  Classical models of spin-lattice coupling are at present unable to accurately
reproduce results for numerous properties of ferromagnetic materials, such as
heat transport coefficients or the sudden collapse of the magnetic moment in
hcp-Fe under pressure. This inability has been attributed to the absence of a
proper treatment of effects that are inherently quantum mechanical in nature,
notably spin-orbit coupling. This paper introduces a time-dependent,
non-collinear tight binding model, complete with spin-orbit coupling and vector
Stoner exchange terms, that is capable of simulating the Einstein-de Haas
effect in a ferromagnetic $\textrm{Fe}_{15}$ cluster. The tight binding model
is used to investigate the adiabaticity timescales that determine the response
of the orbital and spin angular momenta to a rotating, externally applied $B$
field, and we show that the qualitative behaviours of our simulations can be
extrapolated to realistic timescales by use of the adiabatic theorem. An
analysis of the trends in the torque contributions with respect to the field
strength demonstrates that SOC is necessary to observe a transfer of angular
momentum from the electrons to the nuclei at experimentally realistic $B$
fields. The simulations presented in this paper demonstrate the Einstein-de
Haas effect from first principles using a Fe cluster.
