  The static electric dipole polarizability of $\mathrm{Na_N}$ clusters with
even N has been calculated in a collective, axially averaged and a
three-dimensional, finite-field approach for $2\le N \le 20$, including the
ionic structure of the clusters. The validity of a collective model for the
static response of small systems is demonstrated. Our density functional
calculations verify the trends and fine structure seen in a recent experiment.
A pseudopotential that reproduces the experimental bulk bond length and atomic
energy levels leads to a substantial increase in the calculated
polarizabilities, in better agreement with experiment. We relate remaining
differences in the magnitude of the theoretical and experimental
polarizabilities to the finite temperature present in the experiments.
