  We present an implementation of the optimised effective potential (OEP)
scheme for the exact-exchange (EXX) and random phase approximation (RPA) energy
functionals and apply these methods to a range of bulk materials. We calculate
the Kohn-Sham (KS) potentials and the corresponding band gaps and compare them
to the potentials obtained by standard local density approximation (LDA)
calculations. The KS gaps increase upon going from the LDA to the OEP in the
RPA and finally to the OEP for EXX. This can be explained by the different
depth of the potentials in the bonding and interstitial regions. To obtain the
true quasi-particle gaps the derivative discontinuities or $G_0W_0$ corrections
need to be added to the RPA-OEP KS gaps. The predicted $G_0W_0$@RPA-OEP
quasi-particle gaps are about 5% too large compared to the experimental values.
However, compared to $G_0W_0$ calculations based on local or semi-local
functionals, where the errors vary between different materials, we obtain a
rather consistent description among all the materials.
