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 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 DX, DY and DZ 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.
In either convention, if |E/D| = 1/3, then the sign of D is has no effect on the actual values of DX, DY and DZ. 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 Eh/e/a02, and its SI unit is V/m2. Occasionally, V/Å2 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