  In the framework of phenomenological static and QCD-motivated model
potentials for heavy quarkonium, we compute the $\bar{b}c$ mass spectrum as
well as its $\QTR{rm}{1S}$ hyperfine splitting using the recently introduced
shifted $l$- expansion technique. We also predict the leptonic constant \
$f_{B_{c}}$ of the lightest pseudoscalar $B_{c}$ and $f_{B_{c}^{\ast}}$ of the
vector $B_{c}^{\ast}$ states taking into account the one-loop and two-loop QCD
corrections. Further, we use the scaling relation to predict the leptonic
constant of the R{rm}{nS}$-states of the $\bar{b}c$ system. For the sake of
comparison, we use the same fitting parameters of our previous model
potentials. From our results we conclude that shifted $l$-expansion method has
the same accuracy, convergence and status as our previous work.
