Purifying transformation

Imposing condition (1), we need $N_{\rm p} \geq 3$:

Solve for the coefficients $\{ \alpha_i^{(n)} \}$ to obtain

• Recover an existing $O(N)$ method based upon the purifying transformation.
• Repeating the procedure for positive-definite density-matrices leads to the Ordejón-Mauri functional$^{,}$.