  The pentakis dodecahedron, the dual of the truncated icosahedron, consists of
60 edge-sharing triangles. It has 20 six- and 12 five-fold coordinated
vertices, with the former forming a dodecahedron, and each of the latter
connected to the vertices of one of the 12 pentagons of the dodecahedron. When
spins mounted on the vertices of the pentakis dodecahedron interact according
to the nearest-neighbor antiferromagnetic Heisenberg model, the two different
vertex types necessitate the introduction of two exchange constants. As the
relative strength of the two constants is varied the molecule interpolates
between the dodecahedron and a molecule consisting only of quadrangles. The
competition between the two exchange constants, frustration, and an external
magnetic field results in a multitude of ground-state magnetization and
susceptibility discontinuities. At the classical level the maximum is ten
magnetization and one susceptibility discontinuities when the 12 five-fold
vertices interact with the dodecahedron spins with approximately one-half the
strength of their interaction. When the two interactions are approximately
equal in strength the number of discontinuities is also maximized, with three
of the magnetization and eight of the susceptibility. At the full quantum
limit, where the magnitude of the spins equals 1/2, there can be up to three
ground-state magnetization jumps that have the total z spin component changing
by \Delta S^z=2, even though quantum fluctuations rarely allow discontinuities
of the magnetization. The full quantum case also supports a \Delta S^z=3
discontinuity. Frustration also results in nonmagnetic states inside the
singlet-triplet gap. These results make the pentakis dodecahedron the molecule
with the most discontinuous magnetic response from the quantum to the classical
level.
