General description

The tight binding (TB) model is a quantum mechanical method for electronic structure description within the one-particle and adiabatic approximations[1]. The Hamiltonian (H) of a system and other operators in TB are expanded in a minimal, localised, atom centered basis set resulting in the smallest possible matrix sizes (as in ASA-LMTO). In the semiempirical TB models a further assumption is made for implicit basis functions. The H and overlap (S) matrix elements

$$H_{RLR'L'} = \langle \phi_{RL} \vert V_{R''L''}\vert\phi_{R'L'}\rangle \rightarrow V_{RLR'L'}(\vert R-R'\vert)Y_{RLR'L'}(\phi_{R-R'},\theta_{R-R'})$$ (1)

$$S_{RLR'L'} = \langle \phi_{RL} \vert\phi_{R'L'}\rangle \rightarrow Vs_{RLR'L'}(\vert R-R'\vert)Y_{RLR'L'}(\phi_{R-R'},\theta_{R-R'})$$ (2)

are thus are never explicitly calculated, instead they are approximated by parametrised radial pair functions and tabulated angular factors¹. Usually only the 1 and 2-centre integrals are given in this manner and the 3-centre integrals are completely ignored for reasons of sanity or because they are deemed negligble, for example in metals. While localised atom based functions are not quite orthogonal a further approximation is often made such that the implicit function are orthogonal, further simplifying the underlying eigenvalue problem. Often only the bonding part of the energy is treated quantum mechanically and most of the repulsive part is combined with the core (ion) repulsion term in a parametrised classical pairwise potential, however there are exceptions, for example adding Hubbard terms will act as an approximation to the correlation energy and in magnetic models the exchange is included through parametrised onsite energy shifts. These are usually included in self consistent models. Another important ingredient, and one unique to our TB model, is the more detailed description of the charge density through multipole expansions[3].

The significant amount of approximations leads to what may be called the most transparent and simplest to implement quantum mechanical model. This however comes with the cost and responsibility of providing an appropriate parameterisation. The empirical parameterisation on the other hand provides freedom to fit important quantities to desired values and thus often gives a significantly better description of the modelled system than more advanced ab initio methods. This combined with the lower computational cost and also the significantly better forces/energy consistency of TB means that system size and time length dependent quantities (e.g. diffusion coefficient) can be calculated quantum mechanically on a small computing cluster, within acceptable statistical errors, given a capable implementation.

The advantage over the much faster classical methods stems from the quantum nature of TB and the ability to properly describe the formation and dissociation of bonds (including H-bonds on the same footing), charge transfers and polarisability, magnetism and also negative Cauchy pressures occuring in certain intermetallics.