Pairwise additive approximation

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. 

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.)... 

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. 

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)-. 

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. 

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 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. 

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