Geometric constraints on the octahedral rotation system In Fig. 9a we show the BO$_{6}$ octahedral unit as the building block for the perovskite structure. We consider in Fig. 9b, an ideal cubic structure with an axial ratio c / a defined to be unity. In this case all of the B-O bonds and O-B-O bonds are equal yielding the ideal six-coordinate octahedron. In Fig. 9c we schematically show the effect of applying bi-axial compressive strain to a 3x3x3 supercell of the primitive (and cubic) 5-atom perovskite cell. We enforce that the B-O bond lengths and O-B-O angles in each individual octahedron remain identical. [This assumption is good for small strains, and agrees with Pauling’s rule on parsimony.] In order to accommodate the change in the in-plane a lattice constant, the octahedra rotate about the c -axis by an amount γ . In the out-of-plane direction, if the octahedra do not change their rotational pattern, then by the requirement that the out-of-plane lattice constant c increases (from elastic theory), there results a mismatch δc between the repeat unit cell corresponding to c / a =1 (dashed line), and that with c / a > 1 (yellow line). This mismatch that occurs when the octahedra along c maintain their original rotation (in this case zero) and bond lengths, requires that the magnitude of the octahedral rotation pattern about an axis parallel to the plane of the strain changes to make the repeat unit cells coincide with one another. The need to change the octahedral bond angles, then leads to the abrupt, first-order phase transition. Interestingly the origin for this isosymmetric constraint can be completely attributed to the corner connectivity of the octahedral units and their antiferrodistortive nature that is compatible with a three-fold axis but incompatible with the bi-axial constraint imposed on the film by the substrate. 20