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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}

\begin{document}	

\title{Weight-dependent exchange-correlation functionals for molecules: the local-density approximation}

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

\begin{abstract}
We report a first generation of local, weight-dependent exchange-correlation density-functional approximations (DFAs) for molecules.
These density-functional approximations for ensembles (eDFAs) incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
They are specially designed for the computation of double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) to ensembles.
The resulting eDFAs, dubbed eLDA, which are 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 on the prototypical H$_2$ molecule.
\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 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 the user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.

TD-DFT is an in-principle exact theory which formal foundation relies 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 analogs with the same one-electron density.

However, TD-DFT is far from being perfect as, in practice, 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 substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Moreover, because it was so popular, it has been studied in excruciated details, and 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 xc functional.
The development of range-separated hybrids \cite{Tawada_2004,Yanai_2004} provides an effective solution to this problem.
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}.

One possible solution to access double excitation 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 exchange-correlation kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.

Maybe surprisingly, a possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
Density-functional theory for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988,Gross_1988a,Oliveira_1988} is a viable alternative currently under active development which follow such a strategy. \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 has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, \cite{Gross_1988a} and excitation energies can be quite easily extracted from the total ensemble energy.
Although the formal foundation of eDFT has been set three decades ago, \cite{Gross_1988,Gross_1988a,Oliveira_1988} the practical developments of eDFT have been rather slow.
We believe that it is due to the lack of accurate approximations for eDFT.
In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation has never been developed for atoms and molecules in the context of eDFT.
The present contribution is a first step towards this goal.

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

%%%%%%%%%%%%%%%%%%%%
%%% THEORY %%%
%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theo}
As mentioned above, eDFT is based on the so-called Gross-Oliveria-Kohn (GOK) variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy
\begin{equation}
	\E{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \E{}{(I)}
\end{equation}
built from an ensemble of $\Nens$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\Nens-1)}$, and normalized, monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, 
\begin{align}
	& \sum_{I=0}^{\Nens-1} \ew{I} = 1,
	&
	& \ew{0} \ge \ldots \ge \ew{\Nens-1}.
\end{align}
One of the key feature of eDFT in the present context is that one can easily extract individual excitation energies 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}
where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}{\bw}[\n{}{}]$ such that 
\begin{equation}
	\E{}{\bw} = \min_{\n{}{}} \qty[ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} ]
\end{equation}
where $\vext(\br{})$ is the external potential.
In the KS formulation of eDFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
\begin{equation}
	\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
\end{equation}
where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}{}[\n{}{}]$ are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively with
\begin{equation}
\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{}'+ \E{\xc}{\bw}[\n{}{}]
\end{split}
\end{equation}
Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \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 $\E{\xc}{\bw}[\n{}{}]$.

The present work deals with the explicit construction of $\E{\xc}{\bw}[\n{}{}]$ at the LDA level in the case of the two-state ensemble (\ie,  $\Nens = 2$).

%%%%%%%%%%%%%%%%%%
%%% FUNCTIONAL %%%
%%%%%%%%%%%%%%%%%%
\section{Functional}
\label{sec:func}

We adopt the usual decomposition, and write down the weight-dependent exchange-correlation functional as
\begin{equation}
	\e{xc}{\ew{}}(\n{}{}) = \e{x}{\ew{}}(\n{}{}) + \e{c}{\ew{}}(\n{}{}),
\end{equation}
where $\e{x}{\ew{}}(\n{}{})$ and $\e{c}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively.
The construction of these two functionals is described below.
Here, we restrict our study to spin-unpolarized systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$.

The present weight-dependent eDFA is specifically designed for the calculation of double excitations within eDFT.
As mentioned previously, we consider a two-state ensemble including the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the two-electron glomium system.
All these states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome where the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2011b} for more details about this paradigm.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent exchange functional}
\label{sec:Ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The reduced (\ie, per electron) HF energy for these two states is
\begin{subequations}
\begin{align}
	\e{HF}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3},
	\\
	\e{HF}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.	
\end{align}
\end{subequations}
These two energies can be conveniently decomposed as 
\begin{equation}
	\e{HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{H}{(I)}(\n{}{}) + \e{x}{(I)}(\n{}{}),
\end{equation}
with
\begin{subequations}
\begin{align}
	\kin{s}{(0)}(\n{}{}) & = 0,
	&
	\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3}.	
	\\
	\e{H}{(0)}(\n{}{}) & = \frac{8}{3\pi^{1/3}} \n{}{1/3},
	&
	\e{H}{(1)}(\n{}{}) & = \frac{352}{105\pi^{1/3}} \n{}{1/3}.	
	\\
	\e{x}{(0)}(\n{}{}) & = - \frac{4}{3\pi^{1/3}} \n{}{1/3},
	&
	\e{x}{(1)}(\n{}{}) & = - \frac{176}{105\pi^{1/3}} \n{}{1/3}.	
\end{align}
\end{subequations}

