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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
\newcommand{\UL}{Instituut-Lorentz, Universiteit Leiden, P.O.~Box 9506, 2300 RA Leiden, The Netherlands}
\newcommand{\VU}{Division of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands}
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\begin{document}	

\title{Weight dependence of local exchange-correlation functionals in two-electron systems}

\author{Clotilde \surname{Marut}}
	\affiliation{\LCPQ}
\author{Bruno \surname{Senjean}}
	\affiliation{\UL}
	\affiliation{\VU}
\author{Emmanuel \surname{Fromager}}
	\affiliation{\LCQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\begin{abstract}
Gross-Oliveira-Kohn (GOK) ensemble density-functional theory (GOK-DFT) is a time-independent formalism which allows to compute excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within GOK-DFT.
However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous derivative discontinuity contribution to the excitation energies.
In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) \bruno{but it can be applied to larger systems as well right ? thanks to your shift} specifically designed for the computation of double excitations within GOK-DFT.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida,Ulrich_2012}
At a relatively low computational cost (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
Importantly, setting up a TD-DFT calculation for a given system is an almost pain-free process from a user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.

Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations relie on the Runge-Gross theorem. \cite{Runge_1984}
The Kohn-Sham (KS) formalism of TD-DFT transfers the complexity of the many-body problem to the xc functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analog which have both the exact same one-electron density.

However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made for the xc functional.
One of its issues 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.
Moreover, because it is so popular, it has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.

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.
The development of range-separated hybrids provides an effective solution to this problem. \cite{Tawada_2004,Yanai_2004} 
From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Elliott_2011}
Although these double excitations are usually experimentally dark (which means they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007,Loos_2019}

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}
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}
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 excited states) 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.
We believe that it is partly due to the lack of accurate approximations for GOK-DFT.
In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules.
The present contribution is a small step towards this goal.

When one talks about constructing functionals, the local-density approximation (LDA) is never far away.
The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
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}
In particular, we combine 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, and automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}

%The paper is organised as follows.
%In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
%Section \ref{sec:func} provides details about the construction of the weight-dependent xc LDA functional.
%The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:resdis}. 
%Finally, we draw our conclusions in Sec.~\ref{sec:ccl}. 
%Unless otherwise stated, atomic units are used throughout.

%%%%%%%%%%%%%%%%%%%%
%%% THEORY %%%
%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theo}

Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
The corresponding ensemble energy
\begin{equation}
	\E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)}
\end{equation}
fulfils the variational principle
as follows\cite{Gross_1988a}
\begin{eqnarray}\label{eq:ens_energy}
	\E{}{\bw}  = \min_{\hGam{\bw}} \Tr[\hGam{\bw} \hH],
\end{eqnarray}
where $\hH = \hT + \hWee + \hVne$ contains the kinetic, electron-electron and nuclei-electron interaction potential operators, respectively, $\Tr$ denotes the trace and $\hGam{\bw}$ is a trial density matrix of the form
\begin{eqnarray}
	\hGam{\bw} = \sum_{I=0}^{\nEns - 1} \ew{I} \dyad*{\overline{\Psi}^{(I)}},
\end{eqnarray}
where $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1}$ is a set of $\nEns$ orthonormal trial wave functions. 
The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1} = \lbrace \Psi^{(I)} \rbrace_{0 \le I \le \nEns-1}$.
Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states.
One of the key feature of the GOK ensemble is that individual excitation energies are extracted from the ensemble energy via differentiation with respect to individual weights:
\begin{equation}
	\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)}.
\end{equation}

Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles allows to rewrite the exact variational expression for the ensemble energy as\cite{Gross_1988a}
\begin{equation}
\label{eq:Ew-GOK}
	\E{}{\bw} = \min_{\n{}{}} \qty{ \F{}{\bw}[\n{}{}] + \int \vne(\br{}) \n{}{}(\br{}) d\br{} },
\end{equation}
where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional
(the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles).
In the KS formulation, this functional is decomposed as
\begin{equation}
	\F{}{\bw}[\n{}{}] 
	= \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
	= \Tr[ \hgam{\bw} \hT ] + \Tr[ \hgam{\bw} \hWee ],
\end{equation}
where
$\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional,
\begin{equation}
	\hgam{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}}
\end{equation}
is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{0 \le I \le \nEns-1}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le \nOrb}$, and 
\begin{equation}
\label{eq:exc_def}
\begin{split}
	\E{\Hxc}{\bw}[\n{}{}] 
	& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
	\\
	& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' 
	+ \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}
\end{split}
\end{equation}
is the ensemble Hartree-exchange-correlation (Hxc) functional.
Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIE) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$.

From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
\begin{equation}
\begin{split}
	\label{eq:dEdw}
	\pdv{\E{}{\bw}}{\ew{I}} 
	& = \E{}{(I)} - \E{}{(0)} 
	\\
	& = \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})},
\end{split}
\end{equation}
where 
\begin{align}
	\label{eq:nw}
	\n{}{\bw}(\br{}) & = \sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}}{}(\br{}),
	\\
	\label{eq:nI}
	\n{\Det{I}{\bw}}{}(\br{}) & = \sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
\end{align}
are the ensemble and individual one-electron densities, respectively, 
\begin{equation}
\label{eq:KS-energy}
	\Eps{I}{\bw} = \sum_{p}^{\nOrb} \ON{p}{(I)} \eps{p}{\bw}
\end{equation}
is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$]. 
The latters are determined by solving the ensemble KS equation
\begin{equation}
\label{eq:eKS}
	\qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
\end{equation}
where $\hHc(\br{}) = -\nabla^2/2  + \vne(\br{})$, and
\begin{equation}
	\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})} 
	= \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
\end{equation}
is the Hxc potential, with
\begin{subequations}
\begin{align}
	\fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})} 
	& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}',
	\\
	 \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} 
	 & = \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
\end{align}
\end{subequations}
Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
Note that the individual densities $\n{\Det{I}{\bw}}{}(\br{})$ defined in Eq.~\eqref{eq:nI} do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density. 
Nevertheless, 
these densities can still be extracted in principle exactly 
from the KS ensemble as shown by Fromager~\cite{Fromager_2020}.

In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as
\begin{equation}
	\e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}),
\end{equation}
where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% COMPUTATIONAL DETAILS %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
The self-consistent GOK-DFT calculations have been performed with the \texttt{QuAcK} software, freely available on \texttt{github}, where the present functional has been implemented. 
For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
For all calculations, we use the aug-cc-pVXZ (X = D, T, and Q) Dunning's family of atomic basis sets.
Numerical quadratures are performed with the \texttt{numgrid} library using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001}
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. 
Although we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hydrogen molecule}
\label{sec:H2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-independent exchange functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by 
\begin{align}
	\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
	&
	\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
\end{align}
In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state and the lowest doubly-excited state of configuration $1\sigma_u^2$, which has an autoionising resonance nature. \cite{Bottcher_1974}
The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$.
Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/2$.
Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$.

\begin{figure}
	\includegraphics[width=\linewidth]{Ew_H2}
	\caption{
	\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of the weight of the double excitation $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
	\label{fig:Ew_H2}
	}
\end{figure}

\begin{figure}
	\includegraphics[width=\linewidth]{Om_H2}
	\caption{
	\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) as a function of the weight of the double excitation $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
	\label{fig:Om_H2}
	}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent exchange functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Second, in order to remove this spurious curvature of the ensemble energy (which is mostly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error), represented in Fig.~\ref{fig:Cx_H2},
\begin{equation}
	\e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
\end{equation}
with 
\begin{equation}
\label{eq:Cxw}
	\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]
\end{equation}
and 
\begin{subequations}
\begin{align}
	\alpha & = + 0.575\,178,
	\\
	\beta & = - 0.021\,108,
	\\
	\gamma & = - 0.367\,189,
\end{align}
\end{subequations}
makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is strictly forbidden by the GOK variational principle. \cite{Gross_1988a} 
However, it is important to ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$.
Therefore, by construction, the weight-dependent correction vanishes for these two limiting weight values (see Fig.~\ref{fig:Cx_H2}).
Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
\begin{figure}
	\includegraphics[width=0.8\linewidth]{Cx_H2}
	\caption{
	$\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}].
	\label{fig:Cx_H2}
	}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-independent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is now slightly concave) and improved excitation energies, especially at small weights, where the SVWN5 excitation energy is almost spot on. 

