Calculation of ESR spectra and related Fokker-Planck forms by the use of the Lanczos algorithm

The applicability of the Lanczos algorithm in the general ESR (and NMR) line shape problem is investigated in detail. This algorithm is generalized to permit tridiagonalization of complex symmetric matrices characteristic of this problem. It is found to yield very accurate numerical solutions with at least order of magnitude reductions in computation time compared to previous methods. It is shown that this great efficiency is a function of the sparsity of the matrix structure in these problems as well as the efficiency of selecting an approximation to the optimal basis set for representing the line shape problem as distinct from actually solving for the eigenvalues. Furthermore, it is found to aid in the analysis of truncation to minimize the basis set (MTS), which becomes nontrivial in complex problems, although the efficiency of the method is not very strongly dependent upon the MTS. It is also found that typical Fokker–Planck equations arising from stochastic modeling of molecular dynamics have the property of being representable by complex–symmetric matrices that are very sparse, so calculation of associated correlation functions can be very effectively implemented by the Lanczos algorithm. It is pointed out that large problems leading to matrices of very large dimension can be efficiently handled by the Lanczos algorithm.