||We use first-principles calculations to study the structural, electronic, vibrational and optical properties of single-wall carbon nanotubes (SWNT) with diameters of about 4 Å. These SWNT have recently been synthesized inside zeolite channels. There are three possible SWNT with such diameters and they are the zigzag (5,0), the armchair (3,3), and the chiral (4,2). 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 calculated dielectric functions and breathing mode frequencies agree with experimentally measured absorption spectrum and Raman spectrum respectively, giving strong support that these 4 Å tubes are indeed present inside the zeolite crystals. The same calculations for 5 Å tube (6,0) suggest that it cannot be present in appreciable quantities. The effect of nitrogen and oxygen on the electronic properties of the 4 Å tubes are examined. We find that substitutional nitrogen can account for the strong visible photoluminescence (PL) observed in these tubes. Interaction of the tubes with oxygen indicates that they are stable at room temperature. Calculations on the intercalation of carbon nanotubes with alkali atoms suggest that it is theoretically possible for Li to form a single-atom chain inside the interior of these tubes. The Li binding energy exhibits strong chirality dependence, which is traced to the different electron affinity of the tubes. The Li concentration can be very high when the zeolite template is included explicitly in the calculations.