This chapter derives tight-binding models from the Hohenberg–Kohn–Sham density functional theory of total energy. The approach taken is to expose the successive approximations that can be invoked, starting with the neglect of three-centre integrals, and ending with the second moment approximation to the density of states, each leading to a slightly simpler tight-binding model. A rich variety of models can be derived on the way with applications to transition metals, semiconductors or even insulators. Attention is given to the role of self-consistency. The first comprehensive and self-contained description of bond-order potentials is given.
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