A detailed theoretical analyste of the band structure and Davydov splittings in the p and β bands of isotactic polystyrene and polyvinylnaphthalene has been undertaken. It is shown that singlet band splittings are large and have different symmetry characteristics from those of crystalline benzene and naphthalene. The triplet bandwidths are, however, very small. The rate of triplet-triplet annihilation is shown to be considerably less than that characteristic of a monomeric aromatic crystal such as anthracene or naphthalene and therefore phosphorescence from the pure polymer is a distinct possibility. The role of aperiodic potentials on the bandwidths is briefly discussed.