On the stability of perovskite and post-perovskite polymorphs of MgSiO$_3$

The relative stability of 10 distorted perovskite structures of MgSiO$_3$ have been investigated by first principles and contrasted with that of the newly found {\it Cmcm} post- perovskite. The electronic structure of these polymorphs was analyzed and simple relationships between magnitudes of polyhedral distortions, bond-lengths, and band-gaps were found. Up to approximately 95 GPa, the {\it Pnma} phase is the most stable, has the largest band gap, and can accommodate the largest volume reduction with the smallest distortion of the SiO$_6$ octahedra. At higher pressures the post-perovskite polymorph is the most stable one. This phase transition is accompanied by an increase of octahedral volume and band gap. These results do not support a transition from {\it Pnma} to another distorted perovskite structure prior to the transition to the {\it Cmcm} polymorph as proposed by some experiments.