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Manuscript/FarDFT.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-13 20:40:48 +0100
%% Created for Pierre-Francois Loos at 2019-11-14 21:16:19 +0100
%% Saved with string encoding Unicode (UTF-8)
@book{Ulrich_2012,
Address = {New York},
Author = {Ullrich, C.},
Date-Added = {2019-11-14 21:05:44 +0100},
Date-Modified = {2019-11-14 21:05:54 +0100},
Publisher = {Oxford University Press},
Series = {Oxford Graduate Texts},
Title = {Time-Dependent Density-Functional Theory: Concepts and Applications},
Year = {2012}}
@article{Maitra_2017,
Author = {N. T. Maitra},
Date-Added = {2019-11-14 21:04:04 +0100},
Date-Modified = {2019-11-14 21:04:04 +0100},
Journal = {J. Phys. Cond. Matt.},
Keywords = {10.1088/1361-648X/aa836e},
Pages = {423001},
Title = {Charge Transfer In Time-Dependent Density Functional Theory},
Volume = {29},
Year = {2017}}
@article{Runge_1984,
Author = {Runge, E. and Gross, E. K. U.},
Date-Added = {2019-11-14 21:00:31 +0100},
Date-Modified = {2019-11-14 21:02:02 +0100},
Doi = {10.1103/PhysRevLett.52.997},
Journal = PRL,
Pages = {997--1000},
Title = {Density-Functional Theory for Time-Dependent Systems},
Volume = 52,
Year = 1984,
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.52.997}}
@article{Boblest_2014,
Author = {S. Boblest and C. Schimeczek and G. Wunner},
Date-Added = {2019-09-09 09:42:01 +0200},
@ -310,14 +343,17 @@
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1007/s00214-018-2352-7}}
@article{Deur_2018b,
@article{Deur_2019,
Author = {K. Deur and E. Fromager},
Date-Added = {2018-12-08 18:03:14 +0100},
Date-Modified = {2018-12-09 14:19:11 +0100},
Date-Modified = {2019-11-14 21:15:19 +0100},
Doi = {10.1063/1.5084312},
Journal = {arXiv},
Pages = {094106},
Title = {Ground and excited energy levels can be extracted exactly from a single ensemble density-functional theory calculation},
Volume = {1812.02461},
Year = {2018}}
Volume = {150},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5084312}}
@book{NISTbook,
Address = {New York},
@ -2727,12 +2763,12 @@
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.95.035120}}
@article{Deur_2018a,
@article{Deur_2018,
Abstract = {Gross\textendash{}Oliveira\textendash{}Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excitedstate densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [Phys. Rev. B 95, 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre\textendash{}Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.},
Archiveprefix = {arXiv},
Author = {Deur, Killian and Mazouin, Laurent and Senjean, Bruno and Fromager, Emmanuel},
Date-Added = {2018-10-24 22:38:52 +0200},
Date-Modified = {2018-12-11 14:07:04 +0100},
Date-Modified = {2019-11-14 21:14:35 +0100},
Doi = {10.1140/epjb/e2018-90124-7},
File = {/Users/loos/Zotero/storage/2398CIXN/Deur et al. - 2018 - Exploring weight-dependent density-functional appr.pdf},
Journal = {Eur. Phys. J. B},

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Manuscript/FarDFT.tex
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\author{Clotilde \surname{Marut}}
\affiliation{\LCPQ}
\author{Emmanuel Fromager}
\email{fromagere@unistra.fr}
\affiliation{\LCQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract}
@ -126,16 +125,36 @@ Their accuracy is illustrated by computing on the prototypical H$_2$ molecule.
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.
At a relatively low computational cost (at least compared to the other excited-state methods), TD-DFT can provide accurate transition energies for low-lying excited states in organic molecules.
Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida,Ulrich_2012}
At a relatively low computational cost (at least compared to the other excited-state methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
Importantly, setting up a TD-DFT calculation for a given system is an almost pain-free process from the user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.
Indeed, TD-DFT is a in-principle exact theory which recast the many-body problem by transferring its complexity to the xc functional.
However, TD-DFT is far from being perfect, and, in practice, approximations must be made for the xc functional.
One of its issues actually originates directly from the choice of the xc functional, and more specifically, the possible substantial variations in the quality of the excitation energy for two different choices of xc functionals.
TD-DFT is an in-principle exact theory which formal foundation relies on the Runge-Gross theorem. \cite{Runge_1984}
The Kohn-Sham (KS) formalism of TD-DFT transfers the complexity of the many-body problem to the xc functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analogs with the same one-electron density.
However, TD-DFT is far from being perfect as, in practice, approximations must be made for the xc functional.
One of its issues actually originates directly from the choice of the xc functional, and more specifically, the possible substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Moreover, because it was so popular, it has been studied in excruciated details, and researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
Practically, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation is that double excitations are completely absent from the TD-DFT spectra.
Moreover, TD-DFT has problems with charge-transfer and Rydberg excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the xc functional.
For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the xc functional.
The development of range-separated hybrids \cite{Tawada_2004,Yanai_2004} provides an effective solution to this problem.
From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Elliott_2011}.
One possible solution to access double excitation within TD-DFT is provided by spin-flip TD-DFT which describes double excitations as single excitations from the lowest triplet state. \cite{Huix-Rotllant_2010,Krylov_2001,Shao_2003,Wang_2004,Wang_2006,Minezawa_2009}
However, spin contamination might be an issue. \cite{Huix-Rotllant_2010}
In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has been proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} (see also Refs.~\onlinecite{Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012}).
In this approach the exchange-correlation kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.
Maybe surprisingly, a possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
Density-functional theory for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988,Gross_1988a,Oliveira_1988} is a viable alternative currently under active development which follow such a strategy. \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018,Gould_2019,Sagredo_2018,Ayers_2018,Deur_2018,Deur_2019,Kraisler_2013,Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Smith_2016,Senjean_2018}
In the assumption of monotonically decreasing weights, eDFT has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, \cite{Gross_1988a} and excitation energies can be quite easily extracted from the total ensemble energy.
Although the formal foundation of eDFT has been set three decades ago, \cite{Gross_1988,Gross_1988a,Oliveira_1988} the practical developments of eDFT have been rather slow.
We believe that it is due to the lack of accurate approximations for eDFT.
In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation has never been developed for atoms and molecules in the context of eDFT.
The present contribution is a first step towards this goal.
The paper is organised as follows.
In Sec.~\ref{sec:theo}, ...