  $\alpha$-RuCl$_3$ has been extensively studied recently because of potential
bond-dependent Kitaev magnetic exchange interactions and the resulting quantum
spin liquid phase that can be realized therein. It has been known that the
covalency between Ru 4$d$- and Cl $p$-orbitals is crucial to induce large
Kitaev interactions in this compound, therefore replacing Cl into heavier
halogen elements such as Br or I can be a promising way to promote the Kitaev
interaction even further. In a timely manner, there have been reports on
synthesis of $\alpha$-RuBr$_3$ and $\alpha$-RuI$_3$, which are expected to host
the same spin-orbit-entangled orbitals and Kitaev exchange interactions with
$\alpha$-RuCl$_3$. Here in this work we investigate electronic structures of
$\alpha$-RuCl$_3$, $\alpha$-RuBr$_3$, and $\alpha$-RuI$_3$ in a comparative
fashion, focusing on the cooperation of the spin-orbit coupling and on-site
Coulomb repulsions to realize the spin-orbit-entangled pseudospin-1/2 at Ru
sites. We further estimate magnetic exchange interactions of all three
compounds, showing that $\alpha$-RuBr$_3$ can be promising candidates to
realize Kitaev spin liquid phases in solid-state systems.