%%%%%%%%%%%%%%%%%%
%%% FUNCTIONAL %%%
%%%%%%%%%%%%%%%%%%
%\section{Functional}
%\label{sec:func}
%The present work deals with the explicit construction of the (reduced) LDA xc functional $\e{\xc}{\bw}[\n{}{}]$ defined in Eq.~\eqref{eq:exc_def}.
%The generalisation to a larger number of states (in particular the inclusion of the first singly-excited state) is trivial and left for future work.

%The construction of these two functionals is described below.
%Extension to spin-polarised systems will be reported in future work.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Fourth, in the spirit of our recent work, \cite{Loos_2020} we have designed a weight-dependent correlation functional. 
To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}  
As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
Note that the present paradigm is equivalent to the conventional IUEG model in the thermodynamic limit. \cite{Loos_2011b}
We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Weight-dependent exchange functional}
%\label{sec:Ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states are
\begin{subequations}
\begin{align}
	\e{\HF}{(0)}(\n{}{}) & = \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
	\label{eq:eHF_0}
	\\
	\e{\HF}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3} + \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
	\label{eq:eHF_1}
\end{align}
\end{subequations}
%These two energies can be conveniently decomposed as 
%\begin{equation}
%	\e{\HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{\Ha}{(I)}(\n{}{}) + \e{\ex}{(I)}(\n{}{}),
%\end{equation}
%with
%\begin{subequations}
%\begin{align}
%	\kin{s}{(0)}(\n{}{}) & = 0,
%	&
%	\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3},
%	\\
%	\e{\Ha}{(0)}(\n{}{}) & = \frac{8}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
%	&
%	\e{\Ha}{(1)}(\n{}{}) & = \frac{352}{105} \qty(\frac{\n{}{}}{\pi})^{1/3},
%	\\
%	\e{\ex}{(0)}(\n{}{}) & = - \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
%	&
%	\e{\ex}{(1)}(\n{}{}) & = - \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.	
%\end{align}
%\end{subequations}
%
%In analogy with the conventional Dirac exchange functional, \cite{Dirac_1930} we write down the exchange functional of each individual state as
%\begin{equation}
%	\e{\ex}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3},
%\end{equation}
%and we then obtain
%\begin{align}
%	 \Cx{(0)} & = - \frac{4}{3} \qty( \frac{1}{\pi} )^{1/3},
%	 &
%	 \Cx{(1)} & = - \frac{176}{105} \qty( \frac{1}{\pi} )^{1/3}.
%\end{align}
%We can now combine these two exchange functionals to create a weight-dependent exchange functional for a two-state ensemble
%\begin{equation}
%\label{eq:exw}
%	\e{\ex}{\ew{}}(\n{}{}) 
%	= (1-\ew{}) \e{\ex}{(0)}(\n{}{}) + \ew{} \e{\ex}{(1)}(\n{}{})
%	= \Cx{\ew{}} \n{}{1/3}
%\end{equation}
%with
%\begin{equation}
%	\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
%\end{equation}
%Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
Thanks to highly-accurate calculations and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0} and \eqref{eq:eHF_1}, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \cite{Sun_2016,Loos_2020}
\begin{equation}
\label{eq:ec}
	\e{\co}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
\end{equation}
where $a_2^{(I)}$ and $a_3^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2011b}
Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.

