Calculation of ESR spectra and related Fokker–Planck forms by the use of the Lanczos algorithm. II. Criteria for truncation of basis sets and recursive steps utilizing conjugate gradients

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Abstract
The complex symmetric Lanczos algorithm (LA) has proven to be a very efficient means of calculating magnetic resonance line shapes and spectral densities associated with Fokker–Planck forms. However, the relative importance of the various components of the basis set in an accurate representation of the spectrum and the proper number of recursive steps are not easily assessed in practice using the Lanczos algorithm. A systematic and objective procedure for the determination of optimal basis sets and number of recursive steps is developed using a generalization of the conjugate gradient method (CGM) appropriate for the type of complex symmetric matrices occuring in these problems. The relative importance of the individual basis vectors is determined by using the CGM to obtain the ‘‘solution vector’’ from the set of algebraic equations defining the spectrum. This is done at several values of the sweep variable (e.g., the frequency or the magnetic field). The maximum (over these values of sweep variable) for each component of the solution vector is taken to be a measure of the overall importance of the corresponding basis vector in the complete spectrum. Using this method signficant basis set truncation is conveniently possible. The number of recursive steps needed for an accurate representation of the spectrum is easily obtained by monitoring the residual in the approximate solution vector at the center of the spectrum and by recognizing the close relationship between the LA and the CGM. It is this relationship that enables construction of the Lanczos tridiagonal matrix with the CGM which can either be used to calculate the cw ESR spectrum directly or else the eigenvalues. The information obtained from the CGM can be used to ‘‘turbocharge’’ the LA by taking advantage of the nearly optimal basis set and number of recursive steps. Significant savings in computation time are possible, and relative savings are greatest for the most difficult problems. This is illustrated with a variety of examples of slow‐motional cw ESR spectra and of the new two‐dimensional electron‐spin‐echo technique. In keeping with the greater sensitivity of the latter technique to motional dynamics, it is consistently found to require significantly larger optimal basis sets and number of recursive steps for an accurate representation. One of the most challenging problems for both types of spectroscopy is the case of macroscopically oriented samples where the macroscopic director is tilted at an angle relative to the applied static magnetic field, since this removes much of the symmetry in the problem. This case is found to yield to very significant truncation of basis sets, and a new symmetry‐based decoupling of certain basis vectors was found in this study for the particular example of a 90° tilt angle.

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