First principles electric field gradients at A and B site cations across the NaRTiO4 Ruddlesden Popper series

The $n = 1$ Ruddlesden-Popper titanates, NaRTiO$_{4}$ (R = rare-earth), exhibit a structural behaviour where non-centrosymmetry is driven by cooperative oxygen octahedral rotations (OORs) rather than conventional second-order Jahn-Teller distortions. In this work, we present an \textit{ab-initio} investigation of the structural, electronic and hyperfine properties of the entire NaRTiO$_{4}$ series across the two disputed ground states, $Pbcm$ and $P\bar{4}2_1m$, and the high temperature $P4/nmm$ symmetries. Our results reveal an ionic-radius-dependent evolution from a tilt-dominated regime for small rare-earth ions to a distortion-dominated regime for larger cations, leading to an asymptotic regime in which the high-temperature phase becomes increasingly competitive with the ground-state structures as the ionic radius increases. In parallel, the electronic band gap follows a systematic evolution across the series, reflecting the underlying structural changes and the increasing dominance of octahedral distortions at larger ionic radii. The Electric Field Gradient (EFG) tensor reveals that, in the large-radius limit, all symmetries tend locally towards a similar environment. Away from this limit, the EFG tensor for different symmetries progressively diverges, providing a sensitive probe for phase transitions and revealing symmetry-specific fingerprints, particularly for the rare-earth and Ti sites. By establishing these EFG signatures, this work provides a roadmap for experimental techniques, such as Nuclear Magnetic Resonance (NMR) and Perturbed Angular Correlation (PAC), to resolve the ground-state symmetry of these structures.