Hamiltonian chemical shielding

In order to discuss the origin of these terms we need to allow the spins to have anisotropic shielding tensors. Molecular tumbling in solution makes the chemical shielding in the direction of the external magnetic field a stochastic function of time and acts therefore as a relaxation mechanism, called the chemical shielding anisotropy (CSA) mechanism. The Hamiltonian for each of the two spins, analogous to Eq. (5), contains therefore two... 

It is only the zz-component that is of interest since after this transformation this is the component that is aligned with Bq, CTxz and cjyz make only second-order contributions. Hence the Hamiltonian representing the chemical shielding is now given by... 

The effect of the magnetic field experienced by the nuclei is seen in the chemical shielding of the nuclear resonance. It is often convenient to distinguish an isotropic part, independent of the molecular orientation, and an anisotropic part, known as the CSA. Under MAS, the anisotropic part becomes time dependent, unless the spinning frequency is sufficiently high, in which case it vanishes. The chemical shift Hamiltonian under MAS may be expressed as... 

The intensities I can be identified with the intensities of the nth spinning side bands of the chemical-shielding difference tensor [74]. In the time-dependent interaction frame the Hamiltonian is given by... 

The residual chemical shielding contributions are suppressed by the supercycle shown in Fig. 26. The average Hamiltonian of the DRAWS scheme (Eq. (107)) consists of zero-quantum and double-quantum terms with similar magnitudes (Eqs. (108) and (109)) and shows a rather strong dependence on the angles connecting PAS and RAS of the dipolar couphng tensor. 

Reduced sensitivity to RF mismatch and chemical shielding contributions is also provided by the general pulse sequence shown in Fig. 32a, which allows selections of wanted and suppression of unwanted interaction terms in the average Hamiltonian by adjusting the number n of phase-shifted elements that are implemented over N rotation periods. The Hamiltonian of such a Cn... 

The explicit periodicity of the Bloch wave function poses a major challenge for the calculation of aforementioned chemical shielding tensor in the solid state. Under these circumstances, the calculation of NMR parameters requires the inclusion of a macroscopic magnetic field described by a nonperiodic vector potential, yielding a new Hamiltonian ... 

In discussing the results of ah initio calculations of chemical shielding, it is helpful to have some notion of the underlying theory. We present the elements of the necessary theory here, in a form somewhat expanded from what has been done before and in a way sufficient to allow the reader to fill in details if desired. The approach involves standard perturbation theory, and although the idea of an effective spin Hamiltonian may be new to some, its presentation in the form of a perturbation theory result should make it easier to understand. The presentation should allow one to realize the physical underpinnings of the theory and to appreciate the inherent computational complexities. And it will allow us to make some illuminating comments later on when examples are presented. 

The quantity a is known as the chemical shielding tensor associated with that particular nuclear site. The tensorial character of a implies that B/oc is in general in a direction different from that of Bo, which reflects the anisotropy of the molecular environment of the considered nucleus. As this is a purely magnetic interaction, analogous to the Zeeman one, the Hamiltonian Hcs is given by ... 

Up until this point, we have implicitly assumed time independence of the interactions and their corresponding Hamiltonian operators. This assumption is valid for a rigid stationary sample. However, when the sample is spun rapidly, each of the internal Hamiltonians becomes time-dependent. For example, Jfz,cs becomes time-dependent when there is chemical shielding anisotropy due to the fact that the orientations of the chemical shielding tensors relative to the applied magnetic field change as the sample... 

In order to understand some of the essential features of the RR experiment, it is convenient to assume negligible chemical shielding anisotropy. Under these conditions, o) and e CS-perturbed Zeeman angular frequencies, are independent of time. In addition, it is convenient to use the spherical tensor notation to describe the direct dipolar interaction. Thus, the total Hamiltonian is... 

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