  Spin models like the Heisenberg Hamiltonian effectively describe the
interactions of open-shell transition-metal ions on a lattice and can account
for various properties of magnetic solids and molecules. Numerical methods are
usually required to find exact or approximate eigenstates, but for small
clusters with spatial symmetry, analytical solutions exist, and a few
Heisenberg systems have been solved in closed form. This paper presents a
simple, generally applicable approach to analytically solve isotropic spin
clusters, based on adapting the basis to both total-spin and point-group
symmetry to factor the Hamiltonian matrix into sufficiently small blocks. We
demonstrate applications to small rings and polyhedra, some of which are
straightforward to solve by successive spin-coupling for Heisenberg terms only;
additional interactions, such as biquadratic exchange or multi-center terms
necessitate symmetry adaptation.
