# Spin-Orbit Interaction Hamiltonian

EasySpin supports spin systems with any number of electron spins and nuclear spins. Additionally each electron spin can be coupled via spin-orbit interaction to an orbital angular momentum. The total Hamiltonian is

where all parts of the Hamiltonian described in the spin Hamiltonian section are summarized in HSpin. Two terms are new:

Crystal Field and Zeeman Interaction of the orbital angular momenta

An orbital angular momentum L is in fact very similar to an interger spin. Even though in most text books one would find the previous statement in reverse order, it might be the natural approach for EPR spectroscopist, more or less, familiar with Spin Hamiltonians. Major differences are in fact the energy scales. The Hamiltonian for an orbital angular momentum is: The first term describes the crystal-field interaction using the extended Stevens operators.

The second term is sometimes called normal Zeeman Hamiltonian. (Historically the splitting of orbital energie levels was denoted normal Zeeman effect and that of spin energy levels anormal Zeeman effect) The normal Zeeman Hamiltonian has reversed sign compared to the Electron Zeeman Hamiltonian. The interaction with the magnetic field is isotropic. σ denotes the orbital reduction factor. Electron density on the ligands as well as mixing of orbital wave functions might lead to σ slightly reduced from 1. However, also projection coefficients to effective orbital angular moment can be absorbed in σ.

Spin-Orbit Interaction of electron spins with orbital angular momenta
The interaction of an electron spin with an orbital angular momentum can be described by the following Hamiltonian: with the spin-orbit interaction constant λ and the orbital reduction factor σ.

For lighter elements, including 3d transition metals, with several free electrons, we can define a total orbital angular momentum and a total spin (Russel-Saunders coupling). For very heavy elements, like uranium, spin-orbit coupling dominates and we can consider . For intermediate ions, e.g. lanthanides, one might use Russel-Saunders coupling, but include higher orders in the spin-orbit coupling: