Quantifying hidden symmetry in the tetragonal CH₃NH₃PbI₃ perovskite

The assignment of an exact space group to the tetragonal CH₃NH₃PbI₃ perovskite structure is experimentally challenging and controversial in the literature. One issue is that symmetry depends on the relevant spatial and time scales due to the easy rotation of the CH₃NH₃⁺ ions within the Pb-I cage, which is not captured in a static density functional theory calculation. As a result, the structure does not have any symmetry. Nevertheless, it is clear that some kind of approximate symmetries are present, evidenced by the quasi-I4cm and quasi-I4/mcm structures commonly used in calculations. In this work we have developed a methodology to quantify the hidden symmetry of these structures using group theory, to enable use of symmetries in understanding spectroscopy. We study the approximate symmetry of vibrational modes, as well as of the dielectric, elastic, electro-optic, Born effective charge, and Raman tensors and the dynamical matrix, including analysis of degenerate representations and validity of classes. Comparing to each sub-point group, our results show that the quasi-I4cm is best described by C_2v, whereas the quasi-I4/mcm is well described by the expected C_4v. Our methodology can useful generally for any approximately symmetric material.