titou just corrected part of his mistakes
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@ -62,6 +62,7 @@
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% matrices/operator
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\newcommand{\br}[1]{\boldsymbol{r}_{#1}}
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\newcommand{\bx}[1]{\boldsymbol{x}_{#1}}
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\newcommand{\bw}{{\boldsymbol{w}}}
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bS}{\boldsymbol{S}}
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@ -134,6 +135,7 @@ We report a first generation of local, weight-dependent correlation density-func
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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).
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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.
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Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
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\titou{Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.}
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\end{abstract}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -394,18 +396,29 @@ For implementation purposes, we will use in the rest of this work
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(one-electron reduced) density matrices
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as basic variables, rather than Slater determinants. If we expand the
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ensemble KS (spin) orbitals [from which the latter determinants are constructed] in an atomic orbital (AO) basis,
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\beq
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\SO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
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\titou{\beq
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\SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
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\eeq
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then the density matrix of the
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where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatial degrees of freedom, and
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\beq
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s(\omega)
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=
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\begin{cases}
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\alpha(\omega), & \text{for spin-up electrons,} \\
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\text{or} \\
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\beta(\omega), & \text{for spin-down electrons,}
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\end{cases}
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\eeq
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}then the density matrix of the
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determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
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\beq
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\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
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\eeq
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where the summation runs over the spinorbitals that are occupied in
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$\Det{(K)}$. Note that, as the theory is applied later on to spin-polarized
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systems, we drop spin indices in the density matrices, for convenience.
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where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$.
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\trashPFL{Note that, as the theory is applied later on to spin-polarized
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systems, we drop spin indices in the density matrices, for convenience.}
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\manu{Is the latter sentence ok with you?}
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\titou{I don't think we need it anymore. What do you think?}
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The electron density of the $K$th KS determinant can then be evaluated
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as follows:
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\beq
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@ -849,45 +862,45 @@ This is a necessary condition for being able to model the ensemble
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correlation derivatives with respect to the weights [last term
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on the right-hand side of Eq.~(\ref{eq:exact_ener_level_dets})].
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Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous IUEG paradigm. \cite{Loos_2013,Loos_2013a}
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\manu{Let us stress that, in a FUEG like ringium, the interacting and
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Let us stress that, in a FUEG like ringium, the interacting and
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noninteracting densities match individually for all the states within the
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ensemble
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[these densities are all equal to the uniform density], which means that
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(these densities are all equal to the uniform density), which means that
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so-called density-driven correlation
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effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.}
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effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.
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Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model.
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The present weight-dependent eDFA is specifically designed for the
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calculation of excited-state energies within GOK-DFT.
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In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
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(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
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All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
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All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring where the electrons are confined.
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We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
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Generalization to a larger number of states is straightforward and is left for future work.
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To ensure the GOK variational principle, \cite{Gross_1988a} the
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tri-ensemble weights must fulfil the following conditions:
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\manu{\titou{$0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$}. The
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constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
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to consider an equi-bi-ensemble
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for which $\ew{1}=1/2$. This possibility is excluded with your
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inequalities. The correct constraints are given in Ref.~\cite{Deur_2019}
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and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and
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$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.}
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\manu{
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Just in case, starting from
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\beq
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\begin{split}
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0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2})
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\\
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\end{split}
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\eeq
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we obtain
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\beq
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0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2
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\eeq
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which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq
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1/3$.
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}
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tri-ensemble weights must fulfil the following conditions: \cite{Deur_2019}
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\titou{$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$}.
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%The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
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%to consider an equi-bi-ensemble
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%for which $\ew{1}=1/2$. This possibility is excluded with your
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%inequalities. The correct constraints are given in Ref.~\cite{Deur_2019}
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%and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and
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%$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.}
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%\manu{
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%Just in case, starting from
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%\beq
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%\begin{split}
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%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2})
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%\\
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%\end{split}
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%\eeq
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%we obtain
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%\beq
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%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2
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%\eeq
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%which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq
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%1/3$.
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%}
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%%% TABLE 1 %%%
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\begin{table*}
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\caption{
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@ -936,7 +949,7 @@ The weight-dependence of the correlation functional is then carried exclusively
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Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
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\begin{equation}
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\label{eq:becw}
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\manu{\e{c}{\bw}(\n{}{})\rightarrow}\be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
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\titou{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
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\end{equation}
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where
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\begin{equation}
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@ -959,14 +972,14 @@ where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
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a_3^\text{LDA} & = 2.408779.
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\end{align}
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\end{subequations}
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\manu{Note that the strategy described in Eq.~(\ref{eq:becw}) is general and
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can be applied to real (higher-dimension) systems.} In order to make the
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Note that the strategy described in Eq.~(\ref{eq:becw}) is general and
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can be applied to real (higher-dimensional) systems. In order to make the
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connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may
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recast Eq.~\eqref{eq:becw} as
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\begin{equation}
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\label{eq:eLDA}
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\begin{split}
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\be{c}{\bw}(\n{}{})
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\titou{\e{c}{\bw}(\n{}{})}
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& = \e{c}{\text{LDA}}(\n{}{})
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\\
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& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
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@ -975,30 +988,26 @@ recast Eq.~\eqref{eq:becw} as
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or, equivalently,
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\begin{equation}
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\label{eq:eLDA_gace}
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\begin{split}
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\be{c}{\bw}(\n{}{})
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& = \e{c}{\text{LDA}}(\n{}{})
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\\
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& + \sum_{K>0}\int_0^{\ew{K}}
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\titou{\e{c}{\bw}(\n{}{})}
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= \e{c}{\text{LDA}}(\n{}{})
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+ \sum_{K>0}\int_0^{\ew{K}}
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\qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K,
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\end{split}
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\end{equation}
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where the $K$th correlation excitation energy (per electron) is integrated over the
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ensemble weight $\xi_K$ at fixed (uniform) density $n$.
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Eq.~(\ref{eq:eLDA_gace}) nicely highlights the centrality of the LDA in the present eDFA.
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In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
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ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$.
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Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present eDFA.
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In particular, $\titou{\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$.
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Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
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Finally, we note that, by construction,
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\manu{
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\begin{equation}
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\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \be{c}{(J)}(\n{}{}(\br{})) - \be{c}{(0)}(\n{}{}(\br{})).
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\titou{\pdv{\e{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).}
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\end{equation}
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Manu: I guess that the "overlines" and the dependence in $\bf r$ of the
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densities on the RHS should be removed. The final expression should be
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\beq
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\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).
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\eeq
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}
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%Manu: I guess that the "overlines" and the dependence in $\bf r$ of the
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%densities on the RHS should be removed. The final expression should be
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%\beq
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%\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).
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%\eeq
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%}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:comp_details}
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@ -1113,6 +1122,7 @@ This is definitely a very pleasing outcome, which additionally shows that, even
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\end{figure}
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%%% %%% %%%
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\titou{T2: there is a micmac with the derivative discontinuity as it is only defined at zero weight. We should clean up this.}
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It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
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To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage (with respect to FCI) on the excitation energies obtained at the KS-eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium.
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The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
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