The Complete Spin Hamiltonian
Zero-Field Splitting
So far we have discussed the case for one electron spin in magnetic fields of different origins. For systems with more than one electron spin (S > 1/2) an additional energy term, reflecting the strong dipole-dipole interactions between the electrons, has to be included in the spin Hamiltonian3. Examples for such systems are transition metal ions with up to five unpaired d-electrons (e.g. high-spin Mn2+) or spin triplets in organic molecules. The zero-field splitting term, which is active also in absence of the external magnetic field, is written as
$$ \mathcal {H}_\mathrm{zfs}= {\mathbf{S} \mathbf{D} \mathbf{S}}\qquad (22) $$
where D is the traceless zero-field interaction or fine structure tensor. The zero-field interaction can be much larger than the EZI, depending on the symmetry of the ligand field and the coupling of the electron spins.
Nuclear Quadrupole Interaction
The last term in the spin Hamiltonian that will be considered here is found for nuclei with a nuclear spin quantum number I > 1/2. The physical origin of this term is the interaction of the electric quadrupole moment of these nuclei with the electric field gradient. This field gradient arises from uneven distributions of electric charges around the nucleus. The NQI is given by
$$ \mathcal {H}_\mathrm{nqi}= {\mathbf{I}\mathbf{Q}\mathbf{I}}\qquad (23) $$
where Q is the traceless nuclear quadrupole tensor. The NQI can be formally treated in the same way as the ZFS. It's impact on the EPR spectrum is, however, much smaller and the term can often be neglected.
Summary of Energy Terms
The energy terms introduced in the preceding sections and describing the interactions of the electron and nuclear spins among themselves and with their environment can be added to form the complete spin Hamiltonian:
$$ \mathcal {H}_\mathrm{sp}= \frac{\beta_e}{h}{\mathbf{B}_0 \mathbf{g} \mathbf{S}+ \mathbf{S} \mathbf{D} \mathbf{S}+ \mathbf{S} \mathbf{A} \mathbf{I} - g_n} \frac {\beta_n}{h}{\mathbf{I} \mathbf{B}_0 + \mathbf{I} \mathbf{Q} \mathbf{I}}\qquad (24) $$
Depending on the electron and nuclear spin quantum numbers of the interacting ions and their relative magnitudes, terms can be neglected (e.g. the zero-field splitting for S = 1/2). If the electron is coupled to more than one nuclear spin, the Hamiltonian can be extended by adding the appropriate terms. Knowledge of the complete spin Hamiltonian including the magnitude of the interactions and the relative orientations of PAS's of the interaction matrices allows for the calculation of the CW and pulse EPR spectra. The procedure is illustrated in the last chapter for Cu2+.
3 The D tensor may also contain second-order correction terms from the spin-orbital coupling.