JOURNAL OF MAGNETIC RESONANCE 83,65-78 ( 1989) Ultraslow Motions and Asymptotic Lineshapes in ESR GIORGIOJ.MORO* AND ULDERICO SECRET * Istituto di Chimica Fisica. Universitri di Parma, Viale delle Scienze, 43100 Parma, Italy, and t Dipartimento di Chimica Fisica, Universitri di Padova. Via Loredan 2, 35131 Padova, Italy Received May 19, 1987; revised October 11,1988 The characteristic shape of the ESR spectra under ultraslow motion conditions, that is, when the rate ofthe rotational motion is much smaller than the magnetic anisotropies, is analyzed starting from the axial g-tensor problem with the diffusion model. By means of an effective relaxation frequency which takes into account both the intrinsic linewidth and the contribution of the dynamical processes, the asymptotic profiles and the corre- sponding scaling procedures of the spectrum are derived. By resorting to the adiabatic approximation the treatment is extended to radicals having axial hyperfine interactions, with the same asymptotic profiles predicting the shape of individual peaks in ultraslow motion spectra. The approximate spectrum calculated from the asymptotic lineshapesis compared with the numerical solution of the stochastic Liouville equation for a typical nitroxide radical. The early convergence of the outer peaks supports the use of their widths as probes of the correlation time and a simple procedure, which takes into account the correct scaling laws, is proposed for the analysis of experimental spectra. The model dependence of the ultraslow motion spectra is also considered, with a comparison be- tween the diffusion equation and the strong collision operator. o 1989 Academic PI~SS, IX. The stochastic Liouville equation provides the most general description of ESR spectra ( 1) . For a given set of tensors characterizing the magnetic interactions at the molecular level, the ESR spectrum can be predicted over the entire range of rotational motion rates. The comparison between theoretical and experimental spectra allows not only the determination of the time scale of the relevant dynamical process but also the distinction among different regimes of molecular motion. For example, strong collision dynamics for the reorientation motion can be differentiated from rotational diffusion (1-3). However, the theoretical simulation of ESR spectra re- quires numerical solution of the stochastic Liouville equation. Although a very effi- cient numerical procedure based on the Lanczos algorithm has been recently pro- posed (4, 5)) the method has not found intensive application. The difficulties reside mainly in the complexity of the matrix representation of stochastic Liouville opera- tors associated with different types of radicals or environments (see the appendixes of Ref. (6) for a general treatment which includes also the nonsecular contributions of the Hamiltonian). The alternative is provided by the analysis of ESR spectra in limiting cases where the use of simplified descriptions is legitimate. A well-studied case is, of course, the fast tumbling limit where the Redfield theory applies ( 7). In this contribution we shall consider the opposite case, that is, the ultraslow motion limit defined by the 65 0022-2364189 $3.00 Copyright 0 1989 by Academic press, Inc. All rights of repmduction in any form reserved.