Buckingham potentials

The Buckingham potential can be thought of as a generalized Bom-Mayer potential with an exponent other than 1 assigned to the attractive potential. One form is 

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If computing time does not play the major role that it did in the early 1980s, the [12-6] Lennard-Jones potential is substituted by a variety of alternatives meant to represent the real situation much better. MM3 and MM4 use a so-called Buckingham potential (Eq. (28)), where the repulsive part is substituted by an exponential function ... 

Several formulations in which the term in the standard Lennard-Jones formulation is replaced by a theoretically more realistic exponential expression have been proposed. These include the Buckingham potential ... 

Fig. 4.36 A drawback of the Buckingham potential is that it becomes steeply attractive at short distances.

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 ). 

Another short-range force that occurs in a solid is the weak van der Waals attraction between electron orbitals. There are a number of expressions for the short-range potential that takes both these factors into account. One commonly used expression is the Buckingham potential ... 

A second and repulsive energy term must be introduced to take account of the electron-electron repulsion that arises at very short interatomic distances. Several models are used to describe this repulsive term. Often used is the Buckingham potential, which, however, includes both attractive and repulsive components ... 

Sometimes the Buckingham potential is used, rather than the Lennard-Jones ... 

At short range, a plot of In V versus r will be a straight line with a slope (equal to a) and an intercept (equal to A) and thus the value of A and a for Buckingham potential can be determined. The point where the potential V becomes zero would fix the value of B (when n = 6 or any other fixed value). The function can be made more flexible by modifying as... 

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. 

McKean 182> considered the matrix shifts and lattice contributions from a classical electrostatic point of view, using a multipole expansion of the electrostatic energy to represent the vibrating molecule and applied this to the XY4 molecules trapped in noble-gas matrices. Mann and Horrocks 183) discussed the environmental effects on the IR frequencies of polyatomic molecules, using the Buckingham potential 184>, and applied it to HCN in various liquid solvents. Decius, 8S) analyzed the problem of dipolar vibrational coupling in crystals composed of molecules or molecular ions, and applied the derived theory to anisotropic Bravais lattices the case of calcite (which introduces extra complications) is treated separately. Freedman, Shalom and Kimel, 86) discussed the problem of the rotation-translation levels of a tetrahedral molecule in an octahedral cell. 

The repulsion increases exponentially, and it is steeper than the bond length deformation potential. The attractive force is usually modeled by a 1/r6 term, while various possibilities exist for the repulsion. The functions used in modem programs include, apart from the Morse potential (Eq. 2.13), the Lennard-Jones potential (Eq. 2.25)[401 (e.g., AMBER1411), the Buckingham potential (Eq. 2.26)[421 (e.g., MOMEC[81), or a modification thereof, the Hill potential (Eq. 2.27) (e.g., MM2, MM3[1,2,231). 

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... 

Fig. 9. Potential function [U(r), which is the same as Ubb] for the amide hydrogen bond, with e = 5-5 kcal mole"1. Curve A is the analogue of eq. 13, using a Buckingham potential with repulsive part Arep and attractive part Aatt Curve B is eq. 13. In both cases, S(r) is added to either the Buckingham or Lennard-Jones terms to obtain U(r) (Poland and Scheraga, 1967).

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). 

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