TABLE III: CCM LSUB m results for the ground-state energy and excitation energy gap of spin-half antiferromagnetic Heisenberg chains of length N = 12 with periodic boundary conditions. LSUB m m = E$_{g}$/N Gap 4 − 0.421064786 0.589781571 6 − 0.42329432 0.440170513 8 − 0.423975557 0.375801129 10 − 0.424415368 0.348833726 12 − 0.448949243 0.355847514 J 1 – J 2 model at J 2 /J 1 = 0 . 5 may be reproduced using a N´ eel model state. Using a similar model state in 2D, exact results for a nearest-neighbour CAVO model were reproduced using high-order CCM in the limits J $^{′}$/J → 0 (4-site plaquette) and J $^{′}$/J → ∞ (2-site dimers). These results lead on naturally to a “brute-force” approach for solving finite-sized lattices, i.e., without using any of the translational or point-group symmetries of the finite lattice. Indeed, we have shown here that high-order CCM SUB n - m results for the antiferromagnetic Heisenberg model (with periodic 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 were found to agree with exact-diagonalisation (ED) results for the ground-state energy and excitation energy gap agree to at least six decimal places. We note that results for the case of finite clusters for s = 1 were obtained via new high- order formalism for the CCM excited state outlined in Refs. [52, 54]. Furthermore, new code [55] has been written to implement this new formalism for the excited state for s ≥ 1. The agreement between ED results for the ground-state energy and excitation energy gap for chains of up to N = 6 for s = 1 is an excellent test of the validity of this new code. These solutions for the finite chains were found to be stable numerically. Hence, we have “proven the principle” that the high-order CCM code may be used directly to study (relatively small- sized) finite lattices by a somewhat “brute-force” approach. Indeed, this approach is still somewhat inefficient in comparison to exact diagonalisations (ED) because ED results use translational and point-group symmetries of the finite lattice in order to reduce the size of the matrix to be diagonalised, and so ED may to go to much larger lattice sizes. 11