Au-Au Metallic Bonding Potential: The second-moment approximation to the tight-binding potential (TB-SMA)

The second-moment approximation to the tight-binding potential (TB-SMA) 89 is applied throughout all chapters of this dissertation to describe Au-Au interactions. Simple pairwise potentials such as the 12-6 Lennard-Jones potential fail to properly describe many of the properties (e.g., vacancy formation energies, surface structure, and relaxation properties) of transition metals. 89 Semi-empirical potentials, whose functional forms are derived from electronic structure considerations and then fit to experimental data, are better suited for simulations of transition metals. For instance, TB-SMA contains a many-body term that is modeled after the square-root dependence of the band energy on the second moment electron density of state:

where EiB is the many-body energy of atom i.
TB-SMA also contains a pairwise repulsive term given by

The total TB-SMA energy is then

Values for the parameters A, ξ , p, q, and r0 for Au are shown in Table 3.1, and are obtained from fits to the Au experimental cohesive energy, lattice parameter, and elastic constant [Cleri, F.; Rosato, V. Phys. Rev. B 1993, 48, 22–33].
An energy cutoff, rcut, of 5.8 A, is applied such that any pair of Au atoms separated by a distance greater than rcut do not interact. Differentiating equation 3.1 yields an expression for force that depends on the electron density, ρ, of atoms i and j, where

Thus, the total force acting on each atom is calculated in two stages, with each stage looping over atom i’s neighbors. This amounts to an additional computational cost compared to pairwise interaction models, where only a single loop over atom i’s neighbors is performed.

Ref: http://etd.library.vanderbilt.edu/available/etd-03212013-122215/unrestricted/French.pdf