  We review and expand on our work to impose constraints on the effective Kohn
Sham (KS) potential of local and semi-local density functional approximations.
In this work, we relax a previously imposed positivity constraint, which
increased the computational cost and we find that it is safe to do so, except
in systems with very few electrons. The constrained minimisation leads
invariably to the solution of an optimised effective potential (OEP) equation
in order to determine the KS potential. We review briefly our previous work on
this problem and demonstrate with numerous examples that despite well-known
mathematical issues of the OEP with finite basis sets, our OEP equations are
well behaved. We demonstrate that constraining the screening charge of the
Hartree, exchange and correlation potential not only corrects its asymptotic
behaviour but also allows the exchange and correlation potential to exhibit
nonzero derivative discontinuity, a feature of the exact KS potential that is
necessary for the accurate prediction of band-gaps in solids but very hard to
capture with semi-local approximations.
