Axilrod-Teller three body

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 ... 

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

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. 

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. 

B. M. Axilrod and E. Teller, "Interaction of the van der Waals type between three atoms," J. Chem. Phys., 11, 299-300 (1943) see also the pedagogical article by C. Farina, F. C. Santos, and A. C. Tort, "A simple way of understanding the non-additivity of van der Waals dispersion forces," Am. J. Phys., 67, 344-9 (1999) for the step from two-body to three-body interactions. 

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. 

Alternative, three-body functions are provided by the triple-dipole formalism of Axilrod and Teller, which takes the form ... 

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]. 

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. 

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. 

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)... 

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... 

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. 

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