Interdisciplinary Applied Mathematics

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FIGURE 16.2. The variation of the Lennard-Jones potential (in units of e) and the corresponding    force    (in    units    of    e/a)    as    a    function    of    the    intermolecular


distance.


interact via a cut and shifted Lennard-Jones interatomic potential function defined by


V (r)



4e [(y)12-(y)6] -VLJ(rc),



Г < Гс,



(16.8)



0,



Г > Гс,


where rc (= a/21/6) is the truncation distance, VLJ(rc) is the value of the Lennard-Jones potential at the point of truncation, and a and are the Lennard-Jones distance and energy parameters. We note that in this case, the interactions between the atoms are purely repulsive. Therefore, this potential is used in those cases in which we want the atoms to purely repel each other.


5. Buckingham Potential. The Buckingham potential is given by


Cr


V(r) = Aexp(-Br)—g-,    (16.9)


where A, B, and C6 are empirical constants. The major difference from the Lennard-Jones potential is that the repulsion term now has an exponential dependence on the distance, which is shown to be more realistic than the Lennard-Jones potential (Born and Mayer, 1932). However, compared to the Lennard-Jones potential, the Buckingham potential is much more expensive to evaluate.


6. Coulomb Potential. The Coulomb potential accounts for the electrostatic interactions between particles when charges are present, e.g., for ions or polyatomic molecules with partial charges. The Coulomb potential between two particles is given by


V (r)



1 Я1Я2 4ncr cq r ’



(16.10)


where qi, q2 are the charges of the two particles, and eQ is the vacuum permittivity.


Table 16.1 gives a summary of the pairwise (i.e., two-body) intermolecular interaction potential schemes that are commonly used in MD simulation.

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