Pairwise interaction

All of the contributions to the energy function presented above assume that pairwise interactions are sufficient to describe the situation within a molecule or molecular system. Whether or not multi-centered interactions are negligible is controversial. On the other hand, failure or success of a force field with its functional form and corresponding parameter set is not a matter of mathematics... 

The potential energy is written as a series of pairwise interaction terms ... 

Since the solvent molecules, the polymer segments, and the lattice sites are all assumed to be equal in volume, reaction (8.A) impUes constant volume conditions. Under these conditions, AU is needed and what we have called Aw might be better viewed as the contribution to the internal energy of a pairwise interaction AUp jj., where the subscript reminds us that this is the contribution of a single pair formation by reaction A. 

Gmneisen showed that a number of properties of monatomic solids could be conelated if the pairwise interaction between atoms was of die form... 

Figure 2 A particle i interacts mainly with particles that are within the cutoff radius The neighbor list contains only those particles that are within a sphere of radius rj > Particles outside this sphere will not contribute to the force or energy affecting particleThe use of a neighbor list that is periodically updated during the simulation reduces the computer time required in calculating pairwise interactions.

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

The internal (potential) energy is a direct sum of energies, which is normally given as a sum over pairwise interactions (i.e. van der Waals and electrostatic contributions in a force field description). 

The packing energy of an organic crystal can be easily calculated by a lattice sum over pairwise interactions. The potential parameters for these calculations are summarized in Table 15. The packing energy is usually a quite accurate estimate of the crystal sublimation energy. 

The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

Background. Consider that pairwise interactions between active network chains (13), commonly termed trapped entanglements, do not significantly affect the stress in a specimen deformed in simple tension or compression. Then, according to recent theory (16,17), the shear modulus for a network in which all junctions are trifunctional is given by an equation which can be written in the form (13) ... 

According to the equations derived by Miller and Macosko (24), outlined in the Appendix, the probability that a short segment selected at random in the network is part of an elastically active chain is (P /p)2 where p is the final extent of the curing reaction. If an entanglement results from pairwise interactions between chains, as proposed (13), then Te in eq 2 can be equated to (Pxi/p)4, which is the probability that two interacting chains are active. 

---