Buckingham potential, interaction

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 ). 

A number of techniques have been employed to model the framework structure of silica and zeolites (Catlow Cormack, 1987). Early attempts at calculating the lattice energy of a silicate assumed only electrostatic interactions. These calculations were of limited use since the short-range interactions had been ignored. The short-range terms are generally modelled in terms of the Buckingham potential,... 

Benzene-benzene interactions were modeled with a Buckingham potential that was shown to yield reasonable predictions of the properties of liquid and solid benzene. Benzene-zeolite interactions were modeled by a short-range Lennard-Jones term and a long-range electrostatic term. In total, 16 benzene molecules were simulated in a unit cell of zeolite Y, corresponding to a concentration of 2 molecules per supercage. Calculations ran for 24 ps (after an initial 24-ps equilibration time) for diffusion at 300 K. 

The Lennard-Jones potential is simpler than the Buckingham potential since it has two rather than three parameters. Computations involving the Lennard-Jones potential are also faster as they do not involve any exponential terms. However, with the performance of the computers currently available, the Buckingham potential, which gives a better description of short-range interactions, may be preferred. 

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 van der Waals interactions are repulsive at short and attractive at long distances. The energy minimum is at the sum of the van der Waals radii. The repulsive component arises from overlap of electron clouds and mutual repulsion of the nuclei, the attractive component arises from interactions between dipoles and multipoles. A number of functions have been used to mimic these components but the most popular fall into two groups, the Lennard-Jones potential (shown in Eq. 17.9.1 in the 6-12 form) and the Buckingham potential (Eq. 17.9.2). 

The simplest interaction potential for a pair of atoms i and j separated by rij is the Buckingham potential ... 

The structure and dynamics of the lattice were simulated by using empirical potentials of pair wise interactions. The interaction between ion cores is assumed to be long-range purely Coulombic. The interaction between electron shells has two components a long-range purely Coulombic interaction and a short-range interaction described by the Buckingham potential. 

The next set of terms usually added allow for van der Waals-type interactions between the ions. The most popular form of the potential in solid state modelling is the Buckingham potential A exp —rlp)—Clr but there are other forms in use such as the Lennard-Jones — the general form and potentials derived... 

Here a and d are the number of atoms in the acceptor and the donor, respectively, Ry is the distance between atoms i and j and and are the van der Waals and electrostatic potentials, respectively. The van der Waals potential is often represented by a Lennard-Jones potential (Eq. 8) or by a Buckingham potential (Eq. 9). The parameters a, fi, y and o are obtained from solid-state crystal data. The leading term in the electrostatic potential is the Coulomb interaction (first term in Eq. 10), where D is the effective dielectric constant (usually < D <2). Other terms may be added to represent induced polarization, etc. [40]. The geometries of the two components of the cluster are obtained from microwave or electron diffraction data or from quantum chemical calculations. It is assumed that these geometries do not change upon adduct formation. An initial guess is made for the structure of the adduct, and then the relative positions of the two (or more) components are varied until a local energy minimum is obtained. 

From a historical point of view, rare gases have been fundamental for the development of models. Although the first proposed model turned out not very realistic at a later analysis even for these simple systems, still they provided a framework for many models of everyday usage, such as the LJ or Buckingham potentials. In polyatomic systems, only at very large separations can the interaction be described by multipolar terms located at the center of the distributions. At short to medium distances, a most important range for condensed phases, multipolar multicenter expansion are used, whereby the centers may be located at the position of the nuclei or not. 

Buckingham potential function - molecular interaction fields (O steric interaction fields)... 

A steric interaction field is obtained by calculating the van der Waals interaction energy Evdw between probe and target in each grid point [Kim, 1992b]. Different potential energy functions were proposed to model van der Waals interactions between atoms. The most common are Lennard-Jones potential, Buckingham potential, and Hill potential [Leach, 1996]. 

Configurationally biased Monte Carlo techniques [63-65] have made it possible to compute adsorption isotherms for linear and branched hydrocarbons in the micropores of a siliceous zeolite framework. Apart from Monte Carlo techniques, docking techniques [69] have also been implemented in some available computer codes. Docking techniques are convenient techniques that determine, by simulated annealing and subsequent freezing techniques, local energy minima of adsorbed molecules based on Lennard-Jones-or Buckingham-type interaction potentials. 

Note that this contribution to overall energy does not include other through-space effects such as van der Waals interactions. To take account of these effects, interactions between atoms which are separated from each other by greater than 1,4 distances are usually split into van der Waals and electrostatic components. There are many ways of describing van der Waals interactions the most common methods employ either the 6-12 (Lennard-Jones) potential or the Buckingham potential as shown below ... 

Typical analytical representations for nonbonded interactions in Eq. (2) are Leimard-Jones (n=12, w=6) or 9-6 (where =9 and m=6). Alternatively, the nonbonded interactions have been represented by the Buckingham potential ... 

Lennard-Jones potential As two atoms approach one another there is the attraction due to London dispersion forces and eventually a van der Waals repulsion as the interatomic distance r gets smaller than the equilibrium distance. A well-known potential energy function to describe this behavior is the Lennard-Jones (6-12) potential (LJ). The LJ (6-12) potential represents the attractive part as r-6-dependent whereas the repulsive part is represented by an r n term. Another often used nonbonded interaction potential is the Buckingham potential which uses a similar distance dependence for the attractive part as the LJ (6-12) potential but where the repulsive part is represented by an exponential function. 

A Buckingham potential replaces the twelfth power term with an exponential, which is a better theoretical description of the repulsion expected between electron clouds. In both MM2 and MM3, an exponential-6 equation is used. This is a modified Hill equation, which is a particular formulation that contains only two adjustable parameters for the interaction between any two atoms. Equation [9] is for MM3. For MM2 the exponential part was slightly harder, with 12.50 instead of 12.00 in the exponent. 

Vessal et al. (1989, 1993) have used a four-range Buckingham potential to model the short range interactions between different ions. The different components of the potential are as follows ... 

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