  Recent work on approximating ground states of Heisenberg spin clusters by
projected Hartree-Fock theory (PHF) is extended to a cluster-based ansatz
(cPHF). Whereas PHF variationally optimizes a site-spin product state for the
restoration of spin- and point-group symmetry, cPHF groups sites into discrete
clusters and uses a cluster-product state as the broken-symmetry reference.
Intracluster correlation is thus already included at the mean-field level and
intercluster correlation is introduced through symmetry projection. Variants of
cPHF differing in the broken and restored symmetries are evaluated for ground
states and singlet-triplet gaps of antiferromagnetic spin rings for various
cluster sizes, where cPHF in general affords a significant improvement over
ordinary PHF, although the division into clusters lowers the cyclical symmetry.
On the other hand, certain two- or three-dimensional spin arrangements permit
cluster groupings compatible with the full spatial symmetry. We accordingly
demonstrate that cPHF yields approximate ground states with correct spin and
point-group quantum numbers for honeycomb lattice fragments and symmetric
polyhedra.
