Dispersion energy, Axilrod-Teller

TE mechanism (p. 734) polarization catastrophe (p. 738) three-body polarization amplifier (p. 738) Axilrod-Teller dispersion energy (p. 741) van der Waals radius (p. 742) van der Waals surface (p. 742) supramolecular chemistry (p. 744) hydrogen bond (p. 746)... 

Bc3 cluster the 3-body forces cannot be approximated solely by the Axilrod-Teller term. The reasons for the satisfactory approximation of many-body energy by the Axilrod-Teller term in the bulk phases of the rare gases were discussed by Meath and Aziz . As follows from precise calculations of the 3-body interaction energy in the Hcg , Neg and Ara trimers, both the Axilrod-Teller and the exchange energies are important. Nevertheless, in some studies of many-body interactions, the exchange effects are still neglected and the many-body contribution is approximated by only dispersion terms, for example see... 

We wish to end this section by saying that similarly as in the two-body case, nonadditive induction, induction-dispersion, and dispersion terms have well defined asymptotic behaviors from the multipole expansions of the intermolecular interaction operators. For instance, the leading term in the multipole expansion of the three-body dispersion energy for three atoms in a triangular geometry is given by the famous Axilrod-Teller-Muto formula311,312,... 

For multi-molecular assemblies one has to consider whether the total interaction energy can be written as the sum of pairwise interactions. The first-order electrostatic interaction is exactly pairwise additive, the dispersion only up to second order (in third order a generally small three-body Axilrod-Teller term appears [73]) while the induction is not at all pairwise it is non-linearly additive due to the interference of electric fields from different sources. Moreover, for polar systems the inducing fields are strong enough to change the molecular wave functions significantly. 

Most of the potential energy surfaces reviewed so far have been based on effective pair potentials. It is assumed that the parameterization is such as to account for nonadditive interactions, but in a nonexplicit way. A simple example is the use of a charge distribution with a dipole moment of 2.ID in the ST2 model. However, it is well known that there are significant non-pairwise additive interactions in liquid water and several attempts have been made to include them explicitly in simulations. Nonadditivity can arise in several ways. We have already discussed induced dipole interactions, which are a consequence of the permanent diple moment and polarizability of the molecules. A second type of nonadditive interaction arises from the deformation of the molecules in a condensed phase. Some contributions from such terms are implicitly included in calculations based on flexible molecule potentials. Other contributions arises from electron correlation, exchange, and similar effects. A good example is the Axilrod-Teller three-body dispersion interaction ... 

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. 

The 3-body electron correlation energy, AE3orr(A3), at large distances can be represented as the Axilrod-Teller 3-body dispersion energy [40]... 

One of the third-order energy terms represents a correction to the dispersion energy. The correction, as shown by Axilrod and Teller, has a three-body character. The part connected to the interaction of three distant instantaneous dipoles on A. B and C reads as... 

The dispersion interaction in the third-order perturbation theory contributes to the three-body non-additivity and is called the Axilrod-Teller energy. The term represents a correlation effect. Note that the effect is negative for three bodies in a linear configuration. 

The other non-additive effect which has attracted most attention is the Axilrod-Teller triple-dipole dispersion energy, which for three atoms or rotationally-averaged molecules A, B and C is[30]... 

The trouble is that, without saying so, we have assumed that only molecule A feels the effects of other molecules. But we know better. Each molecule polarizes its neighbor, if only a little bit, and that affects how the neighbor molecule will then interact with the next one. The Axilrod-Teller correction estimates the effect of this indirect interaction by applying third-order perturbation theory to the dispersion energy of three spherical molecules. In Section 10.1 we see how second-order perturbation theory predicts the attraction between two polarizable particles. It takes third-order terms to treat the effect of a third particle, and the derivation is lengthy, so let s just jump to the result this time ... 

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