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Pierre-Francois Loos 2020-04-10 21:55:16 +02:00
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FarDFT.nb
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Manuscript/FarDFT.tex
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@ -120,7 +120,7 @@
\newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}} \newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}}
\begin{document} \begin{document}
\title{Weight dependence of local exchange-correlation functionals: double excitations in two-electron systems} \title{Weight Dependence of Local Exchange-Correlation Functionals: Double Excitations in Two-Electron Systems}
\author{Clotilde \surname{Marut}} \author{Clotilde \surname{Marut}}
\affiliation{\LCPQ} \affiliation{\LCPQ}
@ -695,11 +695,11 @@ To investigate the weight dependence of the xc functional in the strong correlat
For this particular geometry, the doubly-excited state becomes the lowest excited state. 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 considering again the aug-cc-pVTZ basis set, we design a GIC-S functional for this system at $\RHH = 3.7$ bohr. 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}]. 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). 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 LDA exchange 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. 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}. Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), 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} As a reference value, we 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 reached 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. 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. Nonetheless, the excitation energy is still off by 3 eV.
@ -709,7 +709,7 @@ The fact that HF exchange yields better excitation energy hints at the effect of
%%% TABLE I %%% %%% TABLE I %%%
\begin{table} \begin{table}
\caption{ \caption{
Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 3.7$ bohr obtained with the aug-cc-pVTZ basis set for various methods and combinations of xc functionals. Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} at $\RHH = 3.7$ bohr obtained with the aug-cc-pVTZ basis set for various methods and combinations of xc functionals.
\label{tab:BigTab_H2st} \label{tab:BigTab_H2st}
} }
\begin{ruledtabular} \begin{ruledtabular}
@ -739,6 +739,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Helium atom} \subsection{Helium atom}
\label{sec:He} \label{sec:He}
@ -748,14 +749,16 @@ As a final example, we consider the \ce{He} atom which can be seen as the limiti
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 \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. 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. 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. Consequently, we 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 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 the blue curve in Fig.~\ref{fig:Cxw}).
The parameters of the GIC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) 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} (blue curve in Fig.~\ref{fig:Cxw}).
In other words, the ghost-interaction hole is deeper. 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 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.
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 (or $0.22$ eV) 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 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. As a final comment, let us stress that the present protocol does not rely on high-level calculations as the sole requirement for constructing the GIC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
%%% TABLE I %%% %%% TABLE I %%%
@ -790,13 +793,47 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.} \fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
\end{table} \end{table}
%%% TABLE I %%%
%\begin{table}
%\caption{
%Excitation energies (in eV) associated with the lowest double excitation of \ce{HNO} obtained with the aug-cc-pVDZ basis set for various methods and combinations of xc functionals.
%\label{tab:BigTab_H2st}
%}
%\begin{ruledtabular}
%\begin{tabular}{llcccc}
% \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
% \cline{1-2} \cline{3-4}
% \tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
% \hline
% HF & & & & & \\
% HF & VWN5 & & & & \\
% S & & 1.72 & 4.00 & 2.86 & 3.99 \\
% S & VWN5 & & & & \\
% GIC-S & & 3.99 & 3.99 & 3.99 & 3.99 \\
% GIC-S & VWN5 & 4.05 & 4.03 & 4.04 & 4.03 \\
% \hline
% S & PW92 & & & & 4.00\fnm[1] \\
% PBE & PBE & & & & 4.13\fnm[1] \\
% SCAN & SCAN & & & & 4.24\fnm[1] \\
% B97M-V & B97M-V & & & & 4.33\fnm[1] \\
% PBE0 & PBE0 & & & & 4.24\fnm[1] \\
% \hline
% \mc{5}{l}{Theoretical best estimate\fnm[2]} & 4.32 \\
%\end{tabular}
%\end{ruledtabular}
%\fnt[1]{Square gradient minimization (SGM) approach from Ref.~\onlinecite{Hait_2020} obtained with the aug-cc-pVTZ basis set. SGM is theoretically equivalent to MOM.}
%\fnt[2]{Theoretical best estimate from Ref.~\onlinecite{Loos_2019} obtained at the (extrapolated) FCI/aug-cc-pVQZ level.}
%\end{table}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%% %%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:ccl} \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, and we hope to be able to report on this in the near 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 further on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%% %%% ACKNOWLEDGEMENTS %%%