  We present bound state masses of the self-conjugate and non-self-conjugate
mesons in the context of the Schr\"{o}dinger equation taking into account the
relativistic kinematics and the quark spins. We apply the usual interaction by
adding the spin dependent correction. The pseudoscalar and vector decay
constants of the $B_{c}$ meson and the unperturbed radial wave function at the
origin are also calculated. We have obtained a local equation with a complete
relativistic corrections to a class of three attractive static interaction
potentials of the general form $% V(r)=-Ar^{-\beta}+\kappa r^{\beta}+V_{0},$
with $\beta =1,1/2,~3/4$ which can also be decomposed into scalar and vector
parts in the form $% V_{V}(r)=-Ar^{-\beta}+(1-\epsilon)\kappa r^{\beta}$ and $%
V_{S}(r)=\epsilon \kappa r^{\beta}+V_{0};$ where $0\leq \epsilon \leq 1.$ The
energy eigenvalues are carried out up to the third order approximation using
the shifted large-N-expansion technique.
