Interdisciplinary Applied Mathematics

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TABLE 16.1. Summary of the commonly employed intermolecular potentials.





со, r < a,

V(r) = < a <r < Act, 0, r > Act

(J, €


со, r < a,

V(r) =

[-f exp[-*(£-l)], r>cr,

(J, €, Z



V(r) = 4e [(-)12-(-)6l L r t J



V(r) = A exp (—Br) — Щг

A, B, C&


лп 1 V(r) =

47Г€г€о Г

11, 12,Cr


. U e[{°-)w-{°-T]-VLJ{rc), r < rc, V(r) = <

1 0, r >rc

C, е,Гс

and bulk modulus of diamond, it is widely used in atomistic simulation of solids.

16.1.2 Calculation of the Potential Function

The calculation of the potential function and the associated force accounts for most of the computational cost in an atomistic simulation. Therefore, efficient algorithms for the calculation of the potential and force are essential to any atomistic simulation program. Here we give a brief overview of the potential calculation, and the force can usually be calculated in a similar way.    Depending    on    the    nature    of    the    potential,    the    algorithm    for

potential calculation can be quite different. It is useful to divide the potentials into two categories, i.e., those for the short-range interactions and those for the long-range interactions. For the short-range interactions between particles, the potential energy usually decreases to essentially zero for an intermolecular distance of 1 nm to 2 nm or even smaller. Most intermolecular potentials (e.g., the Lennard-Jones potential) belong to this category. For the long-range interactions between particles, the potential energy decreases very slowly and is typically not negligible even at a very large distance, e.g., tens of nanometers. The Coulomb potential belongs to this category.

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