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Manuscript/eDFT.bib
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@ -1,13 +1,61 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-09-09 09:45:30 +0200
%% Created for Pierre-Francois Loos at 2020-02-15 17:43:19 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2019,
Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
Date-Added = {2020-02-15 17:43:18 +0100},
Date-Modified = {2020-02-15 17:43:18 +0100},
Doi = {10.1021/acs.jctc.8b01205},
Journal = {J. Chem. Theory Comput.},
Number = {3},
Pages = {1939--1956},
Title = {Reference Energies for Double Excitations},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
@article{Runge_1984,
Author = {Runge, E. and Gross, E. K. U.},
Date-Added = {2020-02-15 17:43:03 +0100},
Date-Modified = {2020-02-15 17:43:03 +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{Gori-Giorgi_2010,
Author = {P. Gori-Giorgi and M. Seidl},
Date-Added = {2020-02-15 17:39:27 +0100},
Date-Modified = {2020-02-15 17:39:40 +0100},
Doi = {10.1039/c0cp01061h},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {14405},
Title = {Density functional theory for strongly-interacting electrons: Perspectives for Physics and Chemistry},
Volume = {12},
Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1039/c0cp01061h}}
@article{Gagliardi_2017,
Author = {L. Gagliardi and D. G. Truhlar and G. Li Manni and R. K. Carlson and C. E. Hoyer and J. Lucas Bao},
Date-Added = {2020-02-15 17:37:18 +0100},
Date-Modified = {2020-02-15 17:37:46 +0100},
Doi = {10.1021/acs.accounts.6b00471},
Journal = {Acc. Chem. Res.},
Pages = {66},
Volume = {50},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.accounts.6b00471}}
@article{Boblest_2014,
Author = {S. Boblest and C. Schimeczek and G. Wunner},
Date-Added = {2019-09-09 09:42:01 +0200},

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Manuscript/eDFT.tex
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@ -147,40 +147,48 @@ Their accuracy is illustrated by computing single and double excitations in one-
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials. \cite{ParrBook}
The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}{}(\br)$, the latter being a much simpler quantity than the many-electron wave function.
Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964,Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials. \cite{ParrBook}
The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function.
The complexity of the many-body problem is then transferred to the xc functional.
Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}
The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge.
Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003,Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tapavicza_2008,Levine_2006}
The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge. \cite{Gori-Giorgi_2010,Gagliardi_2017}
Another issue, which is partly connected to the previous one, is the description of electronically-excited states.
The standard approach for modeling excited states in DFT is linear response time-dependent DFT (TDDFT). \cite{Casida}
The standard approach for modeling excited states in DFT is linear-response time-dependent DFT (TD-DFT). \cite{Runge_1984,Casida}
In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong.
Moreover, in exact TDDFT, the xc functional is time dependent.
The simplest and most widespread approximation in state-of-the-art electronic structure programs where TDDFT is implemented consists in neglecting memory effects. \cite{Casida}
Moreover, in exact TD-DFT, the xc functional is time dependent.
The simplest and most widespread approximation in state-of-the-art electronic structure programs where TD-DFT is implemented consists in neglecting memory effects. \cite{Casida}
In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time.
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,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}
As a result, double electronic excitations are completely absent from the TD-DFT spectrum, thus reducing further the applicability of TD-DFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
When affordable (\ie, for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
The basic idea is to describe a finite ensemble of states (ground and excited) altogether, \ie, with the same set of orbitals.
Interestingly, a similar approach exists in DFT.
Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988a, Oliveira_1988, Gross_1988b} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979}
In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988a, Oliveira_1988, Gross_1988b} and is a generalization of Theophilou's variational principle for equiensembles. \cite{Theophilou_1979}
In GOK-DFT (\ie, eDFT for excited states), the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies.
It actually accounts for the infamous 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 eDFT 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_2018a,Deur_2018b,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}
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_2018a,Deur_2018b,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_2018a,Deur_2018b,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 this article with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
%\titou{Mention WIDFA?}
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.
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.
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, as well as the different approximations that we apply in order to make the present scheme practical.
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.
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.
Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
Atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -188,7 +196,9 @@ Atomic units are used throughout.
\label{sec:eDFT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{GOK-DFT}\label{subsec:gokdft}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The GOK ensemble energy~\cite{Gross_1988a,Oliveira_1988,Gross_1988b} is defined as follows:
\beq
@ -307,6 +317,7 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
\end{split}
\eeq
%%%%%%%%%%%%%%%%
\subsection{One-electron reduced density matrix formulation}
%%%%%%%%%%%%%%%%
@ -795,6 +806,7 @@ Finally, we note that, by construction,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:comp_details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium.
@ -824,6 +836,7 @@ Concerning the KS-eDFT calculations, two sets of weight have been tested: the ze
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\Nel = 5$).
Similar graphs are obtained for the other $\Nel$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
@ -882,6 +895,7 @@ It would highlight the contribution of the derivative discontinuity.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concluding remarks}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the present article, we have constructed a weight-dependent three-state DFA in the context of ensemble DFT.
This eDFA delivers accurate excitation energies for both single and double excitations.