  We study the suitability of correlator product states for describing
ground-state properties of two-dimensional spin models. Our ansatz for the
many-body wave function takes the form of either plaquette or bond correlator
product states and the energy is optimized by varying the correlators using
Monte Carlo minimization. For the Ising model we find that plaquette
correlators are best for estimating the energy while bond correlators capture
the expected long-range correlations and critical behavior of the system more
faithfully. For the antiferromagnetic Heisenberg model, however, plaquettes
outperform bond correlators at describing both local and long-range
correlations because of the substantially larger number of local parameters
they contain. These observations have quantitative implications for the
application of correlator product states to other more complex systems, and
give important heuristic insights: in particular the necessity of carefully
tailoring the choice of correlators to the system considered, its interactions
and symmetries.
