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.

35 years after its synthesis magnetic observables are calculated for the first time for the molecular nanomagnet Mn12-acetate using a spin-Hamiltonian that contains all spins. Starting from a very advanced DFT parameterization [Phys. Rev. B 89, 214422] we evaluate magnetization and specific heat for this anisotropic system of 12 manganese ions with a staggering Hilbert space dimension of 100,000,000 using the Finite-Temperature Lanczos Method.