The first step to calculate exchange interaction parameters using the Green’s function method is to project the DFT Hamiltonian onto the basis of some site-localized orbitals. These could be Wannier functions or e.g. PAW projectors, which we use in integration with VASP.

Having Hamiltonian HWFnm,σ(k) written in the basis set of localized wavefunctions one can calculate the intersites Green’s function, describing scattering process of an electron with spin σ between m and m' orbitals from i to j site at every k point in reciprocal space, as

Gmm'ij,σ(ε,k) = [ε + EF - HWFnm,σ(k)]-1,

where EF is the Fermi energy.

The last step is to calculate exchange integrals using the following expression:

Jij = -1/(2π)∫Σmm'm''m'''Im(imm'Gm'm''ij,jm''m'''Gm'''mji,),

where integration limits overare from -∞ to EF where m, m' are the orbital indexes, and Gm'm''ij,(↑) is Green’s function integrated over the Brillouin zone and

imm' = BZ[Hmm'ii,(k)-Hmm'ii,(k)],

where imm' is the spin splitting.

The detailed specification of Green’s function method one can find in Ref. [1].

[1] Dm. M. Korotin, V. V. Mazurenko, V. I. Anisimov and S. V. Streltsov, Phys. Rev. B 91, 224405 (2015).

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