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In chemistry, particularly the field of NMR spectroscopy, the interaction between two (nucleic) spins (or so I guess?) is governed by the Hamiltonian:

\begin{align} \mathcal{H}=2\pi\cdot J_{ij}{\vec {S_{i}}}\cdot {\vec {S_{j}}} \label{eq1} \end{align} https://en.wikipedia.org/wiki/J-coupling

This interaction is called $J$-coupling and it is usually heuristically explained by interaction between two electronic spins which is governed by the Pauli principle and the electron-nucleus interaction which in the end favors antiparallel alignment of the nucleic spins over parallel and thereby leads to a splitting of energy levels. Note that mainly chemists cover this topic and their approach is usually very hands-on in the sense that the values of $J_{ij}$ are usually measured on a spectrometer.

Now the above equation looks very similar to the Heisenberg model, governing the general relationship between two spins:

\begin{align} H_{\text{Heis}}=-J\sum _{\langle i,j\rangle }{\vec {S_{i}}}\cdot {\vec {S_{j}}}\qquad {\text{with }}i,j\,{\text{next neighbors}} \label{eq2} \end{align}

https://en.wikipedia.org/wiki/Heisenberg_model_(quantum)

This surely looks similar, here $J$ is the value of the exchange integral, which I would assume is also what defines the scalar coupling tensor $J_{ij}$ in the first equation. However, I am by no means sure those two $J$s are the same and if they are, where does the normalization factor of $2\pi$ coming from? Is it just common among chemists to divide their exchange integrals by $2\pi$? I would be very grateful if someone could shed a little light on that.

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  • $\begingroup$ this is a guess, but pulling out 2$\pi$ would make it more convenient to express the coupling in units of Hz instead of inverse second. $\endgroup$ – wcc Apr 26 '19 at 14:04
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Yes, chemists like J couplings in Hz. The NMR J or scalar coupling and the Heisenberg model occur in very different contexts. In NMR, the scalar coupling is typically more than 1000 times smaller than the Zeeman term in the Hamiltonian. Moreover, the temperature is more than 1000 times larger than the Zeeman term. The NMR J coupling is a residual magnetic dipole interaction, which remains after the main magnetic dipole interaction has been averaged away by motion. It can be of either sign. All of this is very different from the usual situation and concerns in applications of the Heidelberg model.

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