Potential models, pairwise

Perhaps the most important consideration when discussing the development and use of empirical potentials for studying atomic solids is that pairwise potential models are often not very suitable The performance of pairwise potential models can be bad for transition metals and even worse for semiconductors There are a number of reasons why this is so, many of which are due to the fundamental behaviour of pairwise potentials for certain experimental properties. The most oft-quoted properties are as follows ... 

The intennolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44, 50, 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

P(r,i) is the pairwise potential, which, depending upon the model, can be considered tc include electrostatic and repulsive contributions. The second term is a function of th< electron density, and varies with the square root, in keeping with the second-momen approximation. The electron density for an afom includes contributions from the neigh bouring atoms as follows ... 

Finite difference techniques are used to generate molecular dynamics trajectories with continuous potential models, which we will assume to be pairwise additive. The essential idea is that the integration is broken down into many small stages, each separated in time by a fixed time 6t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interactions with other particles. From the force we can determine the accelerations of the particles, which are then combined with the positions and velocities at a time t to calculate the positions and velocities at a time t + 6t. The force is assumed to be constant during the time step. The forces on the particles in their new positions are then determined, leading to new positions and velocities at time t - - 2St, and so on. 

Calculations of forces may be improved in several ways. One is to pursue efforts towards the development of accurate classical, atomic-level force fields. A promising extension along these lines is to add nonadditive polarization effects to the usual pairwise additive description of interatomic interactions. This has been attempted in the past [35-39], but has not brought the expected and long-awaited improvements. This is not so much because polarization effects are not important, or pairwise additive models can account for them accurately in an average sense in all, even highly anisotropic environments. Instead, it seems more likely that the previously developed nonadditive potentials were not sufficiently accurate to offer an enhanced description of those systems in which induction phenomena play a crucial role. 

In the pairwise-potential approximation and the shell model, the lattice energy can be written as... 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

Figure 6.8. Variation with separation distance r of the interaction energy due to van der Waals forces as calculated by the pairwise interaction model, between two different atoms (a) and between two surfaces (b). A, the Hamaker constant, equals n2C p,p2 where C is the constant of the atom-atom pair potential in equation (6.11) (case a) and p 1 and p2 are the number of atoms per unit volume in the...

Recently, detailed molecular pictures of the interfacial structure on the time and distance scales of the ion-crossing event, as well as of ion transfer dynamics, have been provided by Benjamin s molecular dynamics computer simulations [71, 75, 128, 136]. The system studied [71, 75, 136] included 343 water molecules and 108 1,2-dichloroethane molecules, which were separately equilibrated in two liquid slabs, and then brought into contact to form a box about 4 nm long and of cross-section 2.17 nmx2.17 nm. In a previous study [128], the dynamics of ion transfer were studied in a system including 256 polar and 256 nonpolar diatomic molecules. Solvent-solvent and ion-solvent interactions were described with standard potential functions, comprising coulombic and Lennard-Jones 6-12 pairwise potentials for electrostatic and nonbonded interactions, respectively. While in the first study [128] the intramolecular bond vibration of both polar and nonpolar solvent molecules was modeled as a harmonic oscillator, the next studies [71,75,136] used a more advanced model [137] for water and a four-atom model, with a united atom for each of two... 

The simplest model uses atom-atom pairwise potentials, as was done for clusters of molecules with rare gas atoms [45] and with other molecules [46]. The Haas group suggested such models for the anthracene- and perylene ammonia adducts, in which an electrostatic interaction [47] was added to account for the charge distribution in the molecules [40], and also as for the larger donors (dimethylaniline). The potential is written as... 

Fig. 7. Schematic diagrams of the point-charge arrangements in the classic pairwise potential functions. On the left-hand-side and in the centre is shown the 3/3-point (the open cycle represents O and the large solid cycle is H), 3/4-point charge distributions, which are two categories of the 3-point charge models. On the right-hand-side is the 4/5-point (or 4-point) charge model.

The success of various models rests on the correct choice of the pairwise potential energy equation. In this section we will address the potential equations commonly employed for adsorbates used in pore characterization. 

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