This work describes an ab initio perturbed ion (aiPI) study for the determination of the relative stability of normal and inverted perovskite structures ABX(3) (A = Li+, K+, Rb+, Cs+; B = Mg2+, Ca2+, Sr2+, Ba2+; X = F-, H-). Large Slater-type orbitals for representing atomic centers have been used. The inverted BaLiF3 system and ternary hydride BaLiH3 and SrLiH3 perovskites have a lower lattice energy than normal structures. The differences between ionic radii of the different alkalines, alkaline earth cations, and fluorine and hydride anions are capable of explaining the relative stability of normal and inverted structures. The values of the bulk modulus reveal that the inverted structure is less compressible than the normal one for the BaLiF3 structure. A local geometry optimization has been carried out for BaLiF3 and BaLiH3 in order to minimize the effective energy using a large cluster model of 173 ions. Vibrational frequencies (v) associated with the breathing vibrational modes, a(1g), in both A and B sites for these pure systems have been characterized. For the corresponding doped (Ni2+ for Ba2+) crystal structure, this property has been characterized at the 12-fold site, revealing an increment of the v value from the pure to the doped BaLiF3 structure; an opposite trend is observed for the BaLiH3 system. These substitutions present favorable defect reaction energies in both crystal lattices, but the energy change calculated for the direct reaction between NiF2 and BaLiF3 is positive for Ni2+ entering the Ba2+ site. The numerical results are analyzed and compared with experimental data, the geometrical distances obtained by computer simulation being in agreement with the reported experimental values. The validity of the methodology is discussed.