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 energy term describing the zero-field interaction (ZFI) 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 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:

[eqn]

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

Nuclear Quadrupole Interaction

Nuclei with spin I>1/2 have an electric quadrupole moment that can interact with the local electric field gradient at the nucleus. The SH term describing this nuclear quadrupole interaction is

[eqn]

where Q is in frequency units. The Q matrix is symmetric ([eqn]) and can be transformed into diagonal form

[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).