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


%%%% added by Manu %%%%%
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\newcommand{\beq}{\begin{eqnarray}}
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%
\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
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%%%% 

\begin{document}	

\title{Supplementary Material for ``Weight-dependent local density-functional approximations for ensembles''}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Ensemble Hartree--Fock method}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%The Hartree--Fock (HF) ensemble energy can be written 
%as
%\beq\label{eq:eHF_ener}
%&&E^{\bw}_{\rm
%HF}({\bm\kappa})=
%\sum_{pq}\langle
%\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{pq}
%\nonumber\\
%&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
%\varphi_r(\bmk)\varphi_s(\bmk)\rangle
%%\times
%\sum^M_{K=0}w^{(K)}D^{(K)}_{pr}D^{(K)}_{qs},
%\nonumber\\
%\eeq
%where the one- and antisymmetrized two-electron integrals read,
%\beq
%\langle
%\varphi_p({\bmk})\vert\hat{h}\vert
%\varphi_q({\bmk})\rangle=\int d{\bfx}\;
%\varphi_p({\bmk},{\bfx})\hat{h}\varphi_q({\bmk},{\bfx}) 
%\eeq
%with $\hat{h}\equiv-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
%ext}(\bfr)$
%and
%\beq
%&&\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
%\varphi_r(\bmk)\varphi_s(\bmk)\rangle=
%\nonumber\\
%&&\int d{\bfx}_1\int d{\bfx}_2\;
%\varphi_p({\bmk},{\bfx}_1)\varphi_q({\bmk},{\bfx}_2)\frac{1}{\vert
%{\bfr}_1-{\bfr}_2\vert}
%\nonumber
%\\
%&&\times\Big[\varphi_r({\bmk},{\bfx}_1)\varphi_s({\bmk},{\bfx}_2)
%-\varphi_s({\bmk},{\bfx}_1)\varphi_r({\bmk},{\bfx}_2)\Big]
%,
%\eeq
%respectively. Note that we use {\it real algebra} and the shorthand
%notation $\int d{\bfx}\equiv\int
%d{\bfr}\sum_{\sigma}$ for integration over space and
%summation over spin.
%% normalization condition
%%$\sum^M_{K=0}w^{(K)}=1$
%%.
%The antihermitian $\bmk\equiv\{\kappa_{pq}\}_{p>q}$ matrix which appears in the integrals controls 
%the rotation of the spin-orbitals as follows,
%\beq\label{eq:orb_taylor_expansion}
%&&\varphi_p({\bmk},{\bfx})=\sum_q\left[e^{-{\bmk}}\right]_{qp}\varphi_q({\bfx})
%\nonumber\\
%&&=
%\varphi_p({\bfx})+\sum_{q<p}\kappa_{pq}\varphi_q({\bfx})-\sum_{q>p}\kappa_{qp}\varphi_q({\bfx})
%+\mathcal{O}\left({\bmk}^2\right).
%\eeq 
%The (${\bmk}$-independent) one-electron reduced density matrices (1RDMs)
%in Eq.~(\ref{eq:eHF_ener}) are defined in the unrotated molecular
%spin-orbital basis for each (unrotated) determinant $\Phi^{(K)}$ belonging to
%the ensemble as follows: $D^{(K)}_{pr}=\delta_{pr}$ if $\varphi_p$ and
%$\varphi_r$ are both
%occupied in $\Phi^{(K)}$, otherwise $D^{(K)}_{pr}=0$. If the unrotated
%spin-orbitals are the minimizing ensemble HF ones, then the following
%stationarity condition is fulfilled,
%\beq\label{eq:station_cond}
%\left.\dfrac{\partial E^{\bw}_{\rm
%HF}({\bm\kappa})}{\partial \kappa_{lm}}
%\right|_{{\bmk}=0}=0, 
%\eeq
%with $l>m$. Since, according to Eq.~(\ref{eq:orb_taylor_expansion}),
%\beq
%\left.\dfrac{\partial 
%\varphi_p({\bmk},{\bfx})}{\partial \kappa_{lm}}
%\right|_{{\bmk}=0}=\delta_{lp}\varphi_m({\bfx})-\delta_{mp}\varphi_l({\bfx}), 
%\eeq
%Eq.~(\ref{eq:station_cond}) can be written more explicitly as the
%following commutation relation,
%\iffalse%%%%%
%%%%%%% intermediate steps ... %%%%
%\beq
%\sum^M_{K=0}w^{(K)}\sum_qD^{(K)}_{mq}f^{(K)}_{lq}
%-
%\sum^M_{K=0}w^{(K)}\sum_qf^{(K)}_{mq}D^{(K)}_{lq}=0
%\eeq
%% original %%
%\iffalse%%%
%\beq
%&&
%2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
%\nonumber\\
%&&-
%2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
%\nonumber\\
%&&+2\sum_{qrs}\langle \varphi_m\varphi_q\vert\vert
%\varphi_r\varphi_s\rangle
%%\times
%\sum^M_{K=0}w^{(K)}D^{(K)}_{lr}D^{(K)}_{qs}
%\nonumber\\
%&&-2\sum_{qrs}\langle \varphi_l\varphi_q\vert\vert
%\varphi_r\varphi_s\rangle
%%\times
%\sum^M_{K=0}w^{(K)}D^{(K)}_{mr}D^{(K)}_{qs}
%\nonumber\\
%&&=0\eeq   
%\fi%%%% 
%%%%%%%%
%\beq
%&&
%2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
%\nonumber\\
%&&-
%2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
%\nonumber\\
%&&+2\sum_{qrs}\langle \varphi_m\varphi_r\vert\vert
%\varphi_q\varphi_s\rangle
%%\times
%\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}D^{(K)}_{rs}
%\nonumber\\
%&&-2\sum_{qrs}\langle \varphi_l\varphi_r\vert\vert
%\varphi_q\varphi_s\rangle
%%\times
%\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}D^{(K)}_{rs}
%\nonumber\\
%&&=0\eeq    
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
%\fi%%%%
%%%%%%%%
%\beq\label{eq:stat_cond_commut_ind}
%\sum^M_{K=0}w^{(K)}\left[{\bm f}^{(K)},{\bm D}^{(K)}\right]=0,
%\eeq
%where the $K$th Fock matrix elements read
%\beq
%f^{(K)}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
%+
%\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
%\varphi_q\varphi_s\rangle D^{(K)}_{rs}.
%\eeq
%In the minimizing ensemble HF spin-orbital basis, Eq.~(\ref{eq:stat_cond_commut_ind}) reads 
%\beq
%\sum^M_{K=0}w^{(K)}\Big(\nu^{(K)}_m-\nu_l^{(K)}\Big)f^{(K)}_{lm}=0,
%\eeq
%where $\nu^{(K)}_m$ is the occupation of the spin-orbital $\varphi_m$ in the
%determinant $\Phi^{(K)}$.\\
%
%Note that, in more conventional ensemble calculations, the following HF
%energy expression is employed,
%\beq\label{eq:GI_ensHF_ener}
%&&\tilde{E}^{\bw}_{\rm
%HF}({\bm\kappa})=
%\sum_{pq}\langle
%\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle D^{\bw}_{pq}
%\nonumber\\
%&&
%+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
%\varphi_r(\bmk)\varphi_s(\bmk)\rangle
%%\times
%D^{\bw}_{pr}D^{\bw}_{qs},
%\eeq 
%where ${\bm D}^{\bw}=\sum^M_{K=0}w^{(K)}{\bm D}^{(K)}$ is the ensemble
%1RDM. In this case, the stationarity condition simply reads 
%\beq
%\left[{\bm f}^{\bw},{\bm D}^{\bw}\right]=0,
%\eeq
%where the ensemble Fock matrix elements are defined as follows,
%\beq
%f^{\bw}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
%+
%\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
%\varphi_q\varphi_s\rangle D^{\bw}_{rs}.
%\eeq
%The major issue with the expression of the ensemble energy in
%Eq.~(\ref{eq:GI_ensHF_ener}) is the
%ghost-interaction error from which our expression (see
%Eq.~(\ref{eq:eHF_ener})) is free. Note also
%that, by construction, the ensemble energy in Eq.~(\ref{eq:GI_ensHF_ener}) is quadratic in the
%ensemble weights while ours, like the exact one, varies linearly with
%the weights.    
