minor corrections from T2

Manuscript/SI/eDFT-SI.tex

@@ -1,4 +1,4 @@
-\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
+\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
 \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
 
 \usepackage{mathpazo,libertine}
@@ -519,7 +519,7 @@ The total energy of the ground and doubly-excited states are given by the two lo
 		& = \frac{\sqrt{\pi}}{2 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{2})} + \frac{ij}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+2}{2})} ],
 \end{split}
 \end{equation}
-where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function \cite{NISTbook}, and
+where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function, \cite{NISTbook} and
 \begin{equation}
 	\psi_i(\omega) = \sin(\omega/2) \sin^{i-1}(\omega/2),	\quad	i=1,\ldots,M
 \end{equation}
@@ -584,7 +584,7 @@ Based on these highly-accurate calculations, one can write down, for each state,
 	\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 I of the manuscript.
-The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
+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} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
 
 

Manuscript/eDFT.tex

@@ -1,4 +1,4 @@
-\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
+\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
 \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
 
 \usepackage{mathpazo,libertine}
@@ -115,67 +115,69 @@ Their accuracy is illustrated by computing single and double excitations in one-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Introduction.---}
+\section{Introduction}
+\label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials \cite{ParrBook}.
+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}. 
+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}.
+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}. 
+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}.
+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}.
+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}.
+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}. 
+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}.
+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}.
+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.\\
+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}
+\section{Theory}
+\label{sec:theory}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Generalized KS-eDFT}
+\label{sec:geKS}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-}
 
 In eDFT, the ensemble energy
 \begin{equation}
-	\E{\bw}=(1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
+	\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
+In analogy with ground-state generalized KS-DFT, we consider the following partitioning of the ensemble Levy-Lieb functional 
-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\}
+	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.---}
+\subsection{KS-eDFT for excited states}
+\label{sec:KS-eDFT}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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: 
@@ -186,7 +188,8 @@ By definition, the ensemble energy is
 	\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.}
+To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions: 
+$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
 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}
@@ -240,9 +243,10 @@ The self-consistent process described above is carried on until $\max \abs{\bF{\
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Extracting individual energies.---}
+\subsection{Extracting individual energies}
+\label{sec:ind_energy}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-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:
+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)})
 			\\
@@ -261,15 +265,16 @@ The only remaining piece of information to define at this stage is the weight-de
 \alert{Mention LIM?}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Uniform electron gases.---}
+\subsection{Uniform electron gases}
+\label{sec:UEG}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-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}. 
+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}. 
+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}. 
+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}.
+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}. 
+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. 
@@ -294,7 +299,8 @@ This is a necessary condition for being able to model derivative discontinuities
 %%% %%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Density-functional approximations for ensembles.---}
+\subsection{Density-functional approximations for ensembles}
+\label{sec:eDFA}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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.
@@ -307,7 +313,7 @@ Based on highly-accurate calculations (see {\SI} for additional details), one ca
 	\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}.
+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}
@@ -316,7 +322,7 @@ Combining these, one can build a three-state weight-dependent correlation eDFA:
 \end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{LDA-centered functional.---}
+\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}
@@ -331,7 +337,7 @@ 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
+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},
 	&
@@ -356,7 +362,7 @@ 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}.}
+\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}
@@ -381,7 +387,7 @@ Finally, we note that, by construction,
 %%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Computational details.---}
+\section{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.
@@ -389,7 +395,7 @@ In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ an
 \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}.
+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) = 
@@ -405,13 +411,13 @@ For KS and eKS calculations, a Gauss-Legendre quadrature is employed to compute
 
 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 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.---}
+\section{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}.
@@ -448,7 +454,7 @@ It would highlight the contribution of the derivative discontinuity.}
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Concluding remarks.---}
+\section{Concluding remarks}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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.
@@ -457,11 +463,17 @@ Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos
 
 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.
+However, as shown by Senjean and Fromager, \cite{Senjean_2018} one must modify the weights accordingly in order to maintain a constant density.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textit{Acknowledgements.---}
+\section*{Supplementary material}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+See {\SI} for the additional details about the construction of the functionals, raw data and additional graphs.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{acknowledgements}
 E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
+\end{acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \bibliography{eDFT}