First commit

FCI/README.txt

@@ -0,0 +1,13 @@
+On 19 Dec 2013, Peter Knowles sent me various files to allow me to do Full CI calculations.
+
+To compile, I needed to do:
+
+	gfortran -lblas fci.f standard.f
+
+To run, I needed to do:
+
+	./a.out < hf.dat
+
+The resulting output compared well with the file hf.out that Peter K provided.
+
+The program uses single-bar, not double-bar, integrals.

FCI/standard.f

@@ -0,0 +1,27 @@
+      SUBROUTINE GMAINV (QQ,IBASE,LENN)
+C.....SUBSTITUTE FOR DYNAMIC MEMORY ALLOCATION
+C.... THIS ROUTINE MUST BE CALLED WITH QQ FIRST WORD IN COMMON/BIG/
+C....  THIS ROUTINE IS NOT FOOLPROOF
+      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+      PARAMETER (MAXM = 1 100 000 000)
+      COMMON/BIG/Q(MAXM)
+      COMMON /CORCTL/ INTREL,ICORCT(8)
+C...  INTREL MUST BE NUMBER OF INTEGERS PER REAL
+      INTREL = 2
+C
+      LENN = MIN(MAXM,LENN)
+      IBASE = 1
+      RETURN
+      ENTRY FMAIN (QQ,LENN)
+C.....SUBSITUTE FOR MEMORY RELEASE
+      RETURN
+      END
+      FUNCTION SECOND()
+      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+C.....SHOUld RETURN CPU TIME IN SECONDS
+      SAVE T
+      DATA T/0D0/
+      SECOND=T
+      T=T+.01D0
+      RETURN
+      END

Manuscript/SI/eDFT-SI.tex

@@ -0,0 +1,928 @@
+\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
+
+\usepackage{mathpazo,libertine}
+\newcommand{\alert}[1]{\textcolor{red}{#1}}
+\definecolor{darkgreen}{RGB}{0, 180, 0}
+
+\usepackage{hyperref}
+\hypersetup{
+    colorlinks=true,
+    linkcolor=blue,
+    filecolor=blue,      
+    urlcolor=blue,	
+    citecolor=blue
+}
+
+%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_{q<p}\kappa_{pq}\varphi_q({\bfx})-\sum_{q>p}\kappa_{qp}\varphi_q({\bfx})
++\mathcal{O}\left({\bmk}^2\right).
+\eeq 
+The (${\bmk}$-independent) one-electron reduced density matrices (1RDMs)
+in Eq.~(\ref{eq:eHF_ener}) are defined in the unrotated molecular
+spin-orbital basis for each (unrotated) determinant $\Phi^{(K)}$ belonging to
+the ensemble as follows: $D^{(K)}_{pr}=\delta_{pr}$ if $\varphi_p$ and
+$\varphi_r$ are both
+occupied in $\Phi^{(K)}$, otherwise $D^{(K)}_{pr}=0$. If the unrotated
+spin-orbitals are the minimizing ensemble HF ones, then the following
+stationarity condition is fulfilled,
+\beq\label{eq:station_cond}
+\left.\dfrac{\partial E^{\bw}_{\rm
+HF}({\bm\kappa})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=0, 
+\eeq
+with $l>m$. Since, according to Eq.~(\ref{eq:orb_taylor_expansion}),
+\beq
+\left.\dfrac{\partial 
+\varphi_p({\bmk},{\bfx})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=\delta_{lp}\varphi_m({\bfx})-\delta_{mp}\varphi_l({\bfx}), 
+\eeq
+Eq.~(\ref{eq:station_cond}) can be written more explicitly as the
+following commutation relation,
+\iffalse%%%%%
+%%%%%% intermediate steps ... %%%%
+\beq