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Ensemble Hartree--Fock exchange and density-functional
%ghost-interaction correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\beq
%F^{\bw}_{\rm HF}[n]&=&
%\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
%Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
%HF}\left[{\bmg}^{\bw}\right]\right\}
%\nonumber\\
%&=&{\rm
%Tr}\left[\hat{\gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
%HF}\left[{\bmg}^{\bw}[n]\right]
%\eeq
%where
%$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\gamma}^{(K)}$ is an ensemble density matrix operator constructed
%from Slater determinants, the ensemble 1RDM elements are $\gamma_{pq}^{\bw}={\rm
%Tr}\left[\hat{\gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
%and $W_{\rm
%HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
%\varphi_r\varphi_s\rangle
%%\times
%\gamma_{pr}\gamma_{qs}$.\\
%
%In-principle-exact decomposition:
%
%\beq
%F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
%Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
%\eeq
%
%The complementary ensemble Hx energy removes the ghost-interaction
%errors introduced in $W_{\rm
%HF}\left[{\bmg}^{\bw}[n]\right]$:
%\beq
%\overline{E}^{{\bw}}_{\rm
%Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
%HF}\left[{\bmg}^{(K)}[n]\right]
%-W_{\rm
%HF}\left[{\bmg}^{\bw}[n]\right],
%\eeq
%which gives in the canonical orbital basis
%\beq
%&&\overline{E}^{{\bw}}_{\rm
%Hx}[n]=
%\dfrac{1}{2}\sum_{pq}
%\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
%\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
%\nonumber\\
%&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
%-\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
%.\eeq
%\manu{I would guess that, in a uniform system, the GOK-DFT and our
%canonical orbitals are the same. This is nice since we can construct
%in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
%Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
%
%Variational expression for the ensemble energy:
%\beq
%E^{{\bw}}=\underset{\hat{\gamma}^{{\bw}}}{\rm min}\Big\{
%&&{\rm
%Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
%HF}\left[{\bmg}^{\bw}\right]
%+
%\overline{E}^{{\bw}}_{\rm
%Hxc}\left[n_{\hat{\gamma}^{{\bw}}}\right]
%%+E^{{\bw}}_{\rm c}\left[n_{\hat{\gamma}^{{\bw}}}\right]
%\nonumber\\
%&&
%+\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\gamma}^{{\bw}}}({\bfr})
%\Big\}
%\eeq
% 
%Note that, if we use orbital rotations, the gradient of the DFT energy
%contributions can be expressed as follows,
%\beq
%\left.\dfrac{\partial 
%\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
%}{\partial \kappa_{lm}}
%\right|_{{\bmk}=0}=\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
%Hxc}\left[n^{{\bw}}\right]}{\delta
%n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}}
%\right|_{{\bmk}=0}, 
%\eeq
%where   
%\beq
%n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw}
%\eeq
%thus leading to
%\beq
%&&\left.\dfrac{\partial 
%\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
%}{\partial \kappa_{lm}}
%\right|_{{\bmk}=0}=
%\sum_{pq}\gamma_{pq}^{\bw}
%\nonumber\\
%&&\times\left.\dfrac{\partial} 
%{\partial \kappa_{lm}}
%\Big[\left\langle\varphi_p(\bmk)\middle\vert\hat{\overline{v}}^{{\bw}}_{\rm
%Hxc}
%\middle\vert \varphi_q(\bmk)\right\rangle
%\Big]
%\right|_{{\bmk}=0}.
%\eeq
%
%In conclusion, the minimizing canonical orbitals fulfill the following
%hybrid HF/GOK-DFT equation,
%\beq
%&&\left(-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
%ext}({\bfr})+\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]
%+\dfrac{\delta \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]}{\delta
%n({\br})}\right)\varphi^{{\bw}}_p({\bfx})
%\nonumber
%\\
%&&=\varepsilon^{{\bw}}_p\varphi^{{\bw}}_p({\bfx}).
%\eeq
%
%
%Since $\partial \gamma_{pq}^{\bw}/\partial
%w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
%
%\manu{just for me ...
%\beq
%&&+\dfrac{1}{2}
%\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
%\varphi_r\varphi_s\rangle
%%\times
%\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
%\nonumber\\
%&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_q\varphi_p\vert\vert
%\varphi_s\varphi_r\rangle
%%\times
% \gamma^{\bw}_{pr}\left(\gamma_{qs}^{(I)}-\gamma_{qs}^{(0)}\right)
%\nonumber\\
%&&=
%\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
%\varphi_r\varphi_s\rangle
%%\times
%\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
%\nonumber\\
%&&=
%\sum_{pr}\left[\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]\right]_{pr}\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)
%\nonumber\\
%&&=
%\sum_p\left[\hat{u}_{\rm
%HF}\left[\gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right)
%\eeq
%}
%
%\beq
%\dfrac{dE^{\bw}}{dw^{(I)}}=\sum_p\varepsilon^{{\bw}}_p\left(\nu_p^{(I)}-\nu_p^{(0)}\right)+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
%Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
%\eeq
%
%LZ shift in this context: $\varepsilon^{{\bw}}_p\rightarrow
%\overline{\varepsilon}^{{\bw}}_p=\varepsilon^{{\bw}}_p+\overline{\Delta}_{\rm
%LZ}^{{\bw}}$ where
%
%\beq
%N\overline{\Delta}_{\rm
%LZ}^{{\bw}}&=&\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]
%-\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
%Hxc}\left[n^{{\bw}}\right]}{\delta
%n({\br})}n^{{\bw}}({\bfr})
%\nonumber\\
%&&
%-W_{\rm
%HF}\left[{\bmg}^{\bw}\right]
%\eeq 
%
%such that
%\beq
%E^{{\bw}}=\sum^M_{K=0}w^{(K)}\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p.
%\eeq
%
%Thus we conclude that individual energies can be expressed in principle
%exactly as follows,
%
%\beq
%E^{(K)}=\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p+\sum^M_{I>0}\left(\delta_{IK}-w^{(I)}\right)\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
%Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
%\eeq
%%%%%%%%%%%%%%%
%\iffalse%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Generalized GOK-DFT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%The energy to be minimized in a generalized GOK-DFT approach can be
%written as
%\beq
%\eeq
%\fi%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Construction of the density-functional approximations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The density-functional approximations designed in this manuscript are based on highly-accurate energies 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.
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.

