Pairwise nonadditivity

In this section we will review the symmetry-adapted perturbation theory of pairwise nonadditive interactions in trimers. This theory was formulated in Ref. (302). We will show that pure three-body polarization and exchange components can be explicitly separated out and that the three-body polarization contributions through the third-order of perturbation theory naturally separate into terms describing the pure induction, mixed induction-dispersion, and pure dispersion interactions. 

Korona T, Moszynski R, Jeziorski B (1996) Convergence of symmetryadapted perturbation theory expansions for pairwise nonadditive interatomic interactions. J Chem Phys 105 8178-8186... 

On the other hand, in the condensed phases the concept of supermolecules is not useful because every atom or molecule interacts simultaneously with many neighbors. The many-body nature of the induction process, combined with the statistical mechanics of liquids and a complex local field problem have been serious difficulties for a quantitative description of CILS in dense matter. Furthermore, for an accurate modeling, irreducible (i.e., the pairwise nonadditive) contributions of the intermolecular interaction and induction mechanisms may be significant, which complicate the problem even more. Most treatments of CILS in the dense phases have been undertaken in the time domain, based on correlation functions of the type... 

The interaction energies of clusters of molecules can be decomposed into pair contributions and pairwise-nonadditive contributions. The emphasis of this chapter is on the latter components. Both the historical and current investigations are reviewed. The physical mechanisms responsible for the existence of the many-body forces are described using symmetry-adapted perturbation theory of intermolecular interactions. The role of nonadditive effects in several specific trimers, including some open-shell trimers, is discussed. These effects are also discussed for the condensed phases of argon and water. 

The potentials discussed above are pairwise or two-body potentials (i.e., potentials describing dimers). Yet, in many cases such potentials are fitted to thermodynamics data for liquids and solids. In such media the pairwise nonadditive effects are usually quite important. Therefore, potentials of this type are called effective two-body potentials since they approximate the many-body effects by an unphysical distortion of the two-body potential relative to the exact two-body potential. As a consequence, the effective two-body potentials perform poorly in predicting pure dimer properties such as dimer spectra or second virial coefficients. In fact, the effective two-body potentials perform poorly also in predicting trimer properties (although the three-body component dominates the nonadditive effects, cf. section III.C). 

In order to write the exact binding isotherm for this model in terms of y(l, 1), we define the pairwise nonadditivity of the quadruplet potential of average force in analogy with (3.6.44), as... 

The intennolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44, 50, 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

Calculations of forces may be improved in several ways. One is to pursue efforts towards the development of accurate classical, atomic-level force fields. A promising extension along these lines is to add nonadditive polarization effects to the usual pairwise additive description of interatomic interactions. This has been attempted in the past [35-39], but has not brought the expected and long-awaited improvements. This is not so much because polarization effects are not important, or pairwise additive models can account for them accurately in an average sense in all, even highly anisotropic environments. Instead, it seems more likely that the previously developed nonadditive potentials were not sufficiently accurate to offer an enhanced description of those systems in which induction phenomena play a crucial role. 

Most of statistical-mechanical computer simulations are based upon the assumption of pairwise additivity for the total interaction energy, what means to truncate the right side of equation (48) up to the two-body term. The remaining terms of the series, which are neglected in this approach, are often known as the nonadditive corrections. 

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. 

As long as the two Ar atoms are held in equivalent equatorial positions, the interaction with each of them should, in the pairwise additive approximation, result in the same incremental shifts of the asymmetric stretch of CO2. In reality, a minute nonadditivity of shifts amounting to 0.042 cm-1 was observed by Sperhac et al. when the second Ar atom was added. 

Therefore, the numerical integration in the LMBW equation can be performed on a much smaller z-domain compared to the BGY equation. This significantly reduces computational load for the equations having two z-dimensions. Moreover, the LMBW equation does not involve the assumption of a pairwise interaction potential inherent in the BGY equation [14], which constitutes a benefit for further extensions to systems with nonadditive potentials. 

For the sake of consistency of terminology, triads of molecules in which the central unit acts simultaneously as both proton donor and acceptor will be termed sequential to distinguish such configurations from those in which the central molecule acts as double proton donor or double acceptor. A perhaps more quantitative expression of cooperativity is referred to in the literature as nonadditivity. The latter term is commonly taken as the difference between the total interaction energy of an aggregation of molecules on one hand and the sum of all the pairwise interactions on the other. 

The highest nonadditivity, defined by these authors as the difference between the total interaction energy and that computed by summing pairwise interactions, is observed for the full heptamer, where it amounts to 16% of the total binding energy. Fig. 5.16 illustrates... 

The author also considered the charge shifts occurring as a result of pairwise and higher-order interactions. In comparison to changes in Mulliken charges of the central molecule caused by dimerization which are of the order of 0.005-0.030 e, the (3-body) nonadditivity in the shifts amounts to less than 0.010, more commonly 0.002 four-body nonadditivities are typically less than 0.001. The authors conclude that the additive contributions to the electron redistributions are responsible for the bulk of the nonadditivity in the energy. 

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 ... 

One consequence of using the pairwise additive approximation is that if a true pair potential is used to calculate the properties of a liquid or solid, there will be an error due to the omission of the nonadditive contributions. Conversely, if the pairwise additive approximation is made in deriving the pair potential U b, the latter will have partially absorbed some form of average over the many-body forces present, producing an error in the calculated properties of the gas phase where only two-body interactions are important. Because the effective pair potential Uab cannot correctly model the orientation and distance dependence of the absorbed nonadditive contributions, there will also be errors in transferring the effective potential to other condensed phases with different arrangements of molecules. 

Strictly taken, a prerequisite for the discussion of cooperativity or nonadditivity requires the definition of the additive or noncooperative case [50]. Generally, in the field of intermolecular interaction, the additive model is a model based on the concept of pairwise additive interactions. For atomic clusters per definition, but also for molecular clusters, the use of pairwise additive interactions is almost always used in combination with the assumption of structurally frozen interaction partners. Even in cases of much stronger intermolecular interactions the concept of pair potentials modified to that of effective pair potentials is often used. Most of the molecular dynamics calculations of liquids and molecular solids take advantage of this concept. 

Van der Waals forces are nonadditive and are affected by the presence of other interacting bodies in the vicinity. What this means is that the total interaction among a group of molecules or particles will not be a simple sum of the individual pairwise interactions. In fact, in most cases, a molecule interacting with a second molecule in a group not only will experience the force of interaction directly, but will also feel a reflected force due to the... 

Solids of hydrogen halides allow one to relate the observed trends in vibrational spectra to the differences in their pairwise additive and nonadditive effects. The different hydrogen halide crystals show fairly varying properties. At low temperatures HF, HCl, and HBr have the same type of crystal structure consisting of zig-zag... 

The interaction energy of N molecules is not pairwise additive, i.e. is not the sum of the interactions of all possible pairs of molecules. Among the energy corrections up to the second order, the exchange and, first of all, the induction terms contribute to the nonadditivity. The electrostatic and dispersion (in the second order) contributions are pairwise additive. 

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