  It is virtually impossible to evaluate the magnetic properties of large
anisotropic magnetic molecules numerically exactly due to the huge Hilbert
space dimensions as well as due to the absence of symmetries. Here we propose
to advance the Finite-Temperature Lanczos Method (FTLM) to the case of
single-ion anisotropy. The main obstacle, namely the loss of the spin
rotational symmetry about the field axis, can be overcome by choosing symmetry
related random vectors for the approximate evaluation of the partition
function. We demonstrate that now thermodynamic functions for anisotropic
magnetic molecules of unprecedented size can be evaluated.
