continue cleaning

Manuscript/eDFT.tex

@@ -1,18 +1,23 @@
 \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
+\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}
 
-\usepackage{mathpazo,libertine}
 \newcommand{\alert}[1]{\textcolor{red}{#1}}
-\definecolor{darkgreen}{RGB}{0, 180, 0}
+\usepackage[normalem]{ulem}
-
+\newcommand{\titou}[1]{\textcolor{red}{#1}}
-\usepackage{hyperref}
+\newcommand{\manu}[1]{\textcolor{blue}{#1}}
-\hypersetup{
+\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
-    colorlinks=true,
+\newcommand{\trashEF}[1]{\textcolor{blue}{\sout{#1}}}
-    linkcolor=blue,
-    filecolor=blue,      
-    urlcolor=blue,	
-    citecolor=blue
-}
 
 %useful stuff
 \newcommand{\cdash}{\multicolumn{1}{c}{---}}
@@ -59,8 +64,9 @@
 \newcommand{\bw}{{\bm{w}}}
 \newcommand{\bG}{\bm{G}}
 \newcommand{\bS}{\bm{S}}
-\newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}}
+\newcommand{\bGam}[1]{\bm{\Gamma}^{#1}}
-\newcommand{\opGamma}[1]{\hat{\Gamma}^{#1}}
+\newcommand{\bgam}[1]{\bm{\gamma}^{#1}}
+\newcommand{\opGam}[1]{\hat{\Gamma}^{#1}}
 \newcommand{\bh}{\bm{h}}
 \newcommand{\bF}[1]{\bm{F}^{#1}}
 \newcommand{\Ex}[1]{\Omega^{#1}}
@@ -70,10 +76,12 @@
 \newcommand{\ew}[1]{w_{#1}}
 \newcommand{\eG}[1]{G_{#1}}
 \newcommand{\eS}[1]{S_{#1}}
-\newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}}
+\newcommand{\eGam}[2]{\Gamma_{#1}^{#2}}
-\newcommand{\hGamma}[2]{\Hat{\Gamma}_{#1}^{#2}}
+\newcommand{\hGam}[2]{\Hat{\Gamma}_{#1}^{#2}}
-\newcommand{\eHc}[1]{h_{#1}}
+\newcommand{\eh}[2]{h_{#1}^{#2}}
 \newcommand{\eF}[2]{F_{#1}^{#2}}
+\newcommand{\ERI}[2]{(#1|#2)}
+\newcommand{\dbERI}[2]{(#1||#2)}
 
 % Numbers
 \newcommand{\Nel}{N}
@@ -82,6 +90,7 @@
 % AO and MO basis
 \newcommand{\Det}[1]{\Phi^{#1}}
 \newcommand{\MO}[2]{\phi_{#1}^{#2}}
+\newcommand{\SO}[2]{\varphi_{#1}^{#2}}
 \newcommand{\cMO}[2]{c_{#1}^{#2}}
 \newcommand{\AO}[1]{\chi_{#1}}
 
@@ -96,8 +105,6 @@
 \newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
 
 %%% added by Manu %%%
-
-\newcommand{\manu}[1]{{\textcolor{darkgreen}{ Manu: #1 }} }
 \newcommand{\beq}{\begin{equation}}
 \newcommand{\eeq}{\end{equation}}
 \newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
@@ -105,8 +112,8 @@
 \newcommand{\bxi}{\bm{\xi}} 
 \newcommand{\bfx}{{\bf{x}}}
 \newcommand{\bfr}{{\bf{r}}}
+\DeclareMathOperator*{\argmax}{arg\,max}
 \DeclareMathOperator*{\argmin}{arg\,min}
-\newcommand{\blue}[1]{{\textcolor{blue}{#1}}}
 %%%% 
 
