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The Hamiltonian for a single spin S and its interaction with the magnetic field
is in general a sum of terms with
, with p_{B} and p_{S} non-negative integers. p_{B} +p_{S} has to be even to fulfill time-reversal invariance. Only terms with p_{S} ≤ 2S are effective. Mc Gavine et al. formulated such general spin Hamiltonian using tesseral harmonics. Advantages are the following:

- behavior of all terms under axis rotation is straightforward to establish
- the same symmetry selection rules as for the crystal field applies and can be readily adopted.

Next we will derive the coefficients for Hamiltonian's widely used in spin physics.

The commonly used Zeeman Hamiltonian is linear in B and S. Usually it is sufficient to describe the interaction with the external magnetic field. with symmetric g matrix and the Bohr magneton μ