Pairwise effective potential

Cut-and-shifted effective potentials describe the interaction between the adsorbed fluid and the zeolite. The Xe-0 and Xe-Na parameters are unchanged from our previous work on Xe in Na/t (55) and Xe-Ar mixtures in Na/f (30). The Kr-0 and Kr-Na potential parameters were obtained as described in Ref. (29). The CH4-0 and CH4-Na potential surfaces were described by the pairwise Lennard-Jones parameters given by Woods and Rowlinson (39). 

The first goal of the present work is to establish an analytical expression for the optimum Hamaker constant of the shell. It will be also shown that the existence of an optimum size of the side chain for which the viscosity of the paste is minimum can be explained on the basis of this optimum Hamaker constant. Finally, it will be noted that, for concentrated dispersions, the collective behavior of particles interacting via a pairwise attractive potential can lead to an effective repulsion component which may play a role in the stability of the concentrated dispersions. 

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) ... 

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... 

Actually, computational convenience has almost always suggested using pairwise additive potentials for simulations of condensed phases also, though strictly two-body potentials are only acceptable for rarefied gases. The computational convenience of two-body potentials is maintained, however, if non-additive effects are included implicitly, i. e. with the so called two-body effective potentials. All empirical or semi empirical functions whose parameters have been optimized with respect to properties of the system in condensed phase belong to this class. As already observed, this makes these potentials state-dependent, with unpredictable performance under different thermodynamic conditions. 

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. 

Consider the excluded volume interaction between spherical particles with effective pairwise interaction potential... 

It must be pointed out that this expression implicitly contains terms that depend on the orientation of the molecule with respect to the surface and the orientation of a given molecule with respect to its neighbors when the molecules are nonspherical. Equation (3.19) assumes that the potentials are additive and pairwise since it does not include three-body or higher terms, this must be considered as an effective potential [8]. 

Moments and Angular Forces. Our aim in the present setting is to exploit the geometry of the densities of states in order to construct effective potentials for describing the bonding in solids which go beyond the pairwise forms considered thus far. We follow Carlsson (1991) in formulating that description. First, we reiterate that our aim is not the construction of the most up-to-date potentials that... 

Side chains (even when modeled as single-interaction spheres) should have some conformational freedom that reflects their internal mobility in real proteins. Pairwise interaction potentials should be as specific as possible, and in the absence of explicit solvent, a burial potential that reflects the hydrophobic effect may be necessary... 

Typically, effective potential models are parameterized empirically by constraining the model s thermophysical properties to experimental values at ambient conditions, by adjusting those parameters in a series of short simulations [81,94,95] or during a rather longer simulation [50,85]. The most convenient choices are the configurational internal energy and the pressure to fit the energy and size force-field parameters for the pairwise repulsion-attraction terms. Additionally, almost all models have been parametrized to reproduce the structure of water obtained in the 1986 Soper-Phillips neutron diffraction experiments [96]. 

Because many effective pairwise additive potentials do not include the self-energy of the dipoles in their original parameterizations, their reported enthalpy values require corrections to make a proper comparison with experi-... 

The minimum potential energy curve of the water dimer is given in Figure 6 for three different water models. Two are effective models, and one is a nonempirical model for comparison. The nonempirical molecular orbital (NEMO) " surface mimics the true two-body water potential V2 in Eq. [44]. It is rather flat, and its minimum is not as sharply defined as in the effective potentials. Pairwise additive potentials like and TIP4P ° show their... 

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... 

And, to check the effect of the form of the interaction potential, calculations were done for process (10) using a truncated harmonic pairwise-additive potential. 

The Lennard-Jones potential can also be used for interactions between dissimilar atoms. The Berthelot mixing rules can then be used to determine the effective a and e parameters. The effective size parameter in the pairwise interaction potential is determined as the arithmetic average of the sizes of the two atoms ... 

In the case of fluids which consist of simple non-polar particles, such as liquid argon, it is widely believed that Ui is nearly pairwise additive. In other words, the functions for n > 2 are small. Water fails to conform to this simplification, and if we truncate the series after the term, then we have to understand that the potential involved is an effective pair potential which takes into account the higher order-terms. 

The results obtained demonstrate competition between the entropy favouring binding at bumps and the potential most likely to favour binding at dips of the surface. For a range of pairwise-additive, power-law interactions, it was found that the effect of the potential dominates, but in the (non-additive) limit of a surface of much higher dielectric constant than in solution the entropy effects win. Thus, the preferential binding of the polymer to the protuberances of a metallic surface was predicted [22]. Besides, this theory indirectly assumes the occupation of bumps by the weakly attracted neutral macromolecules capable of covalent interaction with surface functions. 

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