H2 res

Manuscript/FarDFT.tex

@@ -70,7 +70,7 @@
 \newcommand{\SD}{\text{S}}
 \newcommand{\VWN}{\text{VWN5}}
 \newcommand{\SVWN}{\text{SVWN5}}
-\newcommand{\MSFL}{\text{MSFL}}
+\newcommand{\LIM}{\text{LIM}}
 \newcommand{\CID}{\text{CID}}
 \newcommand{\Hxc}{\text{Hxc}}
 \newcommand{\Ha}{\text{H}}
@@ -277,7 +277,7 @@ is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orb
 The latters are determined by solving the ensemble KS equation
 \begin{equation}
 \label{eq:eKS}
-	\qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
+	\qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
 \end{equation}
 where $\hHc(\br{}) = -\nabla^2/2  + \vne(\br{})$, and
 \begin{equation}
@@ -319,7 +319,7 @@ Numerical quadratures are performed with the \texttt{numgrid} library using 194
 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 we 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 developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+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}
 Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -331,7 +331,7 @@ Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for wh
 \subsection{Weight-independent exchange functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-First, we compute the ensemble energy of the \ce{H2} molecule 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 local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by 
 \begin{align}
 	\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
 	&
@@ -365,7 +365,7 @@ Note that the exact xc correlation ensemble functional would yield a perfectly l
 \subsection{Weight-dependent exchange functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Second, in order to remove this spurious curvature of the ensemble energy (which is partly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
+Second, in order to remove this spurious curvature of the ensemble energy (which is mostly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
 Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error), represented in Fig.~\ref{fig:Cx_H2},
 \begin{equation}
 	\e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
@@ -385,7 +385,7 @@ 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 full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (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 and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
 As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
 Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
 We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $1$.
@@ -405,7 +405,7 @@ It is interesting to note that, around $\ew{} = 0$, the behavior of Eq.~\eqref{e
 
 Third, we add up correlation 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.
+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. 
 
 %%%%%%%%%%%%%%%%%%
 %%% FUNCTIONAL %%%
@@ -489,7 +489,6 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states
 %	\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
 %\end{equation}
 %Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
-
 Thanks to highly-accurate calculations and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0} and \eqref{eq:eHF_1}, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \cite{Sun_2016,Loos_2020}
 \begin{equation}
 \label{eq:ec}
@@ -570,7 +569,7 @@ Because our intent is to incorporate into standard functionals (which are ``univ
 As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014} 
 The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
  
-Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent LDA reference
+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{}{})
@@ -592,14 +591,16 @@ Equation \eqref{eq:becw} can be recast
 \end{equation}
 which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
 In particular, $\be{\co}{(0)}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
-Consequently, in the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the LDA for ensembles.
+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.
 Also, we note that, by construction,
 \begin{equation}
 \label{eq:dexcdw}
 	\pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}}
 	= \be{\co}{(1)}(n) - \be{\co}{(0)}(n),
 \end{equation}
-which shows that the weight correction is purely linear.
+which shows that the weight correction is purely linear in eVWN5.
+
+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}).
 
 %This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
 %\begin{equation}
@@ -613,18 +614,21 @@ which shows that the weight correction is purely linear.
 %In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir 
 %$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
 
-%%%%%%%%%%%%%%%%%%
+For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
-%%% DISCUSSION %%%
+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) excitation energies, \cite{Senjean_2015,Senjean_2016} which are defined as
-\section{Discussion}
+\begin{equation}
-\label{sec:dis}
+	\Ex{\LIM}{(1)} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
-
+\end{equation}
+as well as the MOM excitation energies.
+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 $\ew{} = 1$.
 
 %%% TABLE I %%%
 \begin{table*}
 \caption{
-Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets.
+Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods, combinations of xc functionals, and basis sets.
-\label{tab:Energies}
+\label{tab:BigTab_H2}
 }
 \begin{ruledtabular}
 \begin{tabular}{llccccc}
@@ -660,6 +664,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 	B3			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		27.77\fnm[2]	\\
 	HF			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		29.18\fnm[2]	\\
 	HF			&					&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.65\fnm[2]	\\
+	\\
 	HF			&	FCI				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.75\fnm[2]	\\
 \end{tabular}
 \end{ruledtabular}