Manu: saving work

Manuscript/eDFT.tex

@@ -333,14 +333,26 @@ _{n=n_{\opGamma{\bw}}}.
 \eeq
 
 For implementation purposes, we will use in the rest of this work 
-(one-electron reduced) density matrices rather than Slater determinants
+(one-electron reduced) density matrices 
-as basic variables. If we denote ${\bmg}^{(K)}$  
+as basic variables, rather than Slater determinants. If we expand the
-
+(spin-) orbitals [from which the latter are constructed] in an atomic
+orbital (AO) basis,
 \beq
-\Gamma_{\mu\nu}^{(K)}=\sum_{p\in (K)}c_{\mu p}c_{\nu p}
+\varphi_p({\bfx})=\sum_{\mu}c_{{\mu p}}\AO{\mu}(\bfx),
+\eeq
+then the density matrix elements obtained from the   
+determinant $\Phi^{(K)}$ can be expressed as follows in the AO basis:
+\beq
+\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
-$\Phi^{(K)}$. 
+$\Phi^{(K)}$. We can then construct the ensemble density matrix
+\beq
+{\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}
+\eeq
+and compute the ensemble density as follows:
+$n^{\bw}({\br})=$
+can be determined. 
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 %\subsection{Hybrid GOK-DFT}
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