Ab initio calculations were performed for Rb2+ and FrNa molecules, using a standard quantum chemistry approach based on pseudopotential, Gaussian basis sets, effective core polarization potentials and full configuration interaction calculations. For each system (Rb2+ and FrNa), the adiabatic potential energy curves and their spectroscopic constants (Re, De, Te, ωe, ωeχe and Be) are extracted. In addition, to the potential energy, accurate permanent and transition dipole moment from the ground and the first excited states to higher excited states of 1Σ+ symmetry have been determined for a wide interval of internuclear distances for FrNa. Moreover, we have localized and analyzed numerous avoided crossings between electronic states of 1,3Σ+, 1,3Π and 1,3Δ symmetries. For the FrNa molecule, these avoided crossings can be explained by the ionic interaction between Fr+ and Na-, and Fr- and Na+. The permanent dipole moment of FrNa has revealed both ionic characters relating to electron transfer and yielding Fr+Na- and Fr-Na+ arrangements. The accurate produced potential energy and dipole moment are of great interest for the formation prediction of these cold alkali diatomic molecules...