  Self-consistent correlation potentials for H$_2$ and LiH for various
inter-atomic separations are obtained within the random phase approximation
(RPA) of density functional theory. The RPA correlation potential shows a peak
at the bond midpoint, which is an exact feature of the true correlation
potential, but lacks another exact feature: the step important to preserve
integer charge on the atomic fragments in the dissociation limit. An analysis
of the RPA energy functional in terms of fractional charge is given which
confirms these observations. We find that the RPA misses the derivative
discontinuity at odd integer particle numbers but explicitly eliminates the
fractional spin error in the exact-exchange functional. The latter finding
explains the accurate total energy in the dissociation limit.
