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 $\{ f_i \}$:

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

• The coefficients $\{ \alpha_i^{(n)} \}$ may vary between states $i$.
• Make $N_{\rm p}$ as small as possible.

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

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