clean up Bruno

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Pierre-Francois Loos 2020-04-11 22:29:09 +02:00
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FarDFT.nb

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\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\cloclo}[1]{\textcolor{purple}{#1}} \newcommand{\cloclo}[1]{\textcolor{purple}{#1}}
\newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashCM}[1]{\textcolor{purple}{\sout{#1}}} \newcommand{\trashCM}[1]{\textcolor{purple}{\sout{#1}}}
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\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France} \newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
\newcommand{\UL}{Instituut-Lorentz, Universiteit Leiden, P.O.~Box 9506, 2300 RA Leiden, The Netherlands} \newcommand{\UL}{Instituut-Lorentz, Universiteit Leiden, P.O.~Box 9506, 2300 RA Leiden, The Netherlands}
\newcommand{\VU}{Division of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands} \newcommand{\VU}{Division of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands}
\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}
@ -639,16 +640,16 @@ The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (gree
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.
Note that this linearity at $\RHH = 3.7$ bohr was Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-independent xc functionals in Ref.~\onlinecite{Senjean_2015}.
also observed using weight-independent xc-functionals in Ref.~\cite{Senjean_2015}.
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}. 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 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 \bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}. For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and eVWN5 correlation at equi-ensemble.
%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}.
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. 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. Nonetheless, the excitation energy is still off by $3$ eV.
The fundamental theoretical reason of such a poor agreement is not clear. 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. The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error.
\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 ?} %\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 %%% %%% TABLE IV %%%
\begin{table} \begin{table}
@ -700,11 +701,10 @@ The parameters of the GIC-S weight-dependent exchange functional (computed with
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 (or $0.22$ eV) off the reference value.\bruno{But also with GIC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...} 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, while the opposite conclusion were %\bruno{But also with GIC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...}
made in Ref.~\cite{Loos_2020} thus strengthening the importance of As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
developing weight-dependent functionals that gives linear ensemble This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
energies, \ie, to get rid of the weight-dependency of the excitation energy.
As a final comment, let us stress again 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. As a final comment, let us stress again 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.