The Spin Hamiltonian

The total spin Hamiltonian supported by EasySpin functions is

with the following Hamiltonian terms

H_EZI(i): Electron Zeeman Interaction (EZI) of electron spin i
H_ZFI(i): Zero-Field Interaction (ZFI) of electron spin i
H_NZI(k): Nuclear Zeeman Interaction (NZI) of nuclear spin k
H_NQI(k): Nuclear Quadrupole Interaction (NQI) of nuclear spin k
H_EEI(i,j): Electron-Electron Interaction (EEI) between electron spins i and j
H_HFI(i,k): Hyperfine Interaction (HFI) between electron spin i and nuclear spin k

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

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

[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] .

Nuclear Zeeman Interaction

In EPR, the chemical shift anisotropy is neglected.

Zero-Field Interaction

The term describing the zero-field splitting is

The D tensor is traceless (sum of diagonal elements is zero) and symmetric (D_i,j = D_j,i).

In its eigenframe, the D tensor is diagonal, and the zero-field operator term 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 assigned 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.

Hyperfine Interaction

The hyperfine interaction term is

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

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

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

Nuclear Quadrupole Interaction

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

The Q matrix is symmetric and can be diagonalized

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

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

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).