systematically from LNO/STO (1.968 ± 0.002 ˚ A) to bulk LNO (1.935 ˚ A) to LNO/LAO (1.916 ± 0.005 ˚ A) as the a - and b -axis parameters of the substrate are reduced. The measured out-of-plane (apical) d$_{Ni}$$_{−}$$_{O}$ , however, are less sensitive to the strain; we obtain 1.933 ± 0.002 ˚ A and 1.950 ± 0.002 ˚ A for LNO/STO and LNO/LAO, respectively. (Supplementary Table S1 provides a full summary of our experimentally determined bond lengths and angles.) Interestingly we find that bi-axial strain only weakly modifies the in-plane Ni-O-Ni bond angle ( θ ) of 165 . 8 ± 0 . 5 ◦ and 164 . 0 ± 2 . 0 ◦ for LNO/STO and LNO/LAO, respectively. In contrast, along the out-of-plane direction θ is highly sensitive to the substrate-induced strain; this bond angle decreases from 175 . 2 ± 0 . 6 ◦ in LNO/LAO to 159 . 9 ± 0 . 6 ◦ in LNO/STO. We note that both the α (or β ) and γ angles contribute to the in-plane bond angles and since we find that empirically the average of these angles is robust to strain, the insensitivity of the equatorial bond angle is not surprising, albeit counterintuitive. 

To investigate the origin of these effects and to determine the change in octahedral rotations for intermediate strain values, we perform density functional theory (DFT) calculations within the local-spin density approximation (LSDA) plus Hubbard U method ( U$_{eff}$ = U − J = 3 eV) as implemented in the Vienna ab initio Simulation Package ($_{vasp}$) [28–31]. We first optimize the internal degrees of freedom for bulk rhombohedral LNO (10-atom unit cell) at the experimental lattice parameters to confirm that we accurately describe the electronic and atomic structure. Consistent with the experimental data, our optimized structure is indeed metallic and consists of equal and alternating phase rotations of the NiO$_{6}$ octahedra about each Cartesian axis with α = β = γ = 5 . 76 $^{◦}$, in good agreement with the 1.5 K structural measurement of 5.37◦ [22]. We next carry out structural optimizations for the two homoepitaxially strained films using our experimentally determined lattice parameters. The substrates are not explicitly simulated in our calculations, but the symmetry reduction imposed by enforcing equal in-plane lattice parameters on the bulk R ¯ 3 c structure is included by optimizing the internal degrees of freedom within a monoclinic (20-atom) unit cell (space group C 2 /c ). The excellent consistency between our experimental and theoretical structures is apparent in the structural data obtained for the nickelate films grown on STO and LAO as shown in Fig. 3a. To quantify the comparison between the two methods, we use a symmetry adapted mode analysis [32, 33] and take scalar products between the DFT and experimentally determined vectors ξ that contain the full irreducible representations of the atomic displacement patterns of the strained nickelate films about the bulk rhombohedral structure. We find that ξ ($_{dft}$) · ξ ($_{exp}$) ∼ 0 . 99 

