  We use the constrained random phase approximation (cRPA) to derive from first
principles the Ru-$t_{2g}$ Wannier function based model for the Kitaev
spin-liquid candidate material $\alpha$-RuCl$_3$. We find the non-local Coulomb
repulsion to be sizable compared to the local one. In addition we obtain the
contribution to the Hamiltonian from the spin-orbit coupling and find it to
also contain non-negligible non-local terms. We invoke strong coupling
perturbation theory to investigate the influence of these non-local elements of
the Coulomb repulsion and the spin-orbit coupling on the magnetic interactions.
We find that the non-local Coulomb repulsions cause a strong enhancement of the
magnetic interactions, which deviate from experimental fits reported in the
literature. Our results contribute to the understanding and design of quantum
spin liquid materials via first principles calculations.
