The potential energy curves (PECs) of the X1+ and a3 electronic states of the BF radical are studied by employing the ab initio quantum chemical method. The calculations are performed by using the complete active space self-consistent field (CASSCF) method followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with the basis set, aug-cc-pV6Z, for internuclear separations from 0.095 to 1.33 nm. The effects of the Davidson modification and core-valence correlation and relativistic corrections on the PECs are included. The way to consider the relativistic correction is to employ the second-order DouglasKroll Hamiltonian approximation. The relativistic correction is carried out at the level of cc-pV5Z basis set. And the core-valence correlation correction is made by using the cc-pCV5Z basis set. With the PECs obtained here, the spectroscopic parameters (De, Re, e, exe, eye, Be and e) are determined and compared with those reported in the literature. With the PECs obtained by the MRCI+Q/aug-cc-pV6Z+CV+DK calculations, the complete vibrational states are computed for each electronic state when J = 0. And the vibrational levels, the inertial rotation constants and the centrifugal distortion constants of the first 20 vibrational states of each electronic state are reported. On the whole, as expected, the spectroscopic parameters and molecular constants closest to the experimental data are determined by the MRCI+Q/ aug-cc-pV6Z+DK+CV calculations for the two electronic states.