Reversible hydrogen incorporation was recently attested [N. Lu, $textit{et al.}$, Nature $textbf{546}$, 124 (2017)] in ${text{SrCo}text{O}_{2.5}}$, the brownmillerite phase (BM) of strontium cobalt oxide (SCO), opening new avenues in catalysis and energy applications. However, existing theoretical studies of BM-SCO are insufficient, and that of ${text{HSrCo}text{O}_{2.5}}$, the newly-reported hydrogenated SCO (H-SCO), is especially scarce. In this work, we demonstrate how the electron-counting model (ECM) can be used in understanding the phases, particularly in explaining the stability of the oxygen-vacancy channels (OVCs), and in examining the Co valance problem. Using density-functional theoretical (DFT) methods, we analyze the crystalline, electronic, and magnetic structures of BM- and H-SCO. Based on our structure search, we discovered stable phases with large bandgaps (> 1 eV) for both BM-SCO and H-SCO, agreeing better with experiments on the electronic structures. Our calculations also indicate limited charge transfer from H to O that may explain the special stability of the H-SCO phase and the reversibility of H incorporation observed in experiments. In contrary to the initial study, our calculation also suggests intrinsic antiferromagnetism (AFM) of H-SCO, showing how the measured ferromagnetism (FM) has possible roots in hole doping.
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