Theoretical electronic structure studies on the anionic and neutral Al13X (X=Li-Cs) clusters have been carried out using a gradient corrected density-functional approach. The alkali atoms occupy the hollow sites on the Al-13 icosahedron surface and the calculated vertical transition energies from the ground state of the anions to various neutral states are found to be close to the recent preliminary experimental data. A detailed investigation of the Born-Oppenheimer surface of Al13K shows that the minimum energy path for the motion of K atoms to the neighboring hollow sites passes through a bridge site and has a barrier of 0.13 eV. If energy could be imparted that will enable the K atoms to overcome the barrier and slide freely on the surface, the cluster will exhibit a small orbital magnetic moment that could be observed at low temperatures.