eDFT_FUEG/Manuscript/eDFT.tex

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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}

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\begin{document}	

\title{Weight-dependent local density-functional approximations for ensembles}

\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Fromager}
\email{fromagere@unistra.fr}
\affiliation{\LCQ}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We report a first generation of local, weight-dependent correlation density-functional approximations (DFAs) that incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT). 
These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
The resulting eDFAs, based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence. 
Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Introduction.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 electronic 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} 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. 
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}.
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}. 
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}.

When affordable (i.e.~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, i.e.,~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_1988, Gross_1988a, Oliveira_1988}, 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. 
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. 
\alert{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_2018a,Gould_2018b,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 Letter with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
\alert{Mention WIDFA?}

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}.
Moreover, the present method relies on exact Hartree-Fock (HF) exchange, eschewing the so-called ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}.
Atomic units are used throughout.\\

\manu{I added some key equations in the following. Will polish the all
thing later on.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Generalized KS-eDFT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
}

In eDFT, the ensemble energy
\begin{equation}
	\E{\bw}=(1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
\end{equation}
is obtained variationally as follows,
In analogy with ground-state generalized KS-DFT, we consider the
following partitioning of 

the ensemble Levy-Lieb functional 
\begin{equation}
	F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
\end{equation}

\begin{equation}
F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{eKS for excited states.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here, we explain how to perform a self-consistent KS calculation for ensembles (eKS) in the context of excited states.
In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including: 
i) the ground state ($I=0$), ii) the first singly-excited state ($I=1$), and iii) the first doubly-excited state ($I=2$).
Generalization to a larger number of states is straightforward and is left for future work.
By definition, the ensemble energy is
\begin{equation}
	\E{\bw} = (1 - \ew{1} - \ew{2}) \E{(0)} + \ew{1} \E{(1)} + \ew{2} \E{(2)}.
\end{equation}
$\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the to the single excitation and double excitation, respectively.
\alert{To ensure the GOK variational principle \cite{Gross_1988a}, the weights must fulfil the following conditions: $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$. T2: I don't understand the asymmetry of the weights in this equation.}
Note that, in order to extract individual energies from a single eKS calculation (see below), the weights must remain independent.
By construction, the excitation energies are
\begin{equation}
\label{eq:Ex}
	\Ex{(I)} = \pdv{\E{(I)}}{\ew{I}} = \E{(I)} - \E{(0)}.
\end{equation}
In the following, the orbitals $\MO{p}{\bw}(\br)$ are defined as linear combination of basis functions $\AO{\mu}(\br)$, such as
\begin{equation}
	\MO{p}{\bw}(\br) = \sum_{\mu=1}^{\Nbas} \cMO{\mu p}{\bw} \, \AO{\mu}(\br).
\end{equation}

Within the self-consistent eKS process, one is looking for the following weight-dependent density matrix:
\begin{equation}
\label{eq:Gamma}
	\bGamma{\bw} = (1-\ew{1}-\ew{2}) \bGamma{(0)} - \ew{1} \bGamma{(1)} - \ew{2} \bGamma{(2)},
\end{equation}
where $\bw = (\ew{1},\ew{2})$ and $\bGamma{(I)}$ is the $I$th-state density matrix with elements
\begin{equation}
\label{eq:eGamma}
	\eGamma{\mu\nu}{(I)} = \sum_{i=1}^{\Nel-I} \cMO{\mu i}{\bw} \cMO{\nu i}{\bw} + \sum_{a=\Nel+1}^{\Nel+I} \cMO{\mu a}{\bw} \cMO{\nu a}{\bw}.
\end{equation}
The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamma{\bw}$ in Eq.~\eqref{eq:Gamma} are obtained by diagonalizing the following Fock matrix
\begin{multline}
\label{eq:F}
	\eF{\mu\nu}{\bw} 
	= \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si} 
	\\
	+ \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)}  \AO{\mu}(\br) \AO{\nu}(\br) d\br,
\end{multline}
which itself depends on $\bGamma{\bw}$.
In Eq.~\eqref{eq:F}, $\bHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$,
\begin{equation}
	(\mu\nu|\la\si) = \iint  \frac{\AO\mu(\br_1) \AO\nu(\br_1) \AO\la(\br_2) \AO\si(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
\end{equation}
are two-electron repulsion integrals,
$\v{c}{\bw}[\n{}(\br)] = \n{}(\br) \e{c}{\bw}[\n{}(\br)]$ and $\e{c}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study.
The one-electron ensemble density is
\begin{equation}
	\n{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
\end{equation}
with a similar expression for $\n{(I)}(\br)$, while the ensemble energy reads 
\begin{multline}
\label{eq:Ew}
	\E{\bw} 
	= \Tr(\bGamma{\bw} \, \bHc) 
	+ \frac{1}{2} \Tr(\bGamma{\bw} \, \bG \, \bGamma{\bw}) 
	\\
	+ \int \e{c}{\bw}[\n{\bw}(\br)] \n{\bw}(\br) d\br.
\end{multline}
The self-consistent process described above is carried on until $\max \abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}} < \tau$, where $\tau$ is a user-defined threshold and $\eS{\mu\nu} = \braket{\AO{\mu}}{\AO{\nu}}$ are elements of the overlap matrix $\bS$. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Extracting individual energies.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following Deur and Fromager \cite{Deur_2018b}, it is possible to extract individual energies, $\E{(I)}$, from the ensemble energy [Eq.~\eqref{eq:Ew}] as follows:
\begin{multline}
	\E{(I)} = \Tr(\bGamma{(I)} \, \bHc)  + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)})
			\\
			+ \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{(I)}(\br) d\br
			+ \LZ{c}{} + \DD{c}{(I)}.
\end{multline}
Note that a \emph{single} eKS calculation is required to extract the three individual energies.
The correlation part of the (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by
\begin{align}
	\LZ{c}{} 	& =  - \int \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
	\\
	\DD{c}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int  \left. \pdv{\e{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br.
\end{align}
Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\e{c}{\bw}(\n{})$.
\alert{Mention LIM?}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Uniform electron gases.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas model (also known as jellium) \cite{ParrBook, Loos_2016}. 
One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}. 
From this point of view, using finite uniform electron gases \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs is much more relevant \cite{Loos_2014a, Loos_2014b, Loos_2017a}. 
Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems. 
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}.