 \begin{document}	
@@ -158,13 +165,13 @@ Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and K
 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. \cite{Levy_1995, Perdew_1983}
-%\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
+%\titou{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_2018,Gould_2019,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 article 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?}
+%\titou{Mention WIDFA?}
 In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
 In other words, the Coulomb interaction used in this work describes particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
 Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
@@ -197,18 +204,18 @@ so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldot
 For simplicity we will assume in the following that the energies are not degenerate. Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{}. In GOK-DFT, the ensemble energy is determined variationally as follows:
 \beq\label{eq:var_ener_gokdft}
 	\E{}{\bw}
- 	= \min_{\opGamma{{\bw}}}\qty{ \Tr[\opGamma{{\bw}}\hh]
+ 	= \min_{\opGam{\bw}}\qty{ \Tr[\opGam{\bw} \hh]
-	+ \E{Hx}{\bw} \qty[\n{\opGamma{\bw}}{}]
+	+ \E{Hx}{\bw} \qty[\n{\opGam{\bw}}{}]
-	+ \E{c}{\bw}  \qty[\n{\opGamma{\bw}}{}]
+	+ \E{c}{\bw}  \qty[\n{\opGam{\bw}}{}]
 	},
 \eeq
 where $\Tr$ denotes the trace and the trial ensemble density matrix operator reads
 \beq
-	\opGamma{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
+	\opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
 \eeq
 The determinants (or configuration state functions) $\Phi^{(K)}$ are all constructed from the same set of (ensemble Kohn--Sham) orbitals that is optimized variationally and the trial ensemble density is simply the weighted sum of the individual densities: 
 \beq\label{eq:KS_ens_density}
-	\n{\opGamma{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Phi^{(K)}}{}(\br{}).
+	\n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Phi^{(K)}}{}(\br{}).
 \eeq
 As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange and correlation energies are described with density functionals that are \textit{weight dependent}. 
 We focus here on the (exact) Hx part which is defined as follows:
@@ -217,9 +224,9 @@ We focus here on the (exact) Hx part which is defined as follows:
 \eeq
 where the KS wavefunctions fulfill the ensemble density constraint
 \beq
-	\sum_{K\geq 0} \ew{K} \n{\Det{(K)}[\n{}{}]}{}(\br{})=n(\br{}).
+	\sum_{K\geq 0} \ew{K} \n{\Det{(K)}[\n{}{}]}{}(\br{}) = \n{}{}(\br{}).
 \eeq 
-The (approximate) description of the correlation part is discussed in Sec.~\ref{sec:eDFA}.\\
+The (approximate) description of the correlation part is discussed in Sec.~\ref{sec:eDFA}.
 