+\sum^M_{K=0}w^{(K)}\sum_qD^{(K)}_{mq}f^{(K)}_{lq}
+-
+\sum^M_{K=0}w^{(K)}\sum_qf^{(K)}_{mq}D^{(K)}_{lq}=0
+\eeq
+% original %%
+\iffalse%%%
+\beq
+&&
+2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
+\nonumber\\
+&&-
+2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
+\nonumber\\
+&&+2\sum_{qrs}\langle \varphi_m\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{lr}D^{(K)}_{qs}
+\nonumber\\
+&&-2\sum_{qrs}\langle \varphi_l\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{mr}D^{(K)}_{qs}
+\nonumber\\
+&&=0\eeq   
+\fi%%%% 
+%%%%%%%
+\beq
+&&
+2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
+\nonumber\\
+&&-
+2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
+\nonumber\\
+&&+2\sum_{qrs}\langle \varphi_m\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}D^{(K)}_{rs}
+\nonumber\\
+&&-2\sum_{qrs}\langle \varphi_l\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}D^{(K)}_{rs}
+\nonumber\\
+&&=0\eeq    
+%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%
+\fi%%%%
+%%%%%%%
+\beq\label{eq:stat_cond_commut_ind}
+\sum^M_{K=0}w^{(K)}\left[{\bm f}^{(K)},{\bm D}^{(K)}\right]=0,
+\eeq
+where the $K$th Fock matrix elements read
+\beq
+f^{(K)}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
++
+\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle D^{(K)}_{rs}.
+\eeq
+In the minimizing ensemble HF spin-orbital basis, Eq.~(\ref{eq:stat_cond_commut_ind}) reads 
+\beq
+\sum^M_{K=0}w^{(K)}\Big(\nu^{(K)}_m-\nu_l^{(K)}\Big)f^{(K)}_{lm}=0,
+\eeq
+where $\nu^{(K)}_m$ is the occupation of the spin-orbital $\varphi_m$ in the
+determinant $\Phi^{(K)}$.\\
+
+Note that, in more conventional ensemble calculations, the following HF
+energy expression is employed,
+\beq\label{eq:GI_ensHF_ener}
+&&\tilde{E}^{\bw}_{\rm
+HF}({\bm\kappa})=
+\sum_{pq}\langle
+\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle D^{\bw}_{pq}
+\nonumber\\
+&&
++\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
+\varphi_r(\bmk)\varphi_s(\bmk)\rangle
+%\times
+D^{\bw}_{pr}D^{\bw}_{qs},
+\eeq 
+where ${\bm D}^{\bw}=\sum^M_{K=0}w^{(K)}{\bm D}^{(K)}$ is the ensemble
+1RDM. In this case, the stationarity condition simply reads 
+\beq
+\left[{\bm f}^{\bw},{\bm D}^{\bw}\right]=0,
+\eeq
+where the ensemble Fock matrix elements are defined as follows,
+\beq
+f^{\bw}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
++
+\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle D^{\bw}_{rs}.
+\eeq
+The major issue with the expression of the ensemble energy in
+Eq.~(\ref{eq:GI_ensHF_ener}) is the
+ghost-interaction error from which our expression (see
+Eq.~(\ref{eq:eHF_ener})) is free. Note also
+that, by construction, the ensemble energy in Eq.~(\ref{eq:GI_ensHF_ener}) is quadratic in the
+ensemble weights while ours, like the exact one, varies linearly with
+the weights.    
