\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{txfonts} \usepackage[ colorlinks=true, citecolor=blue, breaklinks=true ]{hyperref} \urlstyle{same} %useful stuff \newcommand{\cdash}{\multicolumn{1}{c}{---}} \newcommand{\mc}{\multicolumn} \newcommand{\mr}{\multirow} \newcommand{\fnm}{\footnotemark} \newcommand{\fnt}{\footnotetext} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} % functionals, potentials, densities, etc \newcommand{\eps}{\epsilon} \newcommand{\e}[2]{\eps_\text{#1}^{#2}} \renewcommand{\v}[2]{v_\text{#1}^{#2}} \newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}} \newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}} \newcommand{\n}[1]{n^{#1}} \newcommand{\DD}[2]{\Delta_\text{#1}^{#2}} \newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}} % energies \newcommand{\EHF}{E_\text{HF}} \newcommand{\Ec}{E_\text{c}} \newcommand{\Ecat}{E_\text{cat}} \newcommand{\Eneu}{E_\text{neu}} \newcommand{\Eani}{E_\text{ani}} \newcommand{\EPT}{E_\text{PT2}} \newcommand{\EFCI}{E_\text{FCI}} % matrices \newcommand{\br}{\bm{r}} \newcommand{\bw}{\bm{w}} \newcommand{\bG}{\bm{G}} \newcommand{\bS}{\bm{S}} \newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}} \newcommand{\bH}{\bm{H}} \newcommand{\bHc}{\bm{H}^\text{c}} \newcommand{\bF}[1]{\bm{F}^{#1}} \newcommand{\Ex}[1]{\Omega^{#1}} \newcommand{\E}[1]{E^{#1}} % elements \newcommand{\ew}[1]{w_{#1}} \newcommand{\eG}[1]{G_{#1}} \newcommand{\eS}[1]{S_{#1}} \newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}} \newcommand{\eHc}[1]{H_{#1}^\text{c}} \newcommand{\eF}[2]{F_{#1}^{#2}} % Numbers \newcommand{\Nel}{N} \newcommand{\Nbas}{K} % Ao and MO basis \newcommand{\MO}[2]{\phi_{#1}^{#2}} \newcommand{\cMO}[2]{c_{#1}^{#2}} \newcommand{\AO}[1]{\chi_{#1}} % units \newcommand{\IneV}[1]{#1~eV} \newcommand{\InAU}[1]{#1~a.u.} \newcommand{\InAA}[1]{#1~\AA} \newcommand{\SI}{\textcolor{blue}{supplementary material}} \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}} \newcommand{\eeq}{\end{eqnarray}} % \newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector \newcommand{\bmg}{\bm{\gamma}} % orbital rotation vector \newcommand{\bfx}{\bf{x}} \newcommand{\bfr}{\bf{r}} %%%% \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_{qp}\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 ensemble correlation derivative $\DD{c}{(I)}$ 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 ensemble correlation derivative $\DD{c}{(I)}$ 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 ensemble correlation derivative $\DD{c}{(I)}$ 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 ensemble correlation derivative $\DD{c}{(I)}$ 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 ensemble correlation derivative $\DD{c}{(I)}$ 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 ensemble correlation derivative $\DD{c}{(I)}$ 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}