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

Deriving models for the Kitaev spin-liquid candidate material $\alpha$-RuCl$_3$ from first principles