Chemical Shielding Anisotropy

From ISMARpedia

Chemical shielding is an anisotropic interaction characterized by a shielding tensor, which is a 3 x 3 matrix which can be diagonalized to yield a tensor with three principal components. The CS tensor is often visualized as a simple ellipsoid, as shown below.

The principal components are designated such that \sigma_{11} \le \sigma_{22} \le \sigma_{33}. There are other conventions for denoting the chemical shielding tensor, including the Herzfeld-Berger and Haeberlen conventions. The former convention denotes the shielding tensor is terms of the isotropic chemical shielding, σiso, the span, Ω and the skew, κ. The span describes the breadth of the tensor and is measured in ppm, while the skew is a measure of the axial symmetry and is a dimensionless parameter ranging from +1 to -1 (see definitions above). The latter convention utilizes the anisotropy, Δσ, and the asymmetry parameter, η, which are defined as:

\Delta \sigma = \sigma_{zz} - \frac{\sigma_{xx} - \sigma_{yy}}{2} and \eta = \frac{\sigma_{yy} - \sigma_{xx}}{\delta}

where δ is the reduced anisotropy, defined as δ = σzz − σiso, and the components are defined such that | σzz − σiso | > = | σxx − σiso | > = | σyy − σiso | .

The isotropic chemical shielding is referenced against the bare nucleus, which is assigned an absolute shielding value of 0 ppm. In practice, most NMR experiments involve the use of the chemical shift scale, where a standard reference compound is arbitrarily chosen and assigned some arbitrary value (e.g., for 1H and 13C, the chemical shift standard is TMS, which is assigned a chemical shift value of δiso = 0 ppm. Chemical shifts can be compared to absolute chemical shielding values via:

\delta_{iso} = \frac{\sigma_{ref} - \sigma}{1 - \sigma_{ref}} \approx \sigma_{ref} - \sigma

Many common NMR nuclei (e.g., 1H, 13C, 15N, 19F, etc.) have well-defined absolute chemical shielding scales. An example is shown below for 31P:

In solution NMR experiments, the rapid tumbling of the molecules commonly average the chemical shift tensor, and only the isotropic chemical shift is measured. In solid samples, the presence of chemical shielding anisotropy often generates broad powder patterns. A hypothetical example comparing solution and solid-state 13C NMR spectra is shown below:

The chemical shift tensors correspond to molecular symmetry, and are hence rich in structural information. Some examples of three different powder pattern types are shown below:

The broad powder pattern shapes, while very useful, also contribute to problems of low signal to noise and poor resolution (i.e., it is difficult to easily separate sites which are close in chemical shift). The technique of Magic-Angle Spinning (MAS) can be used to partially or completely average CSA powder patterns. MAS spectra of sites with large CSAs are generally composed of large arrays of spinning sidebands; however, the peak corresponding to the isotropic shift remains in the same position. For example, pictured below are three 31P NMR spectra: the top one is a static NMR spectrum (stationary sample), and the bottom two spectra are acquired at two different spinning speeds with 5 times less scans.

Notice that: (i) the signal to noise is greatly increased in the MAS spectra; (ii) the isotropic peak remains in the same position; (iii) the manifold of the spinning sidebands loosely resembles the static NMR pattern. It is possible to obtain the chemical shift tensor by simulating the static pattern, or by perform a Herzfeld-Berger analysis of the spinning sidebands.

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