Potential energy surfaces pairwise additive

In the last section, we summarized the different contributions to the potential energy for the interactions between an adsorbate molecule (or atom) and an atom on the solid surface. To calculate the interaction energy between the adsorbate molecule and all atoms on the surface, pairwise additivity is generally assumed. The task is then to sum the interactions, pairwise, with all atoms on the surface, by integration. 

Most of the potential energy surfaces reviewed so far have been based on effective pair potentials. It is assumed that the parameterization is such as to account for nonadditive interactions, but in a nonexplicit way. A simple example is the use of a charge distribution with a dipole moment of 2.ID in the ST2 model. However, it is well known that there are significant non-pairwise additive interactions in liquid water and several attempts have been made to include them explicitly in simulations. Nonadditivity can arise in several ways. We have already discussed induced dipole interactions, which are a consequence of the permanent diple moment and polarizability of the molecules. A second type of nonadditive interaction arises from the deformation of the molecules in a condensed phase. Some contributions from such terms are implicitly included in calculations based on flexible molecule potentials. Other contributions arises from electron correlation, exchange, and similar effects. A good example is the Axilrod-Teller three-body dispersion interaction ... 

Figure 11. Structures of clusters of atoms calculated with pairwise additive Lennard-Jones potentials for n = 7 to 19 atoms. The structures are believed to be the absolute minima in the potential energy surface and correspond to 0 K classical calculations. The shaded atom corresponds to the particle being added to the preceding cluster. Clusters in the same column gain the same number of nearest neighbors with the addition of the shaded atom.

Consider now the simplest case of interacting atoms, where the potential energy surface has dimension 1x1. Thus, it assumes the pairwise additive form... 

The simple addition of pairwise atom-atom interactions was used to described the potential energy surface. Each one of them was represented by means of Morse functions, where the relevant parameters were the same already employedl l (see Table I there). 

The predicted vibrationally averaged structures are consistent with experimental observations. The predicted HF vibrational (red) shifts for the pairwise-additive potential energy surfaces are close to the experimental values but differ in a systematic fashion. The agreement is improved by the inclusion of three-body nonpairwise terms. The results from one study are listed for comparison with experiment in Table 5. 

The common feature observed in both DFT and GCMC simulations is that these results overpredict the amount adsorbed in the reduced pressure region greater than about 0.2. This seems to indicate that the fluid—fluid interaction energy is overestimated in the presence of a solid surface, and therefore the usual assumption of pairwise additivity of fluid-fluid and solid-fluid potential energies is questionable. One way of resolving this issue is the application of the following quadratic equation for the potential of one molecule [83] ... 

The density profiles obtained from atomic models (platinum [49] and rigid mercury surfaces [40]), where water-metal interactions are described by pairwise additive atomatom interaction potentials, are similar in shape. Height and width are correlated with the depth and force constant of the interaction potential. A similar correlation between peak height and interaction energy holds for water near a variety of different smooth model surfaces (see Ref. 139 and references therein). 

Zhu and Philpott" employed a combination of relatively simple anisotropic and isotropic pairwise additive potential energy functions for a number of metal surfaces. They also provide a collective version of their potentials. 

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... 

Consider the same N molecules at rest in a perfectly ordered infinite crystalline array with one molecule in the asymmetric unit. All molecules are equal, and surface or truncation effects are neglected, so the following discussion refers to a bulk crystal. Assuming for the moment that the intermolecular potential is pairwise additive, e.g. in the atom-atom potential approximation, the packing potential energy, PPE, or the interaction energy of any reference molecule m with the surrounding molecules n, is a sum of molecule-molecule terms, each of which is in turn a sum of atom-atom terms ... 

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