The reduced (i.e.~per electron) HF energy for these three states is:
\begin{subequations}
\begin{align}
	\e{HF}{(0)}(n) & = \frac{\pi^2}{8} n^2 + n,
	\\
	\e{HF}{(1)}(n) & = \frac{\pi^2}{2}n^2 + \frac{4}{3} n,		
	\\	 
	\e{HF}{(2)}(n) & = \frac{9\pi^2}{8}n^2 + \frac{23}{15} n.	
\end{align}
\end{subequations}
All these states have the same (uniform) density $n = 2/(2\pi R)$ where $R$ is the radius of the ring on which the electrons are confined.

The total energy of the ground and doubly-excited states are given by the two lowest eigenvalues of the Hamiltonian $\bH$ with elements
\begin{equation}
\begin{split}
	H_{ij} 
		& = \int_0^\pi \qty[ \frac{\psi_i(\omega)}{R} \frac{\psi_j(\omega)}{R} + \frac{\psi_i(\omega)\psi_j(\omega)}{2R\sin(\omega/2)} ] d\omega
		\\
		& = \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
\begin{equation}
	\psi_i(\omega) = \sin(\omega/2) \sin^{i-1}(\omega/2),	\quad	i=1,\ldots,M
\end{equation}
are (non-orthogonal) explicitly-correlated basis functions with overlap matrix elements
\begin{equation}
	S_{ij} 
		= \int_0^\pi \psi_i(\omega)\psi_j(\omega) d\omega
		= \sqrt{\pi} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+2}{2})}.
\end{equation}
Thanks to this explicitly-correlated basis, the convergence rate of the energy is exponential with respect to $M$.
Therefore, high accuracy is reached with a very small number of basis functions.
Here, we typically use $M=10$.
For the singly-excited state, one has to modify the basis functions as 
\begin{equation}
	\psi_i(\omega) = \cos(\omega/2) \sin^{i-1} (\omega/2),
\end{equation}
and its energy is obtained by the lowest root of the Hamiltonian in this basis, and the matrix elements reads
\begin{align}
	H_{ij} & = \frac{\sqrt{\pi}}{4 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{3})} + \frac{3ij+i+j-1}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+4}{2})} ],
	\\
	S_{ij} & = \frac{\sqrt{\pi}}{2} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+4}{2})}.
\end{align}
The numerical values of the correlation energy for various $R$ are reported in Table \ref{tab:Ref} for the three states of interest.

%%% 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{turnpage}
\begin{squeezetable}
\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}{lcddddddddddd}
	State	&	$I$		&	\mc{11}{c}{Ring's radius $R = 1/(\pi n)$}	\\
							\cline{3-13}
			&			&	\tabc{$0$}	&	\tabc{$1/10$}	&	\tabc{$1/5$}	&	\tabc{$1/2$}	&	\tabc{$1$}	&	\tabc{$2$}	&	\tabc{$5$}	&	\tabc{$10$}	&	\tabc{$20$}	&	\tabc{$50$}	&	\tabc{$100$}	\\
			\hline
	Ground state		&	$0$		&	0.013708			&	0.012859	&	0.012525	&	0.011620	&	0.010374	&	0.008558	&	0.005673	&	0.003697	&	0.002226	&	0.001046	&	0.000567	\\
	Singly-excited state		&	$1$		&	0.0238184	&	0.023392			&	0.022979	&	0.021817	&	0.020109	&	0.017371	&	0.012359	&	0.008436	&	0.005257	&	0.002546	&	0.001399		\\
	Doubly-excited state		&	$2$		&	0.018715	&	0.018653	&	0.018576	&	0.018300	&	0.017743	&	0.016491	&	0.013145	&	0.009670	&	0.006365	&	0.003231	&	0.001816	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}
\end{squeezetable}
\end{turnpage}

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_1^{(I)}\,n}{n + a_2^{(I)} \sqrt{n} + a_3^{(I)}},
\end{equation}
where $a_2^{(I)}$ and $a_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
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} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.


%%% FIG 2 %%%
\begin{figure*}
	\includegraphics[height=0.325\linewidth]{EvsL_2}
	\includegraphics[height=0.325\linewidth]{EvsL_3}
	\includegraphics[height=0.325\linewidth]{EvsL_4}
	\includegraphics[height=0.325\linewidth]{EvsL_5}
	\includegraphics[height=0.325\linewidth]{EvsL_6}
	\includegraphics[height=0.325\linewidth]{EvsL_7}
	\caption{
	Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the box length $L$ for various methods.
	See main text for additional details.}
	\label{fig:EvsL}
\end{figure*}
%%% %%% %%%

%%% FIG 3 %%%
%\begin{figure*}
%	\includegraphics[height=0.325\linewidth]{EvsN_0125}
%	\includegraphics[height=0.325\linewidth]{EvsN_025}
%	\includegraphics[height=0.325\linewidth]{EvsN_05}
%	\includegraphics[height=0.325\linewidth]{EvsN_1}
%	\includegraphics[height=0.325\linewidth]{EvsN_2}
%	\includegraphics[height=0.325\linewidth]{EvsN_4}
%	\includegraphics[height=0.325\linewidth]{EvsN_8}
%	\caption{
%	Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the number of electrons $N$ for various methods and box length $L$.
%	}
%	\label{fig:EvsL}
%\end{figure*}
%%% %%% %%%

