  We have studied the properties of the prototype hybrid organic-inorganic
perovskite $\text{CH}_3^{}\text{NH}_3^{}\text{PbI}_3^{}$ using relativistic
density functional theory (DFT). For our analysis we introduce the concept of
$\text{CH}_3^{}\text{NH}_3^+$ "pair modes", that is, characteristic relative
orientations of two neighboring $\text{CH}_3^{}\text{NH}_3^+$ cations. In our
previous work [Phys. Rev. B \textbf{94}, 045201 (2016)] we identified two
preferential orientations that a single $\text{CH}_3^{}\text{NH}_3^+$ cation
adopts in a unit cell. The total number of relevant pairs can be reduced from
the resulting 196 combinations to only 25 by applying symmetry operations. DFT
results of several $2\!\times\!2\!\times\!2$ supercell models reveal the
dependence of the total energy, band gap and band structure on the distribution
of $\text{CH}_3^{}\text{NH}_3^+$ cations and the pair modes. We have then
analyzed the pair-mode distribution of a series of $4\!\times\!4\!\times\!4$
supercell models with disordered $\text{CH}_3^{}\text{NH}_3^+$ cations. Our
results show that diagonally-oriented $\text{CH}_3^{}\text{NH}_3^+$ cations are
rare in optimized $\text{CH}_3^{}\text{NH}_3^{}\text{PbI}_3^{}$ supercell
structures. In the prevailing pair modes, the $\text{C--N}$ bonds of the two
neighboring $\text{CH}_3^{}\text{NH}_3^+$ cations are aligned approximately
vertically. Furthermore, we fit the coefficients of a pair-mode expansion to
our supercell DFT reference structures. The pair-mode model can then be used to
quickly estimate the energies of disordered perovskite structures. Our
pair-mode concept provides combined atomistic-statistical insight into
disordered structures in bulk hybrid perovskite materials.
