  We investigate curvature effects on geometric parameters, energetics and
electronic structure of zigzag nanotubes with fully optimized geometries from
first-principle calculations. The calculated curvature energies, which are
inversely proportional to the square of radius, are in good agreement with the
classical elasticity theory. The variation of the band gap with radius is found
to differ from simple rules based on the zone folded graphene bands. Large
discrepancies between tight binding and first principles calculations of the
band gap values of small nanotubes are discussed in detail.