%%% TABLE II %%%
\begin{table*}
	\caption{
	\label{tab:OptGap_2}
	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 2-boxium (i.e.,~$\Nel = 2$ electrons in a box of length $L$).
	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported. 
	(DNC = KS calculation does not converge.)
	}
	\begin{ruledtabular}
	\begin{tabular}{lclddddddd}
			&				&					&	\mc{7}{c}{2-boxium with a box of length $L$}	\\
													\cline{4-10}
	Method	&	$\bw$		&	State			&	\pi/8			&	\pi/4			&	\pi/2			&	\pi		&	2\pi		&	4\pi		&	8\pi		\\
	\hline				
	FCI		&				&	$\E{(0)}$		&	168.1946	&	44.0662		&	12.0035		&	3.4747	&	1.0896	&	0.3719	&	0.1367	\\
			&				&	$\E{(1)}$		&	330.2471	&	85.0890		&	22.5112		&	6.2247	&	1.8355	&	0.5845	&	0.2006	\\
			&				&	$\E{(2)}$		&	809.9972	&	204.9840	&	52.4777		&	13.7252	&	3.7248	&	1.0696	&	0.3300	\\
			&				&	$\Ex{(1)}$		&	162.0525	&	41.0228		&	10.5078		&	2.7500	&	0.7458	&	0.2125	&	0.0639	\\
			&				&	$\Ex{(2)}$		&	641.8026	&	160.9177	&	40.4743		&	10.2505	&	2.6352	&	0.6977	&	0.1933	\\
	\hline
			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
	\hline
	CIS		&				&	$\Ex{(1)}$		&	0.0104		&	0.0102		&	0.0099		&	0.0092	&	0.0077	&	0.0051	&	0.0022	\\
	\\
	TDHF	&				&	$\Ex{(1)}$		&	0.0019		&	0.0021		&	0.0023		&	0.0027	&	0.0029	&	0.0023	&	0.0011	\\
	\\
	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0099		&	0.0088		&	0.0058		&	-0.0041	&	-0.0316	&	-0.0467	&	\tabc{DNC}	\\
	\\
	TDLDA	&				&	$\Ex{(1)}$		&	0.0015		&	0.0006		&	-0.0018		&	-0.0106	&	-0.0370	&	-0.0518	&	\tabc{DNC}	\\
	\\
	eHF	&	$(0,0)$			&	$\E{(0)}$		&	0.0137	&	0.0131	&	0.0119	&	0.0098	&	0.0067	&	0.0033	&	0.0010	\\
			&				&	$\E{(1)}$		&	0.0685	&	0.0678	&	0.0664	&	0.0637	&	0.0588	&	0.0500	&	0.0372	\\
			&				&	$\E{(2)}$		&	-0.0052	&	-0.0055	&	-0.0058	&	-0.0055	&	-0.0022	&	0.0086	&	0.0271	\\
			&				&	$\Ex{(1)}$		&	0.1082	&	0.1069	&	0.1044	&	0.0998	&	0.0911	&	0.0736  &	\tabc{DNC}	\\
			&				&	$\Ex{(2)}$		&	0.0345	&	0.0336	&	0.0322	&	0.0306	&	0.0302	&	0.0321	&	\tabc{DNC}	\\
	\\
	eHF	&	$(1/3,1/3)$		&	$\E{(0)}$		&	0.0565	&	0.0557	&	0.0540	&	0.0509	&	0.0455	&	0.0372	&	0.0269	\\
			&				&	$\E{(1)}$		&	0.0560	&	0.0553	&	0.0538	&	0.0510	&	0.0461	&	0.0384	&	0.0287	\\
			&				&	$\E{(2)}$		&	0.0371	&	0.0366	&	0.0357	&	0.0342	&	0.0316	&	0.0277	&	0.0223	\\
			&				&	$\Ex{(1)}$		&	0.0529	&	0.0517	&	0.0494	&	0.0456	&	0.0418	&	0.0409	&	\tabc{DNC}	\\
			&				&	$\Ex{(2)}$		&	0.0340	&	0.0330	&	0.0314	&	0.0288	&	0.0274	&	0.0303	&	\tabc{DNC}	\\
	\\
	KS-eLDA	&	$(0,0)$		&	$\E{(0)}$		&	-0.0397	&	-0.0391	&	-0.0380	&	-0.0361	&	-0.0323	&	-0.0236	&	\tabc{DNC}\\
			&				&	$\E{(1)}$		&	0.0215	&	0.0213	&	0.0210	&	0.0200	&	0.0159	&	0.0102	&	\tabc{DNC}\\
			&				&	$\E{(2)}$		&	-0.0426	&	-0.0425	&	-0.0419	&	-0.0387	&	-0.0250	&	-0.0045	&	\tabc{DNC}\\
			&				&	$\Ex{(1)}$		&	0.0612	&	0.0604	&	0.0590	&	0.0561	&	0.0483	&	0.0337	&	\tabc{DNC}\\
			&				&	$\Ex{(2)}$		&	-0.0029	&	-0.0034	&	-0.0039	&	-0.0025	&	0.0074	&	0.0191	&	\tabc{DNC}\\
			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	\tabc{DNC}	\\
			&				&	$\DD{c}{(1)}$	&	0.0064	&	0.0056	&	0.0043	&	0.0022	&	-0.0007	&	-0.0037	&	\tabc{DNC}\\
			&				&	$\DD{c}{(2)}$	&	0.0159	&	0.0147	&	0.0126	&	0.0093	&	0.0046	&	-0.0009	&	\tabc{DNC}\\
	\\
	KS-eLDA	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0031	&	0.0036	&	0.0044	&	0.0054	&	0.0042	&	-0.0025	&	\tabc{DNC}\\
			&				&	$\E{(1)}$		&	0.0090	&	0.0087	&	0.0083	&	0.0076	&	0.0070	&	0.0071	&	\tabc{DNC}\\
			&				&	$\E{(2)}$		&	-0.0005	&	-0.0009	&	-0.0015	&	-0.0023	&	-0.0030	&	-0.0026	&	\tabc{DNC}\\
			&				&	$\Ex{(1)}$		&	0.0058	&	0.0052	&	0.0039	&	0.0022	&	0.0028	&	0.0096	&	\tabc{DNC}\\
			&				&	$\Ex{(2)}$		&	-0.0036	&	-0.0045	&	-0.0058	&	-0.0077	&	-0.0072	&	0.0000	&	\tabc{DNC}\\
			&				&	$\DD{c}{(0)}$	&	-0.0074	&	-0.0067	&	-0.0055	&	-0.0036	&	-0.0010	&	0.0019	&	\tabc{DNC}\\
			&				&	$\DD{c}{(1)}$	&	-0.0010	&	-0.0011	&	-0.0014	&	-0.0017	&	-0.0021	&	-0.0022	&	\tabc{DNC}\\
			&				&	$\DD{c}{(2)}$	&	0.0084	&	0.0079	&	0.0069	&	0.0053	&	0.0031	&	0.0003	&	\tabc{DNC}\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}

