In this chapter we first outline Kohn's derivation of a variational principle for a generalised energy functional which includes a penalty functional to impose the idempotency constraint. We show that this functional is non-analytic at its minimum and therefore incompatible with efficient minimisation algorithms, using conjugate gradients as an example.
We then outline an original scheme to use well-behaved penalty functionals
to approximately impose the idempotency constraint. The density-matrix
which minimises these generalised energy functionals is therefore only an
approximation to the true ground-state density-matrix, but the resulting
error in the total energy can be corrected to obtain accurate estimates of
the true ground-state energy.