axial ratio of  c/a =1. a, The  B O$_{6}$ octahedral building block in perovskite oxides.  b, Ideal  reference structure for a perovskite with cubic symmetry. The ideal, cornershared octahedral  units, are shown to readily tile the unit cell the  x  and  z -directions.  c, The effect of bi-axial  compressive strain causes the  a -axis parameter to shrink, and therefore increases the  c  parameter  due to elasticity. If we fix the B-O bond lengths about their ideal length as in ( b ), then the  octahedral must rotate by an amount along the  z -direction. The geometric mismatch  δc  between  the ideal and elongated unit cells then drives the octahedral to rotate about an in-plane axis, in  order to preserve the original B-O bond lengths and O-B-O internal angles.  Electronic band structure calculations.  We next plot the electronic band structures for the two strained films on SrTiO$_{3}$ and LaAlO$_{3}$  using the experimental nickelate structures along the high symmetry lines in a non-primitive  (2 2 1) unit cell of the monoclinic structure. This supercell doubles the number of bands,  however, it makes the identification of the orbital contributions to the band structure more  transparent. In Figure S6a we show the calculated band structure for the experimentally  determined nickelate film on the LaAlO$_{3}$ substrate and compare it to the bulk experimental  Figure 9: Geometric illustration for the origin of the isosymmetric phase transition at the axial ratio of c / a =1. a , The BO$_{6}$ octahedral building block in perovskite oxides. b, Ideal reference structure for a perovskite with cubic symmetry. The ideal, cornershared octahedral units, are shown to readily tile the unit cell in the x and z -directions. c, The effect of bi-axial compressive strain causes the a -axis parameter to shrink, and therefore increases the c parameter due to elasticity. If we fix the B-O bond lengths about their ideal length as in ( b ), then the octahedral must rotate by an amount γ along the z -direction. The geometric mismatch δc between the ideal and elongated unit cells then drives the octahedral to rotate about an in-plane axis, in order to preserve the original B-O bond lengths and O-B-O internal angles. 21