  The icosahedron has a ground state magnetization discontinuity in an external
magnetic field when classical spins mounted on its vertices are coupled
according to the antiferromagnetic Heisenberg model. This is so even if there
is no magnetic anisotropy in the Hamiltonian. The discontinuity is a
consequence of the frustrated nature of the interactions, which originates in
the topology of the cluster. Here it is found that the addition of the next
order isotropic spin exchange interaction term in the Hamiltonian, the
biquadratic exchange interaction, significantly enriches the classical ground
state magnetic response. For relatively weak biquadratic interaction new
discontinuities emerge, while for even stronger the number of discontinuities
for this small molecule can go up to seven, accompanied by a susceptibility
discontinuity. These results demonstrate the possibility of using a small
entity like the icosahedron as a magnetic unit whose ground state spin
configuration and magnetization can be tuned between many different
non-overlapping regimes with the application of an external field.
