reformating equations
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@ -178,7 +178,7 @@ The present eLDA functional is specifically designed to compute double excitatio
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The paper is organised as follows.
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In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
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Section \ref{sec:func} provides details about the construction of the weight-dependent xc LDA functional.
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The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}.
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The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:resdis}.
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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Unless otherwise stated, atomic units are used throughout.
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@ -418,13 +418,13 @@ Combining these, we build a two-state weight-dependent correlation functional:
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Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}.
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The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.}
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\begin{ruledtabular}
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\begin{tabular}{ldd}
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& \tabc{Ground state} & \tabc{Doubly-excited state} \\
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& \tabc{$I=0$} & \tabc{$I=1$} \\
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\begin{tabular}{lll}
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& \tabc{Ground state} & \tabc{Doubly-excited state} \\
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& \tabc{$I=0$} & \tabc{$I=1$} \\
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\hline
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$a_1$ & -0.023\,818\,4 & -0.014\,463\,3 \\
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$a_2$ & +0.005\,409\,94 & -0.050\,601\,9 \\
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$a_3$ & +0.083\,076\,6 & +0.033\,141\,7 \\
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$a_1$ & $-0.023\,818\,4$ & $-0.014\,463\,3$ \\
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$a_2$ & $+0.005\,409\,94$ & $-0.050\,601\,9$ \\
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$a_3$ & $+0.083\,076\,6$ & $+0.033\,141\,7$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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@ -552,8 +552,8 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o
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%%%%%%%%%%%%%%%
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%%% RESULTS %%%
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%%%%%%%%%%%%%%%
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\section{Results}
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\label{sec:res}
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\section{Results and discussion}
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\label{sec:resdis}
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Here, we consider as testing ground the minimal-basis \ce{H2} molecule.
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We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr).
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This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry.
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@ -561,6 +561,11 @@ We have then access to the individual densities of the ground and doubly-excited
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Moreover, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities.
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In other words, there is no density-driven error and the only error that we are going to observe is the functional-driven error (and this is what we want to study).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Results}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The bonding and antibonding orbitals of the \ce{H2} molecule are given by
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\begin{subequations}
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\begin{align}
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@ -703,7 +708,6 @@ Alternatively to Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}, o
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\end{equation}
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(This is what one would do in practice, \ie, by performing a KS ensemble calculation.)
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We will label these energies as $\tE{}{\ew{}}$ to avoid confusion with the expressions reported in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}.
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\begin{widetext}
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For HF, we have
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\begin{equation}
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\label{eq:bEwHF}
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@ -711,10 +715,12 @@ For HF, we have
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\tE{\HF}{\ew{}}
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& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
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+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
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+ \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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+ (1-\ew{})^2 (2\eJ{11}- \eK{11}) + \ew{}^2 (2\eJ{22}- \eK{22}) + 2 (1-\ew{})\ew{} (2 \eJ{12} - \eK{12}),
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& + \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} + (1-\ew{})^2 (2\eJ{11}- \eK{11})
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\\
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& + \ew{}^2 (2\eJ{22}- \eK{22}) + 2 (1-\ew{})\ew{} (2 \eJ{12} - \eK{12}),
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\end{split}
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\end{equation}
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which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term.
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@ -725,14 +731,17 @@ In the case of the LDA, it reads
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\tE{\LDA}{\ew{}}
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& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
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+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
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+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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\\
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& + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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+ \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
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\\
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& + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
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\\
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& + (1-\ew{}) \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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+ \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
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\\
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& + \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
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\end{split}
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\end{equation}
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which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term.
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@ -743,25 +752,33 @@ For eLDA, the ensemble energy can be decomposed as
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\tE{\eLDA}{\ew{}}
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& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
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+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
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+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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\\
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& + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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+ \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
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\\
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& + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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+ \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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& + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
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\\
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& + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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+ \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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+ (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ]
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+ \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ]
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\\
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& + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12}
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+ \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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+ \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ],
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& + \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
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% & + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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% \\
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% & + \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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% \\
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% & + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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% \\
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% & + \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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% \\
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% & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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% \\
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% &
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% + (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ]
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% \\
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% & + \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ]
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% \\
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% & + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12}
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% + \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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% & + \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ],
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\end{split}
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\end{equation}
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which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term.
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@ -782,36 +799,55 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew
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\begin{subequations}
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\begin{align}
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\begin{split}
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\eps{1}{\ew{},\LDA}
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& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
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+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
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+ \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
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\\
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& + \frac{1}{2} \int \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(0)}(\br{}) d\br{},
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\\
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& + \frac{1}{2} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{},
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\end{split}
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\\
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\begin{split}
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\eps{2}{\ew{},\LDA}
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& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
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+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
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+ \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{},
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\\
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& + \frac{1}{2} \int \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(1)}(\br{}) d\br{},
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\\
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& + \frac{1}{2} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
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\end{split}
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\end{align}
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\end{subequations}
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\begin{subequations}
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\begin{align}
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\begin{split}
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\eps{1}{\ew{},\eLDA}
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& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
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+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
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+ \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
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\\
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& + \frac{1}{2} \int \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(0)}(\br{}) d\br{},
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\\
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& + \frac{1}{2} \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{},
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\end{split}
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\\
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\begin{split}
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\eps{2}{\ew{},\eLDA}
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& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2\ew{} \eJ{22}
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+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
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+ \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{}.
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\\
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& + \frac{1}{2} \int \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(1)}(\br{}) d\br{}.
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\\
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& + \frac{1}{2} \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}.
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\end{split}
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\end{align}
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\end{subequations}
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\end{widetext}
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The derivative discontinuity is modelled by the last term of the right-hand-side of Eq.~\eqref{eq:dEdw}.
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Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Discussion}
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\label{sec:dis}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Numerical results are reported in Table \ref{tab:Energies}.
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