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 ${\mathrm{CrI}}_3$ and ${\mathrm{NiI}}_2$, 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 ${\mathrm{CrI}}_3$ and ${\mathrm{NiI}}_2$, and generalize the findings to other magnetic monolayers dominated by $d^8$ and $d^3$ electronic configurations of the magnetic atoms.

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• Magnetic anisotropy
• Magnetic coupling
• Magnetic interactions
• Magnetism
• Spin-orbit coupling