Properties of $4\mathrm{\AA }$ carbon nanotubes from first-principles calculations

Local-density-functional method is used to study the electronic properties of single-wall carbon nanotubes with very small diameters of about $4\mathrm{\AA },$ which have recently been synthesized inside zeolite channels. Our calculated results indicate that both the (5,0) and (3,3) tubes are metallic, while the (4,2) tube is semiconducting with a small indirect band gap. The metallicity of (5,0) is due to strong ${\sigma }^{*}$ and ${\pi }^{*}$ mixing induced by the large curvature of the tube. The calculated dielectric functions and breathing mode frequencies agree with experimentally measured absorption spectrum and Raman spectrum, respectively, giving strong support that these small radius tubes are indeed present inside the zeolite crystals.