Semiclassical Theory of the Effects of Internal Motions on the Linewidths in Electron Spin Resonance Spectra

A number of classical dynamical models are developed for describing the effects of internal rotational motions and solvent-complex formation on the ESR hyperfine linewidths of free radicals in solution. These dynamical processes can lead to linewidth effects because they cause time-dependent modulations of the isotropic hyperfine interactions of the different magnetic nuclei. An alternating linewidth effect, which has been observed in a number of recent studies, is predicted to result when there is an out-of-phase correlation between the hyperfine splittings ai(t) of equivalent nuclei. By equivalent nuclei here are meant those for which the time-average splittings 〈ai(t) 〉Av are equal, and an out-of-phase correlation is one in which an increase in the instantaneous splitting from Nucleus i, ai(t), is correlated with a decrease in the splitting aj(t) from nucleus j. This out-of-phase correlation can result from a coupling of the mechanical motions of different rotating groups and also from the effects that changes in orientations of uncorrelated rotors may have on redistributing electron spin density in the molecule. Continuous motion cases are treated by Brownian-motion theory using coupled and uncoupled internal rotors and torsional oscillators. Random jump models between discrete states are also considered. It is shown that all these models can give rise to an alternating linewidth effect provided that certain definite relationships of the proper type exist for the coupling of the motions or the variations of the spin densities. For many of these models the correlation a1(t)=a2(t) for all t, or complete equivalence of both nuclei, may also result when different relationships exist. The correlation of the splittings of more than two groups of completely equivalent nuclei will, in general, lead to more complex linewidth effects.