  We consider an implementation of the adiabatic spin dynamics approach in a
tight-binding description of the electronic structure. The adiabatic
approximation for spin-degrees of freedom assumes that the faster electronic
degrees of freedom are always in a quasi-equilibrium state, which significantly
reduces the numerical complexity in comparison to the full electron dynamics.
Non-collinear magnetic configurations are stabilized by a constraining field,
which allows to directly obtain the effective magnetic field from the negative
of the constraining field. While the dynamics are shown to conserve energy, we
demonstrate that adiabatic spin dynamics does not conserve the total spin
angular momentum when the lengths of the magnetic moments are allowed to
change, which is confirmed by numerical simulations. Furthermore, we develop a
method to extract an effective two-spin exchange interaction from the energy
curvature tensor of non-collinear states, which we calculate at each time step
of the numerical simulations. We demonstrate the effect of non-collinearity on
this effective exchange and limitations due to multi-spin interactions in
strongly non-collinear configurations beyond the regime where the Heisenberg
model is valid. The relevance of the results are discussed with respect to
experimental pump-probe experiments that follow the ultra-fast dynamics of
magnetism.
