The application of the "model trust region" modification of the Levenberg-Marquardt minimization algorithm to the analysis of one-dimensional CW EPR and multidimensional Fourier-transform (FT) EPR spectra especially in the slow-motion regime is described. The dynamic parameters describing the slow motion are obtained from least-squares fitting of model calculations based on the stochastic Liouville equation (SLE) to experimental spectra. The trust-region approach is inherently more efficient than the standard Levenberg-Marquardt algorithm, and the efficiency of the procedure may be further increased by a separation-of-variables method in which a subset of fitting parameters is independently minimized at each iteration, thus reducing the number of parameters to be fitted by nonlinear least squares. A particularly useful application of this method occurs in the fitting of multicomponent spectra, for which it is possible to obtain the relative population of each component by the separation-of-variables method. These advantages, combined with recent improvements in the computational methods used to solve the SLE, have led to an order-of-magnitude reduction in computing time, and have made it possible to carry out interactive, real-time fitting on a laboratory workstation with a graphical interface. Examples of fits to experimental data will be given, including multicomponent CW EPR spectra as well as two- and three-dimensional FT EPR spectra. Emphasis is placed on the analytic information available from the partial derivatives utilized in the algorithm, and how it may be used to estimate the condition and uniqueness of the fit, as well as to estimate confidence limits for the parameters in certain cases.
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We gratefully acknowledge extensive contributions of Drs. Richard H. Crepeau and Mingtao Ge and David J. Schneider to the development of earlier versions of the CW EPR and nonlinear-least-squares programs described here and their many useful discussions during the course of this work. Computations and program development were performed at the Cornell Theory Center and the Cornell Materials Science Center. This work was supported by NIH Grants RR07126 and GM25862 and by NSF Grants CHE9313167 and DMR9210638.