Manu: first fully written version of the exact theory section

Manuscript/eDFT.tex

@@ -97,6 +97,7 @@
 \newcommand{\eeq}{\end{eqnarray}}
 \newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
 \newcommand{\bmg}{\bm{\Gamma}} % orbital rotation vector
+\newcommand{\bxi}{\bm{\xi}} 
 \newcommand{\bfx}{\bf{x}}
 \newcommand{\bfr}{\bf{r}}
 \DeclareMathOperator*{\argmin}{arg\,min}
@@ -206,12 +207,12 @@ Tr}$ denotes the trace and the trial ensemble density matrix operator reads
 \opGamma{{\bw}}=\sum_{K\geq 0}w_K\ket{\Phi^{(K)}}\bra{\Phi^{(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
+\beq\label{eq:KS_ens_density}
 n_{\opGamma{\bw}}(\br)=\sum_{K\geq 0}w_Kn_{\Phi^{(K)}}(\br).
 \eeq
 As readily seen from Eq.~(\ref{eq:var_ener_gokdft}), both Hartree-exchange and 
 correlation energies are described with density functionals that are {\it weight-dependent}. We focus here on the (exact) Hx part which is defined as follows:
-\beq
+\beq\label{eq:exact_ens_Hx}
 {E}^{{\bw}}_{\rm
 Hx}[n]=\sum_{K\geq 0}w_K\bra{\Phi^{(K)}[n]}\hat{W}_{\rm
 ee}\ket{\Phi^{(K)}[n]}
@@ -227,8 +228,112 @@ In practice, one is not much interested in ensemble energies but rather in excit
 E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w_K\right)\dfrac{\partial
 E^{{\bw}}}{\partial w_K},
 \eeq
-where, according to the variational ensemble energy expression of Eq.~(\ref{eq:var_ener_gokdft}),  
+where, according to the {\it variational} ensemble energy expression of Eq.~(\ref{eq:var_ener_gokdft}), the derivative in $w_K$ can be evaluated from the minimizing KS wavefunctions $\Phi^{(K)}=\Phi^{(K),\bw}$ as follows:  
+\beq\label{eq:deriv_Ew_wk}
+&&\dfrac{\partial
+E^{{\bw}}}{\partial w_K}=\bra{\Phi^{(K)}}\hat{h}\ket{\Phi^{(K)}}-\bra{\Phi^{(0)}}\hat{h}\ket{\Phi^{(0)}}
+\nonumber\\
+&&+\Bigg[\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
+Hx}\left[n\right]}{\delta
+n({\br})}\left(n_{\Phi^{(K)}}(\br)-n_{\Phi^{(0)}}(\br)\right)
+%\nonumber\\
+%&&
++
+%\left.       
+\dfrac{\partial {E}^{{\bw}}_{\rm
+Hx}\left[n\right]}{\partial w_K}
+%\right|_{n=n_{\opGamma{\bw}}}
+\nonumber\\
+&&+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
+c}\left[n\right]}{\delta
+n({\br})}\left(n_{\Phi^{(K)}}(\br)-n_{\Phi^{(0)}}(\br)\right)
+%\nonumber\\
+%&&
++
+%\left.
+\dfrac{\partial {E}^{{\bw}}_{\rm
+c}\left[n\right]}{\partial w_K}
+%\right|
+\Bigg]_{n=n_{\opGamma{\bw}}}
+.
+\nonumber\\
+\eeq
+The Hx contribution to Eq.~(\ref{eq:deriv_Ew_wk}) can be rewritten as follows: 
+\beq\label{eq:_deriv_wk_Hx}
+\left.\dfrac{\partial
+}{\partial \xi_K}
+\left({E}^{{\bxi}}_{\rm
+Hx}\left[n^{\bxi,\bxi}\right]
+-
+{E}^{{\bw}}_{\rm
+Hx}\left[n^{\bw,\bxi}\right]
+\right)\right|_{\bxi=\bw},
+\eeq
+where $\bxi\equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and
+\beq
+n^{\bw,\bxi}(\br)=\sum_{K\geq 0}w_Kn_{\Phi^{(K),\bxi}}(\br).
+\eeq 
+Since, for given ensemble weight $\bw$ and $\bxi$ values, the ensemble densities $n^{\bxi,\bxi}$ and $n^{\bw,\bxi}$ are generated from the {\it same} KS potential (which is unique up to a constant), it comes
+from the exact expression in Eq.~(\ref{eq:exact_ens_Hx}) that 
+\beq
+{E}^{{\bxi}}_{\rm
+Hx}\left[n^{\bxi,\bxi}\right]=\sum_{K\geq 0}\xi_K\bra{\Phi^{(K),\bxi}}\hat{W}_{\rm
+ee}\ket{\Phi^{(K),\bxi}}
+\eeq  
+with $\xi_0=1-\sum_{K>0}\xi_K$
+and
+\beq
+{E}^{{\bw}}_{\rm
+Hx}\left[n^{\bw,\bxi}\right]=\sum_{K\geq 0}w_K\bra{\Phi^{(K),\bxi}}\hat{W}_{\rm
+ee}\ket{\Phi^{(K),\bxi}},
+\eeq  
+thus leading, according to Eqs.~(\ref{eq:deriv_Ew_wk}) and (\ref{eq:_deriv_wk_Hx}), to the simplified expression
+\beq\label{eq:deriv_Ew_wk_simplified}
+&&\dfrac{\partial
+E^{{\bw}}}{\partial w_K}=\bra{\Phi^{(K)}}\hat{H}\ket{\Phi^{(K)}}-\bra{\Phi^{(0)}}\hat{H}\ket{\Phi^{(0)}}
+\nonumber\\
+&&+\Bigg[
+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
+c}\left[n\right]}{\delta
+n({\br})}\left(n_{\Phi^{(K)}}(\br)-n_{\Phi^{(0)}}(\br)\right)
+%\nonumber\\
+%&&
++
+%\left.
+\dfrac{\partial {E}^{{\bw}}_{\rm
+c}\left[n\right]}{\partial w_K}
+%\right|
+\Bigg]_{n=n_{\opGamma{\bw}}}
+.
+\nonumber\\
+\eeq
+Since the ensemble energy can be evaluated as follows:
+\beq
+E^{{\bw}}=\sum_{K\geq 0}w_K\bra{\Phi^{(K)}}\hat{H}\ket{\Phi^{(K)}}+{E}^{{\bw}}_{\rm
+c}\left[n_{\opGamma{\bw}}\right],
+\eeq 
+with $\Phi^{(K)}=\Phi^{(K),\bw}$ [note that, when the minimum is reached
+in Eq.~(\ref{eq:var_ener_gokdft}), $n_{\opGamma{\bw}}=n^{\bw,\bw}$], 
+we finally recover from Eqs.~(\ref{eq:KS_ens_density}) and
+(\ref{eq:indiv_ener_from_ens}) the {\it exact} expression of Ref.~\cite{} for the $I$th energy level:
+\beq
+E^{(I)}&=&\bra{\Phi^{(I)}}\hat{H}\ket{\Phi^{(I)}}+{E}^{{\bw}}_{\rm
+c}\left[n_{\opGamma{\bw}}\right]
+\nonumber\\
+&&+
+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
+c}\left[n_{\opGamma{\bw}}\right]}{\delta
+n({\br})}\left(n_{\Phi^{(I)}}(\br)-n_{\opGamma{\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_{\opGamma{\bw}}}.
+\eeq
+%%%%%%%%%%%%%%%
 %\subsection{Hybrid GOK-DFT}
+%%%%%%%%%%%%%%%
 
 
 \subsection{Approximations}