On the calculation of cross‐correlation spectral density functions within the model‐free approach

Comprehensive calculations are performed with the aim of arriving at a ready‐to‐use form for cross‐correlation spectral density functions. Axial symmetry is assumed for tensors associated with the two interactions involved in the cross‐correlation spectral density. Moreover, the derived formulas stem from the hypotheses of the “model‐free” approach, i.e., a (generally fast) local motion superposed to a (generally slow) molecular tumbling. The restriction of fast local motions imposes the definition of two different order parameters S and S′ associated with the principal direction of the two considered interactions and it is the product of these order parameters, which is present in the general expression of a spectral density function. The latter includes two terms. The first one corresponds to fast local motions and involves the factor 1/2(3 cos2α − 1), where α is the angle made by the symmetry axes of the two considered tensors. The second term concerns only molecular tumbling sensed by a (virtual) director. Finally, it is shown that, consistently with the model‐free approach, the product SS′ can be expressed as 1/2(3 cos2α − 1)S2 so that 1/2(3 cos2α − 1) factorizes the spectral density function.