The Spin Hamiltonian

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

[eqn]

with the following 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

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

[eqn]

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

Zero-Field Interaction

For a spin S > 1/2, the term describing the zero-field splitting is

[eqn]

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

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

[eqn]

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

[eqn]
[eqn]

Conventionally, the three principal axes are labeled x, y and z such that [eqn]. 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

[eqn]

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

[eqn]

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

[eqn]

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

[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 interactions between two electrons is

[eqn]

The tensor [eqn] describes the exchange interaction between the two electron spins as well as their magnetic dipolar interaction.

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

[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. One common convention is to choose the eigenframe such that the three values are ordered |Q1| ≤ |Q2| < |Q3|.

[eqn] is traceless, which means

[eqn]

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

[eqn]
[eqn]
With the ordering convention above, e2Qq/h can be positive or negative, and η is between 0 and 1. [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).