Ab initio investigations are performed for the FrRb using a standard quantum chemistry approach based on pseudopotentials for atomic core representations, Gaussian basis sets and full configuration interaction calculations. A diabatisation procedure based on the effective Hamiltonian theory and an effective metric is used to produce the quasi-diabatic potential energy. Firstly, we determine the adiabatic and quasi-diabatic potential energies and the spectroscopic constants for the ground and several excited states of 1,3Σ+, 1,3Π and 1,3Δ symmetries. Their predicted accuracy is discussed by comparing our well depths and equilibrium positions with the available theoretical results. Secondly, we calculate the permanent dipole moments for a wide range of internuclear distances. The adiabatic permanent dipole moments for the 151Σ+ electronic states have revealed ionic characters related to electron transfer and yielding both Fr−Rb+ and Fr+Rb− arrangements. Finally, the transition dipole moments between neighbor electronic states revealed many peaks around the avoided crossing positions.