Many-body Tersoff potential

Many body potentials e.g. Sutton-Chen, Tersoff, " Brenner can be used to describe metals and other continuous solids such as silicon and carbon. The Brenner potential has been particularly successful with fullerenes, carbon nanotubes and diamond. Erhart and Albe have derived an analytical potential based on Brenner s work for carbon, silicon and silicon carbide. The Brenner and Tersolf potentials are examples of bond order potentials. These express the local binding energy between any pair of atoms/ions as the sum of a repulsive term and an attractive term that depends on the bond order between the two atoms. Because the bond order depends on the other neighbours of the two atoms, this apparently two-body potential is in fact many-body. An introduction and history of such potentials has recently been given by Finnis in an issue of Progress in Materials Science dedicated to David Pettifor. For a study of solid and liquid MgO Tangney and Scandolo derived a many body potential for ionic systems. 

In 1988, Tersoff [7] introduced an analytical expression for a many-body potential energy function based on bond order, which was able to accommodate reactive dynamies in a straightforward manner. In this formalism, the interaction energy Eij between a pair of atoms i and j is given by... 

The functional form of the Tersoff potential model is motivated by the structural chanistry of covalent systems. The primary basis for this model is that the strength of a bond (i.e. bond order) depends on the local environment (i.e. coordination number). Most predominantly covalent systans have open structures, as an atom with fewer neighbours will form stronger bonds as compared to a close-packed structure where an atom has many neighbours. Tersoff states that the energy is modelled as a sum of pair-like interactions, where, however the coefficient of the attractive term in the pair-like potential (which plays the role of bond order) depends on the local environment, giving a many-body potential. The empirical interatomic potential function for multi-component systems as proposed by Tersoff is as shown below. 

A related potential form, which was primarily developed to reproduce, structural energetics of silicon, was introduced by Tersoff and was based on ideas discussed by Abell . The binding energy in the AbeH-Tersoff expression is written as a sum of repulsive and attractive two-body interactions, with the attractive contribution being modified by a many-body term. 

An expression of the type (7.101), which gives the bond order explicitly in terms of the positions of the neighbouring atoms, is called a bond order potential (BOP). Angularly dependent bond order potentials were first derived heuristically for the elemental semiconductors by TersofF (1988). We will see in the next chapter that a many-body expansion for the bond order may be derived exactly within the model. 

In the early 1990s, Brenner and coworkers [163] developed interaction potentials for model explosives that include realistic chemical reaction steps (i.e., endothermic bond rupture and exothermic product formation) and many-body effects. This potential, called the Reactive Empirical Bond Order (REBO) potential, has been used in molecular dynamics simulations by numerous groups to explore atomic-level details of self-sustained reaction waves propagating through a crystal [163-171], The potential is based on ideas first proposed by Abell [172] and implemented for covalent solids by Tersoff [173]. It introduces many-body effects through modification of the pair-additive attractive term by an empirical bond-order function whose value is dependent on the local atomic environment. The form that has been used in the detonation simulations assumes that the total energy of a system of N atoms is ... 

The potential energy of the system is constructed as a sum of individual bond energies. The interactions are truncated using a cutolf function of the interatomic distance ry. The expressions for the repulsive pair potential Vji(rij) and the attractive pair potential F (ry) have been taken from the original Tersoff potential, but the bond order term by modulating the strength of the attractive potential contribution is expressed by a neural network. This many-body term depends on the local environment of the bonds. There is one separate NN for each bond in the system. For each of these bond, each atom bonded either to atom i or j provides an input vector for the NN of the bond ij. As discussed in the previous section, a major... 

B. W. Dodson, Development of a many-body Tersoff-type potential for silicon. Phys. Rev. B 35, 2795-2798(1987)... 

Classically-based simulations do not work for metals because of the very delocalized bonding characteristic of nearly-free electrons in the substrate. Pair potentials also fail to woik for semiconductors that have strongly directional covalent bonding. For example, a pair potential would predict a close-packed structure for Si instead of the experimentally-observed diamond structure. Only three-body or n-body interatomic PES s can yield the diamond lattice structure, although that structure may contain zero-energy lattice defects [88Terl], Since n-body interatomic potentials contain too many free parameters, in practice potential ejq)ansions usually incorporate terms with no more than three bodies. The two most popular three-body interatomic PES s are the Stilhnger-Weber (SW) and Tersoff potentials. 

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