Axilrod-Teller term

The effect of the Axilrod-Teller term (also known as the triple-dipole correction) is to make the interaction energy more negative when three molecules are linear but to weaken it when the molecules form an equilateral triangle. This is because the linear arrangement enhances the correlations of the motions of the electrons, whereas the equilateral arrangement reduces it. 

Three-body and higher terms are sometimes incorporated into solid-state potentials. The Axilrod-Teller term is the most obvious way to achieve this. For systems such as the alkali halides this makes a small contribution to the total energy. Other approaches involve the use of terms equivalent to the harmonic angle-bending terms in valence force fields these have the advantage of simplicity but, as we have already discussed, are only really appropriate for small deviations from the equilibrium bond angle. Nevertheless, it can make a significant difference to the quality of the results in some cases. 

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... 

Non-pairwise additivity. A significant component of the energy V (1,2,3) of three interacting atoms is given by the sum of the pair potentials, V(l,2) + F(l,3) + F(2,3). However, it is now generally accepted that the so-called Axilrod-Teller term, a long-range, irreducible (classical)... 

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. 

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. 

A simpler potential of the form of Eq. (10) has been used by Pearson et al. to model Si and SiC surfaces . The two-body term is of the familiar Lennard-Jones form while the three-body interaction is modeled by an Axilrod-Teller potential . The physical significance of this potential form is restricted to weakly bound systems, although it apparently can be extended to model covalent interactions. 

In spite of a reference to an article by Y. Muto78 in Axilrod s 1951 article78 (see footnote), and the fact that the ddd energy is often referred to as the Axilrod-Teller-Muto term, the author has not been able to consult Muto s article, because the volume and year given appear inconsistent. 

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,... 

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 ... 

Non-additive dispersion interactions, usually treated implicitly in models of polar systems, should be explicitly considered for non polar systems. The first of these contributions is the well known Axilrod-Teller [123] term... 

In the third order of long-range perturbation theory for a system of three atoms A, B and C, the leading nonadditive dispersion term is the Axilrod-Teller-Mutd triple-dipole interaction [58. 59]... 

A notable example of a potential that does include many-body terms is the Barker-Fisher-Watts potential for argon, which combines a pairwise potential with an Axilrod-Teller triple... 

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 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 ... 

These results suggest that three-body terms that are repulsive at the equilibrium geometry must be included to bring the calculated rotational constants in line with experiment. Hutson and coworkers have investigated the effect of several three-body terms on reducing the discrepancies with experiment. Specifically, the inclusion of terms accounting for (i) the anisotropic Axilrod-Teller triple dipole (DDD) interaction, and (ii) the... 

ATM = Axilrod-Teller-Muto term CCD = coupled-clu.s-ter doubles CCSD = coupled-cluster singles and doubles CP = counterpoise (technique) DCBS = dimer-centered basis set HS = Hirschfelder-Silbey theory SAPT = symmetry-adapted perturbation theory SRS = symmetrized Rayleigh-Schrd-dinger series. 

Nonadditivity of dispersion interactions is introduced in DFT-D3 using Axilrod-Teller-Muto three-body terms that are initially repulsive (they have the sign opposite to that of ,y, see Eq. (11.1)). When DFT-D3 was tested [35], these three-body terms were found to (slightly) worsen the performance of most of the density functionals in the S22 benchmark (see Section 11.2.3 for a detailed discussion on related databases and benchmarks). This negative impact of three-body terms was attributed to the phenomenon found previously by Tkatchenko and von Lilienfeld [49] and these terms were initially excluded in DFT-D3. When the S12L database was introduced, however, the importance of the Axilrod-Teller-Muto term was recognized [31, 50]. Apparently, while repulsive three-body terms make the accuracy of DFT-D3 for small near-equifibrium complexes worse, they strongly improve it for supramolecular systems. So the inclusion of three-body terms seems to be profound when supramolecular and condensed systems are of interest. 

Axilrod and Teller investigated the three-body dispersion contribution and showed that the leading term is ... 

The mechanism of dispersion nonadditivity was proposed over 50 years ago by Axilrod and Teller [76] and independently by Muto [77]. It is referred to as the correlation of three instantaneous dipoles. To better appreciate the behavior of this term, let us consider the same two extreme configurations of a trimer as for the TE nonadditivity described earlier. In the equilateral triangle the three monomers cooperate in correlating with each other i.e. when a third monomer gets close, it sees the other two conveniently pre-correlated. In contrast, for the collinear approach of a third monomer this pre-correlation takes place in the wrong direction. Since pair dispersion interaction is attractive, the nonadditivity is repulsive for the equilateral trimer and attractive in the collinear form. 

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... 

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