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Ab initio methodology for magnetic exchange parameters: \\Generic four-state energy mapping on Heisenberg spin Hamiltonian

The recent development in the field of 2D magnetic materials urges reliable theoretical methodology for determination of magnetic properties. Among the available methods, ab initio four-state energy mapping based on Density Functional Theory stands out as a powerful technique to calculate the magnetic exchange interaction in the Heisenberg spin model. Although the required formulae were explained in earlier works, the considered Hamiltonian in those studies always corresponded to the specific case that J-matrix is anti-symmetric, which may be misleading in other cases. Therefore, using the most general form of the Heisenberg spin Hamiltonian, we here derive the generic formulae. With a proper choice of four different magnetic states, a single formula governs all elements of the exchange interaction matrix for any considered pair of spin sites.