Spin Hamiltonian
The complete Hamiltonian H of a molecular system including space and spin coordinates of electrons and nuclei can be very complex. The quantum-mechanical description of magnetic resonance is considerably simplified by the introduction of the spin Hamiltonian Hsp, obtained by averaging of the full Hamiltonian over the lattice coordinates and over the spin coordinates of the paired electrons.
The spin Hamiltonian is expressed in linear (e.g. interaction of the electron spin with the static field B0) or bilinear terms (e.g. coupling between electron and nuclear spins) of spin angular momentum operators and contains a few phenomenological constants which can be related to the properties of the spin system (e.g. nuclear gn factors). The finite dimension of the Hilbert space for an electron spin S coupled to n nuclear spins I1, I2, ..., In is given by
$$ d_{HS} = \left(2S+1\right)\prod^n_{k=1}\left(2I_k+1\right)\qquad (A. 1) $$
The spin quantum number S reflects either the true electron spin for systems where the contributions of orbital angular momentum to the ground state are small (e.g. organic radicals) or an effective spin of a subsystem with 2S+1 states (e.g. ground state of rare earth ions with large spin-orbit coupling).