  Quantum size effects in armchair graphene nano-ribbons (AGNR) with hydrogen
termination are investigated via density functional theory (DFT) in Kohn-Sham
formulation. "Selection rules" will be formulated, that allow to extract
(approximately) the electronic structure of the AGNR bands starting from the
four graphene dispersion sheets. In analogy with the case of carbon nanotubes,
a threefold periodicity of the excitation gap with the ribbon width (N, number
of carbon atoms per carbon slice) is predicted that is confirmed by ab initio
results. While traditionally such a periodicity would be observed in electronic
response experiments, the DFT analysis presented here shows that it can also be
seen in the ribbon geometry: the length of a ribbon with L slices approaches
the limiting value for a very large width 1 << N (keeping the aspect ratio
small N << L) with 1/N-oscillations that display the electronic selection
rules. The oscillation amplitude is so strong, that the asymptotic behavior is
non-monotonous, i.e., wider ribbons exhibit a stronger elongation than more
narrow ones.
