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% functionals, potentials, densities, etc
\newcommand { \eps } { \epsilon }
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\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} }
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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 %%%%%
\newcommand { \manu } [1]{ { \textcolor { blue} { Manu: #1 } } }
\newcommand { \beq } { \begin { eqnarray} }
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%
\newcommand { \bmk } { \bm { \kappa } } % orbital rotation vector
\newcommand { \bmg } { \bm { \gamma } } % orbital rotation vector
\newcommand { \bfx } { \bf { x} }
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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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Ensemble Hartree--Fock method}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%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%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section * { Construction of the density-functional approximations}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}
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where $ \omega = \theta _ 1 - \theta _ 2 $ is the interelectronic angle, $ \Gamma ( x ) $ is the Gamma function, \cite { NISTbook} and
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\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 {
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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.
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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}
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\e { c} { (I)} (n) = \frac { a_ 1^ { (I)} \, n} { n + a_ 2^ { (I)} \sqrt { n} + a_ 3^ { (I)} } ,
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\end { equation}
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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}
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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*}
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\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}
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\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.
}
\label { fig:EvsL}
\end { figure*}
%%% %%% %%%
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%%% FIG 3 %%%
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\begin { figure*}
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\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}
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\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*}
%%% %%% %%%
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%%% TABLE II %%%
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\begin { table*}
\caption {
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\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.)
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}
\begin { ruledtabular}
\begin { tabular} { lclddddddd}
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& & & \mc { 7} { c} { 2-boxium with a box of length $ L $ } \\
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\cline { 4-10}
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Method & $ \bw $ & State & \pi /8 & \pi /4 & \pi /2 & \pi & 2\pi & 4\pi & 8\pi \\
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\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 \\
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\hline
& & & \mc { 7} { c} { Deviation from FCI} \\
\hline
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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 \\
\\
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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} \\
\\
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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} \\
\\
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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} \\
\\
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} \\
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\end { tabular}
\end { ruledtabular}
\end { table*}
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%%% TABLE III %%%
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\begin { table*}
\caption {
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\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.
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}
\begin { ruledtabular}
\begin { tabular} { lclddddddd}
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& & & \mc { 7} { c} { 3-boxium with a box of length $ L $ } \\
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\cline { 4-10}
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Method & $ \bw $ & State & \pi /8 & \pi /4 & \pi /2 & \pi & 2\pi & 4\pi & 8\pi \\
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\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 \\
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\hline
& & & \mc { 7} { c} { Deviation from FCI} \\
\hline
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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 \\
\\
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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 \\
\\
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eLDA & $ ( 0 , 0 ) $ & $ \E { ( 0 ) } $ & -0.0481 & -0.0478 & -0.0473 & -0.0463 & -0.0446 & -0.0387 & -0.0257 \\
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& & $ \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 \\
\\
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eLDA & $ ( 1 / 3 , 1 / 3 ) $ & $ \E { ( 0 ) } $ & 0.0078 & 0.0080 & 0.0082 & 0.0085 & 0.0081 & 0.0024 & -0.0022 \\
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& & $ \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*}
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%%% TABLE IV %%%
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\begin { table*}
\caption {
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\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.
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}
\begin { ruledtabular}
\begin { tabular} { lclddddddd}
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& & & \mc { 7} { c} { 4-boxium with a box of length $ L $ } \\
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\cline { 4-10}
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Method & $ \bw $ & State & \pi /8 & \pi /4 & \pi /2 & \pi & 2\pi & 4\pi & 8\pi \\
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\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 \\
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\hline
& & & \mc { 7} { c} { Deviation from FCI} \\
\hline
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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 \\
\\
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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 \\
\\
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eLDA & $ ( 0 , 0 ) $ & $ \E { ( 0 ) } $ & -0.0541 & -0.0539 & -0.0537 & -0.0534 & -0.0529 & -0.0504 & -0.0386 \\
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& & $ \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 \\
\\
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eLDA & $ ( 1 / 3 , 1 / 3 ) $ & $ \E { ( 0 ) } $ & 0.0085 & 0.0085 & 0.0084 & 0.0082 & 0.0072 & 0.0021 & -0.0015 \\
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& & $ \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*}
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%%% TABLE V %%%
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\begin { table*}
\caption {
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\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.
