In a recent paper [Phys. Rev. B 90, 125102 ( 2014)], we showed that the random phase approximation with exchange (RPAx) gives accurate total energies for a diverse set of systems including the high and low density regime of the homogeneous electron gas, the N-2 molecule, and the H-2 molecule at dissociation. In this paper, we present results for the van der Waals bonded Ar-2 and Kr-2 dimers and demonstrate that the RPAx gives superior dispersion forces as compared to the RPA. We then show that this improved description is crucial for the bond formation of the Mg-2 molecule. In addition, the RPAx performs better for the Be-2 dissociation curve at large nuclear separation but, similar to the RPA, fails around equilibrium due to the build up of a large repulsion hump. For the strongly correlated LiH molecule at dissociation we have also calculated the RPAx potential and find that the correlation peak at the bond midpoint is overestimated as compared to the RPA and the exact result. The step feature is missing and hence the delocalization error is comparable to the RPA. This is further illustrated by a smooth energy versus fractional charge curve and a poor description of the LiH dipole moment at dissociation.