configuration*

Due to the competition among hopping t, crystal field splitting Δ, electronic correlations (U, J_H), and spin-orbit coupling l, many electronic states are possible for a d⁴ system, such as S = 2, S = 1, and J =0 states. Here, we systematically study the 1D Os-chain systems (OsCl₄, OsOCl₂) ^[1,2] and the “zero-dimensional” K₂OsX₆ (X = F, Cl, and Br) ^[3] with the 5d⁴ electronic configuration. Because of the large bandwidth or large crystal-field splitting Δ (between different t_2g orbitals), the J = 0 state is suppressed in those 1D Os-chains, resulting in an S = 1 spin-1 system ^[1,2]. However, in K₂OsX₆, the well-separated OsX₆ octahedra lead to the cubic crystal-field limit and result in dramatically decreasing hopping t among nearest neighbor Os-Os sites. In this case, the J = 0 nonmagnetic state develops in this system ^[3]. Our results provide guidance to experimentalists and theorists working on this interesting family of osmium halides.