As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e.~a circle) \cite{Loos_2012, Loos_2013a, Loos_2014b}. 
The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform. 
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary. 
This is a necessary condition for being able to model derivative discontinuities. 

%%% TABLE 1 %%%
\begin{table*}
	\caption{
	\label{tab:OG_func}
	Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
%	\begin{ruledtabular}
		\begin{tabular}{lcddd}
			\hline\hline
			State					&	$I$		&	\tabc{$a^{(I)}$}	&	\tabc{$b^{(I)}$}	&	\tabc{$c^{(I)}$}	\\
			\hline
			Ground state			&	$0$		&	-0.0137078			&	0.0538982			&	0.0751740			\\
			Singly-excited state	&	$1$		&	-0.0238184			&	0.00413142			&	0.0568648			\\
			Doubly-excited state	&	$2$		&	-0.00935749			&	-0.0261936			&	0.0336645			\\
			\hline\hline
		\end{tabular}
%	\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Density-functional approximations for ensembles.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
All these states have the same (uniform) density $\n{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.

Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e.~per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
\begin{equation}
\label{eq:ec}
	\e{c}{(I)}(\n{}) = \frac{a^{(I)}\,\n{}}{\n{} + b^{(I)} \sqrt{\n{}} + c^{(I)}},
\end{equation}
where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
Combining these, one can build a three-state weight-dependent correlation eDFA:
\begin{equation}
\label{eq:ecw}
	\e{c}{\bw}(\n{}) =  (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}) + \ew{1} \e{c}{(1)}(\n{}) + \ew{2} \e{c}{(2)}(\n{}).
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{LDA-centered functional.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
\begin{equation}
\label{eq:becw}
	\be{c}{\bw}(\n{}) =  (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}) + \ew{1} \be{c}{(1)}(\n{}) + \ew{2} \be{c}{(2)}(\n{}),
\end{equation}
where
\begin{equation}
	\be{c}{(I)}(\n{}) = \e{c}{(I)}(\n{}) + \e{c}{\text{LDA}}(\n{}) - \e{c}{(0)}(\n{}).
\end{equation}
The local-density approximation (LDA) correlation functional,
\begin{equation}
	\e{c}{\text{LDA}}(\n{}) = a^\text{LDA} \, F\qty[1,\frac{3}{2},c^\text{LDA}, \frac{a^\text{LDA}(1-c^\text{LDA})}{b^\text{LDA}} {\n{}}^{-1}],
\end{equation}
specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function \cite{NISTbook}, and
\begin{align}
	a^\text{LDA} & = - \frac{\pi^2}{360},
	&
	b^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
	& 
	c^\text{LDA} & = 2.408779.
\end{align}
Equation \eqref{eq:becw} can be recast
\begin{equation}
\label{eq:eLDA}
\begin{split}
	\be{c}{\bw}(\n{}) 
	& =  \e{c}{\text{LDA}}(\n{}) 
	\\
	& + \ew{1} \qty[\e{c}{(1)}(\n{})-\e{c}{(0)}(\n{})] + \ew{2} \qty[\e{c}{(2)}(\n{})-\e{c}{(0)}(\n{})],
\end{split}
\end{equation}
which nicely highlights the centrality of the LDA in the present eDFA. 
In particular, $\be{c}{(0,0)}(\n{}) = \e{c}{\text{LDA}}(\n{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
Finally, we note that, by construction,
\begin{equation}
	\left. \pdv{\be{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} = \be{c}{(J)}[\n{\bw}(\br)] - \be{c}{(0)}[\n{\bw}(\br)].
\end{equation}
\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors \cite{Gould_2018b}.}
%%% FIG 1 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{EvsL_5}
	\caption{
	\label{fig:EvsL}
	Error with respect to FCI in single and double excitation energies for 5-boxium for various methods and box length $L$.
	Graphs for additional values of $\Nel$ can be found as {\SI}.
	\alert{T2: we could combine Figs. 1 and 2 into a single figure.}
	}
\end{figure}
%%% %%% %%%

