Long- and Short-Range Anion Order in SrTiO$_{3-x}$H$_x$ Perovskite Oxyhydrides: DFT+$U$ Sensitivity and HSE06 Cluster Expansion

Anion ordering in perovskite oxyhydrides can remain significant even in disordered states, particularly at non-dilute hydrogen concentrations. Nevertheless, hydride substitution poses challenges for accurate simulations based on density functional theory due to configurational complexity and Ti 3$d$ reduction. Here, we develop a cluster expansion (CE) framework for SrTiO$_{3-x}$H$_x$ incorporating HSE06 hybrid-DFT energetics. We first demonstrate that calculated mixing energies and ordering stability are highly sensitive to the choice of DFT+$U$, with maximum variations on the order of 100 meV/anion. We benchmark ordering energetics against HSE06 calculations and identify $U$ = 3.3 eV as an HSE06 proxy, which enables extensive configurational exploration while limiting costly HSE06 calculations to key configurations for efficient learning of ordering energetics. Together, ground-state orderings, correlations between octahedral configurations and structural stability, and MC sampling of CE models all support a strong preference for the O$_4$H$_2$ cis configuration in SrTiO$_{3-x}$H$_x$, in which two hydride ions occupy first-nearest-neighbor anion sites. This cis-type preference was overlooked in previous ATiO$_{3-x}$H$_x$ studies, despite its sizable stabilization of ~200 meV per hydride comparable to reported anion-migration and polaron-formation energies. This study addresses both the previously underexplored sensitivity of CE-based ordering analyses to DFT+$U$ and anion-ordering in perovskite oxyhydrides.