The Pairwise Additive Approximation

When one is calculating the lattice energy of a molecular crystal, or the potential energy of a liquid, or indeed the energy of any ensemble of N molecules relative to their energy when completely separated, it is usual to assume that the energy is equal to the sum of the interactions between every pair of molecules in the ensemble (the pairwise additive approximation)-. 

However, this energy should be expressed formally as an expansion 

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In the C °o limit, all the sites are bound the average correlation g(C is determined by the mth-order correlation function, which is 5 for the cyclic and 5 for the open linear system. This is true within the pairwise additive approximation for direct interaction, and neglecting long-range correlations. 

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. 

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. 

In many cases, it is reasonable to expect that the sum of two-body interactions will be much greater than the sum of the three-body terms which in turn will be greater than the sum of the four-body terms and so on. Retaining only the two-body terms in equation IAI. 5.3) is called the pairwise additivity approximation. This approximation is quite good so the bulk of our attention can be focused on describing the two-body interactions. However, it is now known that the many-body terms cannot be neglected altogether, and they are considered briefly in section A 1.5.2.6 and section A 1.5.3.5. 

The model is a McMillan-Mayer (MM)-level Hamiltonian model. Friedman characterizes models of this type as follows With MM-models it is interesting to see whether one can get a model that economically and elegantly agrees with all of the relevant experimental data for a given system success would mean that we can understand all of the observations in terms of solvent-averaged forces between the ions. However, it must be noted that there is no reason to expect the MM potential function to be nearly pairwise additive. There is an upper Imund on the ion concentration range within which it is sensible to compare the model with data for real systems if the pairwise addition approximation is made. ... 

Another major limitation is the pairwise additive approximation, which is introduced to decrease the computational demand. In this approximation, the interaction energy between one atom and the rest of the atoms is calculated as a sum of pairwise (one atom to one atom) interactions thus certain polarization effects are not explicitly included in the force field (Stote et al. 1999). This can lead to subtle differences between calculated and experimental results. 

Computational constraints impose spatial and temporal limitations on simulated systems. The number of atoms considered is typically in the 10 -10 range. The corresponding cross-sectional length of the interface varies between 2 and 4 nm and each lamella is 2 to 5 nm wide. For the aqueous phase, this is equivalent to approximately 7-18 water diameters. The spatial extent of the system is primarily limited by the rapidly growing number of intermolecular interactions. In the pairwise additive approximation, this number is N x (N — l)/2, where N is the number of atoms in the system. In practice, pair interactions of an atom with other atoms are usually truncated spherically. The largest possible truncation distance is half the shortest box edge. 

Although the pairwise-additive approximation of equations (34) and (35) is adequate for many purposes (and in good correspondence with empirical steric concepts), it should be emphasized that exchange repulsions are inherently a collective response of the entire N-electron system, rather than a set of pairwise changes (equation 35) that can be treated as independent and additive. 

The effective force felt between pairs of molecules in a condensed phase such as a liquid is influenced by the presence of nearby molecules. A simple example of this arises due to molecular polarization one molecule could polarize another, whose interaction with a third molecule is then altered. In such a case the forces experienced by a group of molecules must all be determined synchronously, posing what is, in most applications, a formidable computational problem. To circumvent this complexity the pairwise additive approximation is often introduced, e.g., treating the polarization in an averaged manner, thus allowing the force on a particular molecule to be formulated simply as the sum of the forces (see Section 1.1) between this and all other molecules that are considered separately. 

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

FIG. 5 Comparison of full numerical solution and pairwise-additivity approximation for the force on the nonadsorbed sphere shown in Fig. 4. (From Ref. 13.)... 

Nevertheless, these methods are mostly applied with fixed charges (even if these are chosen in a sophisticated way) and with pairwise additivity approximation as well as with the neglect of nuclear quantum effects. Suggestions for polarizable models appeared in literature mainly for water [23], The quality of potential parameterization varies from system to system and from quantity to quantity, raising the question of transferability. Spontaneous events like reactions cannot appear in simulations unless the event is included in the parameterization. Despite these problems, it is possible to reproduce important quantities as structural, thermodynamic and transport properties with traditional MD (MC) mainly due to the condition of the nanosecond time scale and the large system size in which the simulation takes place [24],... 

