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ccl and bruno

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Pierre-Francois Loos 2020-04-14 21:41:12 +02:00
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Manuscript/FarDFT.tex
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@ -648,12 +648,10 @@ For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the exci
As expected from the linearity of the ensemble energy, 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.
The fundamental theoretical reason of such a poor agreement is not clear.
The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error. For additional comparison, we provide the
excitation energy
calculated by short-range multiconfigurational DFT in Ref.~\cite{Senjean_2015}, using the (weight-independent) srLDA functional~\cite{Toulouse_2004}
and setting the range-separation parameter to $\mu = 0.4$ bohr$^{-1}$.
The excitation energy improves by 1 eV compared
to the weight-independent S-VWN5 functional, thus showing that treating the long-range part of the electron-electron repulsion by wavefunction theory plays a significant role.
The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error.
For additional comparison, we provide the excitation energy calculated by short-range multiconfigurational DFT in Ref.~\onlinecite{Senjean_2015}, using the (weight-independent) srLDA functional \cite{Toulouse_2004} and setting the range-separation parameter to $\mu = 0.4$ bohr$^{-1}$.
The excitation energy improves by $1$ eV compared to the weight-independent SVWN5 functional, thus showing that treating the long-range part of the electron-electron repulsion by wave function theory plays a significant role.
%\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?}
%%% TABLE IV %%%
@ -682,13 +680,13 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
B3 & LYP & & & & 5.55 \\
HF & LYP & & & & 6.68 \\
\hline
\mc{2}{l}{srLDA($\mu = 0.4$) \fnm[2]} & 6,39 & 6,55 & 6,47 & \\
\mc{2}{l}{srLDA ($\mu = 0.4$) \fnm[2]} & 6.39 & 6.55 & 6.47 & \\
\hline
\mc{5}{l}{Accurate\fnm[3]} & 8.69 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{KS calculation does not converge.}
\fnt[2]{short-range multi-configurational DFT / aug-cc-pVQZ calculations performed in Ref.~\cite{Senjean_2015}}
\fnt[2]{Short-range multiconfigurational DFT/aug-cc-pVQZ calculations from Ref.~\onlinecite{Senjean_2015}.}
\fnt[3]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
\end{table}
%%% %%% %%% %%%
@ -787,15 +785,16 @@ Excitation energies (in hartree) associated with the lowest double excitation of
%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
In the present article, we have discussed the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron systems (\ce{He} and \ce{H2}) specifically designed for the computation of double excitations within GOK-DFT, a time-\textit{independent} formalism thanks to which one can extract excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
We have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing most of the ghost-interaction error).
Although the weight-dependent correlation functional developed in this paper (eVWN5) performs systematically better than their weight-independent counterpart (VWN5), the improvement remains rather small.
To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure,
\ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead
of the self-consistent one.
Density- and state-driven errors \cite{Gould_2019,Fromager_2020} can also be calculated to provide additional insights about the present results.
This is left for future work.
Although the weight-dependent functionals developed in this paper perform systematically
better than their
weight-independent counterparts, the improvement remains small.
To better understand the reasons of this small improvement,
it will be particularly interesting to investigate
the error due to the self-consistent procedure,
\ie, by checking the difference in the excitation energy when the {\it exact} ensemble density (built with the exact individual densities) is used instead
of the converged one. Density-driven as well as state-driven errors~\cite{Gould_2019,Fromager_2020} can also be calculated to provide more flavours about the results obtained in this paper. This is left for future work.
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.
%%%%%%%%%%%%%%%%%%%%%%%%