# Is it possible to calculate/estimate the value of the J parameter to be used in the Heisenberg/Ising Hamiltonians?

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

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

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

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

• 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.
– Camps
Commented Jul 19, 2020 at 22:18
• 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. Commented Jul 19, 2020 at 22:34
• Is this related? mattermodeling.stackexchange.com/a/104/88 Commented Jul 26, 2020 at 3:27
• @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! Commented Mar 23, 2021 at 15:02
• @NikeDattani Coincidentally, I learnt the procedure mentioned in the question quite recently. I can write up an answer. Commented Mar 23, 2021 at 20:08

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.