It is first argued that nuclear energy calculations with a density-dependent force may be carried out in a basis which is not self-consistent. The underlying formalism is outlined and the rearrangement energy is shown to be due, in this approach, to the survival of an unlinked diagram which would be cancelled in the absence of density dependence. The formalism is then applied in first order perturbation theory to the calculation of the binding energies of the closed shell nuclei 4He, 16O and 40Ca and of the single-particle energies of 5He, 17O and 41Ca. The force employed throughout is the Skyrme III force, but with a model-density dependent term replacing the three-body term. The binding energies obtained are generally in good agreement with empirical values. As for the single-particle energies, we obtain qualitative agreement with available experimental data, and contributions due to rearrangement which are in agreement with the estimates given by Elton.