%%% TABLE III %%%
\begin{table*}
	\caption{
	\label{tab:OptGap_3}
	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 3-boxium (i.e.,~$\Nel = 3$ electrons in a box of length $L$).
	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
	}
	\begin{ruledtabular}
	\begin{tabular}{lclddddddd}
			&				&					&	\mc{7}{c}{3-boxium with a box of length $L$}	\\
													\cline{4-10}
	Method	&	$\bw$		&	State			&	\pi/8			&	\pi/4			&	\pi/2			&	\pi		&	2\pi		&	4\pi		&	8\pi		\\
	\hline				
	FCI		&				&	$\E{(0)}$		&	475.6891	&	125.7776	&	34.8248	&	10.3536	&	3.3766	&	1.2126	&	0.4721	\\
			&				&	$\E{(1)}$		&	702.8330	&	183.3370	&	49.5922	&	14.2255	&	4.4269	&	1.5105	&	0.5606	\\
			&				&	$\E{(2)}$		&	1379.3128	&	353.5967	&	92.7398	&	25.3135	&	7.3546	&	2.3203	&	0.7990	\\
			&				&	$\Ex{(1)}$		&	227.1438	&	57.5594		&	14.7674	&	3.8720	&	1.0504	&	0.2979	&	0.0885	\\
			&				&	$\Ex{(2)}$		&	903.6236	&	227.8191	&	57.9150	&	14.9599	&	3.9780	&	1.1077	&	0.3269	\\
	\hline
			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
	\hline
%	CIS		&				&	$\Ex{(1)}$		&	0.0163	&	0.0161	&	0.0157	&	0.0149	&	0.0133	&	0.0102	&	0.0057	\\
%	\\
%	TDHF	&				&	$\Ex{(1)}$		&	0.0013	&	0.0013	&	0.0014	&	0.0014	&	0.0013	&	0.0010	&	0.0007	\\
	\\
	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0162	&	0.0157	&	0.0146	&	0.0110	&	-0.0049	&	-0.0344	&	-0.0378	\\
	\\
	TDLDA	&				&	$\Ex{(1)}$		&	0.0262	&	0.0264	&	0.0264	&	0.0269	&	0.0273	&	0.0206	&	-0.0116	\\
	\\
	eHF	&	$(0,0)$			&	$\E{(0)}$		&	0.0327	&	0.0316	&	0.0297	&	0.0261	&	0.0203	&	0.0126	&	0.0055	\\
			&				&	$\E{(1)}$		&	0.1051	&	0.1038	&	0.1013	&	0.0965	&	0.0875	&	0.0708	&	0.0440	\\
			&				&	$\E{(2)}$		&	0.0841	&	0.0829	&	0.0806	&	0.0766	&	0.0704	&	0.0625	&	0.0517	\\
			&				&	$\Ex{(1)}$		&	0.1532	&	0.1516	&	0.1486	&	0.1428	&	0.1320	&	0.1095	&	0.0698	\\
			&				&	$\Ex{(2)}$		&	0.1322	&	0.1307	&	0.1279	&	0.1229	&	0.1149	&	0.1012	&	0.0774	\\
	\\
	eHF	&	$(1/3,1/3)$		&	$\E{(0)}$		&	0.0886	&	0.0874	&	0.0850	&	0.0805	&	0.0725	&	0.0593	&	0.0404	\\
			&				&	$\E{(1)}$		&	0.0880	&	0.0865	&	0.0836	&	0.0783	&	0.0692	&	0.0552	&	0.0374	\\
			&				&	$\E{(2)}$		&	0.1209	&	0.1200	&	0.1182	&	0.1147	&	0.1078	&	0.0953	&	0.0747	\\
			&				&	$\Ex{(1)}$		&	0.0802	&	0.0785	&	0.0754	&	0.0699	&	0.0611	&	0.0529	&	0.0397	\\
			&				&	$\Ex{(2)}$		&	0.1131	&	0.1120	&	0.1100	&	0.1062	&	0.0997	&	0.0930	&	0.0770	\\
	\\
	KS-eLDA	&	$(0,0)$		&	$\E{(0)}$		&	-0.0481	&	-0.0478	&	-0.0473	&	-0.0463	&	-0.0446	&	-0.0387	&	-0.0257	\\
			&				&	$\E{(1)}$		&	0.0343	&	0.0336	&	0.0321	&	0.0292	&	0.0220	&	0.0084	&	0.0008	\\
			&				&	$\E{(2)}$		&	0.0277	&	0.0267	&	0.0247	&	0.0216	&	0.0187	&	0.0208	&	0.0209	\\
			&				&	$\Ex{(1)}$		&	0.0824	&	0.0814	&	0.0794	&	0.0755	&	0.0666	&	0.0471	&	0.0266	\\
			&				&	$\Ex{(2)}$		&	0.0759	&	0.0745	&	0.0720	&	0.0679	&	0.0633	&	0.0595	&	0.0467	\\
			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	\\
			&				&	$\DD{c}{(1)}$	&	0.0100	&	0.0092	&	0.0077	&	0.0051	&	0.0012	&	-0.0034	&	-0.0072	\\
			&				&	$\DD{c}{(2)}$	&	0.0244	&	0.0231	&	0.0208	&	0.0168	&	0.0108	&	0.0029	&	-0.0050	\\
	\\
	KS-eLDA	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0078	&	0.0080	&	0.0082	&	0.0085	&	0.0081	&	0.0024	&	-0.0022		\\
			&				&	$\E{(1)}$		&	0.0172	&	0.0162	&	0.0144	&	0.0112	&	0.0064	&	0.0019	&	0.0004		\\
			&				&	$\E{(2)}$		&	0.0645	&	0.0636	&	0.0621	&	0.0590	&	0.0530	&	0.0420	&	0.0300		\\
			&				&	$\Ex{(1)}$		&	0.0094	&	0.0083	&	0.0062	&	0.0027	&	-0.0018	&	-0.0004	&	0.0026		\\
			&				&	$\Ex{(2)}$		&	0.0567	&	0.0557	&	0.0539	&	0.0506	&	0.0449	&	0.0397	&	0.0323		\\
			&				&	$\DD{c}{(0)}$	&	-0.0115	&	-0.0107	&	-0.0094	&	-0.0072	&	-0.0038	&	0.0005	&	0.0045		\\
			&				&	$\DD{c}{(1)}$	&	-0.0015	&	-0.0016	&	-0.0018	&	-0.0022	&	-0.0028	&	-0.0033	&	-0.0032		\\
			&				&	$\DD{c}{(2)}$	&	0.0129	&	0.0123	&	0.0113	&	0.0094	&	0.0066	&	0.0028	&	-0.0013		\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}


%%% TABLE IV %%%
\begin{table*}
	\caption{
	\label{tab:OptGap_4}
	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 4-boxium (i.e.,~$\Nel = 4$ electrons in a box of length $L$).
	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
	}
	\begin{ruledtabular}
	\begin{tabular}{lclddddddd}
			&				&					&	\mc{7}{c}{4-boxium with a box of length $L$}	\\
													\cline{4-10}
	Method	&	$\bw$		&	State			&	\pi/8			&	\pi/4			&	\pi/2			&	\pi		&	2\pi		&	4\pi		&	8\pi		\\
	\hline				
	FCI		&				&	$\E{(0)}$		&		1020.3778	&	270.0849	&	74.9426		&	22.3790	&	7.3595	&	2.6798	&	1.0633	\\
			&				&	$\E{(1)}$		&		1312.2776	&	344.0184	&	93.8936		&	27.3398	&	8.7021	&	3.0600	&	1.1764	\\
			&				&	$\E{(2)}$		&		2183.4399	&	563.5949	&	149.6753	&	41.7213	&	12.5052	&	4.1033	&	1.4749	\\
			&				&	$\Ex{(1)}$		&		291.8998	&	73.9335		&	18.9510		&	4.9608	&	1.3426	&	0.3802	&	0.1131	\\
			&				&	$\Ex{(2)}$		&		1163.0621	&	293.5099	&	74.7326		&	19.3423	&	5.1457	&	1.4235	&	0.4116	\\
	\hline
			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
	\hline
%	CIS		&				&	$\Ex{(1)}$		&	0.0203	&	0.0202	&	0.0200	&	0.0195	&	0.0187	&	0.0167	&	0.0116	\\
%	\\
%	TDHF	&				&	$\Ex{(1)}$		&	0.0008	&	0.0008	&	0.0009	&	0.0009	&	0.0008	&	0.0008	&	0.0007	\\
%	\\
	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0203	&	0.0201	&	0.0195	&	0.0181	&	0.0106	&	-0.0178	&	-0.0369	\\
	\\
	TDLDA	&				&	$\Ex{(1)}$		&	0.0008	&	0.0007	&	0.0004	&	-0.0006	&	-0.0074	&	-0.0360	&	-0.0653	\\
	\\
	eHF	&	$(0,0)$			&	$\E{(0)}$		&	0.0541	&	0.0529	&	0.0504	&	0.0459	&	0.0381	&	0.0266	&	0.0139	\\
			&				&	$\E{(1)}$		&	0.1359	&	0.1346	&	0.1320	&	0.1271	&	0.1186	&	0.1041	&	0.0804	\\
			&				&	$\E{(2)}$		&	0.1393	&	0.1374	&	0.1335	&	0.1260	&	0.1126	&	0.0910	&	0.0604	\\
			&				&	$\Ex{(1)}$		&	0.1900	&	0.1885	&	0.1857	&	0.1806	&	0.1715	&	0.1545	&	0.1189	\\
			&				&	$\Ex{(2)}$		&	0.1934	&	0.1913	&	0.1872	&	0.1795	&	0.1656	&	0.1414	&	0.0990	\\
	\\
	eHF	&	$(1/3,1/3)$		&	$\E{(0)}$		&	0.1167	&	0.1153	&	0.1125	&	0.1073	&	0.0977	&	0.0818	&	0.0592	\\
			&				&	$\E{(1)}$		&	0.1110	&	0.1093	&	0.1059	&	0.0996	&	0.0888	&	0.0723	&	0.0517	\\
			&				&	$\E{(2)}$		&	0.1688	&	0.1669	&	0.1630	&	0.1553	&	0.1400	&	0.1118	&	0.0705	\\
			&				&	$\Ex{(1)}$		&	0.1025	&	0.1008	&	0.0975	&	0.0915	&	0.0816	&	0.0702	&	0.0532	\\
			&				&	$\Ex{(2)}$		&	0.1603	&	0.1584	&	0.1546	&	0.1471	&	0.1328	&	0.1097	&	0.0720	\\
	\\
	KS-eLDA	&	$(0,0)$		&	$\E{(0)}$		&	-0.0541	&	-0.0539	&	-0.0537	&	-0.0534	&	-0.0529	&	-0.0504	&	-0.0386	\\
			&				&	$\E{(1)}$		&	0.0413	&	0.0406	&	0.0390	&	0.0362	&	0.0304	&	0.0159	&	0.0008	\\
			&				&	$\E{(2)}$		&	0.0642	&	0.0622	&	0.0586	&	0.0517	&	0.0399	&	0.0254	&	0.0149	\\
			&				&	$\Ex{(1)}$		&	0.0954	&	0.0945	&	0.0927	&	0.0896	&	0.0833	&	0.0663	&	0.0394	\\
			&				&	$\Ex{(2)}$		&	0.1182	&	0.1162	&	0.1123	&	0.1051	&	0.0928	&	0.0758	&	0.0534	\\
			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	\\
			&				&	$\DD{c}{(1)}$	&	0.0136	&	0.0127	&	0.0111	&	0.0083	&	0.0038	&	-0.0022	&	-0.0080	\\
			&				&	$\DD{c}{(2)}$	&	0.0330	&	0.0316	&	0.0291	&	0.0248	&	0.0178	&	0.0080	&	-0.0028	\\
	\\
	KS-eLDA	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0085	&	0.0085	&	0.0084	&	0.0082	&	0.0072	&	0.0021	&	-0.0015		\\
			&				&	$\E{(1)}$		&	0.0164	&	0.0152	&	0.0129	&	0.0087	&	0.0020	&	-0.0050	&	-0.0044		\\
			&				&	$\E{(2)}$		&	0.0936	&	0.0917	&	0.0880	&	0.0807	&	0.0664	&	0.0434	&	0.0300		\\
			&				&	$\Ex{(1)}$		&	0.0079	&	0.0067	&	0.0045	&	0.0006	&	-0.0051	&	-0.0071	&	-0.0029		\\
			&				&	$\Ex{(2)}$		&	0.0851	&	0.0832	&	0.0796	&	0.0725	&	0.0593	&	0.0413	&	0.0315		\\
			&				&	$\DD{c}{(0)}$	&	-0.0155	&	-0.0148	&	-0.0134	&	-0.0110	&	-0.0071	&	-0.0017	&	0.0040	\\
			&				&	$\DD{c}{(1)}$	&	-0.0020	&	-0.0021	&	-0.0023	&	-0.0027	&	-0.0034	&	-0.0042	&	-0.0044	\\
			&				&	$\DD{c}{(2)}$	&	0.0175	&	0.0168	&	0.0157	&	0.0137	&	0.0105	&	0.0059	&	0.0004	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}