Combining these, we build a two-state weight-dependent correlation functional:
\begin{equation}
\label{eq:ecw}
	\e{\co}{\ew{}}(\n{}{}) =  (1-\ew{}) \e{\co}{(0)}(\n{}{}) + \ew{} \e{\co}{(1)}(\n{}{}).
\end{equation}

%%% FIG 1 %%%
\begin{figure}
	\includegraphics[width=0.8\linewidth]{fig1}
	\caption{
	Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi^2 \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG.
	The data gathered in Table \ref{tab:Ref} are also reported. 
	}
	\label{fig:Ec}
\end{figure}
%%% %%% %%%

%%% TABLE I %%%
\begin{table}
	\caption{
	\label{tab:Ref}
	$-\e{\co}{(I)}$ as a function of the radius of the glome $R = 1/(\pi^2 \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG.
	}
	\begin{ruledtabular}
	\begin{tabular}{lcc}
					&	\tabc{Ground state}	&	\tabc{Doubly-excited state}	\\
			$R$		&	\tabc{$I=0$}	&	\tabc{$I=1$}	\\
			\hline
			$0$		&	$0.023\,818$	&	$0.014\,463$	\\
			$0.1$	&	$0.023\,392$	&	$0.014\,497$	\\
			$0.2$	&	$0.022\,979$	&	$0.014\,523$	\\
			$0.5$	&	$0.021\,817$	&	$0.014\,561$	\\
			$1$		&	$0.020\,109$	&	$0.014\,512$	\\
			$2$		&	$0.017\,371$	&	$0.014\,142$	\\
			$5$		&	$0.012\,359$	&	$0.012\,334$	\\
			$10$	&	$0.008\,436$	&	$0.009\,716$	\\
			$20$	&	$0.005\,257$	&	$0.006\,744$	\\
			$50$	&	$0.002\,546$	&	$0.003\,584$	\\
			$100$	&	$0.001\,399$	&	$0.002\,059$	\\
			$150$	&	$0.000\,972$	&	$0.001\,458$	\\
	\end{tabular}
	\end{ruledtabular}
\end{table}

%%% TABLE 1 %%%
\begin{table}
	\caption{
	\label{tab:OG_func}
	Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}.
	The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.}
	\begin{ruledtabular}
		\begin{tabular}{lll}
						&	\tabc{Ground state}	&	\tabc{Doubly-excited state}	\\
						&	\tabc{$I=0$}		&	\tabc{$I=1$}				\\
			\hline
				$a_1$	&	$-0.023\,818\,4$	&	$-0.014\,463\,3$	\\
				$a_2$	&	$+0.005\,409\,94$	&	$-0.050\,601\,9$	\\	
				$a_3$	&	$+0.083\,076\,6$	&	$+0.033\,141\,7$	\\
		\end{tabular}
	\end{ruledtabular}
\end{table}
%%% %%% %%% %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{LDA-centered functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Because our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons), we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014} 
The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
 
Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent VWN5 LDA reference
\begin{equation}
\label{eq:becw}
	\be{\co}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
\end{equation}
via the following shift:
\begin{equation}
	\be{\co}{(I)}(\n{}{}) = \e{\co}{(I)}(\n{}{}) + \e{\co}{\VWN}(\n{}{}) - \e{\co}{(0)}(\n{}{}).
\end{equation}

Equation \eqref{eq:becw} can be recast
\begin{equation}
\label{eq:eLDA}
\begin{split}
	\be{\co}{\ew{}}(\n{}{}) 
	& =  \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
	\\
	& =  \e{\co}{\VWN}(\n{}{}) + \ew{} \pdv{\e{\co}{\ew{}}(\n{}{})}{\ew{}},
\end{split}
\end{equation}
which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
In particular, $\be{\co}{(0)}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
Consequently, in the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
Also, we note that, by construction,
\begin{equation}
\label{eq:dexcdw}
	\pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}}
	= \be{\co}{(1)}(n) - \be{\co}{(0)}(n),
\end{equation}
which shows that the weight correction is purely linear in eVWN5.

As shown in Fig.~\ref{fig:Ew_H2}, the SGIC-eVWN5 is slightly less concave than its SGIC-VWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).

