We study exchange interactions in iron and nickel within the density functional theory plus dynamical mean-field theory approach with density-density and SU(2) symmetric Coulomb interaction. In particular, we analyze the representation of exchange interactions through the electron nonlocal Green's functions, connecting the local particle-hole (ph) irreducible interaction vertices. While the neglect of the local ph irreducible vertex corrections, suggested previously within the dual fermion approach [E. A. Stepanov, S. Brener, F. Krien, M. Harland, A. I. Lichtenstein, and M. I. Katsnelson, Phys. Rev. Lett. 121, 037204 (2018)], yields the result corresponding to the generalization of the magnetic force theorem approach, we argue that these vertex corrections are less important for strong localized magnets, such as iron, but become more essential for weak itinerant magnets, such as nickel. At the same time, the overall account of the local vertex and self-energy corrections in the so-called renormalized magnetic force theorem approach is essential for iron, and less important for nickel. The difference of the results of density-density and SU(2) symmetric Coulomb interaction is found to be relatively small, in contrast to the Curie temperature and the value of the local magnetic moment.

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• Exchange interaction
• Magnetic susceptibility
• Magnetism
• Dynamical mean field theory