 In practice, one is not much interested in ensemble energies but rather in excitation energies or individual energy levels (for geometry optimizations, for example). The latter can be extracted exactly as follows~\cite{}: 
 \beq\label{eq:indiv_ener_from_ens}
@@ -236,7 +243,7 @@ where, according to the {\it variational} ensemble energy expression of Eq.~\eqr
 	\\
 	& + \int d\br{} \fdv{\E{c}{\bw}[n]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
 	+ \pdv{\E{c}{\bw}[n]}{\ew{K}}
-	\Bigg\}_{\n{}{} = \n{\opGamma{\bw}}{}}.
+	\Bigg\}_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
 The Hx contribution to Eq.~\eqref{eq:deriv_Ew_wk} can be rewritten as follows: 
@@ -272,29 +279,29 @@ thus leading, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_H
 	\qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
 	+
 	\pdv{\E{c}{\bw} [\n{}{}]}{\ew{K}} 
-	}_{\n{}{} = \n{\opGamma{\bw}}{}}.
+	}_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
 Since the ensemble energy can be evaluated as follows:
 \beq
-	\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGamma{\bw}}{}],
+	\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGam{\bw}}{}],
 \eeq 
-with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGamma{\bw}}{} = \n{}{\bw,\bw}$], 
+with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGam{\bw}}{} = \n{}{\bw,\bw}$], 
 we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 \eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\cite{} for the $I$th energy level:
 \beq\label{eq:exact_ener_level_dets}
 \begin{split}
 	\E{}{(I)} 
-	& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{{\bw}}[\n{\opGamma{\bw}}{}]
+	& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{{\bw}}[\n{\opGam{\bw}}{}]
 	\\
-	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\opGamma{\bw}}{}]}{\n{}{}(\br{})}
+	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\opGam{\bw}}{}]}{\n{}{}(\br{})}
-	\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGamma{\bw}}{}(\br{}) ]
+	\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ]
 	\\
 	&+
 	\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
 	\left.
 	\pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}
-	\right|_{\n{}{} = \n{\opGamma{\bw}}{}}.
+	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
 %%%%%%%%%%%%%%%%
@@ -303,30 +310,19 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 For implementation purposes, we will use in the rest of this work 
 (one-electron reduced) density matrices 
 as basic variables, rather than Slater determinants. If we expand the
-ensemble KS (spin) orbitals [from which the latter are constructed] in an atomic
+ensemble KS spinorbitals [from which the latter are constructed] in an atomic orbital (AO) basis,
-orbital (AO) basis,
 \beq
-%\varphi_{p\sigma}(\br{},\tau)
+	\SO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
-	\varphi^\sigma_p(\br{})=
-%\sigma(\tau)
-\sum_{\mu}c^\sigma_{{\mu p}}\AO{\mu}(\br{}),
 \eeq
 then the density matrix elements obtained from the   
-determinant $\Phi^{(K)}$ can be expressed as follows in the AO basis:
+determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
 \beq
-\bmg^{(K)}\equiv\Gamma_{\mu\nu}^{(K)\sigma}=
+	\bmg^{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
-%\sum_\sigma
-\sum_{\varphi^\sigma_p\in(K)}c^\sigma_{{\mu
-p}}c^\sigma_{{\nu p}},
-%\sum_{p\in (K)}c_{\mu p}c_{\nu p},
 \eeq
-where the summation runs over the spin-orbitals that are occupied in
+where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$. 
-$\Phi^{(K)}$. Note that the density of the $K$th KS state reads
+Note that the density of the $K$th KS state reads
 \beq
-n_{\bmg^{(K)}}(\br{})=
+	\n{\bmg^{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}).
-\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{})\Gamma_{\mu\nu}^{(K)\sigma}.
-%\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br{},
-%\sigma})\AO{\nu}(\br{},\sigma){\Gamma}^{(K)}_{\mu\nu}.
 \eeq 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Manu's derivation %%%
@@ -351,12 +347,14 @@ p}}c^\sigma_{{\nu p}}
 We can then construct the ensemble density matrix
 and the ensemble density as follows:
 \beq
-{\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}\equiv
+	\bmg^{\bw} 
-\Gamma_{\mu\nu}^{\bw\sigma}=\sum_{K\geq 0}w_K \Gamma_{\mu\nu}^{(K)\sigma}
+	= \sum_{K\geq 0} \ew{K} \bmg^{(K)}
+	\equiv \eGam{\mu\nu}{\bw}
+	= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)}
 \eeq
 and
 \beq
-n_{\bmg^\bw}({\br{}})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{}){\Gamma}^{\bw\sigma}_{\mu\nu},
+	\n{\bmg^\bw}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
 \eeq
 respectively. The exact energy level expression in Eq.~\eqref{eq:exact_ener_level_dets} can be
 rewritten as follows:
@@ -365,9 +363,9 @@ rewritten as follows:
 	\E{}{(I)} 
 	& =\Tr[\bmg^{(I)} \bh]
 	+ \frac{1}{2} \Tr[\bmg^{(I)} \bG \bmg^{(I)}]
+	+ \E{c}{{\bw}}[\n{\bmg^{\bw}}{}]
 	\\
-	& + \E{c}{{\bw}}[\n{\bmg^{\bw}}{}]
+	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\bmg^{\bw}}{}]}{\n{}{}(\br{})}
-	+ \int d\br{} \fdv{\E{c}{\bw}[\n{\bmg^{\bw}}{}]}{\n{}{}(\br{})}
 	\qty[ \n{\bmg^{(I)}}{}(\br{}) - \n{\bmg^{\bw}}{}(\br{}) ]
 	\\
 	& + \sum_{K>0} \qty(\delta_{IK} - \ew{K})
@@ -382,27 +380,25 @@ where
 denote the one-electron integrals matrix.
 The individual Hx energy is obtained from the following trace
 \beq
-\Tr(\bmg^{(K)} \, \bG \,
+	\Tr(\bmg^{(K)} \bG \bmg^{(L)}) 
-\bmg^{(L)})=\sum_{\mu\nu\lambda\omega}\sum_{\sigma=\alpha,                                         \beta}\sum_{\tau=\alpha,\beta}G_{\mu\nu\lambda\omega}^{\sigma\tau}
+	= \sum_{\mu\nu\lambda\omega}\sum_{\sigma=\alpha,\beta}\sum_{\tau=\alpha,\beta}G_{\mu\nu\lambda\omega}^{\sigma\tau}
-\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
+\eGam{\mu\nu}{(K)\sigma} \eGam{\lambda\omega}{(L)\tau}
 \nonumber\\
 \eeq
 where the two-electron Coulomb-exchange integrals read
 \beq
-G_{\mu\nu\lambda\omega}^{\sigma\tau}=({\mu}{\nu}\vert{\lambda}{\omega})
+	G_{\mu\nu\lambda\omega} = 
--\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu),
+	\dbERI{\mu\nu}{\la\si} 
+	= \ERI{\mu\nu}{\la\si} - \ERI{\mu\si}{\la\nu},
 \eeq
 with
 \beq
-(\mu\nu|\la\omega) = \iint  \frac{\AO\mu(\br{1}) \AO\nu(\br{1})
+\ERI{\mu\nu}{\la\si} = \iint \frac{\AO{\mu}(\br{1}) \AO{\nu}(\br{1}) \AO{\la}(\br{2}) \AO{\si}(\br{2})}{\abs{\br{1} - \br{2}}} d\br{1} d\br{2}.
-\AO\la(\br{2})                          \AO\omega(\br{2})}{\abs{\br{1} - \br{2}}} d\br{1} d\br{2}
-.
-\nonumber\\
 \eeq
-Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
+%Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
-polarized systems in which $\Gamma_{\mu\nu}^{(K)\beta}=0$ and
+%polarized systems in which $\eGam{\mu\nu}{(K)\beta}=0$ and
-$G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu}\vert{\lambda}{\omega})
+%$G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu}\vert{\lambda}{\omega})
--(\mu\omega\vert\lambda\nu)$. 
+%-(\mu\omega\vert\lambda\nu)$. 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%% Hx energy ...
 %%% Manu's derivation
@@ -410,7 +406,7 @@ $G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu
 \blue{
 \beq
 &&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
-RS\rangle\Gamma^{(K)}_{PR}\Gamma^{(L)}_{QS}
+RS\rangle\eGam{PR}^{(K)}\eGam{QS}^{(L)}
 \nonumber\\
 &&
 =\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
@@ -503,37 +499,27 @@ HF}\left[{\bmg}\right]=\frac{1}{2} \Tr(\bmg \, \bG \, \bmg),
 for the Hx density-functional energy in the variational energy
 expression of Eq.~\eqref{eq:var_ener_gokdft}:  
 \beq
-{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
+	{\bmg}^{\bw} 
-\Big\{
+	\approx \argmin_{\bgam{\bw}} 
-{\rm
+	\qty{ 
-Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
+	\Tr[\bgam{\bw} \bh ] 
-HF}\left[{\bm\gamma}^{\bw}\right]
+	+ W_{\rm HF}[ \bgam{\bw}]
-+
+	+ \E{c}{\bw}[\n{\bgam{\bw}}{}]
-{E}^{{\bw}}_{\rm
+	}.
-c}\left[n_{\bm\gamma^{\bw}}\right]
-%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
-\Big\}.
-\nonumber\\
 \eeq
 The minimizing ensemble density matrix fulfills the following
 stationarity condition
 \beq\label{eq:commut_F_AO}
-{\bm F}^{\bw\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
+	\bF{\bw} \bGam{\bw} \bS = \bS \bGam{\bw} \bF{\bw},
-\Gamma}^{\bw\sigma}{\bm F}^{\bw\sigma},
 \eeq
-where ${\bm S}\equiv S_{\mu\nu}=\braket{\AO{\mu}}{\AO{\nu}}$ is the
+where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the metric and the ensemble Fock-like matrix reads
-metric and the ensemble Fock-like matrix reads
 \beq
-F_{\mu\nu}^{\bw\sigma}=h^\bw_{\mu\nu}+\sum_{\lambda\omega}\sum_{\tau=\alpha,\beta}
+	\eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} + \sum_{\lambda\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw}
-G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
 \eeq 
 with
 \beq
-h^\bw_{\mu\nu}=h_{\mu\nu}+
+	\eh{\mu\nu}{\bw} 
-%\left\langle\AO{\mu}\middle\vert\dfrac{\delta E^\bw_{\rm
+	= \eh{\mu\nu}{} + \int d\br{} \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bmg^\bw}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}).
-%c}[n_{\bmg^\bw}]}{\delta n(\br{})}\middle\vert\AO{\nu}\right\rangle
-\int d\br{}\;\AO{\mu}(\br{})\dfrac{\delta E^\bw_{\rm
-c}[n_{\bmg^\bw}]}{\delta n(\br{})}\AO{\nu}(\br{}).
 \eeq
 
