High-order operators

In systems with electron spin S>1/2, higher-order terms have to be included in the spin Hamiltonian in order to properly describe EPR spectra. The most common parameters are the second-order zero-field splitting parameters D and E and the fourth-order parameters a and F.

In addition to the terms of D, E, a and F, a commonly used complete set of high-order operators are the extended Stevens operators (ESO) [eqn] for [eqn] and [eqn].

The high-order part of the spin Hamiltonian for one electron spin written in terms of ESOs is


where each of the operators [eqn] is hermitian, and the associated coefficients [eqn] are always real.

In contrast to spherical tensor operators (analogous to p+, p0, and p- orbitals), these operators are tesseral (analogous to px, py and pz). Positive k corresponds to cosine tesseral components, and negative k corresponds to sine tesseral components.

Care should be exercised when using literature values for the coefficients, since conventions for the normalization and the phase of these operators vary. Also, many papers contain substantial amounts of typos.

In EasySpin, matrix representations of the extended Stevens operators are provided by the function stev and can be used in all simulation functions.

Relation to conventional parameters

The commonly used second-order zero-field splitting parameters D and E are related to the [eqn] coefficients by


The associated second-order Hamiltonian terms have the forms


The traditional cubic and axial fourth-order parameters a and F are related to the [eqn] coefficients.


The associated fourth-order Hamiltonian terms have the forms


(see Abragam/Bleaney, p. 142, 437; Pilbrow, p. 125, 129). The axes x, y, and z are the fourfold symmetry axes of the cubic part of the crystal field.

Polynomial expressions

The following table lists the most common Stevens operators in terms of polynomials in S+, S- and Sz. A more general list of extended Stevens operators (including odd k) can be found in Altshuler/Kozyrev. Abragram/Bleaney contains only a partial, but compatible list. Ryabov (1999) has devised a formula for computing Stevens operator polynomials for arbitrary 0<=k and -k<=q<=k.


Here are the most important references for extended Stevens operators

Note that there are typos and mistakes both in Altshuler/Kozyrev and in Abragam/Bleaney, as discussed by Rudowicz (2004).