Weblab can calculate the influence of molecular motion on NMR frequencies on all time scales.
While the description of the intermediate motional regime is complicated, the situation becomes simple
in the fast-motion limit, when the inverse correlation time is much larger than the rigid limit value δ0.
In this case the time evolution is governed by an averaged hamiltionian and the explicit time dependent coupling can be replaced by a static average.
In practice the fast limit is applicable, if the spectrum is that of a fully relaxed spin system and the lineshape does not depend on the spacing between
the first and the second NMR pulse of the solid echo pulse sequence.
Fast-motion limit
In the fast-motion limit Weblab simulates spectra for discrete jumps
as well as for homogeneous jump distributions by first calculating the averaged interaction tensor R according to
where j indicates the site and pj the relative population of this site. (In case of homogeneous jump distributions the sum has to be replaced by an integral over the distribution function.) Like the original tensor the averaged tensor is symmetric and traceless again, yet possessing other principal values and possibly a different orientation.
The spectrum is obtained by summing up the contributions to the NMR frequencies for all possible orientations. In case of a powder sample, where integration can be done over the complete unit sphere, the spectrum can be found analytically, speeding up calculation considerably. In case of oriented samples, the integration is restricted to a regime around a director defined by the orientational distribution function and can only be performed numerically.
Intermediate regime
To cover the intermediate motional regime, the spectrum is computed from the time evolution
of the density operator ϱ(t) by solving the following set of differential equations
with the initial conditions
Hj(t) is the (stochastic) Hamiltonian describing the influence on the spin system due to the interaction at site j including the pulse sequence. Πjk is the rate per unit time for a transition from site k to site j.
The spectrum, finally, is obtained after Fourier transformation of the time domain signal given by
and integration over the unit sphere.
For a detailed description see the references listed under About - Introduction.