Pairwise potentials many-body effects

It is easy to see that Eqs. (17-59), (17-60), and (17-61) are equivalent to Eq. (17-58). It must be noted that Eq. (17-60) expresses the solvation free energy of a molecule with a pairwise additive potential, hence the theory of energy representation described in Section 17.3.4 can be applied without any further approximations. An appropriate choice of E and h(r) will make the contribution E + Ap major in the total excess chemical potential. The free energy change expressed by Eq. (17-61) directly depends on the choice of the standard energy E and involves many-body effects since the solute-solvent interaction is described by Eqm/mm (n, X) at the final state... 

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). 

With the explicit treatment of polarizability, we can abandon the concept of pairwise additive potentials. If the polarizability contains the largest contribution to the many-body effects that are effectively included in pair potentials, we should be able to create much more reliable potentials. The polarizable potentials discussed here are compiled in Table 9. One way of circumventing the effective construction is to explicitly calculate the potential energy between water molecules using quantum chemistry methods. If the polarization is... 

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... 

For noble and transition metals, the interactions between atoms are not pairwise and simple empirical potentials are inappropriate (Barreteau et al. 2000). Therefore incorporating many-body effects into the potentials is essential. Moreover, for magnetism studies, ab initio methods need to be employed, which render global optimization efforts extremely computation-intensive. Therefore, most results we shall quote here will be based on restricted searches of the potential energy surface. 

Molecular dynamics calculations have been performed (35-38). One ab initio calculation (39) is particularly interesting because it avoids the use of pairwise potential energy functions and effectively includes many-body interactions. It was concluded that the structure of the first hydration shell is nearly tetrahedral but is very much influenced by its own solvation. 

Let us now consider systems formed by polar molecules, e.g. HF, H20 and HC1. The HF and HC1 crystals contain one-dimensional bent chains of molecules between which the mutual interactions are relatively weak (Fig. 12). In the case of HF we observe a marked decrease of the intermoleeular distance (ARpp 0.3 A) upon the formation of the solid phase. Ice I has a fairly complicated three-dimensional structure (Fig. 12), dipoles appear at different relative orientations, and the infinite chain is no appropriate model. Nevertheless, the contraction of the intermoleeular distance in the solid state is substantial (ARoo 0-24 A). In both cases, the stabilizing contributions have to be attributed to attractive many-body forces since the changes observed exceed by far the effects to be expected in polar systems with pairwise additive potentials. The same is true for the energy of interaction (Table 12) ... 

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. 

Van der Waals forces result from attractions between the electric dipoles of molecules, as described in Section 1.2. Attractive van der Waals forces between colloidal particles can be considered to result from dispersion interactions between the molecules on each particle. To calculate the effective interaction, it is assumed that the total potential is given by the sum of potentials between pairs of molecules, i.e. the potential is said to be pairwise additive. In this approximation, interactions between pairs of molecules are assumed to be unaffected by the presence of other molecules i.e. many-body interactions are neglected. The resulting pairwise summation can be performed analytically by integrating the pair potential for molecules in a microscopic volume dVi on particle 1 and in volume dVi on particle 2, over the volumes of the particles (Fig. 3.1). The resulting potential depends on the shapes of the colloidal particles and on their separation. In the case of two flat infinite surfaces separated in vacuo by a distance h the potential per unit area is... 

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.). 

Many of the fixes or modifications necessary to make an effective water potential work can be traced back to the influence of polarization of the molecular charge distribution. Recent efforts in the development of water potentials have considered the explicit inclusion of a many-body polarizability term. The problem of including polarization is that it is not decomposable into pairwise additive terms. If one water molecule becomes polarized by an electric field generated by other surrounding water molecules, the extra induced moment will in turn affect the charge distribution of the surrounding water, which in turn will change the induced moment on the central water molecule, and so on. 

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