Sr_14Cu_24O_41Sr\_{14}Cu\_{24}O\_{41} : a complete model for the chain sub-system

A second neighbor $t-J+V$ model for the chain subsystem of the $Sr\_{14}Cu\_{24}O\_{41}$ has been extracted from ab-initio calculations. This model does not use periodic approximation but describes the entire chain through the use of the four-dimensional crystallographic description. Second neighbors interactions are found to be of same order than the first neighbors ones. The computed values of the second neighbors magnetic interaction are coherent with experimental estimations of the intra-dimer magnetic interactions, even if slightly smaller. The reasons of this underestimation are detailed. The computed model allowed us to understand the origin of the chain dimerisation and predicts correctly the relative occurrence of dimers and free spins. The orbitals respectively supporting the magnetic electrons and the holes have been found to be essentially supported by the copper 3d orbitals (spins) and the surrounding oxygen $2p$ orbitals (holes), thus giving a strong footing to the existence of Zhang-Rice singlets