Classical dipole dipole interaction

Classical magnetic dipole-dipole interaction can be written in the form of the two-ion interaction with the matrix parameter

  $$\hat{\mathcal{H}}_{CDD} = - \frac{1}{2} \sum_{n\ne n^{\pr... ..._{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}$$ (11)

where the operators are understood as $\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x$, $\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y$, $\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z$ and

  $$\mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n... ...\alpha\beta}}{\vert\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}\vert^3} \Biggr)$$ (12)