diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 5dfcaa9..82d1c2f 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -71,7 +71,7 @@ \newcommand{\VWN}{\text{VWN5}} \newcommand{\SVWN}{\text{SVWN5}} \newcommand{\LIM}{\text{LIM}} -\newcommand{\CID}{\text{CID}} +\newcommand{\MOM}{\text{MOM}} \newcommand{\Hxc}{\text{Hxc}} \newcommand{\Ha}{\text{H}} \newcommand{\ex}{\text{x}} @@ -584,9 +584,9 @@ Equation \eqref{eq:becw} can be recast \label{eq:eLDA} \begin{split} \be{\co}{\ew{}}(\n{}{}) - & = \e{\co}{\LDA}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})] + & = \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})] \\ - & = \e{\co}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\co}{\ew{}}(\n{}{})}{\ew{}}, + & = \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. @@ -620,9 +620,14 @@ For comparison purposes, we also report the linear interpolation method (LIM) ex \begin{equation} \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 $\ew{} = 1$. +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, +\begin{equation} + \Ex{\MOM}{(1)} = \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 improvment of $0.25$ eV as compared to GIC-SVWN5 %%% TABLE I %%% \begin{table*}