Optical properties of a series of finite sized hydrogenated carbon nanotubes with the smallest diameter of 4 A are studied systematically. Their absorption spectra are calculated with the localized-density-matrix method. The semiempirical MNDO parametric method 3 (PM3) Hamiltonian is employed. The finite optical gaps are predicted for the infinite long single-walled carbon nanotubes. Strong anisotropy characteristics of the dynamic polarizabilities are found for these tubes. The calculated results are in good agreement with the recent experimental findings. Further the compositions of the dipole-induced excitations are examined by projecting the corresponding density matrices onto the Hartree-Fock molecular orbital representation. Unlike the larger diameter carbon nanotubes whose absorption spectra are insensitive to the tube chiralities, the absorption spectra of 4 A single-walled carbon nanotubes depend very much on their chiralities. The chirality of the single-walled 4 A carbon nanotubes synthesized in the channels of the porous zeolites is thus determined to be (5,0) by comparing the calculated and measured absorption spectra.