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Interplay of octahedral rotations and breathing distortions in charge ordering perovskite oxides

We investigate the structure--property relationships in $AB$O$_3$ perovskites exhibiting octahedral rotations and cooperative octahedral breathing distortions (CBD) using group theoretical methods. Rotations of octahedra are ubiquitous in the perovskite family, while the appearance of breathing distortions -- oxygen displacement patterns that lead to approximately uniform dilation and contraction of the $B$O$_6$ octahedra -- are rarer in compositions with a single, chemically unique $B$-site. The presence of a CBD relies on electronic instabilities of the $B$-site cations, either orbital degeneracies or valence-state fluctuations, and often appear concomitant with charge order metal--insulator transitions or $B$-site cation ordering. We enumerate the structural variants obtained from rotational and breathing lattice modes and formulate a general Landau functional describing their interaction. We use this information and combine it with statistical correlation techniques to evaluate the role of atomic scale distortions on the critical temperatures in representative charge ordering nickelate and bismuthate perovskites. Our results provide new microscopic insights into the underlying structure--property interactions across electronic and magnetic phase boundaries, suggesting plausible routes to tailor the behavior of functional oxides by design.