The Hamiltonian for a single spin S and its interaction with the magnetic field is in general a sum of terms with , with pB and pS non-negative integers. pB +pS has to be even to fulfil time-reversal invariance. Only terms with pS ≤ 2S are effective. Mc Gavine et al. formulated such general spin Hamiltonian using tesseral harmonics. Advantages are the following:
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 μB. We can construct now the as: The zero-field terms in the spin Hamiltonian are commonly expressed as: However, the most common form which is quadratic in S and consider the diagonal terms in a trace-free representation is: The corresponding coefficients should now be transformed to the (only p = pS is allowed for pB = 0). With ζ00 = 1 and for even pS it is straightforward to transform the coefficients: D and E in HDE are converted to D. G. McGavin, W. C. Tennant, J. A. Weil J. Magn. Reson., 1990, 87, 92-109.