  Coupled cluster and symmetry projected Hartree-Fock are two central paradigms
in electronic structure theory. However, they are very different. Single
reference coupled cluster is highly successful for treating weakly correlated
systems, but fails under strong correlation unless one sacrifices good quantum
numbers and works with broken-symmetry wave functions, which is unphysical for
finite systems. Symmetry projection is effective for the treatment of strong
correlation at the mean-field level through multireference non-orthogonal
configuration interaction wavefunctions, but unlike coupled cluster, it is
neither size extensive nor ideal for treating dynamic correlation. We here
examine different scenarios for merging these two dissimilar theories. We carry
out this exercise over the integrable Lipkin model Hamiltonian, which despite
its simplicity, encompasses non-trivial physics for degenerate systems and can
be solved via diagonalization for a very large number of particles. We show how
symmetry projection and coupled cluster doubles individually fail over
different correlation limits, whereas models that merge these two theories are
highly successful over the entire phase diagram. Despite the simplicity of the
Lipkin Hamiltonian, the lessons learned in this work will be useful for
building an ab initio symmetry projected coupled cluster theory that we expect
to be accurate over the weakly and strongly correlated limits, as well as the
recoupling regime.
