diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index ccd21bc..fde3d0d 100644 --- a/Manuscript/FarDFT.bib +++ b/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}, - Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package}, - Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}} + Year = {2019}} @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}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 8b4cfa7..4cd204f 100644 --- a/Manuscript/FarDFT.tex +++ b/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} -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. +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. +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} - -Equation \eqref{eq:becw} can be recast +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, Eq.~\eqref{eq:becw} can be recast \begin{equation} \label{eq:eLDA} -\begin{split} - \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} + \e{\co}{\ew{},\eVWN}(\n{}{}) = \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})] \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 VWN5 local correlation functional for ensembles. -Also, we note that, by construction, +In particular, $\e{\co}{\ew{}=0,\eVWN}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$. +We note also that, by construction, we have \begin{equation} \label{eq:dexcdw} - \pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}} - = \be{\co}{(1)}(n) - \be{\co}{(0)}(n), + \pdv{\e{\co}{\ew{},\eVWN}(\n{}{})}{\ew{}} + = \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. - -The GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$. +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}). %%% 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 %%%