  We introduce cluster-based mean-field, perturbation and coupled-cluster
theories to describe the ground state of strongly-correlated spin systems. In
cluster mean-field, the ground state wavefunction is written as a simple tensor
product of optimized cluster states. The cluster-language and the mean-field
nature of the ansatz allows for a straightforward improvement based on
perturbation theory and coupled-cluster, to account for inter-cluster
correlations. We present benchmark calculations on the 2D square $J_1-J_2$
Heisenberg model, using cluster mean-field, second-order perturbation theory
and coupled-cluster. We also present an extrapolation scheme that allows us to
compute thermodynamic limit energies very accurately. Our results indicate
that, even with relatively small clusters, the correlated methods can provide
an accurate description of the Heisenberg model in the regimes considered. Some
ways to improve the results presented in this work are discussed.
