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.