Three-body interaction energies

The essential idea of the Alexander model, a global balance of interaction and stretching energies, can be applied to other situations involving tethered chains besides the good solvent case. In theta or poor solvents, the interaction term must be modified to account for poorer solvent quality. A simple limit is precisely at the theta point [29, 30] where binary interactions effectively vanish (% = 1/2 or v = 0). The leading term in Fim now accounts for three-body interactions ... 

This additional to the two-body interaction energies term originates from three-body interactions and is called the three-body interaction energy. 

A well-known prototype of three-body interaction is the helium trimer [25-28]. The three-body interaction energy is discussed in detail in the above-mentioned works. It is interesting that the results, certainly due to the different approaches used, are not uniform. As demonstrated this by one example, the three-body term is found to be negative at the correlated level for a linear He-trimer (in ref.28.) while other authors (ref 27.) obtained a positive value for the same term. 

Accordingly to most of the results found in earlier works, the three-body interaction energy terms at the correlated level are small. 

Table VI. Charge transfer contributions to the three-body interaction energy at the MP3 and MP4 correlated levels for the studied He-clusters (in jiH, see Figure I. for notations, numbering of atoms are given from left to right))...

A simple and accurate way of truncation is to approximate the effective three-body interaction Iphh(E) (Fig. 19 e) by the sum of the effective two-body interactions Ihh(E) and Fh(E) (Figs. 19 f-h). The true three-body part Iphh(E) only represents a small higher order correction (see e.g. Fig. 181) and may be neglected. The second-order relaxation self-energy (Fig. 18 b),... 

Table 1-10. Decomposition of the pair and three-body interaction energies (in kcal/mol) for the structures corresponding to the global minima of small water clusters...

Hydrogen fluoride trimer has also been used as an appropriate model system for the discussion of three-body forces 64>65). The data reported on (HF)3 indicate a stabilizing effect caused by the three-body interactions. The energy per hydrogen... 

The total of all three-body interactions is -3.8 kcal/mol, as compared to -19.2 kcal/mol for the sum of all two-body interactions. When added together, the total of all pairwise and three-body interactions comes within 0.4 kcal/mol of the total interaction energy of -23.4 kcal/mol in the hexamer. With respect to the individual components, there is very little nonadditivity in ES or EX. The total nonadditivity of some 4 kcal/mol is approximately equally divided between POL and CT. 

The question of cooperativity was further probed by computation of the three-body interaction energy, AEj . This quantity is defined as the difference between the total interaction energy in the entire trimer, and three times the two-body interaction, AE2 (one for each pair of subunits), all computed as This property differs from the prior means of con-... 

Table 5.8 Two and three-body interaction energies (AE, in kcal/mol) in the trimers of HF and HCl, computed with -l-VP basis set .

The two- and three-body interaction energies in the water hexamer were decomposed via the Morokuma procedure, without counterpoise correction, and some of the results are listed in Table 5.13. Beginning with the two-body terms, the results for all adjacent molecules are identical to the data for the 1-2 pair in the first row of the table. This similarity is explained by the fact that all adjacent pairs constitute a single H-bond the concept of dou-... 

Figure 5.20 Anisotropy of two and three-body interaction energies in the water trimer, at SCF and MP2 levels . See Fig. 5.19 for definition of a.

It is instructive to compare the SCF three-body interaction, represented by the solid curve in Fig. 5.21, with the induction energy, with which there is a tendency to approximate it in the literature. In this context, it should be noted that the induction curve is far too attractive, by a factor of more than 2 in the vicinity of 20°. Other characteristics of its shape differ from the full SCF curve or deformation energy in Fig. 5.21 as well. The three-body forces at the correlated MP2 level are very small in magnitude, and insensitive to angular characteristics of the trimer. Most of these conclusions have been verified by later calculations and by symmetry-adapted perturbation theory calculations, although there were a number of discepancies as welF. The issue is not entirely closed. 

In addition to the closed cyclic trimer, the open trimers displayed in Fig. 5.22 in which the central molecule acts as (a) simultaneous donor-acceptor, d-a, (b) double donor, d-d, and (c) double acceptor, a-a were considered as well . In the first case, one would expect positive cooperativity, which is confirmed by an attractive total three-body interaction energy equal to about 10% of the two-body contribution. Most of this three-body term is due to the SCF-deformalion, as in the cyclic structure. The double-donor is bound by only about 1/10... 

AE j yn) at various levels. The data clearly indicate the cooperativity as the binding energy rises from 1.9 kcal/mol (at the SCF level) for the dimer up toward 5.4 as the number of molecules reaches six. (However, the data for the dimer may be misleading as the complex is not cyclic and so contains only a single H-bond.) This study concludes that two and three-body interactions can provide most of the total interaction. Correlation is recommended, but the MP2 level appears satisfactory. 

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