arXiv:1002.4396v1 [cond-mat.str-el] 23 Feb 2010 Abstract Recent developments of high-order CCM have been to extend existing formalism and codes to s ≥ 1 2 for both the ground and excited states, and independently to “generalised” expectation values for a wide range of one- and two-body spin operators. An advantage of the CCM is that the Goldstone linked-cluster theorem is obeyed at all levels of approximation and so it provides results in the infinite lattice limit N → ∞ from the outset. However, recent results have also shown that the CCM can provide exact (symmetry-breaking) results for the spin-half linear-chain J 1 – J 2 at the Majumdar-Ghosh point J 2 /J 1 = 0 . 5 by identifying special solutions of the CCM equations for the usual N´ eel model state. Interestingly, the CCM provides exact (non-symmetry-breaking) results for systems in which small magnetic clusters become de-coupled from each other when the bonds connecting them tend to zero. These exact results involve the identification of “special solutions” of the CCM equations for the N´ eel state. An example of this is given by a spin-half system with nearest-neighbour bonds for an underlying lattice corresponding to the magnetic material CaV 4 O 9 (CAVO) in which one of the two different types of bonds on the lattice tend to zero. Larger finite-sized systems may be considered by appropriate choice of the unit cell and the bonds on it. We show here that exact diagonalisation results for ground-state energy and excitation energy gap for the spin-half and spin-one linear Heisenberg model on chains of length up to N = 12 sites for s = 1 / 2 and N = 6 sites for s = 1 with periodic boundary conditions are reproduced exactly using high-order CCM via this “brute-force” approach; i.e., one in which none of the translational or point-group symmetries of the finite lattice are used. High-Order Coupled Cluster Method (CCM) Formalism 3: Finite-Size CCM D. J. J. Farnell Health Methodology Research Group, School of Community-Based Medicine, Jean McFarlane Building, University Place, University of Manchester, Manchester M13 9PL, United Kingdom (Dated: June 28, 2018) 1