Buckingham term

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 ... 

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

Applications of the Buckingham FI theorem results in the formulation of dimensionless terms called FI ratios. These FI ratios have no relation to the number 3.1416. 

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

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 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 ... 

Vvdw may be represented by a potential of Lennard-Jones form (as in the last term of eq. (11.3)). A common alternative is the Buckingham form ... 

Evaluating this integral and writing the result in terms of the flow characteristic gives the well-known Buckingham equation ... 

The different r " terms in the modified function are not linearly independent and, therefore, inclusion of additional terms does not guarantee the improvement in the fit. Buckingham function also suffers from the fact that as r —> 0,... 

The first three terms, stretch, bend and torsion, are common to most force fields although their explicit form may vary. The nonbonded terms may be further divided into contributions from Van der Waals (VdW), electrostatic and hydrogen-bond interactions. Most force fields include potential functions for the first two interaction types (Lennard-Jones type or Buckingham type functions for VdW interactions and charge-charge or dipole-dipole terms for the electrostatic interactions). Explicit hydrogen-bond functions are less common and such interactions are often modeled by the VdW expression with special parameters for the atoms which participate in the hydrogen bond (see below). 

The potential outside a charge distribution can be expressed in terms of a finite series of the outer moments of the distribution. The expression is obtained through a power series expansion of r1, where r is the distance from the field point to the origin of the distribution, and subsequent integration (Hirshfelder et al. 1954, Buckingham 1978). At a point rf, with components ra, for unit value of 47te0, one obtains... 

For a slowly varying charge distribution, the potential can be expanded in a Taylor series with (0), the potential at the origin of the distribution B, as leading term (Buckingham 1959, 1970, 1978 Jackson 1974) ... 

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,... 

The pi theorem is often associated with the name of Buckingham (4), because he introduced this term in 1914, but the proof of it was accomplished in the course of a mathematical analysis of partial differential equations by Federmann in 1911 see Ref. 5, Chap. 1.1, A Brief Historical Survey. 

In the first derivation of MCD equations by Stephens and Buckingham the perturbed polarizability was expanded in a SOS fashion yielding A, B, and C term parameters (9). We have performed a similar analysis but with an expansion in terms of transition densities (51). In this analysis it was shown that aj1)y can be divided into contributions indexed by two occupied orbitals, by two virtual orbitals, or by one occupied and one virtual orbitals (51). Of these contributions the occ-occ and vir-vir are expected to be larger than the occ-vir terms. By eliminating the occ-vir terms a simpler equation for aafj1)r is obtained that follows the spirit of the approximation to Eq. (64) mentioned in Section II.C.2. It was also shown that this approximate equation can be converted into the form of SOS-type equations that include term parameters (51). 

The assumptions upon which these derivations were based were sufficiently general to assure the utility of these dimensionless groups in both laminar- and turbulent-flow problems. For the former case the results must be the same as are obtainable from the Buckingham relationship, Eq. (6). Perkins and Glick (P2) accordingly rearranged Eq. (6) in terms of these dimensionless groups to obtain... 

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 latter term is a neologism that has received limited acceptance — see D. A. Buckingham and A. M. Sargcson,... 

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