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In $4d/5d$ transition-metal systems, many interesting physical propertiesarise from the interplay of bandwidth, electronic correlations, and spin-orbitinteractions. Here, using {\it ab initio} density functional theory, wesystematically study the double-perovskite-like system K$_2$Os$X_6$ (X = F, Cl,and Br) with a $5d^4$ electronic configuration. Our main result is that the $J= 0$ nonmagnetic insulating state develops in this system, induced by strongspin-orbital coupling. Specifically, the well-separated Os$X_6$ octahedra leadto the cubic crystal-field limit and result in dramatically decreasing hoppingsin nearest neighbor Os-Os sites. In this case, the three degenerate $t_{2g}$orbitals are reconstructed into two ``effective'' $j_{\rm eff}$ ($j_{\rm eff} =1/2$ and $j_{\rm eff} = 3/2$ states) states separated by the strong SOC,opening a gap with four electrons occupying the $j_{\rm eff} = 3/2$ orbitals.Furthermore, the hybridization between the Os $5d$ orbitals and the $X$ ($X$ =F, Cl, and Br) $p$ orbitals increases from F to Br, leading the electrons inK$_2$OsF$_6$ to be more localized than in K$_2$OsCl$_6$ and K$_2$OsBr$_6$,resulting in a smaller bandwidth for K$_2$OsF$_6$ than in the Cl- or Br- cases.Our results provide guidance to experimentalists and theorists working on thisinteresting family of osmium halides.

J = 0 nonmagnetic insulating state in K$_2$Os$X_6$ (X = F, Cl and Br)