Knowing that the exchange functional has the following form
\begin{equation}
	\e{x}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3},
\end{equation}
we 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
\begin{equation}
\begin{split}
	\e{x}{\ew{}}(\n{}{}) 
	& = (1-\ew{}) \e{x}{(0)}(\n{}{}) + \ew{} \e{x}{(1)}(\n{}{})
	\\
	& = \Cx{\ew{}} \n{}{1/3}
\end{split}
\end{equation}
with
\begin{equation}
	\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}
\end{equation}
Amazingly, the weight dependence of the exchange functional can be transfered to the $\Cx{}$ coefficient.
This is obvious but kind of nice.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent correlation functional}
\label{sec:Ec}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Based on highly-accurate calculations, 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
\begin{equation}
\label{eq:ec}
	\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
\end{equation}
where the $a_k^{(I)}$'s 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_2013a, Loos_2014a}
Equation \eqref{eq:ec} provides two state-specific correlation DFAs based on a two-electron system.
Combining these, one can build a two-state weight-dependent correlation eDFA:
\begin{equation}
\label{eq:ecw}
	\e{c}{\ew{}}(\n{}{}) =  (1-\ew{}) \e{c}{(0)}(\n{}{}) + \ew{} \e{c}{(1)}(\n{}{}).
\end{equation}

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

%%% TABLE I %%%
\begin{table}
	\caption{
	\label{tab:Ref}
	$-\e{c}{(I)}$ as a function of the radius of the glome $R$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
	}
	\begin{ruledtabular}
	\begin{tabular}{ldd}
			$R$		&	\mc{2}{c}{State }	\\
						\cline{2-3}
					&	\tabc{Ground state}	&	\tabc{Doubly-excited state}	\\
			\hline
			$0$		&	&	\\
			$1/10$	&	&	\\
			$1/5$	&	&	\\
			$1/2$	&	&	\\
			$1$		&	&	\\
			$2$		&	&	\\
			$5$		&	&	\\
			$10$	&	&	\\
			$20$	&	&	\\
			$50$	&	&	\\
			$100$	&	&	\\
	\end{tabular}
	\end{ruledtabular}
\end{table}

Based on these highly-accurate calculations, 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
\begin{equation}
\label{eq:ec}
	\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
\end{equation}
where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.



%%% 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}
			State					&	$I$		&	\tabc{$a_1^{(I)}$}	&	\tabc{$a_2^{(I)}$}	&	\tabc{$a_3^{(I)}$}	\\
			\hline
			Ground state			&	$0$		&	-0.0238184		&	+0.00575719			&		+0.0830576		\\
			Doubly-excited state	&	$1$		&	-0.0144633		&	-0.0504501			&		+0.0331287		\\
		\end{tabular}
	\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{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{xc}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{xc}{(0)}(\n{}{}) + \ew{} \be{c}{(1)}(\n{}{}),
\end{equation}
where
\begin{equation}
	\be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\LDA}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
\end{equation}
The local-density approximation (LDA) exchange-correlation functional is
\begin{equation}
	\e{xc}{\LDA}(\n{}{}) = \e{x}{\LDA}(\n{}{}) + \e{c}{\LDA}(\n{}{}).
\end{equation}
where we use here the Dirac exchange functional and the VWN5 correlation functional
\begin{subequations}
\begin{align}
	\e{x}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3}
	\\
	\e{c}{\LDA}(\n{}{}) & \equiv \e{c}{\text{VWN5}}(\n{}{}).
\end{align}
\end{subequations}
with $\Cx{\LDA} = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}$.

Equation \eqref{eq:becw} can be recast
\begin{equation}
\label{eq:eLDA}
	\be{xc}{\ew{}}(\n{}{}) 
	=  \e{xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})],
\end{equation}
which nicely highlights the centrality of the LDA in the present eDFA. 
In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\LDA}(\n{}{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.

This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014} 
Within this in-principle-exact formalism, the (weight-dependent) correlation energy of the ensemble is constructed from the (weight-independent) ground-state functional (such as the LDA), yielding Eq.~\eqref{eq:eLDA}.
This is a crucial point as we intend 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).

Finally, we note that, by construction,
\begin{equation}
	\left. \pdv{\be{xc}{\ew{}}[\n{}{}]}{\ew{I}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{xc}{(I)}[\n{}{\ew{}}(\br)] - \be{xc}{(0)}[\n{}{\ew{}}(\br)].
\end{equation}

%%%%%%%%%%%%%%%
%%% RESULTS %%%
%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
Here, we do \ce{H2} because \ce{H2} is very interesting.

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

%%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
PFL acknowledges funding from the \textit{Centre National de la Recherche Scientifique}.
CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship.
\end{acknowledgements}

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

\end{document}
