Lennard-Jones pairwise potentials

For any finite system there is no problem the results are always finite. The only danger, therefore, is the summation to infinity ( lattice sums ), which always ends with the interaction of a part or whole unit cell with an infinite number of distant cells. Let us take such an example in the simplest case of a single atom per cell. Let us assume that the atoms interact by the Lennard-Jones pairwise potential (p. 284) ... 

All beads, including the endbeads in PFPE Zdol, interact with each other by a pairwise, dispersive, tmncated Lennard-Jones (LJ) potential as follows... 

The molecules are represented as objects exerting forces on their neighbors (see, e.g.. Berry et al., 2000, chapter 10), so that a potential energy of pairwise interaction may be defined. As was mentioned in section 5.1, the simplest form of potential energy function usable with atoms is that between hard spheres. A more realistic function, but one which is still reasonably easy to handle mathematically, is the Lennard-Jones (1924) potential, a common form of which is represented by Equation 5.4 ... 

The box does not show a real gas but rather the simulation of a gas. The molecules interact pairwise via a so-called Lennard - Jones (LJ) potential, i.e. 

A comparison of the pairwise contribution to the Barker-Fisher-Watts potential with the Lennard-Jones potential for argon is shown in Figure 4.38. 

Early experimental spectroscopic investigations on Rg- XY complexes resulted in contradictory information regarding the interactions within them and their preferred geometries. Rovibronic absorption and LIF spectra revealed T-shaped excited- and ground-state configurations, wherein the Rg atom is confined to a plane perpendicular to the X—Y bond [10, 19, 28-30]. While these results were supported by the prediction of T-shaped structures based on pairwise additive Lennard-Jones or Morse atom-atom potentials, they seemed to be at odds with results from microwave spectroscopy experiments that were consistent with linear ground-state geometries [31, 32]. Some attempts were made to justify the contradictory results of the microwave and optical spectroscopic studies, and... 

When the interaction is expressed in terms of pairwise atom-atom potential functions, all three components of the van der Waals interactions are grouped together because of their common r 6 distance dependence. A repulsive term is added, while Coulombic interactions may be accounted for separately. In the expression due to Lennard Jones, the repulsion has an r 12 dependence, to give the pairwise potential function... 

Extension to many dimensions provides insight into more sophisticated aspects of the method and into the nature of molecular interactions. In the second stage of this unit, the students perform molecular dynamics simulations of 3-D van der Waals clusters of 125 atoms (or molecules). The interactions between atoms are modeled using the Lennard-Jones potentials with tabulated parameters. Only pairwise interactions are included in the force field. This potential is physically realistic and permits straightforward programming in the Mathcad environment. The entire program is approximately 50 lines of code, with about half simply setting the initial parameters. Thus the method of calculation is transparent to the student. 

Cut-and-shifted effective potentials describe the interaction between the adsorbed fluid and the zeolite. The Xe-0 and Xe-Na parameters are unchanged from our previous work on Xe in Na/t (55) and Xe-Ar mixtures in Na/f (30). The Kr-0 and Kr-Na potential parameters were obtained as described in Ref. (29). The CH4-0 and CH4-Na potential surfaces were described by the pairwise Lennard-Jones parameters given by Woods and Rowlinson (39). 

In aqueous solution, the intermolecular interactions were assumed to be pairwise additive and described by Lennard-Jones (12-6) potentials with added interactions corresponding to point charges. The solid line in Fig. 10.1.1 shows the potential of mean force w(rc) evaluated in a solution of 250 water molecules at T = 25° C [4]. 

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... 

For many purposes, the Lennard-Jones 12-6 potential is considered to be a satisfactory starting point for establishing the pairwise adsorbate-adsorbent and adsorbate-adsorbate interactions. If the adsorbed molecule has a permanent dipole or quadrupole moment, it is necessary to allow for electrostatic interactions and possibly also additional polarization effects. To obtain the required overall fluid-fluid and fluid-solid potential functions, it is customary to assume that the pairwise interactions are additive. With some systems, however, the solid can be regarded as a continuum, and integration is then possible in place of more laborious summation (Steele, 1974). 

The F[p(r)] functional can be separated into an ideal gas term and contributions from the repulsive and attractive forces between the adsorbed molecules (i.e. the fluid-fluid interactions). Hard-sphere repulsion and pairwise Lennard-Jones 12-6 potential are usually assumed and a mean field treatment is generally applied to the long-range attraction. However, the evaluation of the density profile of an inhomogeneous hard-sphere fluid presents a special problem since its free energy is... 

Successful computer modelling of physisorption is dependent, inter alia, on th j validity of the solid-fluid and fluid-fluid potential functions (Cracknell et al 1995). A rigorous description of the attractive and repulsive components is complex (see Nicholson, 1997) however, as was indicated in Chapter 1, the pairwise inter action between a single adsorbate atom and a solid atom is still usually expressed in the form of the 12-6 Lennard-Jones potential. For our present purpose, we may write... 

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... 

In this section we establish simple model expressions that approximately describe narrow EELS resonances near a critical point in the binary approximation [8]. In this analysis we terminate the correlational series, taking into account only the pairwise contributions (the accuracy of this approximation is increased from the band of moderate densities). We use a Lennard-Jones potential... 

---