Tight-binding modelling of materials

The tight-binding method of modelling materials lies between the very accurate, very expensive, ab initio methods and the fast but limited empirical methods. When compared with ab initio methods, tight-binding is typically two to three orders of magnitude faster, but suffers from a reduction in transferability due to the approximations made; when compared with empirical methods, tight-binding is two to three orders of magnitude slower, but the quantum mechanical nature of bonding is retained, ensuring that the angular nature of bonding is correctly described far from equilibrium structures. Tight-binding is therefore useful for the large number of situations in which quantum mechanical effects are significant, but the system size makes ab initio calculations impractical. In this paper we review the theoretical basis of the tight-binding method, and the range of approaches used to exactly or approximately solve the tight-binding equations. We then consider a representative selection of the huge number of systems which have been studied using tight-binding, identifying the physical characteristics that favour a particular tight-binding method, with examples drawn from metallic, semiconducting and ionic systems. Looking beyond standard tight-binding methods we then review the work which has been done to improve the accuracy and transferability of tight-binding, and moving in the opposite direction we consider the relationship between tight-binding and empirical models.