%%% TABLE V %%%
\begin{table*}
	\caption{
	\label{tab:OptGap_5}
	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 5-boxium (i.e.,~$\Nel = 5$ electrons in a box of length $L$).
	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
	}
	\begin{ruledtabular}
	\begin{tabular}{lclddddddd}
			&				&					&	\mc{7}{c}{5-boxium with a box of length $L$}	\\
													\cline{4-10}
	Method	&	$\bw$		&	State			&	\pi/8			&	\pi/4			&	\pi/2			&	\pi		&	2\pi		&	4\pi		&	8\pi		\\
	\hline				
	FCI		&				&	$\E{(0)}$		&	1867.6344	&	493.6760	&	136.7020	&	40.7244		&	13.3763	&	4.8811	&	1.9492	\\
			&				&	$\E{(1)}$		&	2224.11488	&	583.8981	&	159.7957	&	46.7553		&	15.0029	&	5.3399	&	2.0855	\\
			&				&	$\E{(2)}$		&	3289.2022	&	852.4249	&	228.0415	&	64.3597		&	19.6613	&	6.6206	&	2.4547	\\
			&				&	$\Ex{(1)}$		&	356.4804	&	90.2221		&	23.0937		&	6.0308		&	1.6266	&	0.4588	&	0.1363	\\
			&				&	$\Ex{(2)}$		&	1421.56773	&	358.7489	&	91.3395		&	23.6352		&	6.2850	&	1.7395	&	0.5055	\\
	\hline
			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
	\hline
	CIS		&				&	$\Ex{(1)}$		&	0.0230		&	0.0230		&	0.0229		&	0.0229		&	0.0230	&	0.0225	&	0.0182	\\
	\\
	TDHF	&				&	$\Ex{(1)}$		&	0.0005		&	0.0005		&	0.0005		&	0.0005		&	0.0005	&	0.0005	&	0.0006	\\
	\\
	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0230		&	0.0230		&	0.0228		&	0.0223		&	0.0192	&	-0.0015	&	-0.0309	\\
	\\
	TDLDA	&				&	$\Ex{(1)}$		&	0.0005		&	0.0005		&	0.0004		&	0.0000		&	-0.0033	&	-0.0248	&	-0.0650	\\
	\\
	eHF	&	$(0,0)$			&	$\E{(0)}$		&	0.0769	&	0.0755	&	0.0728	&	0.0676	&	0.0584	&	0.0439	&	0.0257	\\
			&				&	$\E{(1)}$		&	0.1642	&	0.1628	&	0.1603	&	0.1556	&	0.1475	&	0.1344	&	0.1120	\\
			&				&	$\E{(2)}$		&	0.1800	&	0.1777	&	0.1732	&	0.1645	&	0.1493	&	0.1260	&	0.0988	\\
			&				&	$\Ex{(1)}$		&	0.2228	&	0.2215	&	0.2189	&	0.2143	&	0.2066	&	0.1933	&	0.1626	\\
			&				&	$\Ex{(2)}$		&	0.2387	&	0.2364	&	0.2318	&	0.2233	&	0.2084	&	0.1850	&	0.1495	\\
	\\
	eHF	&	$(1/3,1/3)$		&	$\E{(0)}$		&	0.1426	&	0.1411	&	0.1382	&	0.1326	&	0.1225	&	0.1059	&	0.0823	\\
			&				&	$\E{(1)}$		&	0.1347	&	0.1329	&	0.1295	&	0.1233	&	0.1124	&	0.0957	&	0.0737	\\
			&				&	$\E{(2)}$		&	0.2019	&	0.1996	&	0.1950	&	0.1860	&	0.1696	&	0.1429	&	0.1070	\\
			&				&	$\Ex{(1)}$		&	0.1277	&	0.1260	&	0.1228	&	0.1169	&	0.1071	&	0.0943	&	0.0785	\\
			&				&	$\Ex{(2)}$		&	0.1949	&	0.1926	&	0.1882	&	0.1797	&	0.1643	&	0.1414	&	0.1119	\\
	\\
	KS-eLDA	&	$(0,0)$		&	$\E{(0)}$		&	-0.0587	&	-0.0586	&	-0.0587	&	-0.0588	&	-0.0591	&	-0.0590	&	-0.0506	\\
			&				&	$\E{(1)}$		&	0.0457	&	0.0450	&	0.0435	&	0.0409	&	0.0362	&	0.0241	&	0.0033	\\
			&				&	$\E{(2)}$		&	0.0861	&	0.0838	&	0.0793	&	0.0712	&	0.0571	&	0.0377	&	0.0196	\\
			&				&	$\Ex{(1)}$		&	0.1044	&	0.1036	&	0.1022	&	0.0997	&	0.0953	&	0.0830	&	0.0540	\\
			&				&	$\Ex{(2)}$		&	0.1447	&	0.1424	&	0.1380	&	0.1300	&	0.1162	&	0.0966	&	0.0703	\\
			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	\\
			&				&	$\DD{c}{(1)}$	&	0.0172	&	0.0163	&	0.0147	&	0.0117	&	0.0067	&	-0.0004	&	-0.0080	\\
			&				&	$\DD{c}{(2)}$	&	0.0416	&	0.0402	&	0.0376	&	0.0329	&	0.0253	&	0.0140	&	0.0005	\\
	\\
	KS-eLDA	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0070	&	0.0070	&	0.0068	&	0.0063	&	0.0053	&	0.0015	&	-0.0049	\\
			&				&	$\E{(1)}$		&	0.0162	&	0.0151	&	0.0128	&	0.0086	&	0.0018	&	-0.0066	&	-0.0095	\\
			&				&	$\E{(2)}$		&	0.1080	&	0.1056	&	0.1011	&	0.0925	&	0.0772	&	0.0538	&	0.0325	\\
			&				&	$\Ex{(1)}$		&	0.0092	&	0.0081	&	0.0060	&	0.0022	&	-0.0035	&	-0.0081	&	-0.0047	\\
			&				&	$\Ex{(2)}$		&	0.1010	&	0.0986	&	0.0943	&	0.0862	&	0.0719	&	0.0523	&	0.0373	\\
			&				&	$\DD{c}{(0)}$	&	-0.0196	&	-0.0188	&	-0.0174	&	-0.0148	&	-0.0106	&	-0.0044	&	0.0029	\\
			&				&	$\DD{c}{(1)}$	&	-0.0024	&	-0.0025	&	-0.0027	&	-0.0032	&	-0.0040	&	-0.0050	&	-0.0056	\\
			&				&	$\DD{c}{(2)}$	&	0.0220	&	0.0213	&	0.0201	&	0.0180	&	0.0146	&	0.0093	&	0.0027	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}

