  Spin Hamiltonians, like the Heisenberg model, are used to describe magnetic
properties of exchange-coupled molecules and solids. For finite clusters,
physical quantities such as heat capacities, magnetic susceptibilities or
neutron-scattering spectra, can be calculated based on energies and eigenstates
obtained by exact diagonalization (ED). Utilizing spinrotational symmetry SU(2)
to factor the Hamiltonian with respect to total spin S facilitates ED, but the
conventional approach to spin-adapting the basis is more intricate than
selecting states with a given magnetic quantum number M (the spin z-component),
as it relies on irreducible tensor-operator techniques and spin-coupling
coefficients. Here, we present a simpler technique based on applying a spin
projector to uncoupled basis states. As an alternative to L\"owdin's projection
operator, we consider a group-theoretical formulation of the projector, which
can be evaluated either exactly or approximately using an integration grid. An
important aspect is the choice of uncoupled basis states. We present an
extension of L\"owdin's theorem for s = 1/2 to arbitrary local spin quantum
numbers s, which allows for the direct selection of configurations that span a
complete, linearly independent basis in an S sector upon spin projection. We
illustrate the procedure with a few examples.
