Dipoles many-body forces

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) ... 

KLEIN - You raise an important question. Long-range many body forces are important. Short-range many body forces are also important. Alas, our knowlegde of the latter is very poor. We used the crudest possible model and have neglected both of these effects. Our electron-ammonia solvent potential includes the most important polarization effect namely, that caused by the solvent dipoles on the electron. We have ignored the self-consistent many body polarization of the coupled electron-solvent system. This latter effect has been discussed by Wallqvist, Thirumalai and Berne (J. Chem. Phys., 1987). I refer you to this article for details. 

The van der Waals force is ubiquitous in colloidal dispersions and between like materials, always attractive and therefore the most common cause of dispersion destabilization. In its most common form, intermolecular van der Waals attraction originates from the correlation, which arises between the instantaneous dipole moment of any atom and the dipole moment induced in neighbouring atoms. On this macroscopic scale, the interaction becomes a many-body problem where allowed modes of the electromagnetic field are limited to specific frequencies by geometry and the dielectric properties of the system. 

The earliest quantitative theory to describe van der Waals forces between two colloidal particles, each containing a statistically large number of atoms, was developed by Hamaker, who used pairwise summation of the atom-atom interactions. This approach neglects the multi-body interactions inherent in the interaction of condensed phases. The modem theory for predicting van der Waals forces for continua was developed by Lifshitz who used quantum electrodynamics [19,20] to account for the many-body molecular interactions and retardation within and between materials. Retardation is a reduction of the interaction because of a phase lag in the induced dipole response that increases with distance. 

Specific polarization effects, beyond those modelled by a continuum dielectric model and the movement of certain atoms, are neglected in MIF calculations. Many-body effects are also neglected by use of a pair-wise additive energy function. Polarizable force fields are, however, becoming more common in the molecular mechanics force fields used for molecular dynamics simulations, and MIFs could be developed to account for polarizability via changes in charge magnitude or the induction of dipoles upon movement of the probe. 

Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of so-called retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. 

Recently, a new theoretical method of calculating potential energy and dipole/polarizability surfaces for van der Waals molecules based on symmetry-adapted perturbation theory (sapt) of intermolecular forces (12)— (15) has been developed (16)-(24). In this method, referred to as many-body symmetry-adapted perturbation theory, all physically important contributions to the potential and the interaction-induced properties, such as electrostatics, exchange, induction, and dispersion are identified and computed separately. By making a perturbation expansion in the intermolecular interaction as well as in the intramolecular electronic correlation, it is possible to sum the correlation contributions to the different physical... 

The accuracy of the simulation results depends on a suitable choice of the parameters in the potential functions. On account of equation (23.1), an essential restraint of the calculation method is the pair-wise addition of atomic forces. Although effective pair potentials are used, three-body terms and interactions of higher order are neglected. Consequently, the major many-body contributions to the induced dipole interactions in aqueous ionic systems are not modelled accurately. A further simplification is a common application of... 

Improvements to the DIM model can be effected in two ways increasing the basis, or incorporation of (small) 3-body effects into the polyatomic matrix elements. The latter procedure has been carried out [19], where account was taken of the induced-dipole induced-dipole forces between the pairs of neutrals under the influence of a neighbouring ioa Strictly speaking, this is no longer a DIM procedure however, it may point the way to a systematic way of representing the PES instead of expanding the PES itself in a many-body expansion [20-21], one should first obtain a model hamiltonian for the system and then introduce many-body effects into the matrix elements [21]. 

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... 

An important subset of many-body potentials shown to be important for simulating interfacial systems are those referred to as polarizable force fields.Various aspects of polarizable force fields, especially for use in biomolecular modeling, is explained by Ren et al. in Chapter 3 of this volume. If one treats the fixed charges in Eq. [3] as parameters to be fitted to obtain the best agreement of the condensed phase simulations with experiments, in many cases one finds that the optimal values are considerably different from those obtained from a fit to a molecular (gas phase) dipole moment or from quantum calculations on isolated molecules. This is because in a condensed medium, the local electric field E, (at the location of a particle i) is determined by all the fixed charges and by all the induced dipoles in the system ... 

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