Summary: In the framework of potential models for heavy quarkonium, we compute the mass spectrum of the bottom-charmed \(B_c\) meson system and spin-dependent splittings from the Schrödinger equation using the shifted-large-\(N\) expansion technique. The masses of the lightest vector \(B_c^*\) and pseudoscalar \(B_c\) states as well as the higher states below the threshold are estimated. Our predicted result for the ground state energy is \(6253^{+15}_{-6}\)MeV and are generally in exact agreement with earlier calculations. Calculations of the Schrödinger energy eigenvalues are carried out up to the third order of the energy series. The parameters of each potential are adjusted to obtain best agreement with the experimental spin-averaged data. Our findings are compared with the observed data and with the numerical results obtained by other numerical methods