# The Spin Hamiltonian

EasySpin supports spin systems with any number of electron spins and nuclear spins. The total spin Hamiltonian is

with the following terms

This spin Hamiltonian is a linear function of the magnetic field

with the operators

Electron Zeeman Interaction

The general term describing the interaction between an electron spin and the external magnetic field is

The matrix is usually symmetric, in which case it can be transformed into its diagonal form

via a rotation parameterized by three Euler angles.

, and are the three principal values of the matrix. If is asymmetric, the diagonalization gives complex principal values.

In its diagonal form, the matrix is the sum of an isotropic component and a "g shift" contribution .

Nuclear Zeeman Interaction

The spin Hamiltonian term describing the interaction of a nuclear spin with the external magnetic field is

In EPR, chemical shifts and the chemical shift anisotropy are neglected.

Zero-Field Interaction

For a spin S > 1/2, the energy term describing the zero-field interaction (ZFI) is

In its form commonly used in the spin Hamiltonian, the D tensor is set traceless (sum of diagonal elements is zero) and symmetric ().

In its eigenframe, the D tensor is diagonal, and the zero-field spin Hamiltonian is

The relations between the matrix D in its eigenframe and the commonly used parameters D and E are

Conventionally, the three principal axes are labeled x, y and z such that . In this case, E/D is always positive and lies between 0 and 1/3. If E/D = 1/3, then the sign of D is indeterminate (and inconsequential).

Hyperfine Interaction

The hyperfine interaction term is

Though it can be asymmetric, the matrix is often symmetric and can be transformed to its diagonal form

via a similarity transformation with a orthogonal rotation matrix

The symmetic can be separated into three components, an isotropic, an axial and a rhombic component. In the eigenframe of , they are characterized by the three parameters , and , respectively.

For a spin system with strong anisotropic , the matrices can be significantly asymmetric. In this case, has complex principal values, and 9 parameters are needed to fully specify .

Electron-Electron Interaction

The general term describing the interactions between two electrons is

The tensor describes the total interaction between the two electron spins and includes the isotropic, antisymmetric and symmetric interactions.

For the isotropic exchange interaction, several conflicting conventions are in use in the literature:

EasySpin uses the first one in this list. Therefore, when using values from the literature, make sure to understand which convention was used.