H2 eq done

Manuscript/FarDFT.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-04-09 10:05:15 +0200 
+%% Created for Pierre-Francois Loos at 2020-04-09 12:04:38 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -54,9 +54,7 @@
 	Publisher = {Zenodo},
 	Title = {numgrid: numerical integration grid for molecules},
 	Url = {https://github.com/dftlibs/numgrid},
-	Year = {2019},
+	Year = {2019}}
-	Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package},
-	Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}}
 
 @misc{QuAcK,
 	Author = {P. F. Loos},
@@ -439,7 +437,8 @@
 	Pages = {1884},
 	Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
 	Volume = {51},
-	Year = {1983}}
+	Year = {1983},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.51.1884}}
 
 @article{Schulz_1993,
 	Author = {H. J. Schulz},

Manuscript/FarDFT.tex

@@ -69,6 +69,7 @@
 \newcommand{\LDA}{\text{LDA}}
 \newcommand{\SD}{\text{S}}
 \newcommand{\VWN}{\text{VWN5}}
+\newcommand{\eVWN}{\text{eVWN5}}
 \newcommand{\SVWN}{\text{SVWN5}}
 \newcommand{\LIM}{\text{LIM}}
 \newcommand{\MOM}{\text{MOM}}
@@ -373,7 +374,7 @@ Doing so, we have found that the present weight-dependent exchange functional (d
 with 
 \begin{equation}
 \label{eq:Cxw}
-	\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]
+	\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ],
 \end{equation}
 and 
 \begin{subequations}
@@ -385,11 +386,12 @@ and
 	\gamma & = - 0.367\,189,
 \end{align}
 \end{subequations}
-makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
+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.
-Note 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 strictly forbidden by the GOK variational principle. \cite{Gross_1988a} 
+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 $\ew{} = 1$, which corresponds to a genuine saddle point of the KS equations, as mentioned above.
+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.
+We shall come back to this point later on.
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{Cxw}
@@ -404,9 +406,9 @@ Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq
 \subsection{Weight-independent correlation functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
+Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980}
 For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
-The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is now slightly concave) and improved excitation energies, especially at small weights, where the SVWN5 excitation energy is almost spot on. 
+The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is slightly concave) and improved excitation energies, especially at small weights, where the SVWN5 excitation energy is almost spot on. 
 
 %%%%%%%%%%%%%%%%%%
 %%% FUNCTIONAL %%%
@@ -573,33 +575,27 @@ The weight-dependence of the correlation functional is then carried exclusively
 Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent VWN5 LDA reference
 \begin{equation}
 \label{eq:becw}
-	\be{\co}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
+	\e{\co}{\ew{},\eVWN}(\n{}{}) =  (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
 \end{equation}
-via the following shift:
+via the following global, state-independent shift:
 \begin{equation}
 	\be{\co}{(I)}(\n{}{}) = \e{\co}{(I)}(\n{}{}) + \e{\co}{\VWN}(\n{}{}) - \e{\co}{(0)}(\n{}{}).
 \end{equation}
-
+In the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
-Equation \eqref{eq:becw} can be recast
+Also, Eq.~\eqref{eq:becw} can be recast
 \begin{equation}
 \label{eq:eLDA}
-\begin{split}
+	\e{\co}{\ew{},\eVWN}(\n{}{}) =  \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
-	\be{\co}{\ew{}}(\n{}{}) 
-	& =  \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
-	\\
-	& =  \e{\co}{\VWN}(\n{}{}) + \ew{} \pdv{\e{\co}{\ew{}}(\n{}{})}{\ew{}},
-\end{split}
 \end{equation}
 which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
-In particular, $\be{\co}{(0)}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
+In particular, $\e{\co}{\ew{}=0,\eVWN}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
-Consequently, in the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
+We note also that, by construction, we have
-Also, we note that, by construction,
 \begin{equation}
 \label{eq:dexcdw}
-	\pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}}
+	\pdv{\e{\co}{\ew{},\eVWN}(\n{}{})}{\ew{}}
-	= \be{\co}{(1)}(n) - \be{\co}{(0)}(n),
+	= \e{\co}{(1)}(n) - \e{\co}{(0)}(n),
 \end{equation}
-which shows that the weight correction is purely linear in eVWN5.
+showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
 
 As shown in Fig.~\ref{fig:Ew_H2}, the SGIC-eVWN5 is slightly less concave than its SGIC-VWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
 
@@ -617,23 +613,22 @@ As shown in Fig.~\ref{fig:Ew_H2}, the SGIC-eVWN5 is slightly less concave than i
 
 For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
 In particular, we report the excitation energies obtained with GOK-DFT in the zero-weight limit (\ie, $\ew{} = 0$) and for the equi-ensemble (\ie, $\ew{} = 1/2$).
-For comparison purposes, we also report the linear interpolation method (LIM), \cite{Senjean_2015,Senjean_2016} which is
+For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} 
 a pragmatic way of getting weight-independent
 excitation energies defined as
 \begin{equation}
-	\Ex{\LIM}{(1)} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
+	\Ex{\LIM}{} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
 \end{equation}
 as well as the MOM excitation energies. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
 We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
-MOM excitation energies can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
+They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
 \begin{equation}
-	\Ex{\MOM}{(1)} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
+	\Ex{\MOM}{} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
 \end{equation}
 
 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}).
-The GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$.
 
 %%% TABLE I %%%
 \begin{table*}
@@ -736,14 +731,12 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 \end{table*}
 %%% %%% %%% %%%
 
-
-
 %%%%%%%%%%%%%%%%%%
 %%% CONCLUSION %%%
 %%%%%%%%%%%%%%%%%%
 \section{Conclusion}
 \label{sec:ccl}
-We have studied the weight dependence of the ensemble energy in the framework of GOK-DFT.
+\titou{We have studied the weight dependence of the ensemble energy in the framework of GOK-DFT.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%% ACKNOWLEDGEMENTS %%%