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Four linear scaling tight-binding methods (the density matrix method, bond order potentials, the global density of states method, and the Fermi operator expansion) are described and compared to show relative computational efficiency for a given accuracy. Various example systems are explored: an insulator (carbon in the diamond structure), a semiconductor (silicon), a transition metal (titanium) and a molecule (benzene). The density matrix method proves to be most efficient for systems with narrow features in their energy gaps, while recursion-based moments methods prove to be most efficient for metallic systems.

We present a simple tight-binding parametrization for TiC that is capable of giving qualitatively accurate results for a range of properties. The systematic procedure used to develop these parameters is described, and the transferability of the model is tested. We use the model to account for the existence of substoichiometric titanium carbide, and the variation of its lattice constant with vacancy concentration. We show that the unusual lattice constant trend arises from the fact that the lattice relaxation around a vacancy is highly localized. The physical origin of this localization is the strongly anisotropic strengthening of bonds around a vacancy, which follows from the quantum mechanical nature of the bonding.

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