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}
\begin { ruledtabular}
\begin { tabular} { lclddddddd}
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& & & \mc { 7} { c} { 5-boxium with a box of length $ L $ } \\
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\cline { 4-10}
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Method & $ \bw $ & State & \pi /8 & \pi /4 & \pi /2 & \pi & 2\pi & 4\pi & 8\pi \\
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\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 \\
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\hline
& & & \mc { 7} { c} { Deviation from FCI} \\
\hline
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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 \\
\\
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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 \\
\\
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eLDA & $ ( 0 , 0 ) $ & $ \E { ( 0 ) } $ & -0.0587 & -0.0586 & -0.0587 & -0.0588 & -0.0591 & -0.0590 & -0.0506 \\
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& & $ \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 \\
\\
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eLDA & $ ( 1 / 3 , 1 / 3 ) $ & $ \E { ( 0 ) } $ & 0.0070 & 0.0070 & 0.0068 & 0.0063 & 0.0053 & 0.0015 & -0.0049 \\
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& & $ \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*}
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%%% TABLE VI %%%
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\begin { table*}
\caption {
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\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.
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}
\begin { ruledtabular}
\begin { tabular} { lclddddddd}
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& & & \mc { 7} { c} { 6-boxium with a box of length $ L $ } \\
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\cline { 4-10}
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Method & $ \bw $ & State & \pi /8 & \pi /4 & \pi /2 & \pi & 2\pi & 4\pi & 8\pi \\
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\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 \\
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\hline
& & & \mc { 7} { c} { Deviation from FCI} \\
\hline
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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 \\
\\
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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 \\
\\
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eLDA & $ ( 0 , 0 ) $ & $ \E { ( 0 ) } $ & -0.0626 & -0.0627 & -0.0628 & -0.0632 & -0.0641 & -0.0654 & -0.0612 \\
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& & $ \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 \\
\\
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eLDA & $ ( 1 / 3 , 1 / 3 ) $ & $ \E { ( 0 ) } $ & 0.0046 & 0.0045 & 0.0043 & 0.0039 & 0.0031 & 0.0006 & -0.0067 \\
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& & $ \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*}
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%%% TABLE VII %%%
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\begin { table*}
\caption {
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\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.
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}
\begin { ruledtabular}
\begin { tabular} { lclddddddd}
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& & & \mc { 7} { c} { 7-boxium with a box of length $ L $ } \\
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\cline { 4-10}
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Method & $ \bw $ & State & \pi /8 & \pi /4 & \pi /2 & \pi & 2\pi & 4\pi & 8\pi \\
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\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 \\
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\hline
& & & \mc { 7} { c} { Deviation from FCI} \\
\hline
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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 \\
\\
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eHF & $ ( 0 , 0 ) $ & $ \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 ) } $ & 0.2473 & 0.2458 & 0.2425 & 0.2366 & 0.2263 & 0.2099 & 0.1825 \\
& & $ \Ex { ( 2 ) } $ & 0.3196 & 0.3180 & 0.3137 & 0.3054 & 0.2904 & 0.2676 & 0.2286 \\
\\
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 \\
\\
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eLDA & $ ( 0 , 0 ) $ & $ \E { ( 0 ) } $ & -0.0664 & -0.0666 & -0.0667 & -0.0672 & -0.0684 & -0.0707 & -0.0702 \\
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& & $ \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 \\
\\
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eLDA & $ ( 1 / 3 , 1 / 3 ) $ & $ \E { ( 0 ) } $ & 0.0014 & 0.0013 & 0.0012 & 0.0009 & 0.0003 & -0.0013 & -0.0079 \\
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& & $ \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}
