  The magnetic structures of small clusters of Fe, Mn, and Cr supported on a
Cu(111) surface have been studied with non-collinear first principles theory.
Different geometries such as triangles, pyramids and wires are considered and
the cluster sizes have been varied between two to ten atoms. The calculations
have been performed using a real space linear muffin-tin orbital method. The Fe
clusters are found to order ferromagnetically regardless of the cluster
geometry. For Mn and Cr clusters, antiferromagnetic exchange interactions
between nearest-neighbours are found to cause collinear antiferromagnetic
ordering when the geometry allows it. If the antiferromagnetism is frustrated
by the cluster geometry non-collinear ordering is found. A comparison between
the calculated structures and ground states obtained from simplified Heisenberg
Hamiltonians show that the exchange interaction varies for different atoms in
the clusters as a result of the different local structure.
