Manu: first version of the theory section. Will polish the manuscript now
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@ -362,7 +362,7 @@ n_{\bmg^\bw}({\br})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,\sigma}
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\eeq
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respectively. The exact energy level expression in Eq.~(\ref{eq:exact_ener_level_dets}) can be
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rewritten as follows:
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\beq
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\beq\label{eq:exact_ind_ener_rdm}
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E^{(I)}&&={\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
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@ -445,58 +445,33 @@ Note that this approximation, where the ensemble density matrix is
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optimized from a non-local exchange potential [rather than a local one,
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as expected from Eq.~(\ref{eq:var_ener_gokdft})] is applicable to real
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(three-dimension) systems. As readily seen from
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Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost-interaction} errors will be
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introduced in the ensemble HF interaction energy:
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Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost interactions}~\cite{}
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and curvature~\cite{} will be
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introduced in the Hx energy:
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\beq
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W_{\rm
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HF}\left[{\bmg}^\bw\right]&=&\frac{1}{2}\sum_{K\geq 0}w^2_K
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\Tr(\bmg^{(K)} \,
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\bG \, \bmg^{(K)})
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\nonumber\\
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&&+\sum_{L>K\geq 0}w_Kw_L\Tr(\bmg^{(K)} \,
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\bG \, \bmg^{(L)}).
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\eeq
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These errors will be removed when computing individual energies
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according to Eq.~(\ref{eq:exact_ind_ener_rdm}).\\
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In order to remove ghost interactions from the variational energy
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expression used in the first step, we then employ the (in-principle-exact)
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expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
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step, the response of the individual density matrices to weight
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variations (last term on the right-hand side of
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Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
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procedure can be summarized as follows,
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and
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In order to compute (approximate) energy levels within generalized
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GOK-DFT we use a two-step procedure. The first step consists in
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optimizing variationally the ensemble density matrix according to
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Eq.~(\ref{eq:var_princ_Gamma_ens}) with an approximate Hxc ensemble
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functional where (i) the ghost-interaction correction functional $\overline{E}^{{\bw}}_{\rm
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Hx}[n]$ in
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Eq.~(\ref{eq:exact_GIC}) is
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neglected, for simplicity, and (ii) the weight-dependent correlation
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energy is described at the local density level of approximation.
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At this
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level of approximation, the two correlation functionals $\overline{E}^{{\bw}}_{\rm
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c}[n]$ and ${E}^{{\bw}}_{\rm
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c}[n]$ are actually identical and can be expressed as
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Turning to the density-functional ensemble correlation energy, the
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following eLDA will be employed:
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\beq\label{eq:eLDA_corr_fun}
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{E}^{{\bw}}_{\rm
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c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)).
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c}[n]=\int d\br\;n(\br)\;\epsilon_{c}^{\bw}(n(\br)),
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\eeq
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More
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details about the construction of such a functional will be given in the
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following.
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\beq
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E^{(I)}&&\approx{\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
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\bmg^{(I)})
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\nonumber\\
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&&+{E}^{{\bw}}_{\rm
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c}\left[n_{\bmg^{\bw}}\right]
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+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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c}\left[n_{\bmg^{\bw}}\right]}{\delta
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n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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\nonumber\\
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&&+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
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,
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\eeq
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thus leading to the final implementable expression [see Eq.~(\ref{eq:eLDA_corr_fun})]
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where the correlation energy per particle is {\it weight-dependent}. Its
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construction from a finite uniform electron gas model is discussed
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in detail
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in Sec.~\ref{sec:eDFA}. Combining Eq.~(\ref{eq:exact_ind_ener_rdm}) with
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Eq.~(\ref{eq:eLDA_corr_fun}) leads to our final energy level expression
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within eLDA:
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\beq
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E^{(I)}&&\approx{\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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@ -508,8 +483,10 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
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\nonumber\\
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&&
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+\int d\br\,\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
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c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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+\int d\br\,
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n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
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c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}
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\nonumber\\
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&&
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+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
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@ -517,6 +494,33 @@ c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bm
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c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
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\eeq
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%%%% REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
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%\iffalse%%%%
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\blue{
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Indeed,
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\beq
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\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&\sum_{K\geq
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0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&
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{\bmg}^{{\bw}}+\sum_{K,L\geq
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0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
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\nonumber\\
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&&+\sum_{K, L >0
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}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&\neq&{\bmg}^{{\bw}}
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.
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\eeq
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}
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%%%% End -- REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
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%\fi%%%
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\blue{$================================$}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory (old)}
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@ -1299,31 +1303,6 @@ E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Gra
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%% REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
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\iffalse%%%%
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Indeed,
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\beq
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\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&\sum_{K\geq
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0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&
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{\bmg}^{{\bw}}+\sum_{K,L\geq
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0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
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\nonumber\\
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&&+\sum_{K, L >0
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}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&\neq&{\bmg}^{{\bw}}
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.
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\eeq
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%%%% End -- REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
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\fi%%%
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