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 *H ^{WF}_{nm,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

*G ^{mm'}_{ij,σ}*(

*ε*,

**k**) = [

*ε + E*(

_{F}- H^{WF}_{nm,σ}**k**)]

^{-1},

where *E _{F}* is the Fermi energy.

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

* J_{ij}* = -1/(2π)∫

*dε*Σ

_{mm'm''m'''}Im(*∆*

_{i}^{mm'}

*G*^{m'm''}_{ij,}

_{↓}

*∆*_{j}^{m''m'''}*G*^{m'''m}_{ji,}

_{↑}),^{}

where integration limits over* dε *are from *-∞ *to *E _{F }, *where

*m, m'*are the orbital indexes, and

*G*^{m'm''}_{ij,}

_{↓}_{(↑) }is Green’s function integrated over the Brillouin zone and

*∆ _{i}^{mm'} = *∫

_{BZ}[

*H*

^{mm'}_{ii,}_{↑}(

**k**)-

*H*

^{mm'}_{ii,}_{↓}(

**k**)],

where *∆ _{i}^{mm'} *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).