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Manu: repolished the introduction

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Emmanuel Fromager 2020-02-28 17:13:55 +01:00
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Manuscript/eDFT.tex
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@ -160,15 +160,16 @@ Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1
The description of strongly multiconfigurational ground states (often The description of strongly multiconfigurational ground states (often
referred to as ``strong correlation problem'') still remains a referred to as ``strong correlation problem'') still remains a
challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017} challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017}
Another issue, which is partly connected to the previous one, is the description of electronically-excited states. Another issue, which is partly connected to the previous one, is the
description of low-lying quasi-degenerate states.
The standard approach for modeling excited states in DFT is The standard approach for modeling excited states in a DFT framework is
linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012} linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012}
In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong. In this case, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, which may break down when electron correlation is strong.
Moreover, in exact TDDFT, the xc energy is in fact an xc action \cite{Vignale_2008} which is a Moreover, in exact TDDFT, the xc energy is in fact an xc {\it action} \cite{Vignale_2008} which is a
functional of the time-dependent density $n\equiv n(\br,t)$ and, as functional of the time-dependent density $n\equiv n(\br,t)$ and, as
such, has memory. Standard implementations of TDDFT rely on such, it should incorporate memory effects. Standard implementations of TDDFT rely on
the adiabatic approximation where memory effects are neglected. In other the adiabatic approximation where these effects are neglected. In other
words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012} words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019} As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
@ -180,11 +181,11 @@ and excited states altogether, \ie, with the same set of orbitals.
Interestingly, a similar approach exists in DFT. Referred to as Interestingly, a similar approach exists in DFT. Referred to as
Gross--Oliveira--Kohn (GOK) DFT\cite{Gross_1988a,Gross_1988b,Oliveira_1988}, it was proposed at the end of the 80's as a generalization Gross--Oliveira--Kohn (GOK) DFT\cite{Gross_1988a,Gross_1988b,Oliveira_1988}, it was proposed at the end of the 80's as a generalization
of Theophilou's DFT for equiensembles. \cite{Theophilou_1979} of Theophilou's DFT for equiensembles. \cite{Theophilou_1979}
In exact GOK-DFT, the ensemble xc energy not only a functional of the In GOK-DFT, the ensemble xc energy is a functional of the
density but also a density but also a
function of the ensemble weights. Note that, unlike in conventional function of the ensemble weights. Note that, unlike in conventional
Boltzmann ensembles~\cite{Pastorczak_2013}, the weights (each state in the ensemble Boltzmann ensembles~\cite{Pastorczak_2013}, the ensemble weights [each state in the ensemble
is assigned a given and fixed weight) are allowed to vary is assigned a given and fixed weight] are allowed to vary
independently in a GOK ensemble. independently in a GOK ensemble.
The weight dependence of the xc functional plays a crucial role in the The weight dependence of the xc functional plays a crucial role in the
calculation of excitation energies. calculation of excitation energies.
@ -193,12 +194,12 @@ It actually accounts for the derivative discontinuity contribution to energy gap
%\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?} %\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
Even though GOK-DFT is in principle able to Even though GOK-DFT is in principle able to
tackle near-degenerate situations and multiple-electron excitation describe near-degenerate situations and multiple-electron excitation
processes, it has not 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} 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 One of the reason is the lack, not to say the absence, of reliable
density-functional approximations for ensembles (eDFAs) in the literature. density-functional approximations for ensembles (eDFAs).
The most recent works on this topic are still fundamental and The most recent works dealing with this particular issue are still fundamental and
exploratory, as they rely either on simple (but nontrivial) model exploratory, as they rely either on simple (but nontrivial) model
systems systems
\cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019} \cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019}
@ -210,7 +211,7 @@ discontinuity problem that ocurs when crossing an integral number of
electrons can be recast into a weight-dependent ensemble electrons can be recast into a weight-dependent ensemble
one. \cite{Senjean_2018,Senjean_2020} one. \cite{Senjean_2018,Senjean_2020}
The present work is an attempt to answer this question, The present work is an attempt to address this problem,
with the ambition to turn, in the forthcoming future, GOK-DFT into a 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. (low-cost) practical computational method for modeling excited states in molecules and extended systems.
Starting from the ubiquitous local-density approximation (LDA), we Starting from the ubiquitous local-density approximation (LDA), we
@ -220,8 +221,8 @@ extracted. The present eDFA, \trashEF{is specially designed for the computation
single and double excitations within GOK-DFT}, which can be seen as a natural 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. 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 As a proof of concept, we apply this general strategy to
ensemble correlation energies only [we use the orbital-dependent exact ensemble correlation energies [that we combine with
ensemble exchange energy for convenience] in the particular case of ensemble exact exchange energies] in the particular case of
\emph{strict} one-dimensional (1D) and \emph{strict} one-dimensional (1D) and
spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b} spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
In other words, the Coulomb interaction used in this work describes In other words, the Coulomb interaction used in this work describes
@ -234,9 +235,8 @@ In these extreme conditions, where magnetic effects compete with Coulombic force
The paper is organized as follows. The paper is organized as follows.
Exact and approximate formulations of GOK-DFT are discussed in Section Exact and approximate formulations of GOK-DFT are discussed in Section
\ref{sec:eDFT}, with a particular emphasis on the extraction of \ref{sec:eDFT}, with a particular emphasis on the calculation of
individual energy levels and the calculation of individual exact exchange individual energy levels.
energies.
In Sec.~\ref{sec:eDFA}, we detail the construction of the In Sec.~\ref{sec:eDFA}, we detail the construction of the
weight-dependent local correlation functional specially designed for the weight-dependent local correlation functional specially designed for the
computation of single and double excitations within GOK-DFT. computation of single and double excitations within GOK-DFT.