Properties of low-lying states in some high-nuclearity Mn, Fe, and V clusters: Exact studies of Heisenberg models

Using an efficient numerical scheme that exploits spatial symmetries and spin parity, we have obtained the exact low-lying eigenstates of exchange Hamiltonians for the high-nuclearity spin clusters, ${\mathrm{Mn}}_{12},$ ${\mathrm{Fe}}_8,$ and ${\mathrm{V}}_{15}.$ The largest calculation involves the ${\mathrm{Mn}}_{12}$ cluster that spans a Fock space of a hundred million. Our results show that the earlier estimates of the exchange constants need to be revised for the ${\mathrm{Mn}}_{12}$ cluster to explain the level ordering of low-lying eigenstates. In the case of the ${\mathrm{Fe}}_8$ cluster, correct level ordering can be obtained, which is consistent with the exchange constants for the already known clusters with butterfly structure. In the ${\mathrm{V}}_{15}$ cluster, we obtain an effective Hamiltonian that reproduces exactly, the eight low-lying eigenvalues of the full Hamiltonian.