange functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-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 
+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 
 \begin{align}
 	\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
 	&
@@ -390,6 +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}).
+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 $0.025$. 
 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.
@@ -692,13 +693,14 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 
 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 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}.
+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.
+It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see 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.
+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.
+In other words, the ghost-interaction ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
 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. 
+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. 
 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.
@@ -743,13 +745,14 @@ 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 an auto-ionising resonance state, extremely high in energy and lying in the continuum. \cite{Madden_1963}
+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}
 In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate 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 functionals 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 \ce{H2} (see Fig.~\ref{fig:Cxw}).
-The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or Slater-Dirac exchange.
+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}).
+In other words, the ghost-interaction hole is deeper.
+The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
 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.
 As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight.
 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.
@@ -793,7 +796,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
 \section{Conclusion}
 \label{sec:ccl}
 
-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.
+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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%% ACKNOWLEDGEMENTS %%%