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Ensemble Hartree--Fock exchange and density-functional
+ghost-interaction correction}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\beq
+F^{\bw}_{\rm HF}[n]&=&
+\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]\right\}
+\nonumber\\
+&=&{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]
+\eeq
+where
+$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\gamma}^{(K)}$ is an ensemble density matrix operator constructed
+from Slater determinants, the ensemble 1RDM elements are $\gamma_{pq}^{\bw}={\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
+and $W_{\rm
+HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\gamma_{pr}\gamma_{qs}$.\\
+
+In-principle-exact decomposition:
+
+\beq
+F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
+Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
+\eeq
+
+The complementary ensemble Hx energy removes the ghost-interaction
+errors introduced in $W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]$:
+\beq
+\overline{E}^{{\bw}}_{\rm
+Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
+HF}\left[{\bmg}^{(K)}[n]\right]
+-W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right],
+\eeq
+which gives in the canonical orbital basis
+\beq
+&&\overline{E}^{{\bw}}_{\rm
+Hx}[n]=
+\dfrac{1}{2}\sum_{pq}
+\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
+\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
+\nonumber\\
+&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
+-\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
+.\eeq
+\manu{I would guess that, in a uniform system, the GOK-DFT and our
+canonical orbitals are the same. This is nice since we can construct
+in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
+Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
+
+Variational expression for the ensemble energy:
+\beq
+E^{{\bw}}=\underset{\hat{\gamma}^{{\bw}}}{\rm min}\Big\{
+&&{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
++
+\overline{E}^{{\bw}}_{\rm
+Hxc}\left[n_{\hat{\gamma}^{{\bw}}}\right]
+%+E^{{\bw}}_{\rm c}\left[n_{\hat{\gamma}^{{\bw}}}\right]
+\nonumber\\
+&&
++\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\gamma}^{{\bw}}}({\bfr})
+\Big\}
+\eeq
+ 
+Note that, if we use orbital rotations, the gradient of the DFT energy
+contributions can be expressed as follows,
+\beq
+\left.\dfrac{\partial 
+\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
+}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}, 
+\eeq
+where   
+\beq
+n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw}
+\eeq
+thus leading to
+\beq
+&&\left.\dfrac{\partial 
+\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
+}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=
+\sum_{pq}\gamma_{pq}^{\bw}
+\nonumber\\
+&&\times\left.\dfrac{\partial} 
+{\partial \kappa_{lm}}
+\Big[\left\langle\varphi_p(\bmk)\middle\vert\hat{\overline{v}}^{{\bw}}_{\rm
+Hxc}
+\middle\vert \varphi_q(\bmk)\right\rangle
+\Big]
+\right|_{{\bmk}=0}.
+\eeq
+
+In conclusion, the minimizing canonical orbitals fulfill the following
+hybrid HF/GOK-DFT equation,
+\beq
+&&\left(-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
+ext}({\bfr})+\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]
++\dfrac{\delta \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}\right)\varphi^{{\bw}}_p({\bfx})
+\nonumber
+\\
+&&=\varepsilon^{{\bw}}_p\varphi^{{\bw}}_p({\bfx}).
+\eeq
+
+
+Since $\partial \gamma_{pq}^{\bw}/\partial
+w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
+
+\manu{just for me ...
+\beq
+&&+\dfrac{1}{2}
+\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
+\nonumber\\
+&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_q\varphi_p\vert\vert
+\varphi_s\varphi_r\rangle
+%\times
+ \gamma^{\bw}_{pr}\left(\gamma_{qs}^{(I)}-\gamma_{qs}^{(0)}\right)
+\nonumber\\
+&&=
+\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
+\nonumber\\
+&&=
+\sum_{pr}\left[\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]\right]_{pr}\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)
+\nonumber\\
+&&=
+\sum_p\left[\hat{u}_{\rm
+HF}\left[\gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right)
+\eeq
+}
+
+\beq
+\dfrac{dE^{\bw}}{dw^{(I)}}=\sum_p\varepsilon^{{\bw}}_p\left(\nu_p^{(I)}-\nu_p^{(0)}\right)+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
+\eeq
+
+LZ shift in this context: $\varepsilon^{{\bw}}_p\rightarrow
+\overline{\varepsilon}^{{\bw}}_p=\varepsilon^{{\bw}}_p+\overline{\Delta}_{\rm
+LZ}^{{\bw}}$ where
+
+\beq
+N\overline{\Delta}_{\rm
+LZ}^{{\bw}}&=&\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]
+-\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}n^{{\bw}}({\bfr})
+\nonumber\\
+&&
+-W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
+\eeq 
+
+such that
+\beq
+E^{{\bw}}=\sum^M_{K=0}w^{(K)}\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p.
