Lennard-Jones interactions describing potentials between atoms

Figure 7.13 Left interaction potential and force between an atom at the apex of the tip and an atom in the surface. Tip-surface interactions can be described by a summation of these potentials over all combinations of atoms from the tip and the surface. Right interaction potential between the tip, approximated as a sphere, and a plane surface, valid in the non-contact mode of force microscopy. To stress the long-range character of the non-contact potential, the Lennard-Jones interaction potential between two atoms has been included as well (dotted line).

The OPLS model is an example of pair potential where non-bonded interactions are represented through Coulomb and Lennard-Jones terms interacting between sites centred on nuclei (equation (51). Within this model, each atomic nucleus has an interaction site, except CH groups that are treated as united atoms centered on the carbon. It is important to note that no special functions were found to be needed to describe hydrogen bonding and there are no additional interaction sites for lone pairs. Another important point is that standard combining rules are used for the Lennard-Jones interactions such that An = (Ai As )1/2 and Cu = (C Cy)1/2. The A and C parameters may also be expressed in terms of Lennard-Jones o s and e s as A = 4ei Oi and C ... 

June et al. (12) used TST as an alternative method to investigate Xe diffusion in silicalite. Interactions between the zeolite oxygen atoms and the Xe atoms were modeled with a 6-12 Lennard-Jones function, with potential parameters similar to those used in previous MD simulations (11). Simulations were performed with both a rigid and a flexible zeolite lattice, and those that included flexibility of the zeolite framework employed a harmonic term to describe the motion of the zeolite atoms, with a force constant and bond length data taken from previous simulations (26). 

Several other functional forms, besides the Lennard-Jones and Buckingham potentials, have been used to describe the non-bonded interactions. For example, Kihara (1953) used a form in which the repulsive potential becomes infinite at very short distances, about -jfo- that at which the minimum of the function occurs if the distance, at which the potential becomes infinite, is reduced to zero, then the Lennard-Jones form results (Rowlinson, 1965). Kitaigorodskii (1961) derived another function from the Buckingham form. By defining r0 as the distance between the atoms at which U is a minimum, and letting z = r/r0, a = br0, and C72/3 be the value of 17 at r = 2r0/3 he obtained... 

The first term on the right hand side of Eq. (18-2) represents the van der Waals interactions between QM and MM atoms and is described classically using the same Lennard-Jones 6-12 potential used in the classical AMBER force fields ... 

Next, intermolecular potential functions are developed to describe the interactions between the reacting system and a solvent molecule. For aqueous solutions, the potential functions are based on numerous ab initio calculations for complexes of the substrate and a water molecule. The potentials vary with Tj and are represented in our work through Coulomb and Lennard-Jones interactions between sites normally coincident with the atoms. 

In principle a variety of equations could be used to describe van der Waals interactions. The Lennard-Jones 6-12 potential has been used in many molecular mechanics formulations. However, in general, the repulsive part of the 6-12 curve is too steep or hard to describe interactions between atoms in organic molecules over a wide range of distances. Work by Lifson and co-workers showed that a power of 9 or 10 was better than 12 for organic compounds, and such values are sometimes used. The 12 power is often used in protein calculations, not because it is accurate, but because it is fast to compute from the attractive term. 

The simplest simulated system is a Stockmayer fluid structureless particles characterized by dipole-dipole and Lennard-Jones interactions, moving in a box (size L) with periodic botmdary conditions. The results described below were obtained using 400 such particles and in addition a solute atom A which can become an ion of charge q embedded in this solvent. The long range nature of the electrostatic interactions is handled within the effective dielectric environment seheme. In this approach the simulated system is taken to be surrounded by a continuum dielectric environment whose dielectric constant e is to be chosen self consistently with that eomputed from the simulation. Accordingly, the electrostatic potential between any two partieles is supplemented by the image interaction associated with a spherical dielectric boundary of radius (taken equal to L/2) placed so that one of these... 

In our simulations, all potentials between atoms, solid as well as liquid, are described by the standard pair-wise Lennard-Jones 12-6 interactions ... 

Historically, one of the most common methods for describing the van der Waals interaction between atoms ( and J is the Lennard-Jones 12-6 potential in equation (10)... 

