We report density functional calculations of the atomic and electronic structure of the spinel phases ZnRh2O4 and ZnIr2O4 as well as crystalline ZnO lightly doped (1 at.%) with Rh and Ir ions using the B3LYP hybrid functional. Calculations for the spinels show band gaps (similar to 3 eV) and lattice parameters (similar to 2% difference) in reasonable agreement with experimental data. Incorporation of the transition metals into ZnO induces local distortions in the lattice and the appearance of metal d levels in the low gap region and near the conduction band minimum, with a d-d splitting of about 2 eV, which reduces the effective transparency of the material. Addition of a hole to the simulation cell of both spinels and doped ZnO leads to charge localization in the neighbourhood of Rh/Ir accompanied by local lattice deformations to form a small polaron which may lead to low hole mobility. We calculate polaron diffusion barriers in the spinels and obtain values around 0.02-0.03 eV. These very low barrier energies suggest that at high Rh/Ir concentrations polaron hopping is not going to be detected at room temperature. (C) 2013 Elsevier B.V. All rights reserved.