+\eeq
+
+Thus we conclude that individual energies can be expressed in principle
+exactly as follows,
+
+\beq
+E^{(K)}=\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p+\sum^M_{I>0}\left(\delta_{IK}-w^{(I)}\right)\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
+\eeq
+%%%%%%%%%%%%%%
+\iffalse%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Generalized GOK-DFT}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The energy to be minimized in a generalized GOK-DFT approach can be
+written as
+\beq
+\eeq
+\fi%%%%%%%%%% 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Construction of the density-functional approximations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The density-functional approximations designed in this manuscript are based on highly-accurate energies for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
+We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
+
+The reduced (i.e.~per electron) HF energy for these three states is:
+\begin{subequations}
+\begin{align}
+	\e{HF}{(0)}(n) & = \frac{\pi^2}{8} n^2 + n,
+	\\
+	\e{HF}{(1)}(n) & = \frac{\pi^2}{2}n^2 + \frac{4}{3} n,		
+	\\	 
+	\e{HF}{(2)}(n) & = \frac{9\pi^2}{8}n^2 + \frac{23}{15} n.	
+\end{align}
+\end{subequations}
+All these states have the same (uniform) density $n = 2/(2\pi R)$ where $R$ is the radius of the ring on which the electrons are confined.
+
+The total energy of the ground and doubly-excited states are given by the two lowest eigenvalues of the Hamiltonian $\bH$ with elements
+\begin{equation}
+\begin{split}
+	H_{ij} 
+		& = \int_0^\pi \qty[ \frac{\psi_i(\omega)}{R} \frac{\psi_j(\omega)}{R} + \frac{\psi_i(\omega)\psi_j(\omega)}{2R\sin(\omega/2)} ] d\omega
+		\\
+		& = \frac{\sqrt{\pi}}{2 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{2})} + \frac{ij}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+2}{2})} ],
+\end{split}
+\end{equation}
+where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function \cite{NISTbook}, and
+\begin{equation}
+	\psi_i(\omega) = \sin(\omega/2) \sin^{i-1}(\omega/2),	\quad	i=1,\ldots,M
+\end{equation}
+are (non-orthogonal) explicitly-correlated basis functions with overlap matrix elements
+\begin{equation}
+	S_{ij} 
+		= \int_0^\pi \psi_i(\omega)\psi_j(\omega) d\omega
+		= \sqrt{\pi} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+2}{2})}.
+\end{equation}
+Thanks to this explicitly-correlated basis, the convergence rate of the energy is exponential with respect to $M$.
+Therefore, high accuracy is reached with a very small number of basis functions.
+Here, we typically use $M=10$.
+For the singly-excited state, one has to modify the basis functions as 
+\begin{equation}
+	\psi_i(\omega) = \cos(\omega/2) \sin^{i-1} (\omega/2),
+\end{equation}
+and its energy is obtained by the lowest root of the Hamiltonian in this basis, and the matrix elements reads
+\begin{align}
+	H_{ij} & = \frac{\sqrt{\pi}}{4 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{3})} + \frac{3ij+i+j-1}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+4}{2})} ],
+	\\
+	S_{ij} & = \frac{\sqrt{\pi}}{2} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+4}{2})}.
+\end{align}
+The numerical values of the correlation energy for various $R$ are reported in Table \ref{tab:Ref} for the three states of interest.
+
+%%% FIG 1 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{Ec}
+	\caption{
+	$\e{c}{(I)}$ [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^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
+\end{equation}
+where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
+The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
+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=0.25\linewidth]{EvsL_2}
+	\includegraphics[height=0.25\linewidth]{EvsL_3}
+	\includegraphics[height=0.25\linewidth]{EvsL_4}
+	\includegraphics[height=0.25\linewidth]{EvsL_5}
+	\includegraphics[height=0.25\linewidth]{EvsL_6}
+	\includegraphics[height=0.25\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.
