  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 Schr\"{o}dinger 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 $%
6253_{-6}^{+15}$ $MeV$ and are generally in exact agreement with earlier
calculations. Calculations of the Schr\"{o}dinger energy eigenvalues are
carried out up to third order of the energy series. The parameters of each
potential are adjusted to obtain best agreement with the experimental
spin-averaged data (SAD). Our findings are compared with the observed data and
with the numerical results obtained by other numerical methods.
