\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
\usepackage{libertine}

\usepackage[
	colorlinks=true,
    citecolor=blue,
    breaklinks=true
	]{hyperref}
\urlstyle{same}

\newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\cloclo}[1]{\textcolor{purple}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashCM}[1]{\textcolor{purple}{\sout{#1}}}

\newcommand{\cmark}{\color{green}{\text{\ding{51}}}}
\newcommand{\xmark}{\color{red}{\text{\ding{55}}}}

%useful stuff
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\ra}{\rightarrow}
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
\newcommand{\mc}{\multicolumn}
\newcommand{\mr}{\multirow}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}

% operators
\newcommand{\hH}{\Hat{H}}
\newcommand{\hHc}{\Hat{h}}
\newcommand{\hT}{\Hat{T}}
\newcommand{\bH}{\Hat{T}}
\newcommand{\hVext}{\Hat{V}_\text{ext}}
\newcommand{\hWee}{\Hat{W}_\text{ee}}

% functionals, potentials, densities, etc
\newcommand{\eps}{\epsilon}
\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
\newcommand{\kin}[2]{t_\text{#1}^{#2}}
\newcommand{\E}[2]{E_\text{#1}^{#2}}
\newcommand{\bE}[2]{\overline{E}_\text{#1}^{#2}}
\newcommand{\be}[2]{\overline{\eps}_\text{#1}^{#2}}
\newcommand{\bv}[2]{\overline{f}_\text{#1}^{#2}}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}

% energies
\newcommand{\EHF}{E_\text{HF}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\Ecat}{E_\text{cat}}
\newcommand{\Eneu}{E_\text{neu}}
\newcommand{\Eani}{E_\text{ani}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EFCI}{E_\text{FCI}}

% matrices
\newcommand{\br}{\bm{r}}
\newcommand{\bw}{\bm{w}}
\newcommand{\bG}{\bm{G}}
\newcommand{\bS}{\bm{S}}
\newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}}
\newcommand{\bHc}{\bm{h}}
\newcommand{\bF}[1]{\bm{F}^{#1}}
\newcommand{\Ex}[1]{\Omega^{#1}}

% elements
\newcommand{\ew}[1]{w_{#1}}
\newcommand{\eG}[1]{G_{#1}}
\newcommand{\eS}[1]{S_{#1}}
\newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}}
\newcommand{\hGamma}[2]{\Hat{\Gamma}_{#1}^{#2}}
\newcommand{\eHc}[1]{h_{#1}}
\newcommand{\eF}[2]{F_{#1}^{#2}}

% Numbers
\newcommand{\Nel}{N}
\newcommand{\Nbas}{K}

% Ao and MO basis
\newcommand{\MO}[2]{\phi_{#1}^{#2}}
\newcommand{\cMO}[2]{c_{#1}^{#2}}
\newcommand{\AO}[1]{\chi_{#1}}

% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal/mol}

\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}
\email{fromagere@unistra.fr}
\affiliation{\LCQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email[Corresponding author: ]{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.
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 in organic molecules.
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.
Indeed, TD-DFT is a in-principle exact theory which recast the many-body problem by transferring its complexity to the xc functional.
However, TD-DFT is far from being perfect, and, 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 energy 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.
Practically, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation is that double excitations are completely absent from the TD-DFT spectra.
Moreover, TD-DFT has problems with charge-transfer and Rydberg excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the xc functional.

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}
Here is the theory.

%%%%%%%%%%%%%%%%%%
%%% 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{subequations}
\begin{align}
	\kin{s}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3},
	\\
	\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.	
\end{align}
\end{subequations}

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

Based on highly-accurate calculations (see below), 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{xc}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
\end{equation}
where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
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} 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 $R = 1/(\pi n)$ for 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.
	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 ring $R$ for 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.
	}
	\begin{ruledtabular}
	\begin{tabular}{ldd}
			$R$		&	\mc{2}{c}{State }	\\
						\cline{2-3}
					&	\tabc{Ground state}	&	\tabc{Doubly-excited state}	\\
			\hline
			$0$		&	&	\\
			$1/10$	&	&	\\
			$1/5$	&	&	\\
			$1/2$	&	&	\\
			$1$		&	&	\\
			$2$		&	&	\\
			$5$		&	&	\\
			$10$	&	&	\\
			$20$	&	&	\\
			$50$	&	&	\\
			$100$	&	&	\\
			$150$	&	&	\\
			$200$	&	&	\\
	\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{c_1^{(I)}}{1 + c_2^{(I)} \n{}{-1/6} + c_3^{(I)} \n{}{-1/3}},
\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{$c_1^{(I)}$}	&	\tabc{$c_2^{(I)}$}	&	\tabc{$c_3^{(I)}$}	\\
			\hline
			Ground state			&	$0$		&		&				&			\\
			Doubly-excited state	&	$1$		&		&				&			\\
		\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}{\text{LDA}}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
\end{equation}
The local-density approximation (LDA) exchange-correlation functional is
\begin{equation}
	\e{xc}{\text{LDA}}(\n{}{}) = \e{x}{\text{LDA}}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}).
\end{equation}
 
Equation \eqref{eq:becw} can be recast
\begin{equation}
\label{eq:eLDA}
	\be{xc}{\ew{}}(\n{}{}) 
	=  \e{xc}{\text{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, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\text{LDA}}(\n{}{})$.
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{J}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{xc}{(J)}[\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 would like to thank Emmanuel Fromager for enlightening discussions.
He also 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}
