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Manuscript/FarDFT.bib
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@ -1,13 +1,34 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-04-09 22:26:01 +0200
%% Created for Pierre-Francois Loos at 2020-04-10 15:29:54 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Hait_2020,
Author = {D. Hait and M. Head-Gordon},
Date-Added = {2020-04-10 15:18:47 +0200},
Date-Modified = {2020-04-10 15:20:34 +0200},
Doi = {10.1021/acs.jctc.9b01127},
Journal = {J. Chem. Theory Comput.},
Pages = {1699--1710},
Title = {Excited state orbital optimization via minimizing the square of the gradient: General approach and application to singly and doubly excited states via density functional theory},
Volume = {16},
Year = {2020}}
@article{Madden_1963,
Author = {R. P. Madden and K. Codling},
Date-Added = {2020-04-10 15:13:19 +0200},
Date-Modified = {2020-04-10 15:14:37 +0200},
Doi = {10.1103/PhysRevLett.10.516},
Journal = {Phys. Rev. Lett.},
Title = {New Autoionizing Atomic Energy Levels in He, Ne, and Ar},
Volume = {10},
Year = {1963}}
@article{Becke_1988a,
Author = {A. D. Becke},
Date-Added = {2020-04-09 22:22:05 +0200},
@ -17,7 +38,8 @@
Pages = {3098},
Title = {Density-functional exchange-energy approximation with correct asymptotic behavior},
Volume = {38},
Year = {1988}}
Year = {1988},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.38.3098}}
@article{Lee_1988,
Author = {C. Lee and W. Yang and R. G. Parr},
@ -28,7 +50,8 @@
Pages = {785},
Title = {Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density},
Volume = {37},
Year = {1988}}
Year = {1988},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.37.785}}
@article{Burges_1995,
Author = {A. Burgers and D. Wintgen and J.-M. Rost},

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Manuscript/FarDFT.tex
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\newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}}
\begin{document}
\title{Weight dependence of local exchange-correlation functionals in two-electron systems}
\title{Weight dependence of local exchange-correlation functionals: double excitations in two-electron systems}
\author{Clotilde \surname{Marut}}
\affiliation{\LCPQ}
@ -137,7 +137,7 @@
Gross-Oliveira-Kohn (GOK) ensemble density-functional theory (GOK-DFT) is a time-independent formalism which allows to compute excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within GOK-DFT.
However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous derivative discontinuity contribution to the excitation energies.
In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) \bruno{but it can be applied to larger systems as well right ? thanks to your shift} specifically designed for the computation of double excitations within GOK-DFT.
In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) specifically designed for the computation of double excitations within GOK-DFT.
\end{abstract}
\maketitle
@ -743,12 +743,17 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As a final example, we consider the \ce{He} atom which can be seen as the limiting form of the \ce{H2} molecule for very short bond lengths.
In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state which is extremely high in energy known to lie in the continuum. \cite{Burges_1995}
In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimates an excitation energy of $2.126$ hartree.
In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lying in the continuum. \cite{Madden_1963}
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.
This is why we have 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 functions 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 {H2} (see Fig.~\ref{fig:Cxw}).
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 Fig.~\ref{fig:Cxw}).
The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or Slater-Dirac 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.
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.
%%% TABLE I %%%
\begin{table}
@ -779,7 +784,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\mc{5}{l}{Accurate\fnm[1]} & 2.126 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Explicitly-correlated calculation from Ref.~\onlinecite{Burges_1995}.}
\fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
\end{table}
%%%%%%%%%%%%%%%%%%
@ -787,12 +792,14 @@ Excitation energies (in hartree) associated with the lowest double excitation of
%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
\titou{We have studied the weight dependence of the ensemble energy in the framework of GOK-DFT.}
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.
%%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
PFL thanks Radovan Bast and Anthony Scemama for technical assistance, as well as Julien Toulouse for stimulating discussions on double excitations.
CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship.
%PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}