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Studying magnetic systems, two frequently used approximations are the Heisenberg and Ising models (a discussion about these approximations can be read here):

\begin{equation} \tag{Heisenberg} \hat{H}_H=-\sum_{\langle i j\rangle}J\hat{S}_i\hat{S}_j \end{equation}

\begin{equation} \tag{Ising} \hat{H}_I=-\sum_{\langle ij\rangle}J\hat{S}_i^z\hat{S}_j^z \end{equation}

Is there a theoretical method/way to determine realistic values for the $J$ parameter?

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    $\begingroup$ Thanks @Xivi76. Yes, it is related to the inter-site exchange but I think that it is slightly different as it is just for spins (as individual entities) and not for electrons. $\endgroup$
    – Camps
    Jul 19, 2020 at 22:18
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    $\begingroup$ It's quantifiable in the same way as the hubbard 'U'. Even the Hubbard 'U' is dependent on the spin operator. There are plenty of notes that explain this. $\endgroup$
    – Xivi76
    Jul 19, 2020 at 22:34
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    $\begingroup$ Is this related? mattermodeling.stackexchange.com/a/104/88 $\endgroup$
    – Thomas
    Jul 26, 2020 at 3:27
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    $\begingroup$ @Xivi76 since this is now one of the longest standing unanswered questions, do you think maybe you could write an answer? I'm trying to help clean up the unanswered queue a little bit! $\endgroup$ Mar 23, 2021 at 15:02
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    $\begingroup$ @NikeDattani Coincidentally, I learnt the procedure mentioned in the question quite recently. I can write up an answer. $\endgroup$
    – Xivi76
    Mar 23, 2021 at 20:08

3 Answers 3

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The equation in your question, be it Heisenberg or Ising exchange, can be calculated by Energy mapping analysis. This has to be the most popular paper that discusses this technique. Basically, you consider different spin configurations and map the exchange 'J' into the total energy values from DFT.

In the simplest form, if you have two magnetic atoms in your unit cell, there are four possible spin configurations - $\uparrow$$\uparrow$ , $\uparrow$$\downarrow$, $\downarrow$$\uparrow$,$\downarrow$$\downarrow$, denoted by $E_1, E_2, E_3, E_4$ respectively. The exchange 'J' can be estimated as $1/4* (E_1 - E_2 - E_3 + E_4$). Now, the energy values for the four different configurations are just the total energy values from your DFT calculation. Typically, you would need to constrain your magnetization on the magnetic atoms in your DFT package. If you want to include anisotropic exchange, you need to consider spin-orbit coupling as well.

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I agree with Xivi76. One can use this method to determine J. However, a supercell should be used and the two atoms chosen for the four states should not interact with their periodic image. If the choice of supercell is not large enough one could run into problems.

Here is the link to the original paper. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.84.224429

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For normal DFT use Noodleman’s broken symmetry approach. Noodleman This one may be relevant for magnetic systems: PRB2019

Edited: in the supporting information (p. 9) of JACS 2017, 139, 18448, Kieber-Emmons and co-workers report how to compute the J value in detail. Here one needs the energy of the triplet (S = 1), the broken-symmetry solution for S = 0, and the corresponding expectation values of .

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