+	}
+	\label{fig:EvsL}
+\end{figure*}
+%%% %%% %%%
+
+%%% FIG 1 %%%
+\begin{figure*}
+	\includegraphics[height=0.25\linewidth]{EvsN_0125}
+	\includegraphics[height=0.25\linewidth]{EvsN_025}
+	\includegraphics[height=0.25\linewidth]{EvsN_05}
+	\includegraphics[height=0.25\linewidth]{EvsN_1}
+	\includegraphics[height=0.25\linewidth]{EvsN_2}
+	\includegraphics[height=0.25\linewidth]{EvsN_4}
+	\includegraphics[height=0.25\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 I %%%
+\begin{table*}
+	\caption{
+	\label{tab:OptGap}
+	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 $\Nel = 4$ electrons in a box of length $L$.
+	The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+	}
+	\begin{ruledtabular}
+	\begin{tabular}{lclddddddd}
+			&				&					&	\mc{7}{c}{$L/\pi$}	\\
+													\cline{4-10}
+	Method	&	$\bw$		&	State			&	1/8			&	1/4			&	1/2			&	1		&	2		&	4		&	8		\\
+	\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	\\
+	\\
+	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	&		\\
+	\\
+	TDLDA	&				&	$\Ex{(1)}$		&	0.0015		&	0.0006		&	-0.0018		&	-0.0106	&	-0.0370	&	-0.0518	&		\\
+	\\
+	eDFT	&	$(0,0)$		&	$\E{(0)}$		&	-0.0397	&	-0.0391	&	-0.0380	&	-0.0361	&	-0.0323	&	-0.0236	\\
+			&				&	$\E{(1)}$		&	0.0215	&	0.0213	&	0.0210	&	0.0200	&	0.0159	&	0.0102	\\
+			&				&	$\E{(2)}$		&	-0.0426	&	-0.0425	&	-0.0419	&	-0.0386	&	-0.0249	&	-0.0044	\\
+			&				&	$\Ex{(1)}$		&	0.0612	&	0.0604	&	0.0590	&	0.0561	&	0.0483	&	0.0338	\\
+			&				&	$\Ex{(2)}$		&	-0.0029	&	-0.0034	&	-0.0039	&	-0.0025	&	0.0074	&	0.0191	\\
+			&				&	$\DD{c}{(0)}$	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	&	0.0000	\\
+			&				&	$\DD{c}{(1)}$	&	0.0064	&	0.0056	&	0.0043	&	0.0022	&	-0.0007	&	-0.0037	&	\\
+			&				&	$\DD{c}{(2)}$	&	0.0159	&	0.0147	&	0.0126	&	0.0093	&	0.0046	&	-0.0009	&	\\
+			&				&	$\LZ{c}{}$		&	0.0013	&	0.0024	&	0.0043	&	0.0070	&	0.0100	&	0.0121	&	\\
+	\\
+	eDFT	&	$(1/3,1/3)$	&	$\E{(0)}$		&	0.0031	&	0.0036	&	0.0044	&	0.0054	&	0.0042	&	-0.0025		\\
+			&				&	$\E{(1)}$		&	0.0090	&	0.0087	&	0.0083	&	0.0076	&	0.0070	&	0.0071		\\
+			&				&	$\E{(2)}$		&	-0.0005	&	-0.0009	&	-0.0015	&	-0.0023	&	-0.0030	&	-0.0026		\\
+			&				&	$\Ex{(1)}$		&	0.0058	&	0.0052	&	0.0039	&	0.0022	&	0.0028	&	0.0096		\\
+			&				&	$\Ex{(2)}$		&	-0.0036	&	-0.0045	&	-0.0058	&	-0.0077	&	-0.0072	&	0.0000		\\
+			&				&	$\DD{c}{(0)}$	&	-0.0074	&	-0.0067	&	-0.0055	&	-0.0036	&	-0.0010	&	0.0019	&	\\
+			&				&	$\DD{c}{(1)}$	&	-0.0010	&	-0.0011	&	-0.0014	&	-0.0017	&	-0.0021	&	-0.0022&	\\
+			&				&	$\DD{c}{(2)}$	&	0.0084	&	0.0079	&	0.0069	&	0.0053	&	0.0031	&	0.0003	&	\\
+			&				&	$\LZ{c}{}$		&	0.0007	&	0.0013	&	0.0023	&	0.0040	&	0.0063	&	0.0087	&	\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table*}
+
+%%% TABLE II %%%
+\begin{table*}
+	\caption{
+	\label{tab:OptGap}
+	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 $\Nel = 4$ electrons in a box of length $L$.
