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Manu: starting polishing. Done with the introduction

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Emmanuel Fromager 2020-05-09 09:50:00 +02:00
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22
Manuscript/FarDFT.bib
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@ -9251,3 +9251,25 @@
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5043411}}
@article{Seunghoon_2018,
author = {Lee,Seunghoon and Filatov,Michael and Lee,Sangyoub and Choi,Cheol Ho },
title = {Eliminating spin-contamination of spin-flip time dependent density functional theory within linear response formalism by the use of zeroth-order mixed-reference (MR) reduced density matrix},
journal = {J. Chem. Phys.},
volume = {149},
number = {10},
pages = {104101},
year = {2018},
doi = {10.1063/1.5044202},
URL = {
https://doi.org/10.1063/1.5044202
},
eprint = {
https://doi.org/10.1063/1.5044202
}
}

22
Manuscript/FarDFT.tex
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@ -18,8 +18,9 @@
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\cloclo}[1]{\textcolor{purple}{#1}}
\newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}}
\newcommand{\manu}[1]{\textcolor{magenta}{Manu: #1}}
\newcommand{\manuf}[1]{\textcolor{magenta}{#1}}
\newcommand{\manuf}[1]{\textcolor{magenta}{Manu: #1}}
\newcommand{\trashEF}[1]{\textcolor{magenta}{\sout{#1}}}
\newcommand{\manu}[1]{\textcolor{magenta}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashCM}[1]{\textcolor{purple}{\sout{#1}}}
@ -179,11 +180,13 @@ system and its interacting analog which have both
exactly the same one-electron density.
However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made.
First, within the linear-response approximation, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, \cite{Runge_1984, Casida_1995, Casida_2012} which may not be adequate in certain situations (such as strong correlation).
First, within the \manu{commonly used} linear-response \trashEF{approximation} \manu{regime}, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, \cite{Runge_1984, Casida_1995, Casida_2012} which may not be adequate in certain situations (such as strong correlation).
Second, the time dependence of the functional is usually treated at the local approximation level within the standard adiabatic approximation.
In other words, memory effects are absent from the xc functional which is assumed to be local in time
(the xc energy is in fact an xc action, not an energy functional). \cite{Vignale_2008}
Third and more importantly in the present context, a major issue of TD-DFT actually originates directly from the choice of the xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Third and more importantly in the present context, a major issue of
TD-DFT actually originates directly from the choice of the
\manu{(ground-state)} xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Because of its popularity, approximate TD-DFT has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies.
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 semi-local xc functional.
@ -194,11 +197,16 @@ Although these double excitations are usually experimentally dark (which means t
One possible solution to access double excitations 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}
\manu{Note that a simple remedy based on a mixed reference reduced density
matrix has been recently introduced by Lee {\it et al.} \cite{Seunghoon_2018}}
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 xc kernel is made frequency dependent, which allows to treat doubly-excited states. \cite{Romaniello_2009a,Sangalli_2011,Loos_2019}
Maybe surprisingly, another possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
With a computational cost similar to traditional KS-DFT, DFT for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988a,Gross_1988b,Oliveira_1988} is a viable alternative following such a strategy currently under active development.\cite{Gidopoulos_2002,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}
With a computational cost similar to traditional KS-DFT, DFT for
ensembles (eDFT)
\cite{Theophilou_1979,Gross_1988a,Gross_1988b,Oliveira_1988} is a viable
alternative \trashEF{following} \manu{that follows} such a strategy \manu{and is} currently under active development.\cite{Gidopoulos_2002,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 for excited states has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, the so-called Gross--Oliveria--Kohn (GOK) variational principle. \cite{Gross_1988a}
In short, GOK-DFT (\ie, eDFT for neutral excitations) is the density-based analog of state-averaged wave function methods, and excitation energies can then be easily extracted from the total ensemble energy. \cite{Deur_2019}
Although the formal foundations of GOK-DFT have been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
@ -215,10 +223,10 @@ However, Loos and Gill have recently shown that there exists other UEGs which co
Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
Very recently, \cite{Loos_2020} two of the present authors have taken advantages of these FUEGs to construct a local, weight-dependent correlation functional specifically designed for one-dimensional many-electron systems.
Unlike any standard functional, this first-rung functional automatically incorporates ensemble derivative contributions thanks to its natural weight dependence, \cite{Levy_1995, Perdew_1983} and has shown to deliver accurate excitation energies for both single and double excitations.
Extending this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT.
\manu{In order to} extend this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT.
The paper is organised as follows.
In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is briefly presented.
Section \ref{sec:compdet} provides the computational details.
The results of our calculations for two-electron systems are reported and discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.