  The crystal structure of the MAPbI$_3$ hybrid perovskite forms an intricate
electrostatic puzzle with different ordering patterns of the MA molecules at
elevated temperatures. For this perovskite three published model Hamiltonians
based on the point-dipole (pd) approximation combined with short-range
effective interactions are compared to a recently developed machine-learning
force field. A molecular order parameter is used to consistently compare the
transformation of the anti-ferroelectric ordering in the orthorhombic phase
upon raising the temperature. We show that the ground states and the
order-disorder transition of the three models are completely different. Our
analysis indicates that the long-range order in the low-temperature
orthorhombic phase can be captured by pd-based models with a short cutoff
radius, including the nearest and next-nearest neighbor molecules. By
constructing effective atomic interactions the ordering can already be
described within 6 A radius. By extracting the coupling energetics of the
molecules from density functional theory calculations on
MA$_x$Cs$_{1-x}$PbI$_3$ test systems, we show that the pd-approximation holds
at least for static structures. To improve the accuracy of the pd-interaction
an Ewald summation is applied combined with a distance dependent electronic
screening function.
