The Spin Hamiltonian

The total spin Hamiltonian supported by EasySpin functions is

[eqn]

with the following Hamiltonian terms

This spin Hamiltonian is a linear function of the magnetic field

[eqn]

with the operators

[eqn]
Electron Zeeman Interaction

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

[eqn]

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

[eqn]

via a rotation [eqn] parameterized by three Euler angles.

[eqn]

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

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

[eqn]
Nuclear Zeeman Interaction

In EPR, the chemical shift anisotropy is neglected.

[eqn]
Zero-Field Interaction

In the eigenframe of D, the Hamiltonian term is

[eqn]

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

[eqn]
[eqn]
Hyperfine Interaction

The hyperfine interaction term is

[eqn]

The matrix [eqn] is usually symmetric and can be transformed to its diagonal form

[eqn]

via a similarity transformation with a orthogonal rotation matrix [eqn]

[eqn]

The principal values of [eqn] can be separated into three components, an isotropic contribution [eqn], a dipolar contribution [eqn] and an orthorhombicity parameter [eqn].

[eqn]

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

Electron-Electron Interaction

The general term describing the interaction between two electrons is

[eqn]
Nuclear Quadrupole Interaction

The term describing the nuclear quadrupole interaction is present only of nuclei with I>1/2.

[eqn]

The Q matrix is symmetric and can be diagonalized

[eqn]

where [eqn], [eqn] and [eqn] are the three principal values.

[eqn] is traceless, which means

[eqn]

The relations between the diagonal matrix and the usual parameters [eqn] and [eqn] are

[eqn]
[eqn]
[eqn] is the largest-magnitude component of the EFG (electric field gradient) at the nucleus. Its SI unit is V/m2. [eqn] is the electric quadrupole moment of the nucleus, its SI unit is m2. It is usually given in barn (1 barn = 10-28 m2 = 10-24 cm2).