Hamiltonian of Eq. (3) and we rotate the “up” spins as usual. The Hamiltonian in the new spin coordinates is then given by Eq. (5). Hence, we form bonds (of strength J = 1) between all nearest-neighbour sites in the unit cell. However, we do not “link” the bonds by forming an intermediate bond between the unit cells, and so, in effect, the unit cells become isolated clusters. A technical point is that we set the single Bravais lattice vector to be of magnitude + N for this “brute force” approach; again, even though the clusters/unit cells are not linked by intermediate bonds. Furthermore, we may also provide an additional bond between the first and last sites (again of strength J = 1). In effect, this bond acts as a “boundary condition” that creates a “ring” of sites for the finite lattice. These results should be compared to those of ED with periodic boundary conditions. High-order CCM SUB n - m results for the antiferromagnetic Heisenberg model (with pe- riodic conditions) for finite-sized chains of length N with SUB n - m with n = m = N for s = 1 / 2 and n/ 2 = m = N for s = 1 in Tables I and II, respectively. We remark that CCM results for the ground-state energy and excitation energy gap agree to at least six decimal places with exact diagonalisations obtained using the SpinPack code of Joerg Schulenburg [64]. We may also study the manner in which LSUB m results behave with increasing m for a set value of chain length N . Table III presents LSUB m results for the spin-half Heisenberg chain of length N = 12. The ground-state energies decrease monotonically with increasing LSUB m level of approximation, although no simple extrapolation “rule” may be seen. By contrast, the LSUB m data for the excitation energy gap is only monotonically decreasing up to LSUB10. Indeed, the gap for LSUB10 lies lower than that of the (exact) LSUB12 result. We note again that the LSUB m approximation for the spin-half system and the SUB2 m - m approximation for the spin-one system reproduce exact results for the chains of length N when we set m = N . The CCM results in Tables I to III arise from “special solutions” of the CCM equations in exactly the same manner as for the nearest-neighbour CAVO problem in the limits J $^{′}$/J → 0 and | J $^{′}$/J | → ∞ . However, it is clearly more complicated in this case because we are dealing with unit cells (i.e., finite-sized 1D lattices here) of greater size than 4 sites. Interestingly though, the CCM ground-state equations were found to converge readily to this “special solution” and for a wide range of starting values at all levels of SUB n - m approximation attempted here. For example, setting the initial ket-state correlation coefficients to a small (positive) non-zero value was found to work quite well. Indeed, the ground-state solution 9