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Various properties (geometry, band structure and the totally symmetric vibrational modes) of small diameter single-walled carbon nanotubes (SWCNTs) were investigated by first principles density functional theory (DFT) calculations. We studied 40 different SWCNTs, including 14 chiral ones down to diameters of 0.3 nm. The behavior of small diameter tubes is significantly different from that of the usual, larger diameter nanotubes. The diameter is larger than what is expected from simple folding. The bond lengths and bond angles are not uniform. The strong σ–π rehybridization effect modifies the band structure with respect to the tight binding approximation. The frequency of the radial breathing mode (RBM) shows a softening with decreasing diameter as compared to the usual 1/d dependence and this softening depends strongly on chirality. RBM frequencies are further modified by the coupling with high frequency totally symmetric modes in a non-negligible way for small diameter tubes. These deviations cannot be described by a smooth monotonic function of the diameter. [Copyright &y& Elsevier]