An introduction to the tight binding approximation— implementation by diagonalisation

“Tight binding” has existed for many years as a convenient and transparent model for the description of electronic structure in molecules and solids. It often provides the basis for construction of many body theories such as the Hubbard model and the Anderson impurity model. Slater and Koster call it the tight binding or “Bloch” method and their historic paper provides the systematic procedure for formulating a tight binding model.1 In their paper you will find the famous “Slater–Koster” table that is used to build a tight binding hamiltonian. This can also be found reproduced as table 20–1 in Harrison’s book and this reference is probably the best starting point for learning the tight binding method.2 Building a tight binding hamiltonian yourself, by hand, as in Harrison’s sections 3–C and 19–C is certainly the surest way to learn and understand the method. The rewards are very great, as I shall attempt to persuade you now. More recent books are the ones by Sutton,3 Pettifor4 and Finnis.5 In my development here I will most closely follow Finnis. This is because whereas in the earlier literature tight binding was regarded as a simple empirical scheme for the construction of hamiltonians by placing “atomic-like orbitals” at atomic sites and allowing electrons to hop between these through the mediation of “hopping integrals,” it was later realised that the tight binding approximation may be directly deduced as a rigorous approximation to the density functional theory. This latter discovery has come about largely through the work of Sutton et al.6 and Foulkes;7 and it is this approach that is adopted in Finnis’ book from the outset. In the context of atomistic simulation, it can be helpful to distinguish schemes for the calculation of interatomic forces as “quantum mechanical,” and “non quantum mechanical.” In the former falls clearly the local density approximation (LDA) to density functional theory and nowadays it is indeed possible to make molecular dynamics calculations for small numbers of atoms and a few picoseconds of time using the LDA. At the other end of the scale, classical potentials may be used to simulate millions of atoms for some nanoseconds or more. I like to argue that tight binding is the simplest scheme that is genuinely quantum mechanical. Although you will read claims that the “embedded atom method” and other schemes are LDA-based, tight binding differs from these in that an explicit calculation of the electron kinetic energy is attempted either by diagonalising a hamiltonian, which is the subject of this lecture; or by finding its Green function matrix elements which is the subject of the lecture by Ralf Drautz.8 The enormous advantage of the latter is that calculations scale in the computer linearly with the number of atoms, while diagonalisa-