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6. Penalty Functionals

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.


Peter Haynes