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Penalty-functional form

Condition (3) requires that $P[\rho]$ be expressed as the trace of a polynomial of $\rho({\bf r},{\bf r}')$, and thus of its eigenvalues $\{ f_i \}$:

\begin{displaymath}P[\rho] = \sum_i \sum_{n=0}^{N_{\rm p}} \alpha_i^{(n)} f_i^n \end{displaymath}

In terms of the $\{ f_i \}$ and the expectation values $\{ \tilde{\varepsilon_i} \}$ of ${\hat H}_{\rm KS}$ of the natural orbitals [eigenfunctions of $\rho({\bf r},{\bf r}')$]

\begin{displaymath}E[\rho] - \lambda N[\rho] = 2 \sum_i f_i \left( \tilde{\varepsilon}_i
- \lambda \right) \end{displaymath}

Peter D. Haynes 2001-11-19