Computer experiments particularly use quantum chemical approaches that provide accurate result with intense computational cost. Classical or semiempirical methods on the other hand are able to simulate thousands or up to millions of atoms of a system with pairwise Lennard-Jones (LJ)-type potentials [104-107]. Thus, LJ-type potentials are very accurate for inert gas systems [108], whereas they are unable to describe reactions or they do so by predetermined reactive sites within the molecules of the reactive system [109]. van Duin and coworkers [109-115] developed bond-order-dependent reactive force field technique is called ReaxFF as a solution to the aforementioned problems. Therefore, ReaxFF force field is intended to simulate reactions. They are successfully implemented to study hydrocarbon combustion [112,115,116] that is based on C-H-0 combustion parameters, fuel cell [110,111], metal oxides [117-122], proteins [123,124], phosphates [125,126], and catalyst surface reactions and nanotubes [110-113] based on ReaxFF water parameters [127]. Bond order is the number of chemical bonds between a pair of atoms that depends only on the number and relative positions of other atoms that they interact with [127]. Parameterization of ReaxFFs is achieved using experimental and quantum mechanical data. Therefore, ReaxFF calculations are fairly accurate and robust. The total energy of the molecule is calculated as the combination of bonded and nonbonded interaction energies. 

Rare-gas clusters can be produced easily using supersonic expansion. They are attractive to study theoretically because the interaction potentials are relatively simple and dominated by the van der Waals interactions. The Lennard-Jones pair potential describes the stmctures of the rare-gas clusters well and predicts magic clusters with icosahedral stmctures [139, 140]. The first five icosahedral clusters occur at 13, 55, 147, 309 and 561 atoms and are observed in experiments of Ar, Kr and Xe clusters [1411. Small helium clusters are difficult to produce because of the extremely weak interactions between helium atoms. Due to the large zero-point energy, bulk helium is a quantum fluid and does not solidify under standard pressure. Large helium clusters, which are liquid-like, have been produced and studied by Toennies and coworkers [142]. Recent experiments have provided evidence of... 

The interaction between atoms separated by more than two bonds is described in terms of potentials that represent non-bonded or Van der Waals interaction. A variety of potentials are being used, but all of them correspond to attractive and repulsive components balanced to produce a minimum at an interatomic distance corresponding to the sum of the Van der Waals radii, V b = R — A. The attractive component may be viewed as a dispersive interaction between induced dipoles, A = c/r -. The repulsive component is often modelled in terms of either a Lennard-Jones potential, R = a/rlj2, or Buckingham potential R = aexp(—6r ). 

Xenon has been considered as the diffusing species in simulations of microporous frameworks other than faujasite (10-12, 21). Pickett et al. (10) considered the silicalite framework, the all-silica polymorph of ZSM-5. Once again, the framework was assumed to be rigid and a 6-12 Lennard-Jones potential was used to describe the interactions between Xe and zeolite oxygen atoms and interactions between Xe atoms. The potential parameters were slightly different from those used by Yashonath for migration of Xe in NaY zeolite (13). In total, 32 Xe atoms were distributed randomly over 8 unit cells of silicalite at the beginning of the simulations and calculations were made for a run time of 300 ps at temperatures from 77 to 450 K. At 298 K, the diffusion coefficient was calculated to be 1.86 X 10 9 m2/s. This... 

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 < r < R), where R is the largest radius of the radii of interaction Ry between atoms / and /). 

The last term in the formula (1-196) describes electrostatic and Van der Waals interactions between atoms. In the Amber force field the Van der Waals interactions are approximated by the Lennard-Jones potential with appropriate Atj and force field parameters parametrized for monoatomic systems, i.e. i = j. Mixing rules are applied to obtain parameters for pairs of different atom types. Cornell et al.300 determined the parameters of various Lenard-Jones potentials by extensive Monte Carlo simulations for a number of simple liquids containing all necessary atom types in order to reproduce densities and enthalpies of vaporization of these liquids. Finally, the energy of electrostatic interactions between non-bonded atoms is calculated using a simple classical Coulomb potential with the partial atomic charges qt and q, obtained, e.g. by fitting them to reproduce the electrostatic potential around the molecule. 

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