Pair potentials Lennard-Jones introduced

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

A simple model, which has been quite successful in solids with the diamond or zinc-blende stmcrnre, was introduced by Stillinger and Weber (Stillinger and Weber, 1985). The first term in the potential is the product of a Lennard-Jones-like pair-wise interaction and a cut-off function smoothly terminating the potential at some distance r. The second term is a multi-variable three-body potential written as a separable product of two radial functions and an angular function ... 

In 1924, Lennard-Jones [16] introduced a model for a soft-sphere pair potential of the form... 

In condensed phases such as liquids, two molecules are not isolated but have many other molecules in their vicinity. If these two molecules are dissolved in another solvent medium then the situation may be very complicated, and only by introducing simplifying assumptions is it possible to construct a simple theory of dense media and a solution using Lennard-Jones or similar total intermolecular pair potentials. Some important facts, which should be taken into account before applying any simplifying assumptions, are given below ... 

In Section 1.7, we stressed the fact that any pair potential ever used in the theory of simple fluids has actually been a model pair potential for real particles. Alternatively, we may adopt the point of view that we are developing a theory for model particles, interacting via an exact pair potential. This is the way we introduced the idea of Lennard-Jones particles, which, of course, are not real particles. 

Two general forms have been used for the pair potential v. The first was introduced by Walmsley and Pople (1964) in their treatment of the <1 = 0 lattice frequencies of solid COj. It consists of a 6-12 Lennard-Jones term (between molecular centers) and an orientation-dependent term in the form of a quadrupole-quadrupole interaetion. The seeond form, which has found wide application, consists of a sum over atom-atom interactions, summed over the nonbonded atoms of the two molecules. This type of pair potential function was developed for organic molecules and was used to account for the crystal structures of these systems. Kitaigorodskii (1966) determined the parameters for such potentials in this way. Dows (1962) first applied such a potential to the calculation of the librational lattice modes frequencies of solid ethylene using hydrogen-hydrogen repulsion terms as given by de Boer (1942). The usual form of this type of potential contains 6-exponential atom-atom terms ... 

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