distinct locales on the lattice defined by m or fewer contiguous sites are retained. Another important feature of the method is that the bra and ket states are not always explicitly constrained to be Hermitian conjugates when we make such approximations, although the important Helmann-Feynman theorem is always preserved. We remark that the CCM pro- vides results in the infinite-lattice limit N → ∞ from the outset. Key to understanding the application of the CCM to spin problems is the concept of the unit cell and the Bravais lattice. The unit cell contains a number of sites at specific positions (given by the “primitive” lattice vectors) that are replicated at all possible multiples of the Bravais lattice vectors. Thus, for example, we have a single site in the unit cell for the linear chain, say, at position (0 , 0) and a single Bravais lattice vector ˆ a = (1 , 0) $^{T}$. The lattice is formed by translating the single site in the unit cell by all integer multiples of ˆ a . Two-dimensional lattices have two Bravais lattice vectors. For example the square lattice has a single site in the unit cell and the lattice vectors are ˆ a = (1 , 0) T and ˆ b = (0 , 1) $^{T}$. The triangular lattice is given by vectors ˆ a = (1 , 0) T and ˆ b = (1 / 2 $_{,}$√$_{3}$$_{/}$$_{2)}$ T and so on for other lattices. We see also that the basic building blocks of unit cell, Bravais lattice, and bonds/interactions in the Hamiltonian placed on the lattice gives us a broad canvas to work with. For example, we may form models that interpolate between different lattices (and even different spatial dimensions) by varying the strengths of various bonds that have been carefully placed with respect to the underlying lattice. Hence, the number of possible such quantum spin systems is enormous. Furthermore, the development in the number and complexity of these theoretical models is often driven by the magnetic materials studied in experiment. The Hamiltonian for this spin-half model has nearest-neighbour bonds of strength J 1 and next-nearest-neighbour bonds of strength J 2 . We use a N´ eel model state in which nearest- neighbour spins on the linear chain are anti-parallel. We rotate the spin coordinates of the ‘up’ spins so that notationally they become ‘down’ spins in these locally defined axes. The III. RESULTS A. The Spin-Half J 1 – J 2 Model on the Linear Chain 4