 %%%%%%%%%%%%%%%
@@ -680,8 +666,7 @@ according to Eq.~\eqref{eq:exact_ind_ener_rdm}.\\
 Turning to the density-functional ensemble correlation energy, the
 following eLDA will be employed:  
 \beq\label{eq:eLDA_corr_fun}
-{E}^{{\bw}}_{\rm
+	\E{c}{\bw}[\n{}{}] = \int d\br{} \n{}{}(\br{}) \e{c}{\bw}[\n{}{}(\br{})],
-c}[n]=\int d\br{}\;n(\br{}) \epsilon_{c}^{\bw}(n(\br{})),
 \eeq
 where the correlation energy per particle is {\it weight-dependent}. Its
 construction from a finite uniform electron gas model is discussed
@@ -691,22 +676,18 @@ Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression
 within eLDA: 
 \beq
 \begin{split}
-	\E{}{(I)} & \approx \Tr[\bmg^{(I)} \bh]
+	\E{}{(I)} 
+	& \approx \Tr[\bmg^{(I)} \bh]
 	+ \frac{1}{2} \Tr(\bmg^{(I)} \bG \bmg^{(I)})
 	\\
-	& +\int d\br{} \epsilon^{\bw}_{\rm
+	& + \int d\br{} \e{c}{\bw}(\n{\bmg^{\bw}}{}(\br{})) \n{\bmg^{(I)}}{}(\br{})
-c}(n_{\bmg^{\bw}}(\br{}))\,n_{\bmg^{(I)}}(\br{})
+	\\
-\nonumber\\
+	&
-&&
+	+ \int d\br{} \n{\bmg^{\bw}}{}(\br{}) \qty[ \n{\bmg^{(I)}}{}(\br{}) - \n{\bmg^{\bw}}{}(\br{}) ]
-+\int d\br{}\,
+	\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = n{\bmg^{\bw}}{}(\br{})}
-n_{\bmg^{\bw}}(\br{})\left(n_{\bmg^{(I)}}(\br{})-n_{\bmg^{\bw}}(\br{})\right)
+	\\
-\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
+	& + \int d\br{} \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bmg^{\bw}}{}(\br{})
-c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br{})}
+	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bmg^{\bw}}{}(\br{})}.
-\nonumber\\
-&&
-+\int d\br{}\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br{})\left.
-\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
-c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br{})}.
 \end{split}
 \eeq
 
