The four electronic states (2Σ−, 2Π, 4Σ−, and 4Π) of CuC corresponding to the lowest dissociation limits Cu(2Sg) + C(3Pg) are calculated by using multi-reference configuration interaction method with Davidson correction (MRCI + Q) approach in combination with the effective core potentials (ECPs) basis sets LANLTZ for the Cu atom and 6–311+g(d) basis sets for the C atom. The calculation covers the internuclear distance ranging from 0.04 to 0.54 nm, and the equilibrium bond length Re and the vertical excited energy Te are determined directly. The potential energy curves (PECs) show that the lowest two states are the 4Σ− and 2Π, and 4Σ− is the ground state where the 2Π state is higher than 4Σ− about 0.28 eV. With the potentials, all of the vibrational levels and rotational constants are predicted by numerically solving the radial Schröbinger equation of nuclear motion. Then the spectroscopic data of ωe, ωexe, Be, and αe are obtained after data fitting which are compared with theoretical results currently available.