%This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
%\begin{equation}
%\label{eq:GACE}
%	\E{\xc}{\bw}[\n{}{}] 
%	= \E{\xc}{}[\n{}{}] 
%	+ \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
%\end{equation}
%(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} 
%Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional. 
%In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir 
%$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}

For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
In particular, we report the excitation energies obtained with GOK-DFT in the zero-weight limit (\ie, $\ew{} = 0$) and for the equi-ensemble (\ie, $\ew{} = 1/2$).
For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} which are defined as
\begin{equation}
	\Ex{\LIM}{(1)} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
\end{equation}
as well as the MOM excitation energies. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
MOM excitation energies can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
\begin{equation}
	\Ex{\MOM}{(1)} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
\end{equation}

The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.

%%% TABLE I %%%
\begin{table*}
\caption{
Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods, combinations of xc functionals, and basis sets.
\label{tab:BigTab_H2}
}
\begin{ruledtabular}
\begin{tabular}{llccccc}
	\mc{2}{c}{xc functional} 		&					&	\mc{2}{c}{GOK}	\\
	\cline{1-2}												\cline{4-5}
	exchange	&	correlation		&	Basis			&	 $\ew{} = 0$	&	$\ew{} = 1/2$	&	LIM		&	MOM		\\
	\hline
	HF			&					&	aug-cc-pVDZ		&	38.52			&	30.86			&	34.55	&	28.65	\\
				&					&	aug-cc-pVTZ		&	38.58			&	35.82			&	35.68	&	28.65	\\
				&					&	aug-cc-pVQZ		&	39.12			&	35.94			&	35.64	&	28.65	\\
	\\
	S			&					&	aug-cc-pVDZ		&	38.40			&	27.35			&	23.54	&	26.60	\\
				&					&	aug-cc-pVTZ		&	39.21			&	27.42			&	23.62	&	26.67	\\
				&					&	aug-cc-pVQZ		&	39.78			&	27.42			&	23.62	&	26.67	\\
	\\
	S			&	VWN5			&	aug-cc-pVDZ		&	38.10			&	27.76			&	24.40	&	27.10	\\
				&					&	aug-cc-pVTZ		&	38.54			&	27.81			&	24.46	&	27.17	\\
				&					&	aug-cc-pVQZ		&	38.81			&	27.81			&	24.46	&	27.17	\\
	\\
	GIC-S		&					&	aug-cc-pVDZ		&	26.83			&	26.51			&	26.53	&	26.60	\\
				&					&	aug-cc-pVTZ		&	26.88			&	26.59			&	26.61	&	26.67	\\
				&					&	aug-cc-pVQZ		&	26.82			&	26.60			&	26.62	&	26.67	\\
	\\
	GIC-S		&	VWN5			&	aug-cc-pVDZ		&	28.54			&	26.94			&	27.48	&	27.10	\\
				&					&	aug-cc-pVTZ		&	28.66			&	27.00			&	27.56	&	27.17	\\
				&					&	aug-cc-pVQZ		&	28.64			&	27.00			&	27.56	&	27.17	\\
	\\
	GIC-S		&	eVWN5			&	aug-cc-pVDZ		&	28.78			&	27.10			&	27.56	&	27.27	\\
				&					&	aug-cc-pVTZ		&	28.90			&	27.16			&	27.64	&	27.34	\\
				&					&	aug-cc-pVQZ		&	28.89			&	27.16			&	27.65	&	27.34	\\
	\\
	B88			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.42\fnm[2]	\\
	B3			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		27.77\fnm[2]	\\
	HF			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		29.18\fnm[2]	\\
	HF			&					&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.65\fnm[2]	\\
	\\
	HF			&	FCI				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.75\fnm[2]	\\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Reference \onlinecite{Mielke_2005}.}
\fnt[2]{Reference \onlinecite{Barca_2018a}.}
\end{table*}
%%% %%% %%% %%%

%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
As concluding remarks, we would like to say that what we have done, we think, is awesome.

%%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship.
%PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} 
\end{acknowledgements}

%%%%%%%%%%%%%%%%%%%%
%%% BIBLIOGRAPHY %%%
%%%%%%%%%%%%%%%%%%%%
\bibliography{FarDFT}

\end{document}
