A widely adopted model used in many early MD simulations in statistical mechanics is the Lennard — Jones (6-12) potential, which is considered a reason­able description of van der Waals interactions between closed-shell atoms (noble gas elements, Ne, Ar, Kr, and Xe). This model has two parameters that are fixed by fitting to selected experimental data. One should recognize that there is no one single physi­cal property that can determine the entire potential function. Thus, using different data to fix the model parameters of the same potential form can lead to different simulations, making quantitative compar­isons ambiguous. To validate a model, it is best to calculate an observable property not used in the fitting and compare with experiment. This would provide a test of the transferability of the potential, a measure of robustness of the model. In fitting model parameters, one should use different kinds of properties, for example, an equilibrium or ther­modynamic property and a vibrational property to capture the low — and high-frequency responses (the hope is that this would allow a reasonable inter­polation over all frequencies). Since there is consid­erable ambiguity in what is the correct method of fitting potential models, one often has to rely on agreement with experiment as a measure of the goodness of potential. However, this could be mis­leading unless the relevant physics is built into the model.

For a qualitative understanding of MD essentials, it is sufficient to assume that the interatomic

potential U can be represented as the sum of two — body interactions

U (r1;… rN) V (rj) [7]

><J

where ry = |r, — ry I is the separation distance between particles i and j. V is the pairwise additive interaction, a central force potential that is a func­tion of only the scalar separation distance between the two particles, ry. A two-body interaction energy commonly used in atomistic simulations is the Lennard-Jones potential

V (r) = 4e[(s/r)12 — (s/r)6 ] [8]

where e and s are the potential parameters that set the scales for energy and separation distance, respectively. Figure 4 shows the interaction energy rising sharply when the particles are close to each other, showing a minimum at intermediate separation and decaying to zero at large distances. The interatomic force is also sketched in Figure 4. The particles repel each other when they are too close, whereas at large separa­tions they attract. The repulsion can be understood as arising from overlap of the electron clouds, whereas the attraction is due to the interaction between the induced dipole in each atom. The value of 12 for the first exponent in V(r) has no special significance, as the repulsive term could just as well be replaced by an exponential. The value of 6 for the second expo­nent comes from quantum mechanical calculations (the so-called London dispersion force) and therefore

is not arbitrary. Regardless of whether one uses eqn [8] or some other interaction potential, a short-range repulsion is necessary to give the system a certain size or volume (density), without which the particles will collapse onto each other. A long-range attraction is also necessary for cohesion of the system, without which the particles will not stay together as they must in all condensed states of matter. Both are necessary for describing the physical properties of the solids and liquids that we know from everyday experience.

Pair potentials are simple models that capture the repulsive and attractive interactions between atoms. Unfortunately, relatively few materials, among them the noble gases (He, Ne, Ar, etc.) and ionic crystals (e. g., NaCl), can be well described by pair potentials with reasonable accuracy. For most solid engineering materials, pair potentials do a poor job. For example, all pair potentials predict that the two elastic con­stants for cubic crystals, C12 and C44, must be equal to each other, which is certainly not true for most cubic crystals. Therefore, most potential models for engi­neering materials include many-body terms for an improved description of the interatomic interaction. For example, the Stillinger-Weber potential16 for silicon includes a three-body term to stabilize the tetrahedral bond angle in the diamond-cubic struc­ture. A widely used typical potential for metals is the embedded-atom method17 (EAM), in which the many-body effect is introduced in a so-called embed­ding function.