Potential energy pairwise additivity

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 ... 

The heat current is the amount of energy passing through a unit area. In the classical picture and if the interaction potential is pairwise additive, the energy (hj) for an atom i in a N-particle system is... 

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

The relationship between g(r) and the interparticle potential energy is most easily seen if we assume that the interparticle energy can be factorized into pairwise additive potentials as... 

The intennolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44, 50, 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

Ding YB, Bernardo DN, Kroghjespersen K, Levy RM (1995) Solvation free-energies of small amides and amines from molecular-dynamics free-energy perturbation simulations using pairwise additive and many-body polarizable potentials. J Phys Chem 99(29) 11575—11583... 

Here we present and discuss an example calculation to make some of the concepts discussed above more definite. We treat a model for methane (CH4) solute at infinite dilution in liquid under conventional conditions. This model would be of interest to conceptual issues of hydrophobic effects, and general hydration effects in molecular biosciences [1,9], but the specific calculation here serves only as an illustration of these methods. An important element of this method is that nothing depends restric-tively on the representation of the mechanical potential energy function. In contrast, the problem of methane dissolved in liquid water would typically be treated from the perspective of the van der Waals model of liquids, adopting a reference system characterized by the pairwise-additive repulsive forces between the methane and water molecules, and then correcting for methane-water molecule attractive interactions. In the present circumstance this should be satisfactory in fact. Nevertheless, the question frequently arises whether the attractive interactions substantially affect the statistical problems [60-62], and the present methods avoid such a limitation. 

The assumption of additivity that underlies many empirical intermolecular potentials can be stated more formally as follows. Suppose that A, B, C,... represent chosen molecules in a given spatial configuration. The potential-energy function V(A, B, C,...) (relative to isolated molecules61) will be said to be pairwise additive (and denoted Vpw) if... 

One of the remarkable features of Coulomb s law when applied to nuclei and electrons is its additivity. The potential energy of an assemblage of particles is just the sum of all the pairwise interaetions in the form given in Eq. (1.1). Thus, consider a system with K nuclei, a =, 2,..., K having atomic numbers Z . We also consider the molecule to have N eleetrons. If the molecule is uncharged as a whole, then = N. We will use lower ease Latin letters, /, j,k,..to label electrons and lower case Greek letters, a, f, y,..., to label nuclei. The full potential energy may then be written... 

The aim of the present study is to investigate the validity of the pairwise additivity of two-body and three-body potentials for He2Br2. These results are compared with ah initio calculations" and a simple model of the three-body potential is proposed to determine well depths and equilibrium structures for different isomeric configurations of the complex, as well as the minimum energy pathways through them. Additionally, variational methods are used to calculate the vibrational states of He2Br2. The wavefunctions of the lower states are analyzed in terms of probability distributions of the internal coordinates and the zero-point energy of the vdW cluster is evaluated. 

Non-pairwise additivity. A significant component of the energy V (1,2,3) of three interacting atoms is given by the sum of the pair potentials, V(l,2) + F(l,3) + F(2,3). However, it is now generally accepted that the so-called Axilrod-Teller term, a long-range, irreducible (classical)... 

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