The recently introduced coupled-cluster effective-Hamiltonian (CC-EH) approximation is used to calculate the C6 dispersion constants for selected interatomic potentials. The formal relationship between C6 and an integral over the dynamic polarizability evaluated at imaginary frequencies is used in the calculations, and a discussion of its computational implementation within the CC-EH framework is presented. For the long-range interaction between two ground-state beryllium atoms, the calculated dispersion constant is in good agreement with the full-configuration-interaction result. The applicability of the method to both closed- and open-shell systems is demonstrated by calculations of C6 for interactions involving the ground states of both 1S alkaline-earth-metal and 2S alkali-metal atoms. The results of these calculations are in acceptable agreement with recommended values for most examples. However, significant discrepancies between the present estimates and empirical upper and lower bounds are observed for all systems containing calcium atoms. This observation, as well as previous criticism of calculations that were used in establishing these bounds, suggests that a reexamination of the empirical estimates is warranted.