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We have studied the properties of the prototype hybrid organic-inorganicperovskite $\text{CH}_3^{}\text{NH}_3^{}\text{PbI}_3^{}$ using relativisticdensity functional theory (DFT). For our analysis we introduce the concept of$\text{CH}_3^{}\text{NH}_3^+$ "pair modes", that is, characteristic relativeorientations of two neighboring $\text{CH}_3^{}\text{NH}_3^+$ cations. In ourprevious work [Phys. Rev. B \textbf{94}, 045201 (2016)] we identified twopreferential orientations that a single $\text{CH}_3^{}\text{NH}_3^+$ cationadopts in a unit cell. The total number of relevant pairs can be reduced fromthe resulting 196 combinations to only 25 by applying symmetry operations. DFTresults of several $2\!\times\!2\!\times\!2$ supercell models reveal thedependence of the total energy, band gap and band structure on the distributionof $\text{CH}_3^{}\text{NH}_3^+$ cations and the pair modes. We have thenanalyzed the pair-mode distribution of a series of $4\!\times\!4\!\times\!4$supercell models with disordered $\text{CH}_3^{}\text{NH}_3^+$ cations. Ourresults show that diagonally-oriented $\text{CH}_3^{}\text{NH}_3^+$ cations arerare in optimized $\text{CH}_3^{}\text{NH}_3^{}\text{PbI}_3^{}$ supercellstructures. In the prevailing pair modes, the $\text{C--N}$ bonds of the twoneighboring $\text{CH}_3^{}\text{NH}_3^+$ cations are aligned approximatelyvertically. Furthermore, we fit the coefficients of a pair-mode expansion toour supercell DFT reference structures. The pair-mode model can then be used toquickly estimate the energies of disordered perovskite structures. Ourpair-mode concept provides combined atomistic-statistical insight intodisordered structures in bulk hybrid perovskite materials.

Multi-scale model for disordered hybrid perovskites: the concept of organic cation pair modes