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Manu: saving work in the introduction

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Manuscript/eDFT.tex
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@ -189,29 +189,59 @@ independently in a GOK ensemble.
The weight dependence of the xc functional plays a crucial role in the
calculation of excitation energies.
\cite{Gross_1988b,Yang_2014,Deur_2017,Deur_2019,Senjean_2018,Senjean_2020}
It actually accounts for the infamous derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
It actually accounts for the derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
%\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
Despite its formal beauty and the fact that GOK-DFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently. \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,Senjean_2018,Smith_2016}
The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature.
Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed. \cite{Yang_2014,Yang_2017}
In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open.
A first step towards this goal is presented in the present manuscript with the ambition to turn, in the forthcoming future, GOK-DFT into a practical computational method for modeling excited states in molecules and extended systems.
The present eDFA is specially designed for the computation of single and double excitations within GOK-DFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) for ensemble.
Consequently, we will refer to this eDFA as eLDA in the remaining of this paper.
Even though GOK-DFT is in principle able to
tackle near-degenerate situations and multiple-electron excitation
processes, it has not
been given much attention until quite recently. \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,Senjean_2018,Smith_2016}
One of the reason is the lack, not to say the absence, of reliable
density-functional approximations for ensembles (eDFAs) in the literature.
The most recent works on this topic are still fundamental and
exploratory, as they rely either on simple (but nontrivial) model
systems
\cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019}
or atoms. \cite{Yang_2014,Yang_2017,Gould_2019_insights}
Despite all these efforts, it is still unclear how weight dependencies
can be incorporated into eDFAs. This problem is actually central not
only in GOK-DFT but also in conventional (ground-state) DFT as the infamous derivative
discontinuity problem that ocurs when crossing an integral number of
electrons can be recast into a weight-dependent ensemble
one. \cite{Senjean_2018,Senjean_2020}
In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
In other words, the Coulomb interaction used in this work describes particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
The present work is an attempt to answer this question,
with the ambition to turn, in the forthcoming future, GOK-DFT into a
(low-cost) practical computational method for modeling excited states in molecules and extended systems.
Starting from the ubiquitous local-density approximation (LDA), we
design a weight-dependent ensemble correction based on a finite uniform
electron gas from which density-functional excitation energies can be
extracted. The present eDFA, \trashEF{is specially designed for the computation of
single and double excitations within GOK-DFT}, which can be seen as a natural
extension of LDA, will be referred to as eLDA in the remaining of this paper.
As a proof of concept, we apply this general strategy to
ensemble correlation energies only [we use the orbital-dependent exact
ensemble exchange energy for convenience] in the particular case of
\emph{strict} one-dimensional (1D) and
spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
In other words, the Coulomb interaction used in this work describes
particles\manu{Manu: the sentence sounds weird to me. An interaction does not
describe particles ...} which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
%Early models of 1D atoms using this interaction have been used to study the effects of external fields upon Rydberg atoms \cite{Burnett_1993, Mayle_2007} and the dynamics of surface-state electrons in liquid helium. \cite{Nieto_2000, Patil_2001}
This description of 1D systems also has interesting connections with the exotic chemistry of ultra-high magnetic fields (such as those in white dwarf stars), where the electronic cloud is dramatically compressed perpendicular to the magnetic field. \cite{Schmelcher_1990, Lange_2012, Schmelcher_2012}
In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
The paper is organized as follows.
Section \ref{sec:eDFT} introduces the equations behind GOK-DFT.
In Sec.~\ref{sec:eDFA}, we detail the construction of the weight-dependent local correlation functional specially designed for the computation of single and double excitations within eDFT.
Exact and approximate formulations of GOK-DFT are discussed in Section
\ref{sec:eDFT}, with a particular emphasis on the extraction of
individual energy levels and the calculation of individual exact exchange
energies.
In Sec.~\ref{sec:eDFA}, we detail the construction of the
weight-dependent local correlation functional specially designed for the
computation of single and double excitations within GOK-DFT.
Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eDFA by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eLDA by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
Atomic units are used throughout.