Exact ground state properties of the classical Heisenberg model for giant magnetic molecules

We find the exact ground state energy and magnetic moment for an arbitrary magnetic field H of the classical Heisenberg model of spins on the vertices of an icosidodecahedron. This model provides an accurate description of the magnetic properties of the giant paramagnetic molecule Mo₇₂Fe₃₀ in which 30 Fe³⁺ ions are coupled via antiferromagnetic exchange. The strong frustration of the magnetic interaction in the molecule is relaxed when the angle between nearest-neighbor spins is 120^∘{}. We predict that the magnetic moment is linear with H until saturating at a critical field Hc, and this is consistent with the results of a recent experiment at 0.46 K. We derive our results using a graph-theoretical construction and a special property, three-colorability, of the icosidodecahedron. We also consider spins on the vertices of an octahedron, icosahedron, and dodecahedron.