%%% FIG 2 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{EvsN_1}
	\caption{
	\label{fig:EvsN}
	Error with respect to FCI in single and double excitation energies for $\Nel$-boxium for various methods and number of electrons $\Nel$ at $L=\pi$.
	Graphs for additional values of $L$ can be found as {\SI}.
	}
\end{figure}
%%% %%% %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Computational 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.
In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \Nel \le 7$. 
\alert{Comment on the quality of these density: density- and functional-driven errors?}

These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime \cite{Rogers_2017,Rogers_2016}.
We use as basis functions the (orthonormal) orbitals of the one-electron system, i.e.
\begin{equation}
	\AO{\mu}(x) = 
	\begin{cases}
		\sqrt{2/L} \cos(\mu \pi x/L),	&	\mu \text{ is odd,}
		\\
		\sqrt{2/L} \sin(\mu \pi x/L),	&	\mu \text{ is even,}
	\end{cases}
\end{equation} 
with $ \mu = 1,\ldots,\Nbas$ and $\Nbas = 30$ for all calculations.
For the self-consistent calculations (such as HF, KS or eKS), the convergence threshold has been set to $\tau = 10^{-7}$.
For KS and eKS calculations, a Gauss-Legendre quadrature is employed to compute numerical integrals.

In order to test the present eLDA functional we have performed various sets of calculations.
To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
For the single excitations, we have also performed time-dependent HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations \cite{Dreuw_2005}.
For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also tested.

Concerning the eKS calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textit{Results and discussion.---}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e.~$\Nel = 5$).
Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.
In the weakly correlated regime (i.e.~small $L$), all methods provide accurate estimates of the excitation energies.
When the box gets larger, they start to deviate.
For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
TDLDA yields larger errors at large $L$ by underestimating the excitation energies.
TDA-TDLDA slightly corrects this trend thanks to error compensation. 
Concerning the eLDA functional, our results clearly evidences that the equi-weights [i.e. $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [i.e. $\bw = (0,0)$].
This is especially true for the single excitation which is significantly improved by using state-averaged weights.
The effect on the double excitation is less pronounced.
Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.
This conclusion is verified for smaller and larger number of electrons (see {\SI}).
\alert{Shall I test the one-electron system for self-interaction?}

Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies, for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
The graphs associated with other $L$ values are reported as {\SI}.
Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations. 
As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDHF or TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$, an error of the same order as CIS or TDA-TDLDA.
Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only a few percent.

\alert{Need further discussion on DD and LZ shift. Linearity of energy wrt weights?}

\alert{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.
However, when the box gets larger, there is a strong mixing between different degree of excitations.
In particular, the single and double excitations strongly mix.
This is clearly evidenced if one looks at the weights of the different configurations in the FCI wave function.
In one hand, if one does construct a eDFA with a single state (either single or double), one clearly sees that the results quickly deteriorates when the box gets larger.
On the other hand, building a functional which does mix singles and doubles corrects this by allowing configuration mixing.}

\alert{It might be useful to add eHF results where one switch off the correlation part.
For both zero weight and state-averaged weights?
It would highlight the contribution of the derivative discontinuity.}


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\textit{Concluding remarks.---}
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In the present Letter, 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.
Generalization to a larger number of states is straightforward and will be investigated in future work.
Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.

Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap.
This can be done by constructing DFAs for the one- and three-electron ground state systems, and combining them with the two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and \eqref{eq:ecw}.
However, as shown by Senjean and Fromager \cite{Senjean_2018}, one must modify the weights accordingly in order to maintain a constant density.

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\textit{Acknowledgements.---}
E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
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