Idempotency

• Impose normalisation with a Lagrange multiplier $\lambda$.
• Impose idempotency by adding an extra penalty-functional $P[\rho]$ to the energy $E[\rho]$

i.e. minimise

$$Q[\rho] = E[\rho] - \lambda N[\rho] + P[\rho]$$

Choose $P[\rho]$ such that:

1. $Q[\rho]$ is minimised by the idempotent ground-state density-matrix
2. the eigenvalue spectrum of the Hessian of $Q[\rho]$ is as narrow as possible $\rightarrow$ efficient minimisation
3. $P[\rho]$ may be evaluated in $O(N)$ operations