Spin Hamiltonian and Relaxation Theory

Molecules in ordinary liquids average out all anisotropic spin interactions due to isotropic Brownian motions, and their NMR spectra are governed by the Hamiltonian in units of Hz due to the Zeeman interaction, the isotropic chemical shift (a) and the isotropic indirect spin-spin coupling (7) 

The spin interactions, dipole-dipole (D), nuclear electric quadrupole (Q) and chemical shielding (C.S), may be formally written in terms of irreducible tensors of rank l34 in Hz  

For the heteronuclear dipole-dipole interaction, the spin I S whereas for the homonuclear dipolar or electric quadrupole interaction, I=S. For the anisotropic chemical shielding interaction, the spin operators are 

The coupling tensor Rlm in the laboratory frame is time dependent due to the motions of spin-bearing molecules. It can be expressed in terms of the rotational transformation of the corresponding irreducible components pln in the principal axis system (PAS) to the laboratory frame by 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide 

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