diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index 1ffce86..bef8870 100644 --- a/Manuscript/FarDFT.bib +++ b/Manuscript/FarDFT.bib @@ -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}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 345cb2c..189c566 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -120,7 +120,7 @@ \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''.}