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We present a comprehensive first-principles investigation of the atomic and electronic structures of gallium nitride nanowires, and examine the dependence on nanowire diameter and shape. We consider nanowires in the [0001] growth direction, with diameters ranging from $8\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}35\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$, and investigate the influence of saturating the dangling bonds at the edges of nanowires. We find that unsaturated nanowires are semiconducting and contain dangling bond states in the region of the band gap, the positions of which remain rather constant with varying diameter. Saturating the nanowires with hydrogen removes these states, and the band gap decreases with increasing nanowire diameter. For the unsaturated wires there is a considerable contraction of the Ga-N bond lengths at the edge of the wires of 6.0%--7.4%, while for saturated wires it is 1.5%. We also calculate the heat of formation of the nanowires and find that as the diameter of the nanowire increases, the average relative stability of the nanowire increases, as intuitively expected.

Geometry and diameter dependence of the electronic and physical properties of GaN nanowires from first principles