  State-of-the-art {\em ab initio} techniques have been applied to compute the
potential energy curves of the (BaRb)$^+$ molecular ion in the Born-Oppenheimer
approximation. The long-range coefficients describing the electrostatic,
induction, and dispersion interactions at large interatomic distances are also
reported. The electric transition dipole moments governing the X$^1\Sigma \to
^1\Sigma, ^1\Pi$ have been obtained as the first residue of the polarization
propagator computed with the linear response coupled-cluster method restricted
to single and double excitations. Nonadiabatic radial and angular coupling
matrix elements, as well as the spin-orbit coupling matrix elements have been
evaluated using the multireference configuration interaction method restricted
to single and double excitations with a large active space. With these
couplings, the spin-orbit coupled (relativistic) potential energy curves for
the 0$^+$ and 1 states relevant for the running experiments have been obtained.
Finally, relativistic transition moments and nonadiabatic coupling matrix
elements were obtained from the nonrelativistic results and spin-orbit
eigenvectors. The electronic structure input has been employed in the single
channel scattering calculations of the collisional cross sections between the
Ba$^+$ ion and Rb atom. Both nonrelativistic and relativistic potentials were
used in these calculations. Our results show that the inelastic cross section
corresponding to the charge transfer from the Rb atom to the Ba$^+$ ion is much
smaller than the elastic one over a wide range of energies up to 1 mK. This
suggests that sympathetic cooling of the Ba$^+$ ion by collisions with
ultracold Rb atoms should be possible.
