Manu: first version of the theory section. Will polish the manuscript now

Manuscript/eDFT.tex

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