Manu: saving work

Manuscript/eDFT.tex

@@ -316,7 +316,7 @@ 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
+\beq\label{eq:exact_ener_level_dets}
 E^{(I)}&=&\bra{\Phi^{(I)}}\hat{H}\ket{\Phi^{(I)}}+{E}^{{\bw}}_{\rm
 c}\left[n_{\opGamma{\bw}}\right]
 \nonumber\\
@@ -346,11 +346,45 @@ determinant $\Phi^{(K)}$ can be expressed as follows in the AO basis:
 \Gamma_{\mu\nu}^{(K)}=\sum_{p\in (K)}c_{\mu p}c_{\nu p},
 \eeq
 where the summation runs over the spin-orbitals that are occupied in
-$\Phi^{(K)}$. We can then construct the ensemble density matrix
+$\Phi^{(K)}$. Note that the density of the $K$th KS state reads
+\beq
+n_{\bmg^{(K)}}(\br)=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,
+\sigma})\AO{\nu}(\br,\sigma){\Gamma}^{(K)}_{\mu\nu}.
+\eeq 
+We can then construct the ensemble density matrix
+and the ensemble density as follows:
 \beq
 {\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}
 \eeq
-and compute the ensemble density as follows:
+and
+\beq
+n_{\bmg^\bw}({\br})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,\sigma})\AO{\nu}(\br,\sigma){\Gamma}^{\bw}_{\mu\nu},
+\eeq
+respectively. The exact energy level expression in Eq.~(\ref{eq:exact_ener_level_dets}) can be
+rewritten as follows:
+\beq
+E^{(I)}&&={\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
+where ${\bm
+h}\equiv\langle\AO{\mu}\vert-\frac{1}{2}\nabla_{\br}^2+v_{\rm
+ne}(\br)\vert\AO{\nu}\rangle$ and ${\bm G}\equiv{\bm J}-{\bm K}$ denote
+the Coulomb-exchange
+integrals. 
+%%%%%%%%%%%%%%%%%%%%%
+\iffalse%%%% Manu's derivation ...
 \blue{
 \beq
 n^{\bw}({\br})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
@@ -377,10 +411,7 @@ w}_K
 &=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
 \eeq
 }
-\beq
-n^{\bw}({\br})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,\sigma})\AO{\nu}(\br,\sigma){\Gamma}^{\bw}_{\mu\nu}
-\eeq
-can be determined. 
+\fi%%%%%%%% end
 %%%%%%%%%%%%%%%
 %\subsection{Hybrid GOK-DFT}
 %%%%%%%%%%%%%%%