Three-body potential

The presence of tln-ee-body interactions in the total potential energy leads to an additional temi in the internal energy and virial pressure involving the three-body potential / 2, r, and the corresponding tlnee-... 

Table V. The binding energy for AI7 to AI15 based on the modified three-body potential...

The self-diffusion coefficient calculated for the three body potential is Z) = 1.3 X lO cm /sec. This is to be compared with the experimental value of 2.3 x 10" cmVsec and to the value of 2.25 x 10 cm /sec for the two-body liquid. It could be said that the three-body liquid shows more rigidity in some sense than the two-body liquid. 

The aim of the present study is to investigate the validity of the pairwise additivity of two-body and three-body potentials for He2Br2. These results are compared with ah initio calculations" and a simple model of the three-body potential is proposed to determine well depths and equilibrium structures for different isomeric configurations of the complex, as well as the minimum energy pathways through them. Additionally, variational methods are used to calculate the vibrational states of He2Br2. The wavefunctions of the lower states are analyzed in terms of probability distributions of the internal coordinates and the zero-point energy of the vdW cluster is evaluated. 

Figure 22. Excess internal energy, Eex/N, and virial pressure, PP/p, calculated with the ODS integral equation versus the reduced densities p = pa3, along the isotherms T = 297.6, 350 and 420 K (from bottom to top), by using the two-body potential alone (dotted lines) and the two- plus three-body potentials (solid lines). The experimental data (open circles) are those of Michels et al. [115], Taken from Ref. [129].

The structure of this complex is shown in Fig. 9. Ab initio calculations of the three-body potential determined the relative importance of each term. They showed [78,79] that the total three-body interaction is very anisotropic with respect to the in-plane and out-of-plane rotations of HC1 within the cluster (see Fig. 10). It is instructive to have a closer look at the composition of this three-body term. 

The first analytical three-body potential for A HCl was proposed by Hutson and collaborators [75] on the basis of semiempirical considerations. It consisted of three terms exchange, induction, and dispersion nonadditivities. The dispersion nonadditivity was represented by Eq.(19), and the induction nonadditivity as an interaction of multipoles induced... 

A simple model, which has been quite successful in solids with the diamond or zinc-blende stmcrnre, was introduced by Stillinger and Weber (Stillinger and Weber, 1985). The first term in the potential is the product of a Lennard-Jones-like pair-wise interaction and a cut-off function smoothly terminating the potential at some distance r. The second term is a multi-variable three-body potential written as a separable product of two radial functions and an angular function ... 

Figure 10. The potential energies for the three two-hody potentials are shown in this figure. The three-body potential is a minor perturbation. Note that the CN potential is much deeper and narrower than the other two potentials. This DOF is often decoupled from the other two dynamical DOFs.

So far the method has only been fully tested for one-electron atoms. In the case of hydrogen a complete check is available for a very restricted subset of angular-momentum states, namely LS-coupled collision states with / = X = 0. This is the Temkin—Poet problem (Temkin, 1962 Poet, 1978, 1981). The three-body potentials are separable in the radial coordinates. This enables a convergent numerical solution to be obtained. 

So far, this method has been successfully applied to the development of effective potentials for a series of cations in water [129-132]. In this case, the radius of the spherical cavity that encloses the cation has been fixed so that the three-body potential for the complex cation-two water molecules given by the PCM, can be decomposed into the sum of two-body potentials, also obtained with the PCM... 

The importance of many-body interactions for accurate prediction of phase diagrams of rare gases has been demonstrated.87 A study on liquid mercury that uses an ab initio pair potential and a semiempirical many-body correction to produce an effective three-body potential that was able to predict the liquid-vapour phase diagram well,110 suggests that the main deficiency of the classical pair potential is the neglect of many-body effects. 

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