  We highlight the non-universality of the asymptotic behavior of dispersion
forces, such that a sum of inverse sixth power contributions is often
inadequate. We analytically evaluate the cross-correlation energy Ec between
two pi-conjugated layers separated by a large distance D within the
electromagnetically non-retarded Random Phase Approximation, via a
tight-binding model. For two perfect semimetallic graphene sheets at T=0K we
find Ec = C D^{-3}, in contrast to the "insulating" D^{-4} dependence predicted
by currently accepted approximations. We also treat the case where one graphene
layer is replaced by a thin metal, a model relevant to the exfoliation of
graphite. Our general considerations also apply to nanotubes, nanowires and
layered metals.
