Many-body effects, pairwise interactions

In typical organic crystals, molecular pairs are easily sorted out and ab initio methods that work for gas-phase dimers can be applied to the analysis of molecular dimers in the crystal coordination sphere. The entire lattice energy can then be approximated as a sum of pairwise molecule-molecule interactions examples are crystals of benzene [40], alloxan [41], and of more complex aziridine molecules [42]. This obviously neglects cooperative and, in general, many-body effects, which seem less important in hard closed-shell systems. The positive side of this approach is that molecular coordination spheres in crystals can be dissected and bonding factors can be better analyzed, as examples in the next few sections will show. 

As noted in Chapter 2, computation of charge-charge (or dipole-dipole) terms is a particularly efficient means to evaluate electrostatic interactions because it is pairwise additive. However, a more realistic picture of an actual physical system is one that takes into account the polarization of the system. Thus, different regions in a simulation (e.g., different functional groups, or different atoms) will be characterized by different local polarizabilities, and the local charge moments, by adjusting in an iterative fashion to their mutual interactions, introduce many-body effects into a simulation. 

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... 

Water models can be conveniently divided into three types. In the simple interaction-site models each water molecule is maintained in a rigid geometry and the interaction between molecules is described using pairwise Coulombic and Lennard-Jones expressions. Flexible models permit internal changes in conformation of the molecule. Finally, models have been developed that explicitly include the effects of polarisation and many-body effects. 

As the polarization term in EFP is non-additive, the EFP method captures a majority of many-body effects in H-bonded systems. By definition, the many-body energy is a difference between a total energy of a system and energies of all pairwise interactions. In polar complexes, the many-body interactions are predominantly of... 

It has been suggested that pairwise interactions are insufficient to fold proteins and higher-order terms are necessary [30]. It is of interest to check if the environment models that we use catch cooperative, many-body effects. As an example we consider the cases of vaUne-valine and lysine-lysine interactions. We use Eq. (8) to define the energy of a contact. In the usual pairwise... 

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. 

In 2004 we developed a method referred to as QM/MM-ER [23] for the purpose of computing solvation free energy of a QM solute in MM solvent by combining the hybrid QM/MM with the theory of energy representation. As described in Sect. 6.3 the standard theory of solutions [25] is based on the assumption that the solute-solvent interaction is pairwise additive. However, the QM/MM method involves the many-body interaction that originates from the electron density polarization of the QM subsystem. Hence, we have to make some device to manage the many-body effect within the framework of the energy representation. 

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) ... 

In order to account for hydrodynamic interactions among the suspended particles, Bossis and Brady (1984) used both pairwise additivity of velocities (mobilities) and forces (resistances), discussing the advantages and disadvantages of each method. While their original work did not take explicit account of three- (or more) body effects, the recent formulation of Durlofsky, Brady, and Bossis (1988) does provide a useful procedure for incorporating both the far-field, many-body interactions and near-field, lubrication forces into the grand resistance and mobility matrices. 

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. 

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. 

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