relevant Hamiltonian in rotated coordinates is then given by where 〈 i, j 〉 runs over all nearest-neighbour sites on the lattice counting each pair once and once only and 〈〈 i, k 〉〉 runs over all next-nearest-neighbor sites on the lattice, again counting each pair once and once only. Henceforth we put J 1 = 1 and consider J 2 > 0. The ground-state properties of this system have been studied using methods such as exact diagonalisations [57, 62], DMRG [17, 58–60], CCM [14, 16, 42], and field-theoretical approaches [60] (see Refs. [60, 61] for a general review). We shall not go into detail about this model here except to note that there are two degenerate simple exact dimer-singlet product ground states at the Majumdar-Ghosh point J 2 /J 1 = 0 . 5. We now consider how this model can be treated at the Majumdar-Ghosh point J 2 /J 1 = 0 . 5by the CCM via the identification of a special dimerised solution of the CCM equations for a N´ eel model state. We use a doubled unit cell including two neighbouring sites for a spin-half system on the linear chain at points (0,0,0) and (1,0,0) and a single Bravais vector (2,0,0) T to take into account the symmetry breaking. There are thus two distinct types of two-spin nearest-neighbour ket-state correlation coefficients and again these are denoted as S a 2 and S b 2 for S 2 = S a 2 ∑ i a s + i $_{a}$s + i $_{$_{a}$+1}$+ S b 2 ∑ i b s + i $_{b}$s + i $_{b}$+1 with respect to the CCM ket state of Eq. (2), and where i$_{a}$ runs over all sites with odd-numbered indices and i$_{b}$ runs over all sites with even-numbered indices. The exact ground state at J 2 /J 1 = 0 . 5 is obtained by setting S a 2 = 1 and all other coefficients equal to zero. This result is found to hold for all levels of LSUB m approximation. The exact ground-state energy of E$_{g}$/N = − 0 . 375 J 1 is thus obtained at the point J 2 /J 1 = 0 . 5, as expected. Furthermore, the sublattice magnetisation is found to be zero at this point at all levels of approximation. We find values [54] for the excitation energy gap of 0 . 35250, 0 . 34170, 0 . 30548, 0 . 28732, 0 . 27559, and 0 . 26760 at the LSUB m levels of approximation with m = { 4 , 6 , 8 , 10 , 12 , 14 } . A simple extrapolation of these results in the limit m → ∞ using a quadratic function gives a value for the gap of 0.2310. This result is in agreement with results of exact diagonalisations that predict a gap of 0.234 [63]. 5 H = − J 1 ∑ 〈 i,j 〉 ( s z $_{i}$s z j + 1 2 { s − i s − j + s + i s + j } ) + J 2 ∑ 〈〈 i,k 〉〉 ( s z $_{i}$s z k + 1 2 { s + i s − k + s − i s + k } ) , (6)