Beyond Orbitally Resolved Magnetic Exchange in CrI_{3} and NiI_{2}

The pertinent need for a microscopic understanding of magnetic exchange motivated us to go beyond the existing theories and develop a systematic method to quantify all possible mechanisms that contribute to magnetic exchange for an arbitrary pair of atoms in a given material. We apply it to the archetypal 2D magnetic monolayers CrI_{3} and NiI_{2}, to reveal the previously underrated d_{x^{2}-y^{2}},d_{x^{2}-y^{2}} contribution as either the leading or the second largest contribution to the total magnetic exchange. We proceed to explore the microscopic mechanisms behind all the nonzero orbital contributions in both CrI_{3} and NiI_{2}, and generalize the findings to other magnetic monolayers dominated by d^{8} and d^{3} electronic configurations of the magnetic atoms.