Interdisciplinary Applied Mathematics

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Many-Body Intermolecular Potentials

Though the pairwise potentials have been fairly successful in describing the intermolecular interactions, there is evidence that the three-body interactions (or even higher-order interactions) can be important in some cases (Bobetic and Barker, 1970; Monson et al., 1983; Rittger, 1990b; Rittger, 1990a; Rittger, 1990c). Here we introduce the Tersoff potential, which is a three-body potential.

The    Tersoff potential    is based    on    the    concept    of    bond    order;    i.e.,    the

strength of a bond between two atoms is not constant, but depends on the local environment. The Tersoff potential has the form

v(r) =    + •• •, (16.11)

ij    ij

where R and A mean “repulsive” and “attractive.” The Tersoff potential is not a pair potential    because    Bij    is    not    a    constant.    In fact,    it    is the    bond

order for the bond joining atoms i and j, and is a decreasing function of the “coordination” Gij assigned to the bond; i.e., Bij = B(Gij). Gij is defined


Gij     fc (rik )g(@jik)f (rij — rik )j


where fc(r), f (r), and g(6) are empirical functions. The basic idea is that the bond ij is weakened by the presence of other bonds ik involving atom i. The amount of weakening is determined by the location of the other bonds. The angular terms are introduced to help construct a realistic model. Tersoff potential has been calibrated for silicon (Tersoff, 1988b) and carbon (Tersoff, 1988a). In practice, there are two major problems with this potential. First, since the potential involves a large number of parameters to be calibrated, finding a good parameterization for a given material is rather difficult. Second, the calculation of the potential and the associated force is very expensive. However, because simulations based on it can reproduce many important solid material properties, e.g., the lattice constant

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