  We present results of first-principles calculations of the magnetic
properties of Fe chains deposited on the Re(0001) surface. By increasing the
length of the chain, a transition is found from an almost collinear
antiferromagnetic state for a 5-atom-long chain to a spin spiral state with the
rotational plane slightly tilted from the surface of the substrate for the
15-atom-long chain. It is shown that a classical spin model derived from the ab
initio calculations containing only two-spin interactions supports opposite
chirality of the spin spiral compared to a direct optimization of the spin
configuration within the ab initio method. The differences between the results
of the two methods can be understood by introducing chiral four-spin
interactions in the spin model.
