  New experiments were proposed recently to investigate the regime of cold
atomic and molecular ion-atom collision processes in a special hybrid
neutral-atom--ion trap under high vacuum conditions. The collisional cooling of
laser pre-cooled Ca$^+$ ions by ultracold Na atoms is being studied. Modeling
this process requires knowledge of the radiative lifetime of the excited
singlet A$^1\Sigma^+$ state of the (NaCa)$^+$ molecular system. We calculate
the rate coefficient for radiative charge transfer using a semiclassical
approach. The dipole radial matrix elements between the ground and the excited
states, and the potential curves were calculated using Complete Active Space
Self-Consistent field and M\"oller-Plesset second order perturbation theory
(CASSCF/MP2) with an extended Gaussian basis, 6-311+G(3df). The semiclassical
charge transfer rate coefficient was averaged over a thermal Maxwellian
distribution. In addition we also present elastic collision cross sections and
the spin-exchange cross section. The rate coefficient for charge transfer was
found to be $2.3\times 10^{-16}$ cm$^3$/sec, while those for the elastic and
spin-exchange cross sections were found to be several orders of magnitude
higher ($1.1\times 10^{-8}$ cm$^3$/sec and $2.3\times 10^{-9}$ cm$^3$/sec,
respectively). This confirms our assumption that the milli-Kelvin regime of
collisional cooling of calcium ions by sodium atoms is favorable with the
respect to low loss of calcium ions due to the charge transfer.
