Pair potential approximation

As it will be explained in section 6, the usual way to evaluate the potential energy of a system simulated by Monte Carlo techniques, makes use of the pair potential approximation (although, as it will also be reviewed, several works have already appeared where nonadditivity corrections to the interaction potential have been included). In the pair potential approximation only two body interactions are taken into account. We will briefly explain here how to apply this approximation for the calculation of the potential energy, to the periodic system just described. The interaction potential energy under the pair potential approximation can be written as ... 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

Keeping only the Eij terms yields the so-called pair-potential approximation. The pair interactions include Coulomb dependence), van der... 

In the pair-potential approximation, the potential energy function is given by... 

In eqn (9.6) we revealed the surface energy for an fee (100) surface in the near-neighbor pair potential approximation. Deduce analogous results for the (110) and (111) fee surfaces. 

A widely used model, which goes beyond the pair potential approximation, is that due to Keating (1966) which has been used mainly for the relaxation of random network models (Keating, 1966). It is in concept a molecular mechanics like potential as it consists of a bond stretching and bond bending term (Gaskell, 1991) ... 

We performed MD simulations of low-quartz, low-cristobalite, coesite and stishovite, i.e., virtually all the natural polymorphs of silica so far known. These polymorphs correspond to different pressure-temperature regimes but can also exist at normal pressure and temperature. Our interests are (1) whether the interatomic potential derived from the cluster calculation is applicable to bulk crystals, (2) whether various physical properties of polymorphs are reproduced by the same interatomic potentials, and (3) whether pair-potential approximation is valid for silica. 

Other approaches for computing the surface tension start from the statistical mechanical expression for the Helmholtz free energy or for the pressure. The Kirkwood-Buff formula for the surface tension of a liquid/vapor interface of an atomic liquid described by the pair potential approximation is ... 

Combining tliis witli the Omstein-Zemike equation, we have two equations and tluee unknowns h(r),c(r) and B(r) for a given pair potential u r). The problem then is to calculate or approximate the bridge fiinctions for which there is no simple general relation, although some progress for particular classes of systems has been made recently. 

While smooth pair potentials are the rule in the literature, surface terms have traditionally been discontinuous the only potential using smooth surface terms seems to appear in Lund et al. [19], where the surface term is a function of a smooth approximation to the number of neighbors of a Ca atom.)... 

Quite recently, Pini et al. [56] have reported a new, thermodynamically self-consistent approximation to the OZ relation for a fluid of spherical particles for a pair potential given by a hard-core repulsion and a Yukawa attractive tail (Eq. (6)). The closure to the OZ equation they have proposed has the form... 

Tlie suffices i and J refer to individual atoms and S and Sj to the species of the atoms involved. The summation over j extends over those neighbors of the atom i for which ry, the separation of atoms i and J, is within the cutoff radii of these potentials. The second term in Equation (la) is the attractive many-body term and both V and are empirically fitted pair potentials. A Justification for the square root form of the many-body function is provided in the framework of a second moment approximation of the density of states to the tight-binding theory incorporating local charge conservation in this framework the potentials represent squares of the hopping integrals (Ackland, et al. 1988). 

Equilibrium Theory of Fluid Structure. In all the theoretical work reported herein, we assume that the particles Interact with pair additive forces whose pair potentials can be approximated by... 

It was assumed that the volume V was reduced by the presence of other particles to the free volume V — Nvf) where N is the number of particles. In arriving at the binding energy effect the mean field approximation was used, which says that the soft (negative) part of the pair potential was sampled in an uncorrelated manner as if the system was an ideal gas. The corresponding free energy per particle in the bulk fluid is... 

Here is the soft and normally attractive part of the pair potential. This simple bilinear form of functional lacks correlation effects except that which is introduced by the truncation of the integral at the onset of the inner hard part of the potential. We are then using an extended form of the mean field approximation as did van der Waals in his original... 

MicroEnv is calculated using standard approximations used in classical molecular mechanics (i.e., pair potentials and analytical expressions for strain energy190) ... 

There are other approximations in the literature that may be more appropriate for particular conditions.12,27,29,35 In many colloidal systems the value of <5 is 2 nm < <5 < 10 nm and until h < 2S the pair potential will be dominated by the other types of interparticle interactions, i.e. it is of short range, albeit very steep. 

Until very recently there was no information about how an MM pair potential should look, based upon calculations from the deeper BO level. In the simplest BO level model for an ionic solution the solvent molecules are represented as hard spheres with centered point dipoles and the ions as hard spheres with centered charges. Now there are two sets of calculations, (16,17) by very different approximation methods, for this model where all of the spheres are 3A in diameter, where the dipole moments are near 1 Debye, and where the ions are singly charged. The temperature is 25° and the solvent concentration is about 50M, corresponding to a liquid state. The dielectric constant of the model solvent is believed to be near 9 6. 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

This inverse-square dependence is chosen because the magnitude of the ssa bond integral in Fig. 3.11 decreases approximately as R 2 in the vicinity of kR 5. We are identifying the attractive contribution to the pair potential with covalent bonding as in eqn (3.46). From eqn (4.2) we have... 

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