Three-body attraction

This work was repeated and extended by Lee, Barkn, and Pound who used his method not only to calculate the surface profile and tension of a Lennard-Jones liquid, also to estimate the dffierence in surface ten n between diat liquid and one of a liquid with a pair potential that accuratdy represents the interaction of argon molecules, and the further difference on adding die three-body attractive forces. In this way they were able to show that the experimental surface tension of argon could be matdied to the best estimate firom an accurate pair potentiid, if proper allowanoe is made for the three-body forces ( 6.4 and Fig. 6.5). 

Three-body attachment of electrons seems to provide a more attractive explanation. Several reactions are worthy of consideration ... 

Sorbie and Murrell have developed a function for triatomic systems based on a many body expansion of PE. It has been applied to repulsive potentials as well and has become an attractive option for fitting ah initio PES. It has sufficient flexibility and extendable to higher polyatomic systems and to two-valued surfaces. The potential for triatomic system is decomposed into one-, two- and three-body terms. The potential is given as... 

Others (e.g., Fukashi Sasaki s upper bound on eigenvalues of 2-RDM [2]). Claude Garrod and Jerome Percus [3] formally wrote the necessary and sufficient A -representability conditions. Hans Kummer [4] provided a generalization to infinite spaces and a nice review. Independently, there were some clever practical attempts to reduce the three-body and four-body problems to a reduced two-body problem without realizing that they were actually touching the variational 2-RDM method Fritz Bopp [5] was very successful for three-electron atoms and Richard Hall and H. Post [6] for three-nucleon nuclei (if assuming a fully attractive nucleon-nucleon potential). 

Three body with Keplerian forces or other short-range attractive forces have singular conhgurations when 2 and 3 bodies collide [14]. 

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... 

The coefBcient ( >2 is expressed as a function of the second and third virial coefficients in Eq. 2 from Ref 4. These coefficients depend on the magnitude of the pairwise and three-body interactions respectively. The value of becomes more negative as the pairwise attractive forces between molecules increase. These pairwise forces dominate the three-body interactions in the second term and the log of the compressibility factor, the third term of this equation. As the ln2, which is proportional to By, becomes more negative, becomes 1 and, being in the denominator ofEq. 1, causes the mole fi-action of the liquid in the SCF, 2. to increase. 

This problem is called the three-body problem by the people who study such things (the theoreticians of quantum mechanics). When you have two particles in motion that attract each other, you can describe the situation with an equation. But when you have three particles, and there are attractions and repulsions, and all these particles are in motion, there are too many things going on for one neat equation. The problem is one of clouds a cloud exists and we can point to it and measure it, but to predict in advance just where it will be and what form it will take is not possible. There are too many factors, too many variables, many of which are unknown or unknowable. This problem is at the heart of the probability approach to atomic structure. 

The dispersion energy is the universal attractive glue that leads to the formation of condensed phases. It is additive at second order in perturbation theory, and the form of the three-body term that arises at third order (the tripledipole dispersion term) is also well known from perturbation theory. This Axilrod-Teller term " was the only addition to the pair potential for argon that was required to quantitatively account for its solid and liquid state properties. This may be grounds for optimism that other nonadditive dispersion terms are negligible. Whether this can be extended to less symmetrical organic molecules and their typical crystalline and liquid environments has not yet been established however. 

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