@@ -815,7 +796,7 @@ Finally, we note that, by construction,
 \begin{equation}
 	\left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br{})} = \be{c}{(J)}[\n{}{\bw}(\br{})] - \be{c}{(0)}[\n{}{\bw}(\br{})].
 \end{equation}
-\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
+\titou{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
@@ -823,7 +804,7 @@ Finally, we note that, by construction,
 Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
 Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium.
 In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \Nel \le 7$. 
-\alert{Comment on the quality of these density: density- and functional-driven errors?}
+\titou{Comment on the quality of these density: density- and functional-driven errors?}
 
 These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
 For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
@@ -890,16 +871,16 @@ Even for larger boxes, the discrepancy between FCI and eLDA for double excitatio
 \end{figure}
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-\alert{Need further discussion on DD and LZ shift. Linearity of energy wrt weights?}
+\titou{Need further discussion on DD and LZ shift. Linearity of energy wrt weights?}
 
-\alert{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.
+\titou{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.
 However, when the box gets larger, there is a strong mixing between different degree of excitations.
 In particular, the single and double excitations strongly mix.
 This is clearly evidenced if one looks at the weights of the different configurations in the FCI wave function.
 In one hand, if one does construct a eDFA with a single state (either single or double), one clearly sees that the results quickly deteriorates when the box gets larger.
 On the other hand, building a functional which does mix singles and doubles corrects this by allowing configuration mixing.}
 
-\alert{It might be useful to add eHF results where one switch off the correlation part.
+\manu{It might be useful to add eHF results where one switch off the correlation part.
 For both zero weight and state-averaged weights?
 It would highlight the contribution of the derivative discontinuity.}