For periodic boundary conditions, the manner in which this is achieved for ED is by identifying common states for a given k -value via a complex phase factor e − i k · r for states that a related by a translational vector r on the finite lattice (again: note periodic boundary conditions are assumed). The Hamiltonian then links only those states of common k . Point- group (PG) symmetries/permutations of indices may then used to form states with real components only, thus simplifying the computational problem. In principle, we ought to be able to employ the finite lattice translational and point-group symmetries analogously in order to simplify the finite-size problem with periodic boundary conditions also for the CCM. However, it is unclear how one might do this in practice for the exponentiated S in the ground ket and bra states. The use of translational and PG symmetries for finite-lattice CCM will be the subject of future research. Despite the fact that translational and PG symmetries for the finite lattice were not used directly here, we were still able to treat finite-lattices of size N = 12 for the spin- half case using high-order CCM with only relatively meagre computational resources. (A MacBook with a “Core Duo” processor and 1 GIG RAM was used in this case was used to carry out this calculation.) Still larger lattices are possible using the CCCM code [55], which has been implemented to work in parallel on a cluster of processors. We note that the addition of intermediate bonds that link the isolated clusters in order to form extended lattices of infinite numbers of sites requires only relatively small increases in computational effort for high-order CCM compared to treating the case of isolated magnetic clusters alone. Excellent results were seen here for such a model that interpolated between finite clusters and an extended lattice, namely, for the n.n. CAVO model (e.g., see the results for the ground state energy in Fig. 2). High-order CCM might provide a good choice for the study of a whole range of such “interpolating” models between finite clusters and infinite lattices. [1] F. Coester, Nucl. Phys. 7 , 421 (1958); F. Coester and H. K¨ ummel, ibid. 17 , 477 (1960). [2] J. ˇ Ciˇ zek, J. Chem. Phys. 45 , 4256 (1966); Adv. Chem. Phys. 14 , 35 (1969). [3] R.F. Bishop and K.H. L¨ uhrmann, Phys. Rev. B 17 , 3757 (1978); ibid. 26 , 5523 (1982). [4] H. K¨ ummel, K.H. L¨ uhrmann, and J.G. Zabolitzky, Phys Rep. 36C , 1 (1978). [5] J.S. Arponen, Ann. Phys. (N.Y.) 151 , 311 (1983). 12