# Nuclear Spin Hamiltonian for NMR of Anisotropic Fluid Systems

The relevant nuclear spin Hamiltonian for NMR of anisotropic fluid systems includes beyond the Zeeman term *H _{z}* the four terms introduced in the previous chapter, Section 4.4; these terms are the chemical shift

*H*, the indirect spin-spin coupling

_{a}*H*j , the direct dipolar interaction

*H*and the quadrupolar interaction

_{D},*H*q, present for spins with

*I >*1. As discussed in the previous chapter, Section 4.4.1, in high-field NMR spectroscopy, the energy levels of the nuclear spin Hamiltonian can be determined in first- order perturbation theory, considering the Zeeman term

*H*as the unperturbed Hamiltonian Ho =

_{z}*H*and the remaining terms as the perturbation H

_{z}_{1}=

*H*+

_{a}*H*j +

*H*Hq. Due to the rapid molecular reorientation and diffusional motions occurring in anisotropic fluid systems the contribution of H

_{D}+_{1}for the system's Hamiltonian is actually replaced by a time average of this quantity Hu As discussed in Section 4.4.4 the molecular motions decouple the spins from different molecules in terms of energy levels of the Hamiltonian and due to the 1/r

^{3}dependence of the Hamiltonian terms involving the interaction of two spins r apart, it is frequently possible to separate the spins of a molecule in several groups of few interacting spins each and analyze each group separately. The total Hamiltonian is the sum of the Hamiltonians from each spin group in the molecules with the different molecules equivalent to each other. From Eq. 4.27 the perturbing Hamiltonian terms arising from a group of

*n*spins is

Al these terms have a similar form, they are constituted by the product of spin operators by the *zz* component or the trace of phenomenological second rank tensors related to system properties, these tensor components are

The time averages of the *zz* components of the different second rank tensorial quantities associated with the specific molecular properties contain information on molecular structure and orientational order and can be given in terms of the principal values of the corresponding second rank tensorial
quantities as follows

The quantities V_{k},_{xx},, *Vk, _{y}y*, V

_{k},are the principal values of the tensor

*V*

_{k}, the primed frame [x',

*y', z*is the principal frame of that tensor and the angles

^{/}]*в*and

*ф*are the polar and azimuthal angles defining the orientation of the

*B*field in the principal frame. Introducing the asymmetry parameter

_{0}*n*for the tensor

*V*

_{k},

and considering that the tensor *V _{k}* is traceless,

*V*can be written in a more compact form as

_{k},_{zz}A corresponding development applies to the other tensorial quantities in Eq. 5.1. All those quantities can also be related to a set of orientational order parameters characterizing the anisotropic fluid.

The two terms ^{and} *Jki* are scalars and carry information on the molecular structure, they are the only nonzero terms in isotropic systems and are discussed in 4.4.2. The quantities listed in Eq. 5.2 can in principle be obtained from the energy spectrum of the spin Hamiltonian.