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In a recent article [Phys. Rev. B 90, 125102 (2014)] we showed that therandom phase approximation with exchange (RPAx) gives accurate total energiesfor a diverse set of systems including the high and low density regime of thehomogeneous electron gas, the N$_2$ molecule, and the H$_2$ molecule atdissociation. In this work, we present results for the van der Waals bondedAr$_2$ and Kr$_2$ dimers and demonstrate that the RPAx gives superiordispersion forces as compared to the RPA. We then show that this improveddescription is crucial for the bond formation of the Mg$_2$ molecule. Inaddition, the RPAx performs better for the Be$_2$ dissociation curve at largenuclear separation but, similar to the RPA, fails around equilibrium due to thebuild up of a large repulsion hump. For the strongly correlated LiH molecule atdissociation we have also calculated the RPAx potential and find that thecorrelation peak at the bond mid-point is overestimated as compared to the RPAand the exact result. The step feature is missing and hence the delocalisationerror is comparable to the RPA. This is further illustrated by a smooth energyversus fractional charge curve and a poor description of the LiH dipole momentat dissociation.

Molecular bonding with the RPAx: from weak dispersion forces to strong correlation