Iteration method and meaning of the scattering order

When using the Jacobi iteration method, the scattering order stands for its usual meaning -i.e., the maximum number of scattering events along a given photoelectron path since the photoelectron is excited and until it is collected by the detector.

When using the recursion method, the scattering order represents the order of iteration. Convergence is faster using this method as compared with the Jacobi method -i.e., the scattering order needed for convergence is smaller in the recursion method.

The Jacobi method is always used internally for scattering order below 3.

These two methods take approximately the same computation time for a given scattering order. The recursion method is recommended for its stability, since it prevents divergences that occur when using the Jacobi method (these divergences have a deep mathematical meaning, they are real, independent of the code, and unavoidable within the Jacobi iteration method). However, the recursion method does not permit to connect the results obtained for a given number of iterations to a given maximum number of atomic scattering events along the photoelectron path.

The recursion method used here is based upon the Lanczos method as applied by Haydock and Heine to electronic band structure calculations (Solid State Physics 35 (1980)). The original method has been modified in such a way that multiple emission angles can be dealt with in a simple multiple scattering calculation for each energy, so that the final computational cost varies very little with the number of emission angles under consideration. For further details, see García de Abajo, Van Hove, and Fadley.