%%% TABLE VI %%%
\begin{table*}
	\caption{
	\label{tab:OptGap_6}
	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 6-boxium (i.e.,~$\Nel = 6$ electrons in a box of length $L$).
	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
	}
	\begin{ruledtabular}
	\begin{tabular}{lclddddddd}
			&				&					&	\mc{7}{c}{6-boxium with a box of length $L$}	\\
													\cline{4-10}
	Method	&	$\bw$		&	State			&	\pi/8			&	\pi/4			&	\pi/2			&	\pi		&	2\pi		&	4\pi		&	8\pi		\\
	\hline				
	FCI		&				&	$\E{(0)}$		&	3082.5386	&	813.0910	&	224.3734	&	66.5257	&	21.7454	&	7.9136	&	3.1633	\\
			&				&	$\E{(1)}$		&	3503.4911	&	919.5487	&	251.5842	&	73.6145	&	23.6504	&	8.4487	&	3.3217	\\
			&				&	$\E{(2)}$		&	4762.0921	&	1236.8257	&	332.1993	&	94.3988	&	29.1455	&	9.9582	&	3.7572	\\
			&				&	$\Ex{(1)}$		&	420.9525	&	106.4577	&	27.2108		&	7.0888	&	1.9050	&	0.5351	&	0.1583	\\
			&				&	$\Ex{(2)}$		&	1679.5536	&	423.7347	&	107.8259	&	27.8731	&	7.4001	&	2.0446	&	0.5938	\\
	\hline
			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
	\hline
	CIS		&				&	$\Ex{(1)}$		&	0.0249		&	0.0248		&	0.0250		&	0.0253	&	0.0261	&	0.0272	&	0.0248	\\
	\\
	TDHF	&				&	$\Ex{(1)}$		&	0.0002		&	0.0000		&	0.0003		&	0.0003	&	0.0003	&	0.0003	&	0.0004	\\
	\\
	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0249		&	0.0248		&	0.0250		&	0.0250	&	0.0242	&	0.0114	&	-0.0223		\\
	\\
	TDLDA	&				&	$\Ex{(1)}$		&	0.0002		&	0.0000		&	0.0002		&	0.0000	&	-0.0016	&	-0.0162	&	-0.0612	\\
	\\
	eHF	&	$(0,0)$			&	$\E{(0)}$		&	0.1004	&	0.0989	&	0.0960	&	0.0903	&	0.0802	&	0.0634	&	0.0403	\\
			&				&	$\E{(1)}$		&	0.1908	&	0.1893	&	0.1870	&	0.1824	&	0.1747	&	0.1627	&	0.1422	\\
			&				&	$\E{(2)}$		&	0.2144	&	0.2120	&	0.2073	&	0.1985	&	0.1830	&	0.1595	&	0.1333	\\
			&				&	$\Ex{(1)}$		&	0.2534	&	0.2520	&	0.2498	&	0.2456	&	0.2387	&	0.2281	&	0.2035	\\
			&				&	$\Ex{(2)}$		&	0.2770	&	0.2747	&	0.2701	&	0.2617	&	0.2470	&	0.2249	&	0.1946	\\
	\\
	eHF	&	$(1/3,1/3)$		&	$\E{(0)}$		&	0.1676	&	0.1660	&	0.1631	&	0.1574	&	0.1471	&	0.1300	&	0.1049	\\
			&				&	$\E{(1)}$		&	0.1580	&	0.1561	&	0.1528	&	0.1464	&	0.1352	&	0.1173	&	0.0924	\\
			&				&	$\E{(2)}$		&	0.2294	&	0.2270	&	0.2222	&	0.2130	&	0.1966	&	0.1702	&	0.1336	\\
			&				&	$\Ex{(1)}$		&	0.1534	&	0.1516	&	0.1485	&	0.1425	&	0.1321	&	0.1167	&	0.0991	\\
			&				&	$\Ex{(2)}$		&	0.2249	&	0.2225	&	0.2179	&	0.2091	&	0.1935	&	0.1696	&	0.1404	\\
			\\
	KS-eLDA	&	$(0,0)$		&	$\E{(0)}$		&	-0.0626	&	-0.0627	&	-0.0628	&	-0.0632	&	-0.0641	&	-0.0654	&	-0.0612	\\
			&				&	$\E{(1)}$		&	0.0486	&	0.0477	&	0.0465	&	0.0440	&	0.0400	&	0.0308	&	0.0078	\\
			&				&	$\E{(2)}$		&	0.1017	&	0.0992	&	0.0946	&	0.0862	&	0.0718	&	0.0507	&	0.0271	\\
			&				&	$\Ex{(1)}$		&	0.1112	&	0.1104	&	0.1093	&	0.1072	&	0.1041	&	0.0962	&	0.0690	\\
			&				&	$\Ex{(2)}$		&	0.1643	&	0.1619	&	0.1575	&	0.1494	&	0.1358	&	0.1162	&	0.0884	\\
			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	\\
			&				&	$\DD{c}{(1)}$	&	0.0208	&	0.0199	&	0.0182	&	0.0151	&	0.0098	&	0.0018	&	-0.0075	\\
			&				&	$\DD{c}{(2)}$	&	0.0503	&	0.0488	&	0.0460	&	0.0412	&	0.0330	&	0.0205	&	0.0046	\\
	\\
	KS-eLDA	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0046	&	0.0045	&	0.0043	&	0.0039	&	0.0031	&	0.0006	&	-0.0067		\\
			&				&	$\E{(1)}$		&	0.0157	&	0.0144	&	0.0123	&	0.0080	&	0.0009	&	-0.0091	&	-0.0160		\\
			&				&	$\E{(2)}$		&	0.1167	&	0.1142	&	0.1095	&	0.1007	&	0.0853	&	0.0616	&	0.0355		\\
			&				&	$\Ex{(1)}$		&	0.0112	&	0.0099	&	0.0080	&	0.0041	&	-0.0022	&	-0.0097	&	-0.0093		\\
			&				&	$\Ex{(2)}$		&	0.1121	&	0.1097	&	0.1051	&	0.0968	&	0.0822	&	0.0610	&	0.0423		\\
			&				&	$\DD{c}{(0)}$	&	-0.0237	&	-0.0229	&	-0.0214	&	-0.0188	&	-0.0142	&	-0.0073	&	0.0013	\\
			&				&	$\DD{c}{(1)}$	&	-0.0029	&	-0.0030	&	-0.0032	&	-0.0037	&	-0.0045	&	-0.0057	&	-0.0066	\\
			&				&	$\DD{c}{(2)}$	&	0.0266	&	0.0259	&	0.0246	&	0.0224	&	0.0187	&	0.0130	&	0.0053	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}


