Bond order potentials Tersoff

As in the MD method, PES for KMC can be derived from first-principles methods or using empirical energy functionals described above. However, the KMC method requires the accurate evaluation of the PES not only near the local minima, but also for transition regions between them. The corresponding empirical potentials are called reactive, since they can be used to calculate parameters of chemical reactions. The development of reactive potentials is quite a difficult problem, since chemical reactions usually include the breaking or formation of new bonds and a reconfiguration of the electronic structure. At present, a few types of reactive empirical potentials can semi-quantitatively reproduce the results of first-principles calculations these are EAM and MEAM potentials for metals and bond-order potentials (Tersoff and Brenner) for covalent semiconductors and organics. 

The Tersoff potential [Tersoff 1988] is based on a model known as the empirical bond-order potential. This potential can be written in a form very similar to the Finnis-Sinclair potential ... 

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. 

To overcome this limitation we developed a series of potentials in the late 1980 s and early 1990 s that have become known as reactive empirical bond order (REBO) potentials. These potentials are based on the empirical bond order potential form introduced by Tersoff to describe the static properties of silicon but were tailored by us to incorporate a modicum of chemistry. In Sec. 2.1, after introducing the REBO potential form, we describe our simple models for energetic materials that are based on these potentials. In Sec. 2.2, we provide an overview of the approach taken to implement our simulations of shock-induced chemistry and detonations. 

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. 

Molecular dynamics studies of diatomic model detonations were first carried out by Karo and Hardy in 1977 [14]. They were soon followed by other groups [15, 16]. These early studies employed predissociative potentials, in which the reactant dimer molecules are metastable and can dissociate exothermically. More realistic models, combining an endothermic dissociation of reactants with an exothermic formation of product molecules, were introduced by White and colleagues at the Naval Research Laboratory and U.S. Naval Academy, first using a LEPS (London-Eyring-Polanyi-Sato) three-body potential for nitric oxide [17], and later a Tersoff-type bond-order potential [18] for a generic AB model, loosely based on NO [19, 20]. 

To address this problem, Monte Carlo simulations [32] and MD smdies [33-37] have been carried out before, aU of them using reactive empirical bond-order (REBO) Tersoff-type [38,39] interatomic carbon-carbon potentials developed originally for studying the vapor deposition of diamond [40,41]. Unlike traditional molecular mechanics force fields, the REBO potential allows for the formation and dissociation of covalent chemical bonds by determination of next neighbors and on-the-fly switching... 

Pettifor, D., Oleinik, I. Analytic bond-order potentials beyond Tersoff-Brenner. i. Theory. Phys. Rev. B 59, 8487-8499 (1999). doi 10.1103/PhysRevB.59.8487... 

The form of interatomic potential suggested by Tersoff [25] is an example of the wider family of bond-order potentials [26]. The total energy is written as a sum of pair like terms. 

Two-body potentials are typically a good approximation of neutral atoms such as noble gases, which are dominated by attractive van der Waals forces at separations and with strong repulsion at close distances due to Pauli s exclusion principle. The electronic distributions of other systems, such as covalently bonded materials, are modeled more accurately by more complicated environmentally dependent semiempirical potentials. These include bond-order potentials like Tersoff [11], Brenner [12], and ReaxFF [13] and embedded atom model (EAM) [14] potentials, which are particularly applicable to metallic systems. 

The Tersoff potential was designed specifically for the group 14 elements and extends the basic empirical bond-order model by including an angular term. The interaction energy between two atoms i and j using this potential is ... 

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

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 ... 

When angular-dependent interactions contribute significantly to the bonding, pair potentials like those described above are not sufficient and three-body or higher order terms must be included in the potential energy. That is the case of covalently bonded systems like silicon or transition metals. For the case of silicon two interatomic potentials are widely used the one developed by StiUinger and Weber [20] and the one developed by Tersoff [21]. Other empirical potentials have been developed to include the angular dependence such as the modified embedded atom method (MEAM) [22]. 

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 most widely used family of reactive FFs is based on the concept of bond order it is assumed that the strength of a bond between two atoms is not constant, but depends on the local environment. Examples indude the Tersoff potential,the reactive empirical bond order (REBO and REB02) modd, the adaptive intermolecular reactive empirical bond order (AIREBO) model, the second-moment TB potentials, and the bond-based analytic bond order (BOP) potentials. These models allow for bond formation,... 

Mainly in the field of materials science various types of potentials have been developed based on the concept of the bond order. " Like for reactive force fields also for the application of these potentials a specification of the atomic positions is sufficient. Although many of these potentials like the Tersoff potential, the Stillinger-Weber potential, the Breimer potential and many others have been introduced already one or two decades ago, they are still frequently used in materials simulations, in particular for semiconductors. For metallic systems the embedded atom method (EAM) and the modified embedded atom method (MEAM) introduced by Baskes and coworkers are widely distributed. 

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... 

Fig. 8 Structure of the high-dimensional neural network used to fit the bond order term by in the Tersoff potential. For each bond a set of input vectors is generated. The first hidden layer does not have a fixed size but is duplicated for each input vector. The values of the nodes of this adaptive first hidden layer are then collected and processed by the nodes of a second hidden layer whose size is fixed and independent of the system. In the node of the output layer the bond order term by is obtained. The bias weights are not shown for clarity.

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. 

One particularly successful example of a Tersoff potential is the reactive empirical bond-order (REBO) potential developed by Brenner (26-29) to describe the covalent bonding interactions in carbon and hydrocarbon systems. Originally developed for use in simulating the chemical vapor deposition of diamond (26), the REBO potential has recently been extended to provide more accurate treatment of the energetic, elastic, and vibrational properties of solid carbon and small hydrocarbons (29). 

Because the electrostatic energy terms for an uncharged pure system (such as a bulk structure) is zero, the energy contribution of COMB for such a system is from the short-range interactions [eqn (7.18)] and the formalism is reduced to the Tersoff type of potential. This case is straightforward for the parameterization of the COMB potential, which only fits parameters in the pairwise term and bond-order terms, as shown in Table 7.2. The van der Waals interaction will be considered for hydrocarbon systems. 

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