In this approach it is assumed that the basis set superposition error in the many-body cluster can be approximated by the sum of the Boys-Bemardi function counterpoise corrections for pairs of bodies. Hence the total interactions for an N-body cluster using the pairwise additive function counterpoise correction is given by... 

It should be remembered that hard spheres are not real particles, and (1.7.5) is valid by virtue of definitions (1.7.2) and (1.7.3). Therefore, the pairwise additivity assumption must be viewed as being a built-in feature of the definition of a system of hard spheres. By simple generalization, one can define nonspherical hard particles for which (1.7.1) is fulfilled. Other systems for which the pairwise additivity assumption is presumed to hold are systems of idealized point charges, point dipoles, point quadruples, and the like. A system of real particles such as argon atoms is believed to obey relation (1.7.1) approximately. Although it is now well known that even the simplest molecules do not obey (1.7.1) exactly, it is still considered a useful approximation without which little progress in the theory of liquids, if any, could have been achieved. 

This formal resemblance can be misleading. Equation (3.6.53) is the exact isotherm for a system with direct interactions only. The two independent parameters of the model are Ki and 5. On the other hand, Eq. (3.6.51) has been derived on the basis of the pairwise additivity (or superposition approximation) assumptions (3.6.46) and (3.6.47). We have already seen that this approximation is unjustified for the indirect correlations. Since we know that in hemoglobin direct interactions are negligible, we have concluded that all correlations are due to indirect interactions, therefore (3.5.51) is incorrect. If we insist on expressing the isotherm in terms of the pair correlation function y, 1), we must also include nonadditivity effects [see Eq. (3.6.58) below]. But this is not necessary. A simpler and exact expression can be written in terms of the fundamental parameters of the model. This is essentially Eq. (3.6.37), where the Ki are defined in (3.6.36). 

Current research in water potentials tends to focus on incorporating explicit many-body polarization terms in the water-water energy. This avoids the pairwise additive approach, i.e., the effective media approximation inherent in pairwise additive water potentials, and allows for a better parameterization of the true water-water interaction. Two main avenues for treating polarization effects have developed in the last decade an explicit treatment of classical polarization and fluctuating charge models. The effort expended to find suitable water models will slowly pay off in an enhanced awareness of how to improve current molecular force fields for interactions of other types (e.g., between organic solutes, biomolecules, etc.). 

The choice of the adjustable parameters used in conjunction with classical potentials can result to either effective potentials that implicitly include the nuclear quantization and can therefore be used in conjunction with classical simulations (albeit only for the conditions they were parameterized for) or transferable ones that attempt to best approximate the Born-Oppenheimer PES and should be used in nuclear quantum statistical simulations. Representative examples of effective force fields for water consist of TIP4P (Jorgensen et al. 1983), SPC/E (Berendsen et al. 1987) (pair-wise additive), and Dang-Chang (DC) (Dang and Chang 1997) (polarizable, many-body). The polarizable potentials contain - in addition to the pairwise additive term - a classical induction (polarization) term that explicitly (albeit approximately) accounts for many-body effects to infinite order. These effective potentials are fitted to reproduce bulk-phase experimental data (i.e., the enthalpy at T = 298 K and the radial distribution functions at ambient conditions) in classical molecular dynamics simulations of liquid water. Despite their simplicity, these models describe some experimental properties of liquid... 

It is also possible to approximate xch in terms of the pairwise-additive sum (eI k) of localized steric exchange... 

To first order, the dispersion (a-a) interaction is independent of the structure in a condensed medium and should be approximately pairwise additive. Qualitatively, this is because the dispersion interaction results from a small perturbation of electronic motions so that many such perturbations can add without serious mutual interaction. Because of this simplification and its ubiquity in colloid and surface science, dispersion forces have received the most significant attention in the past half-century. The way dispersion forces lead to long-range interactions is discussed in Section VI-3 below. Before we present this discussion, it is useful to recast the key equations in cgs/esu units and SI units in Tables VI-2 and VI-3. 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

This simple theoryis based on the expectation that, to a reasonable degree of approximation, proton-proton, dipolar contributions to the measured spin-lattice relaxation-rate are pairwise additive and decrease as a simple sixth power of the interproton distance. The simplified version of the dipole-dipole mechanism is summarized in the following two equations for spin i coupled intramolecularly with a group of spins j... 

Next, the statistical thermodynamics of the pairwise surface segment interactions can be performed exactly, i.e. without any additional approximations beyond the assumption of surface pair formation, using ... 

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