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

• H_EZ(i): Electron Zeeman Interaction of electron spin i
• H_ZF(i): Zero-Field Interaction of electron spin i
• H_NZ(k): Nuclear Zeeman Interaction of nuclear spin k
• H_NQ(k): Nuclear Quadrupole Interaction of nuclear spin k
• H_EE(i,j): Electron-Electron Interaction between electron spins i and j
• H_HF(i,k): Hyperfine Interaction between electron spin i and nuclear spin k

This spin Hamiltonian is a linear function of the magnetic field

with the operators

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 parametrized 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 .

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.

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

where x, y and z are the axes of an arbitrary molecule-fixed frame and the 3x3 matrix is the D tensor. Systems with S = 1/2 do not have a zero-field interaction term.

In its form commonly used in the spin Hamiltonian, the D tensor is set to be traceless (sum of diagonal elements is zero) and symmetric (). D tensors calculated using quantum-chemistry programs in general might not be traceless and exactly symmetric.

In its eigenframe, the D tensor is diagonal

where X, Y and Z are now the (molecule-fixed) principal axes of the D tensor. In this frame, the Hamiltonian is

Since the D tensor is set to be traceless, only two parameters are needed to fully specify the three principal values. Conventionally, the two parameters D and E are used. They are related to D_X, D_Y and D_Z via

The ratio E/D is called the rhombicity. The Hamiltonian expressed using D and E is

The Z label is generally assigned to the principal axis corresponding to the absolute largest eigenvalue, assignment of the X and Y labels is a matter of convention. Two conventions are in use.

• For organic triplets, they are assigned such that . In this case, D and E have opposite signs, and E/D is negative and lies between 0 and -1/3.
• For transition metals, they are assigned such that . In this case, D and E have the same sign, and E/D is positive and lies between 0 and 1/3.

In either convention, if |E/D| = 1/3, then the sign of D is has no effect on the actual values of D_X, D_Y and D_Z. See the zfsframes documentation for more details.

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 symmetric 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 .

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 inconsistent 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.

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

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

where , and are the three principal values. One common convention is to choose the eigenframe such that the three values are ordered according to .

is traceless, which means

Instead of the principal values , , and , it is common to encounter the conventional parameters and , related to the principal values via

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

is the largest-magnitude component of the EFG (electric field gradient) tensor at the nucleus. The EFG tensor is the matrix of all second derivatives of the electrostatic potential. The atomic unit of the EFG is E_h/e/a₀², and its SI unit is V/m². Occasionally, V/Ų is used. 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).

The nuclear quadrupole tensor is related to the EFG tensor via