  In a recent article [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 work, 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 mid-point is overestimated as compared to the RPA
and the exact result. The step feature is missing and hence the delocalisation
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.
