Three-body contributions

The three-body contribution may also be modelled using a term of the form i ( AB,tAc,J Bc) = i A,B,c exp(-Q AB)exp(-/i Ac)exp(-7 Bc) where K, a, j3 and 7 are constants describing the interaction between the atoms A, B and C. Such a functional form has been used in simulations of ion-water systems, where polarisation alone does not exactly model configurations when there are two water molecules close to an ion [Lybrand and Kollman 1985]. The three-body exchange repulsion term is thus only calculated for ion-water-water trimers when the species are close together. 

If two-body potentials and the three-body contribution of Li+(H20)2 are taken into account the optimum coordination number for a static Li+(H20)n complex turns out to be 4. For the most stable conformation of Li+(H20)6 they found that two water molecules are bound in a second, outer hydration layer. 

Although experimental evidence seems to support these results, the fact that the optimum coordination number obtained depended to a large extent on the use of three body contributions warns against uncritical belief in the convergency of the expansion [Eq. (49)] and stresses the preliminary nature of the conclusions. When two-body potentials were considered exclusively, Kollman and Kuntz 219>... 

The coefficients M k) describe the (i + k)-body contribution involving i atoms of species 1 and k atoms of 2. At not too high densities, the virial expansion of spectral moments provides a sound basis for the study of the spectroscopic three-body (and possibly higher) effects. We note that theoretically terms like M 30 gj and M g should be included in the expansion, Eq. 3.9. These correspond to homonuclear three-body contributions which, however, were experimentally shown to be insignificant in the rare gases and are omitted, see p. 58 for details. 

It has been argued that, in the low-density limit, intercollisional interference results from correlations of the dipole moments induced in subsequent collisions (van Kranendonk 1980 Lewis 1980). Consequently, intercollisional interference takes place in times of the order of the mean time between collisions, x. According to what was just stated, intercollisional interference cannot be described in terms of a virial expansion. Nevertheless, in the low-density limit, one may argue that intercollisional interference may be modeled as a sequence of two two-body collisions in this approximation, any irreducible three-body contribution vanishes. 

Contrary to numerical simulation, taking the three-body contribution into account in IETs requires that we define an effective pair potential. Making use of the two- and three-body contributions allows one to write the state-dependent effective potential weff(V) under the standard form [10, 112, 117]. 

Since MD results compare favorably to experiment on the small- / part of c (q), it is possible to affirm without ambiguity that the interaction scheme, which consists of combining the AS two-body potential with the AT three-body contribution, is suitable for studying rare gases fluids. 

Figure 24. Isothermal compressibility for Xc at T — 297, 350, and 420 K (from top to buttom) calculated with the ODS scheme. Two-body potential only (solid lines), and two- plus three-body contribution (crosses), compared to data of Michels et al. [115]. Taken from [129].

Since the polarization wave functions pO°10) defining are purely additive, i.e. hpo°i0) = 3> °i0)(2, 3), the two-body term as defined by Eq. (1-209) is equal to as defined by the SRS theory of two-body interactions89. Thus, to extract pure three-body contribution to E one has to subtract the E f term of the two-body SRS theory89. 

The second-order exchange nonadditivity splits into exchange-induction, exL-ind and exchange-dispersion, h-disP> three-body contributions ... 

It is interesting to note that the analysis of the pair and three-body contributions to the total interaction energy quantitatively explains the relative stability of the ionic and neutral minima of the pentamer. This is illustrated in Table 1-12, where... 

Table 1-12. Two- and three-body contributions (in kcal/mol) to the total interaction energies for the ionic and neutral structures of the (H20)4HC1 pentamer...

AExy describes the pairwise interaction between two monomers, and the AEabc term represents the three-body contribution arising between the relaxed-geometry monomers... 

The effect of the three-body contributions upon the frequency shift of the CO2 antisymmetric stretch (in pE ). 

The latter has been employed so far in the calculation of three-body contributions adding to the reference Hamiltonian three perturbation potentials, one for each pair, and allows, as already mentioned, to decompose the total three-body term into physically meaningful contributions, such as repulsion, polarization and dispersion. 

The essential feature of the ideal, or unperturbed, state resides in that two chain atoms do not interact if their separation along the chain sequence is sufficiently large. This will be expressed by saying that the sum of the binary cluster integral / and of a repulsive three-body contribution is zero at the ideal temperature 7=0 [6], We have [3], in k T units. 

Computer simulation of molecular dynamics is concerned with solving numerically the simultaneous equations of motion for a few hundred atoms or molecules that interact via specified potentials. One thus obtains the coordinates and velocities of the ensemble as a function of time that describe the structure and correlations of the sample. If a model of the induced polarizabilities is adopted, the spectral lineshapes can be obtained, often with certain quantum corrections [425,426]. One primary concern is, of course, to account as accurately as possible for the pairwise interactions so that by carefully comparing the calculated with the measured band shapes, new information concerning the effects of irreducible contributions of inter-molecular potential and cluster polarizabilities can be identified eventually. Pioneering work has pointed out significant effects of irreducible long-range forces of the Axilrod-Teller triple-dipole type [10]. Very recently, on the basis of combined computer simulation and experimental CILS studies, claims have been made that irreducible three-body contributions are observable, for example, in dense krypton [221]. 

J. J. Perez, J. H. R. Clarke, and A. Hinchliffe. Three-body contributions to the dipole polarizability of He3 clusters. Chem. Phys. Lett., 704 583-586 (1984). 

We may also rewrite k, to obtain qualitatively the magnitude of the relative three-body contribution to relaxation. 

In the dense phase the intermolecular potential consists mainly of a two-body term to which small three-body contributions should be added. This problem is poorly documented for molecular systems, and the classic example remains that of argon where an effective two-body Lennard-Jones potential accounts fairly well for the thermodynamic data simply as a result of cancellation of errors. For vibrational energy relaxation one is not directly concerned with the whole intermolecular potential, but rather by its vibrationally dependent part. As mentioned earlier, three-body effects are not usually observable and may be masked by inadequate knowledge of the true potential. Nevertheless one can expect some simply observable solvent effects describable by changes of either the intermolecular or the vibrational potentials. 

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