The energy levels of three cerium centres in (CaF₂, SrF₂ and BaF₂ were calculated in the framework of the relativistic Dirac-Hartree-Fock formalism. In this study cerium was taken to be in various ionic states and lattice relaxation was included. The centres were the following: the charge-uncompensated cerium centre (Ce) and two charge-compensated cerium centres. The charge-compensating fluorine ion was situated at the (111) next-nearest-neighbour interstitial site (CeF_nnn) or at the (100) nearest-neighbour interstitial site (CeF_nn). The authors did not work with crystal field parameters, their method being ab initio. The results are compared with a variety of experimental data. Relaxation of the lattice around the cerium centres is crucial for the results. From the location of the energy levels, they conclude that only trivalent and tetravalent cerium centres are stable in the alkaline-earth fluorides. If free electrons are present, only the trivalent cerium centre is stable. The divalent centre without charge compensation is metastable. This is in agreement with experiment, Further, it is found that excitation of trivalent cerium centres by capture of an electron and a hole at the cerium ion is improbable. This agrees with earlier observations of the scintillation decay of Ce³⁺-doped BaF₂.