Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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properties of the material. The amount of published work with Leonard-Jones systems since the creation of MD is unsurpassed by any other potential, and only for that reason, not to mention others, its importance cannot be overemphasized. However, it can be ill-advised to apply this potential for the quantitative study of more complex materials.

Figure 6.5: Schematic representation of pair potentials with unit depth of the potential well : Lenard-Jones 6-12 with corresponding force (solid black line), Morse for two parameter values α M which define the width of the potential well. These curves illustrate the atomic hypothesis that „all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another” [18]. in (6.5), which dominates the interatomic repulsion, is selected for application convenience: as far as the physics is concerned, the exponential term is a more appropriate choice. The simplest potential that takes this into account is the Morse potential: φ ( ¯ r i j ) = ε M exp [ − 2 α M ( ¯ r i j − 1 )] − 2exp [ − α M ( ¯ r i j − 1 )] The adoptable parameter α M defines the slope of the repulsive wall (that is, the width of the potential well; Figure 6.5). The parameters ε M , r 0 and α M are, as in the previous case, usually determined by fitting the material properties that are most relevant to the problem being analyzed. The Morse potential was originally developed for covalent bonds that are strongly spatially oriented, so the interatomic distance is not sufficient for a realistic description of the interaction [22]. Although the inverse-power form of the interatomic potential (6.5) is most often used in heuristic studies, the results of quantum mechanical calculations, as already mentioned, favor the exponential form, which is later used in some particle models (6.26). If electrostatic charges are present, it is necessary to add the appropriate Coulomb potentials φ ( r i j ) = q i q j 4 π ¯ ε p r i j where q i and q j are charges, and ¯ ε p the permittivity. The term ¯ r − 12 i j