From c555a1cf828b4bc31d4a2a01dbef94f6345e7c15 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 14 Apr 2020 21:41:12 +0200 Subject: [PATCH] ccl and bruno --- Manuscript/FarDFT.tex | 31 +++++++++++++++---------------- 1 file changed, 15 insertions(+), 16 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 56ff262..c876f14 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -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. %%%%%%%%%%%%%%%%%%%%%%%%