%%% TABLE VII %%%
\begin{table*}
	\caption{
	\label{tab:OptGap_7}
	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 7-boxium (i.e.,~$\Nel = 7$ electrons in a box of length $L$).
	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
	}
	\begin{ruledtabular}
	\begin{tabular}{lclddddddd}
			&				&					&	\mc{7}{c}{7-boxium with a box of length $L$}	\\
													\cline{4-10}
	Method	&	$\bw$		&	State			&	\pi/8			&	\pi/4			&	\pi/2			&	\pi		&	2\pi		&	4\pi		&	8\pi		\\
	\hline				
	FCI		&				&	$\E{(0)}$		&	4729.98018	&	1244.7753	&	342.1796	&	100.8943	&	32.7728	&	11.8683	&	4.7359	\\
			&				&	$\E{(1)}$		&	5215.3307	&	1367.4316	&	373.4897	&	109.0326	&	34.9524	&	12.4779	&	4.9156	\\
			&				&	$\E{(2)}$		&	6667.18516	&	1733.3319	&	466.4133	&	132.9686	&	41.2715	&	14.2096	&	5.4146	\\
			&				&	$\Ex{(1)}$		&	485.3505	&	122.6563	&	31.3101		&	8.1382		&	2.1796	&	0.6096	&	0.1797	\\
			&				&	$\Ex{(2)}$		&	1937.2050	&	488.5566	&	124.2336	&	32.0743		&	8.4987	&	2.3413	&	0.6787	\\
	\hline
			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
	\hline
	CIS		&				&	$\Ex{(1)}$		&	0.0262		&	0.0264		&	0.0265		&	0.0270		&	0.0283	&	0.0308	&	0.0309	\\
	\\
	TDHF	&				&	$\Ex{(1)}$		&	0.0000		&	0.0001		&	0.0000		&	0.0000		&	0.0000	&	0.0001	&	0.0003	\\
	\\
	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0262		&	0.0264		&	0.0264		&	0.0269		&	0.0273	&	0.0206	&	-0.0116	\\
	\\
	TDLDA	&				&	$\Ex{(1)}$		&	0.0000		&	0.0001		&	0.0000		&	-0.0001		&	-0.0009	&	-0.0107	&	-0.0539	\\
	\\
	eHF	&	$(0,0)$			&	$\E{(0)}$		&	0.1240	&	0.1224	&	0.1194	&	0.1135	&	0.1027	&	0.0842	&	0.0570	\\
			&				&	$\E{(1)}$		&	0.2162	&	0.2150	&	0.2125	&	0.2081	&	0.2006	&	0.1894	&	0.1711	\\
			&				&	$\E{(2)}$		&	0.2437	&	0.2419	&	0.2376	&	0.2291	&	0.2137	&	0.1916	&	0.1650	\\
			&				&	$\Ex{(1)}$		&	0.2473	&	0.2458	&	0.2425	&	0.2366	&	0.2263	&	0.2099	&	0.1825	\\
			&				&	$\Ex{(2)}$		&	0.1796	&	0.1780	&	0.1746	&	0.1685	&	0.1576	&	0.1406	&	0.1202	\\
	\\
	eHF	&	$(1/3,1/3)$		&	$\E{(0)}$		&	0.1918	&	0.1902	&	0.1873	&	0.1817	&	0.1713	&	0.1538	&	0.1273	\\
			&				&	$\E{(1)}$		&	0.1810	&	0.1792	&	0.1758	&	0.1694	&	0.1579	&	0.1392	&	0.1122	\\
			&				&	$\E{(2)}$		&	0.2532	&	0.2514	&	0.2470	&	0.2381	&	0.2220	&	0.1969	&	0.1584	\\
			&				&	$\Ex{(1)}$		&	0.1796	&	0.1780	&	0.1746	&	0.1685	&	0.1576	&	0.1406	&	0.1202	\\
			&				&	$\Ex{(2)}$		&	0.2518	&	0.2501	&	0.2458	&	0.2372	&	0.2217	&	0.1982	&	0.1663	\\
			\\
	KS-eLDA	&	$(0,0)$		&	$\E{(0)}$		&	-0.0664		&	-0.0666		&	-0.0667	&	-0.0672	&	-0.0684	&	-0.0707	&	-0.0702		\\
			&				&	$\E{(1)}$		&	0.0502		&	0.0495		&	0.0482	&	0.0459	&	0.0423	&	0.0355	&	0.0131		\\
			&				&	$\E{(2)}$		&	0.1122		&	0.1104		&	0.1061	&	0.0979	&	0.0836	&	0.0635	&	0.0360		\\
			&				&	$\Ex{(1)}$		&	0.1165		&	0.1161		&	0.1149	&	0.1131	&	0.1108	&	0.1062	&	0.0834		\\
			&				&	$\Ex{(2)}$		&	0.1785		&	0.1769		&	0.1728	&	0.1652	&	0.1520	&	0.1342	&	0.1063		\\
			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000		\\
			&				&	$\DD{c}{(1)}$	&	0.0244	&	0.0235	&	0.0218	&	0.0186	&	0.0130	&	0.0043	&	-0.0065		\\
			&				&	$\DD{c}{(2)}$	&	0.0589	&	0.0574	&	0.0546	&	0.0496	&	0.0410	&	0.0275	&	0.0095		\\
	\\
	KS-eLDA	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0014	&	0.0013	&	0.0012	&	0.0009	&	0.0003	&	-0.0013	&	-0.0079		\\
			&				&	$\E{(1)}$		&	0.0149	&	0.0138	&	0.0115	&	0.0072	&	-0.0001	&	-0.0110	&	-0.0209		\\
			&				&	$\E{(2)}$		&	0.1217	&	0.1198	&	0.1154	&	0.1069	&	0.0917	&	0.0691	&	0.0389		\\
			&				&	$\Ex{(1)}$		&	0.0135	&	0.0125	&	0.0103	&	0.0063	&	-0.0005	&	-0.0096	&	-0.0130		\\
			&				&	$\Ex{(2)}$		&	0.1203	&	0.1185	&	0.1142	&	0.1060	&	0.0914	&	0.0705	&	0.0469		\\
			&				&	$\DD{c}{(0)}$	&	-0.0278	&	-0.0270	&	-0.0255	&	-0.0227	&	-0.0180	&	-0.0105	&	-0.0007	\\
			&				&	$\DD{c}{(1)}$	&	-0.0034	&	-0.0034	&	-0.0037	&	-0.0041	&	-0.0050	&	-0.0063	&	-0.0076	\\
			&				&	$\DD{c}{(2)}$	&	0.0311	&	0.0304	&	0.0291	&	0.0268	&	0.0230	&	0.0168	&	0.0083	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}


\bibliography{../eDFT}

\end{document}