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In this review, the authors present several schemes to calculate various quantities like single-ion anisotropy, DMI, and anisotropic exchange J using DFT. For instance, the DMI term in the y-direction $D_y$ is calculated as follows (extracted from the appendix): Source: https://pubs.rsc.org/en/content/articlelanding/2013/dt/c2dt31662e/unauth

However, I could not find any literature on how to go about calculating the anisotropic J terms like $J_{xy}, J_{xz}, J_{yz}, J_{yx}, ...$. The review 1 gives schema for calculating the diagonal elements $J_{xx}, J_{yy}, J_{zz}$, but not the others.

In this paper, the authors give a scheme for $J_{xz}$ (below eq. 4). But that is basically the same as what 1 gives for $D_y$. I don't think this should be the case.

So, would someone mind clarifying how I could go about calculating these non-diagonal anisotropic J.


EDIT: I missed that 3 states that $D_y=J_{xz}=-J_{zx}$ if the off-diagonal terms are anti-symmetric. But this does not make sense to me. So, if the scheme in that paper is indeed correct, I don't understand why the anisotropic J are the same as DMI up to sign, at least for one block of the anisotropic J matrix...

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    $\begingroup$ +1 welcome back TribalChief! Long time no see! $\endgroup$ Oct 24, 2023 at 1:16

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