It is important to remark that the procedure of increasing the number of Gaussians that describe the local field is not straightforward, since in the AW fitting function, Eq. (4), one needs the second-moment pairs M2LT and M2HT. As M2HT depends on the geometry of motion, a priori information on the effect of motional line narrowing of each Gaussian used to describe the local field in the rigid limit is required.

Therefore, in the next section we present a geometry-sensitive algebraic method, earlier proposed by Terao and coworkers [48], to decompose the dipolar local field. As already mentioned in the last section, the double-Gaussian approximation for the local dipolar field is justified by earlier work of Terao et al. [48], who

BGB324 ic50 proposed check details the decomposition of a CHn dipolar powder spectrum into 2n2n dipolar powder patterns, each one associated with specific proton spin configurations, see Fig. 7. The principal axes and values of the dipolar tensor corresponding to each component as well as their average values for the fast-motion limit can be obtained on the basis of the spatial arrangement of the CH bonds and the motional geometry, as proposed in Ref. [48]. Fig. 7 exemplifies this decomposition for the case of a CH2 group executing a planar three-site jump motion. The full dipolar spectrum can be decomposed into 4 components. However, in the rigid limit the dipolar pattern has only two components, since the proton spin configurations (↑,↑),(↓,↓) and (↑,↓),(↓,↑) lead to the same dipolar pattern, see Fig. 7a. Considering a fast three-site jump with the geometry presented in the inset of Fig. 7b, the dipolar patterns associated with the spin configuration (↑,↓),(↓,↑) narrows to an isotropic peak, while the configurations (↑,↑),(↓,↓) correspond to an axially symmetric Pake pattern, see Fig. 7b. Note that the motionally averaged patterns are sensitive

to the motional geometry, but can be precisely predicted as described in Ref. [48]. Given the second possibility of decomposing the dipolar local field and calculating the dipolar tensor parameters, one can now approximate each component with a Gaussian local field with the same second moment as the dipolar field components associated with each distinct spin configuration. In general, this gives rise to a multi-Gaussian AW approximation, where the motional effect on each component of the dipolar pattern is considered independently. Nevertheless, one should note that for CH2CH2 the lower of the two M2HT is either zero or has a finite value, depending if the motion is evenly uniaxial (3 or more symmetric jump sites) or not. This shows that for CH2CH2 groups one should never need more than two Gaussians, which stands for a two-Gaussian approximation for the local field, as exemplified by the Gaussian powder in the inset of Fig. 7b.