Using a parameter-free, density-functional method that has been developed explicitly for the theoretical treatment of infinite, periodic, isolated, helical polymers we study various polymers related to polythiophene. In particular we discuss how the electronic properties of polythiophene are changed when replacing some of the H atoms by CH₃ group, by incorporating vinylene bridges into the backbone, or when replacing some or all the CH units of the backbone by N atoms. We observe the weakest effects for the methyl-substitution and the strongest for the N-incorporation. The latter leads to an overall downward shift of all bands, but in contrast to the case for polyacetylene, the unrelaxed compound with N atoms does not have N lone-pair orbitals as the highest occupied ones. Instead these occur at somewhat deeper energies. When comparing the aromatic and quinoid forms we found for the pure compound as well as for the methyl-containing one that the gap closes when passing from the one to the other form which was not found for any of the other materials of the present study. Moreover, the energy of the HOMO was found to depend stronger on the bond-length alternation than the energy of the LUMO, ultimately giving that polarons will induce two asymmetrically placed gap states with the energetically lower one appearing deeper in the gap than the other one.