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The assignment of an exact space group to the tetragonal CH$_3$NH$_3$PbI$_3$perovskite structure is experimentally challenging and controversial in theliterature. Average orientation of the methylammonium ion that gives symmetryto the experimental measurement is not captured in a static density functionaltheory calculation, although the quasi-I4cm and quasi-I4/mcm structures arecommonly used in calculations. In this work we have developed a methodology toquantify the hidden symmetry of these structures using group theory, to enableuse of symmetries in understanding spectroscopy and other properties. We studythe approximate symmetry of vibrational modes, including analysis of degeneraterepresentations, as well as the dielectric, elastic, electro-optic, Borneffective charge, and Raman tensors and the dynamical matrix. Comparing to eachsubgroup of the full tetragonal D$_{4h}$, our results show that the quasi-I4cmis best described by the expected corresponding point group C$_{4v}$, whereasthe quasi-I4/mcm (despite corresponding to point group D$_{4h}$) is bestdescribed by the lower symmetry C$_{2v}$. Our methodology can be usefulgenerally for analysis of other soft hybrid materials or any approximatelysymmetric material.

Quantifying Hidden Symmetry in the Tetragonal CH$_3$NH$_3$PbI$_3$ Perovskite