Pairwise interaction potentials

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. 

In most MC (11,12) and MD (12,13) studies, a small number (N) of particles are placed in a cell of fixed volume (V) and the total interaction potential energy (U ) from all pairwise interaction potentials (U j) between particles i and j is calculated ... 

As a test problem for comparing the various methods described above, we have chosen a heptamer island on the (111) surface of an FCC crystal. Partly, this choice is made because it is relatively easy to visualize the saddle point config-mations and partly because there is great interest in the atomic scale mechanism of island diffusion on surfaces (see for example reference 60). The interaction potential is chosen to be a simple function to make it easy for others to verify and extend om results. The atoms interact via a pairwise additive Morse potential... 

Figure 5. Comparison of two different potential energy cnrves for tetrahedron (a), linear (b) and police-nightstick (c,d) orientations of He2Br2. Solid lines are for the snm of three-body CCSD(T) interaction potential, while dotted hnes correspond to the pairwise atom-atom form. The MP4 ab initio valnes are also indicated by filled circles, whereas potential valnes obtained nsing the snm of three-body MP4 interaction HeBr2 potential are shown by open circles.

CCSD(T) HeBr2 interaction potential, dashed lines correspond to the pairwise atom-atom form, while filled circles indicate the MP4 ab initio values. Open circles are for the potential values obtained using the sum of the three-body MP4 potential for HeBr2, at the specific geometries with the same basis set as in the He2Br2 calculations. Fig.Sa represents the potential energy curves as a function of the distance R between the center of masses of Br2 and He2 in the tetrahedron structure. As can be seen both forms represent well the ab initio data at this configuration. In Fig. 5b an one-... 

Theory, on the other hand, should be accurate to within roughly ten percent as far as the pairwise-additive contributions are concerned. Theory neglects the slightly different interaction potentials of two H2 molecules when one is vibrationally excited [151, 294, 295], which may introduce errors of this magnitude. 

In an exact representation of the interaction between a solute and a solvent, i.e., solvation, the solvent molecules must be explicitly taken into account. That is, the solvent is described on a microscopic level, where the individual solvent molecules are considered explicitly. The interaction potential between solvent molecules and between solvent molecules and the solute can, in principle, be found by solving the electronic Schrodinger equation for a system consisting of all the involved molecules. Typically, in practice, a more empirical approach is followed where the interaction potential is described by parameterized energy functions. These potential energy functions (often referred to as force fields) are typically parameterized as pairwise atom-atom interactions. 

In practice, empirical or semi-empirical interaction potentials are used. These potential energy functions are often parameterized as pairwise additive atom-atom interactions, i.e., Upj(ri,r2,..., r/v) = JT. u ri j), where the sum runs over all atom-atom distances. An all-atom model usually requires a substantial amount of computation. This may be reduced by estimating the electronic energy via a continuum solvation model like the Onsager reaction-field model, discussed in Section 9.1. 

Interaction of the noble gas atom with condensed matter is considerably more complicated and is usually approximate simply by summing or integrating potentials pairwise. Such treatments are necessarily crude nevertheless, they allow an appraisal of the general features of an interaction and often provide realistic numerical values as well. Young and Crowell (1962), for example, review theoretical treatments of noble gas adsorption along these lines predicted potentials for adsorption on various forms of carbon, to consider one example, range from a few hundred calories per mole for He to a few kilocalories per mole for Xe, in reasonable agreement with observed heats of adsorption. 

Extraction of the pairwise interaction potential follows one of two procedures, depending on whether the rods in A and B are parallel (9 = 0) or are skewed. As with the reduction of the Lifshitz result for planar isotropic bodies to give the interaction... 

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... 

The average dissociation time (7") for doublets of equal-size particles, in air at 1 atm and 298 K, has been calculated using Eq. [38]. The overall interaction potential between the constituent particles has been obtained by the integration of the Lennard-Jones 6-12 inter-molecular potential, under the assumption of pairwise additivity. The expression for this overall interaction potential in terms of the Hamaker constant can be found elsewhere (1). All the calculations have been performed for a Hamaker constant of 10 11 erg. The time scale of Brownian motion f-1 of the constituent particles was calculated from... 

Figure 10 One-dimensional potential energy parallel to the surface (a) empty surface with a single particle bound with adsorption energy FjiK (b) superposition of the potential energy in (a) with a pairwise interaction potential of partitles on the surface (/>, ) which nia be either attractive or repulsive...

The semiempirical potential functions which have been used in trajectory calculations fall into two categories those based on pairwise interactions between the atoms, and more general interaction potentials which deal more directly with the three-atom reaction surface. In discussing their characteristics it is customary to concentrate on the surface for the collinear configuration since it is possible to represent this function by a contour diagram of the kind shown in Figure 1.13. 

Therefore, the numerical integration in the LMBW equation can be performed on a much smaller z-domain compared to the BGY equation. This significantly reduces computational load for the equations having two z-dimensions. Moreover, the LMBW equation does not involve the assumption of a pairwise interaction potential inherent in the BGY equation [14], which constitutes a benefit for further extensions to systems with nonadditive potentials. 

Classical interaction potentials for molecular systems can be pairwise or take manybody interactions into account, molecules can be rigid or flexible, and both stretching and torsion motions can be considered [28]. Polarizability can be introduced, but for example for water this does not seem to make the potentials mimic reality unambiguously better. The potentials, also called force fields, are fitted to reproduce certain sets of properties, and can fail to reproduce other characteristics. The potentials suffer from problems with transferability for example a water-sulfuric acid potential developed for the two-component mixture does not necessarily describe the interaction between water and sulfuric acid in a three-component mixture of water, sulfuric acid and ammonia. Also, potentials created for bulk liquid do not always work for surface layers or small clusters. 

Lyubartsev has also developed a multiscale parameterisation method that has been used to systematically build a CG model of a DMPC bilayer. Lyubartsev uses an inverse Monte Carlo method to generate the CG parameters from an underlying atomistic simulation. The atomistic simulation trajectory is analysed to generate the radial distribution functions (RDFs) for the CG bead model. These RDFs can be converted into pairwise interaction potentials between the beads. The... 

The second additivity problem concerns the question whether the interaction potential in Van der Waals trimers or multimers (or molecular solids or liquids) is a sum of pairwise intermolecular (A—B) potentials. This question can be considered for each component of the interaction energy. The (first order) electro-... 

The interaction potential in excited states of the HX-Rg clusters has been divided into a non-pairwisc X- -Rgn component and the remaining pairwise additive interactions... 

One important point we should stress, in conjunction with our current interest, is that similar slow relaxation as liquid water is observed in much simpler model systems The binary mixture of Lennard-Jones liquids, which consist of two species of particles, is now studied extensively as a toy model of glass-forming liquids. It is simulated after careful preparation of simulation conditions to avoid crystallization. Also, the modified Lennard-Jones model glass, in which a many-body interaction potential is added to the standard pairwise Lennard-Jones potential, is also studied as a model system satisfying desired features. 

The concept of an atom-atom potential (Kitaigorodsky, 1973) is based on the idea that the interaction potential between two molecules P and P can be approximated by pairwise additive interactions between the constituent atoms, a P and j8 E P, which, in practice, are nearly always taken to be isotropic, i.e., dependent only on the interatomic distances rap ... 

Consider the excluded volume interaction between spherical particles with effective pairwise interaction potential... 

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