EasySpin supports spin systems with any number of electron spins and nuclear spins. The total spin Hamiltonian is
with the following terms
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 parameterized 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 (ZFI) is
In its form commonly used in the spin Hamiltonian, the D tensor is set traceless (sum of diagonal elements is zero) and symmetric ().
In its eigenframe, the D tensor is diagonal, and the zero-field spin Hamiltonian 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 labeled x, y and z such that . 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).
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 symmetic 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 conflicting 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 SH term 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 |Q1| ≤ |Q2| < |Q3|.
is traceless, which means
The relations between the diagonal matrix and the usual parameters and are