+	The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+	}
+	\begin{ruledtabular}
+	\begin{tabular}{lclddddddd}
+			&				&					&	\mc{7}{c}{$L/\pi$}	\\
+													\cline{4-10}
+	Method	&	$\bw$		&	State			&	1/8			&	1/4			&	1/2			&	1		&	2		&	4		&	8		\\
+	\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	\\
+	\\
+	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	\\
+	\\
+	eDFT	&	$(0,0)$		&	$\E{(0)}$		&	-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	\\
+			&				&	$\LZ{c}{}$		&	0.0014	&	0.0026	&	0.0048	&	0.0083	&	0.0127	&	0.0169	&	0.01860	\\
+	\\
+	eDFT	&	$(1/3,1/3)$	&	$\E{(0)}$		&	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		\\
+			&				&	$\LZ{c}{}$		&	0.0007	&	0.0013	&	0.0025	&	0.0044	&	0.0074	&	0.0110	&	0.0143	\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table*}
+
+
+%%% TABLE III %%%
+\begin{table*}
+	\caption{
+	\label{tab:OptGap}
+	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 $\Nel = 4$ electrons in a box of length $L$.
+	The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+	}
+	\begin{ruledtabular}
+	\begin{tabular}{lclddddddd}
+			&				&					&	\mc{7}{c}{$L/\pi$}	\\
+													\cline{4-10}
+	Method	&	$\bw$		&	State			&	1/8			&	1/4			&	1/2			&	1		&	2		&	4		&	8		\\
+	\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	\\
+	\\
+	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	\\
+	\\
+	eDFT	&	$(0,0)$		&	$\E{(0)}$		&	-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	\\
+			&				&	$\LZ{c}{}$		&	0.0014	&	0.0027	&	0.0051	&	0.0091	&	0.0147	&	0.0207	&	0.0245	\\
+	\\
+	eDFT	&	$(1/3,1/3)$	&	$\E{(0)}$		&	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	\\
+			&				&	$\LZ{c}{}$		&	0.0007	&	0.0013	&	0.0025	&	0.0047	&	0.0081	&	0.0127	&	0.0175	\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table*}
+
+%%% TABLE IV %%%
+\begin{table*}
+	\caption{
+	\label{tab:OptGap}
+	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 $\Nel = 4$ electrons in a box of length $L$.
+	The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+	}
+	\begin{ruledtabular}
+	\begin{tabular}{lclddddddd}
+			&				&					&	\mc{7}{c}{$L/\pi$}	\\
+													\cline{4-10}
+	Method	&	$\bw$		&	State			&	1/8			&	1/4			&	1/2			&	1			&	2		&	4		&	8		\\
+	\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	\\
+	\\
+	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	\\
+	\\
+	eDFT	&	$(0,0)$		&	$\E{(0)}$		&	-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	\\
+			&				&	$\LZ{c}{}$		&	0.0015	&	0.0028	&	0.0053	&	0.0096	&	0.0161	&	0.0238	&	0.0297	\\
+	\\
+	eDFT	&	$(1/3,1/3)$	&	$\E{(0)}$		&	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	\\
+			&				&	$\LZ{c}{}$		&	0.0007	&	0.0013	&	0.0026	&	0.0049	&	0.0087	&	0.0141	&	0.0202	\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table*}
+
+%%% TABLE V %%%
+\begin{table*}
+	\caption{
+	\label{tab:OptGap}
+	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 $\Nel = 4$ electrons in a box of length $L$.
