# Spin Hamiltonian expanded in Magnetic Field and Electron Spin

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 fulfill 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:

• 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.
This general spin Hamiltonian can be written as: We used the extended Stevens operators . The magnetic field is expressed in spherical coordinates and denote the spherical harmonics. Prefactors Apq and αpB are tabulated in McGavin et al. is a Wigner 3-j symbol.Wigner 3j symbols are zero unless qB + qS = q and . The parameters describing the spin system are the . They are given in units of Energy (Magnetic Field)pB, which correspond in Easyspin to MHz (mT)pB. The allowed coefficients have p an even integer numbers between |pB-pS| and pB+pS and q integer number between -p and +p. For example for pB = 1 and pS = 3 the following coefficients are allowed:

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

Electron Zeeman Interaction
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:
Zero-Field Hamiltonian
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
Reference
D. G. McGavin, W. C. Tennant, J. A. Weil J. Magn. Reson., 1990, 87, 92-109.