We next extend our analysis to intermediate strain values to investigate the transition region between the experimentally obtained strain values. In this section, we carry out structural relaxations about the theoretical LSDA+ U equilibrium volume rather than at the experimental lattice parameters reported previously. We explore bi-axial strain states by fixing the in-plane lattice constants at each strain value, and performing full structural optimizations of the internal atomic coordinates and c -axis parameter about the LSDA+ U reference structure. In Fig. 4a we show that over the strain range investigated the NiO$_{6}$ octahedra always rotate out-of-phase about each axis with the a $^{−}$a$^{−}$c− rotation pattern with only changes in their magnitude about each axis. In agreement with the experimental data, we find that compressive (tensile) strain predominately induces rotations of the NiO$_{6}$ octahedra about the c -axis (in-plane axes). We attribute this effect to the epitaxial constraint that forces coherency between the film and the substrate, which contracts (elongates) the equatorial bond lengths and elongates (contracts) the apical bonds under compressive (tensile) strain (Fig. 4b). The a $^{−}$a$^{−}$c− octahedral rotation pattern is able to accommodate the bond length distortions by increasing the NiO$_{6}$ rotation about the c -axis at the expense of the in-plane rotations. For the tensile case the amount of in-plane rotation decreases accordingly to sustain the bond length elongation in-plane. We find that there is an additional structural phase that is approximately 2 meV lower in energy for small tensile strains and shows an intriguing small charge disproportionation (CDP). The deviations in rotation angles between the two phases shown in Fig. 4a are negligible, and since our room temperature experiments are likely unable to access this phase, we do not discuss it further. The transition region which captures the crossover from mainly rotations about the c -axis to the in-plane axes occurs near 0.50% strain or when the axial ratio c/a approaches unity (Fig. 4c). At this strain value, standard crystallographic distortion parameters that measure deviations in ideal six-fold coordination through the apical and equatorial bond lengths and the Ni–O–Ni bonds all show a sharp discontinuity and indicate almost ideal octahedra [34]. In the absence of the small CDP phase, we identify this abrupt, first-order phase transition to be isosymmetric [35] since the atomic structure remains monoclinic—the full symmetry of the C 2 /c space group is retained and the same Wyckoff positions are occupied—across the transition. The origin for first-order phase transition about c/a = 1 can be understood on geometric grounds (Supplementary Fig. S5): when the ideal NiO$_{6}$ octahedra is recovered at 0.5% strain, the bi-axial lattice distortion imposed on the film by the substrate leads to a 9