+	The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+	}
+	\begin{ruledtabular}
+	\begin{tabular}{lclddddddd}
+			&				&					&	\mc{7}{c}{$L/\pi$}	\\
+													\cline{4-10}
+	Method	&	$\bw$		&	State			&	1/8			&	1/4			&	1/2			&	1		&	2		&	4		&	8		\\
+	\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	\\
+	\\
+	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	\\
+	\\
+	eDFT	&	$(0,0)$		&	$\E{(0)}$		&	-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	\\
+			&				&	$\LZ{c}{}$		&	0.0015	&	0.0029	&	0.0054	&	0.0100	&	0.0172	&	0.0264	&	0.0344	\\
+	\\
+	eDFT	&	$(1/3,1/3)$	&	$\E{(0)}$		&	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	\\
+			&				&	$\LZ{c}{}$		&	0.0007	&	0.0013	&	0.0026	&	0.0050	&	0.0091	&	0.0151	&	0.0224	\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table*}
+
+
+%%% TABLE VI %%%
+\begin{table*}
+	\caption{
+	\label{tab:OptGap}
+	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 $\Nel = 4$ electrons in a box of length $L$.
+	The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+	}
+	\begin{ruledtabular}
+	\begin{tabular}{lclddddddd}
+			&				&					&	\mc{7}{c}{$L/\pi$}	\\
+													\cline{4-10}
+	Method	&	$\bw$		&	State			&	1/8			&	1/4			&	1/2			&	1		&	2		&	4		&	8		\\
+	\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	\\
+	\\
+	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	\\
+	\\
+	eDFT	&	$(0,0)$		&	$\E{(0)}$		&	-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		\\
+			&				&	$\LZ{c}{}$		&	0.0015	&	0.0029	&	0.0055	&	0.0103	&	0.0180	&	0.0284	&	0.038555	\\
+	\\
+	eDFT	&	$(1/3,1/3)$	&	$\E{(0)}$		&	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	\\
+			&				&	$\LZ{c}{}$		&	0.0007	&	0.0013	&	0.0027	&	0.0051	&	0.0094	&	0.0159	&	0.0242	\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table*}
+
+
+\bibliography{../eDFT}
+
+\end{document}

Manuscript/eDFT.tex

@@ -0,0 +1,469 @@
+\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
+
+\usepackage{mathpazo,libertine}
+\newcommand{\alert}[1]{\textcolor{red}{#1}}
+\definecolor{darkgreen}{RGB}{0, 180, 0}
+
+\usepackage{hyperref}
+\hypersetup{
+    colorlinks=true,
+    linkcolor=blue,
+    filecolor=blue,      
+    urlcolor=blue,	
+    citecolor=blue
+}
+
+%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{\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}{\nonumber\end{eqnarray}}
+%%%% 
+
+\begin{document}	
+
+\title{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}
+We report a first generation of local, weight-dependent correlation density-functional approximations (DFAs) that incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT). 
+These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
+The resulting eDFAs, based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence. 
+Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
+\end{abstract}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\maketitle
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Introduction.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials \cite{ParrBook}.
+The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}(\br)$, the latter being a much simpler quantity than the electronic wave function. 
+The complexity of the many-body problem is then transferred to the xc functional. 
+Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} suffers, in practice, from various deficiencies \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}. 
+The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge. 
+Another issue, which is partly connected to the previous one, is the description of electronically-excited states. 
+
+The standard approach for modeling excited states in DFT is linear response time-dependent DFT (TDDFT) \cite{Casida}.
+In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong. 
+Moreover, in exact TDDFT, the xc functional is time dependent. 
+The simplest and most widespread approximation in state-of-the-art electronic structure programs where TDDFT is implemented consists in neglecting memory effects \cite{Casida}. 
+In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time. 
+As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT \cite{Maitra_2004,Cave_2004,Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}.
+
+When affordable (i.e.~for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above. 
+The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e.,~with the same set of orbitals. 
+Interestingly, a similar approach exists in DFT. 
+Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK) \cite{Gross_1988, Gross_1988a, Oliveira_1988}, and is a generalization of Theophilou's variational principle for equi-ensembles \cite{Theophilou_1979}.
+In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest. 
+This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies. 
+It actually accounts for the infamous derivative discontinuity contribution to energy gaps. 
