We investigate the electronic and band structure for the (8; 0) single-wall carbon nanotube (SWCNT) with a europium (Eu) and a uranium (U) atom outside by using the first-principles method with the density functional theory (DFT). The calculated band structure (BS), total density of state (TDOS), and projected density of state (PDOS) can elucidate the differences between the pure (8; 0) SWCNT and the nuclei outside the SWCNT. The indirect band gaps are obtained when Eu and U atom are put outside the (8; 0) CNT; they are 0.037 eV and 0.036 eV, respectively, which is much smaller than 0.851 eV for pure CNT. Compared with pure (8; 0) SWCNT, the bottom of the conduction band moves down by 0.383 eV and 0.451 eV with the Eu and U outside, and the top of valence band moves up by 0.127 eV and 0.162 eV, respectively. More significantly, the top of the valence band has exceeded the fermi-level. So, a single nucleus changes the semiconductor character of pure nanotube to semi-metal.