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.
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