+\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
+Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018a,Gould_2018b,Sagredo_2018,Ayers_2018,Deur_2018a,Deur_2018b,Kraisler_2013, Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Senjean_2018,Smith_2016}.
+The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature. 
+Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018a,Deur_2018b,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed \cite{Yang_2014,Yang_2017}. 
+In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open. 
+A first step towards this goal is presented in this Letter with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
+\alert{Mention WIDFA?}
+
+In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}.
+Moreover, the present method relies on exact Hartree-Fock (HF) exchange, eschewing the so-called ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}.
+Atomic units are used throughout.\\
+
+\manu{I added some key equations in the following. Will polish the all
+thing later on.\\
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Generalized KS-eDFT}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+}
+
+In eDFT, the ensemble energy
+\begin{equation}
+	\E{\bw}=(1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
+\end{equation}
+is obtained variationally as follows,
+In analogy with ground-state generalized KS-DFT, we consider the
+following partitioning of 
+
+the ensemble Levy-Lieb functional 
+\begin{equation}
+	F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
+\end{equation}
+
+\begin{equation}
+F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
+\end{equation}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{eKS for excited states.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Here, we explain how to perform a self-consistent KS calculation for ensembles (eKS) in the context of excited states.
+In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including: 
+i) the ground state ($I=0$), ii) the first singly-excited state ($I=1$), and iii) the first doubly-excited state ($I=2$).
+Generalization to a larger number of states is straightforward and is left for future work.
+By definition, the ensemble energy is
+\begin{equation}
+	\E{\bw} = (1 - \ew{1} - \ew{2}) \E{(0)} + \ew{1} \E{(1)} + \ew{2} \E{(2)}.
+\end{equation}
+$\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the to the single excitation and double excitation, respectively.
+\alert{To ensure the GOK variational principle \cite{Gross_1988a}, the weights must fulfil the following conditions: $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$. T2: I don't understand the asymmetry of the weights in this equation.}
+Note that, in order to extract individual energies from a single eKS calculation (see below), the weights must remain independent.
+By construction, the excitation energies are
+\begin{equation}
+\label{eq:Ex}
+	\Ex{(I)} = \pdv{\E{(I)}}{\ew{I}} = \E{(I)} - \E{(0)}.
+\end{equation}
+In the following, the orbitals $\MO{p}{\bw}(\br)$ are defined as linear combination of basis functions $\AO{\mu}(\br)$, such as
+\begin{equation}
+	\MO{p}{\bw}(\br) = \sum_{\mu=1}^{\Nbas} \cMO{\mu p}{\bw} \, \AO{\mu}(\br).
+\end{equation}
+
+Within the self-consistent eKS process, one is looking for the following weight-dependent density matrix:
+\begin{equation}
+\label{eq:Gamma}
+	\bGamma{\bw} = (1-\ew{1}-\ew{2}) \bGamma{(0)} - \ew{1} \bGamma{(1)} - \ew{2} \bGamma{(2)},
+\end{equation}
+where $\bw = (\ew{1},\ew{2})$ and $\bGamma{(I)}$ is the $I$th-state density matrix with elements
+\begin{equation}
+\label{eq:eGamma}
+	\eGamma{\mu\nu}{(I)} = \sum_{i=1}^{\Nel-I} \cMO{\mu i}{\bw} \cMO{\nu i}{\bw} + \sum_{a=\Nel+1}^{\Nel+I} \cMO{\mu a}{\bw} \cMO{\nu a}{\bw}.
+\end{equation}
+The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamma{\bw}$ in Eq.~\eqref{eq:Gamma} are obtained by diagonalizing the following Fock matrix
+\begin{multline}
+\label{eq:F}
+	\eF{\mu\nu}{\bw} 
+	= \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si} 
+	\\
+	+ \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)}  \AO{\mu}(\br) \AO{\nu}(\br) d\br,
+\end{multline}
+which itself depends on $\bGamma{\bw}$.
+In Eq.~\eqref{eq:F}, $\bHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) -