Potential energy, pairwise

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. 

In the pairwise-potential approximation and the shell model, the lattice energy can be written as... 

Calculations have been performed on atom clusters by Hoare and Pal (55) and Burton (39) to determine their geometrical and thermodynamic properties using empirical potential functions. In this technique the pairwise potentials V(Rjj) between atoms, a distance Ry apart, are assumed additive to give a potential energy... 

An approximation to the packing energy can be given by the sum of terms due to all the interactions among atoms belonging to adjacent molecules. This approach is the so-called atom-atom potential method [6] where pairwise potentials are empirically determined and the molecular conformation is kept fixed. This method enables is to estimate the packing energy and to justify crystal properties, such as the enthalpy of formation. 

Among the systems for which one cannot reasonably employ pairwise additive potentials are metals. It is well known that only a small fraction of the binding energy of a metal can be accounted for by pairwise potentials [25]. Furthermore, the use of pairwise additive potential functions leads to incorrect relationships between the components of the bulk modulus [15]. The simulation of a metal requires a truly multibody potential function. 

Due to the convergence factor, it is possible to calculate the total energy by a pairwise potential 4> such that... 

The success of various models rests on the correct choice of the pairwise potential energy equation. In this section we will address the potential equations commonly employed for adsorbates used in pore characterization. 

Over the past 15 years, a large number of papers have appeared dealing with computer simulations of water structure, thanks to the increased capacity and availability of fast computers. Simulations are based on accepting a reasonable expression for the pairwise interaction of water molecules, namely, the pair potential (energy) function. Much has been learned from simulation studies, and the results are valuable hints as to what the structure of liquid water may be, but not necessarily as to what the structure must be. Because of computational limitations, the typical sample of water molecules used in such analyses is about 500. If these molecules formed a small droplet, the radius would be five molecules, half of which would be from the outside layer. Thus, one must expect some dramatic surface effects. Furthermore, as discussed below, the calculations are based on a pairwise potential energy function this is probably a most serious limitation, as also discussed below. 

For a pairwise potential, Vy, the energy can be decomposed into contributions due to the nn nearest neighbor interactions, the strain due to deviations from the optimum pair separation in such pairs, and the non-nearest neighbor contribution, as represented, respectively, by the three terms in the expression [208]... 

To proceed further, three major approximations to the theory are made [44] First, that the transition operator can be written as a pairwise summation of elements where the index I denotes surface cells and k counts units of the basis within each cell second, that the element is independent of the vibrational displacement and, third, that the vibrations can all be treated within the harmonic approximation. These assumptions yield a form for w(kf, k ) which is equivalent to the use of the Bom approximation with a pairwise potential between the probe and the atoms of the surface, as above. However, implicit in these three approximations, and therefore also contained within the Bom approximation, is the physical constraint that the lattice vibrations do not distort the cell, which is probably tme only for long-wavelength and low-energy phonons. 

The repulsive energy is calculated from a sum of pairwise potentials r). Harrison argued that the repulsive interaction originates from the overlap of the atomic orbitals and therefore, should scale as [18]... 

Atom-atom potential method The packing energy is expressed by summing all intermolecular interactions, the pairwise potentials are determined especially, and, again, the molecular structure is kept unchanged. 

Regardless of the choice of boundary and/or the inclusion of non-pairwise potentials, the minimum of the potential energy occurs where... 

The potential energy of the crystal lattice is the sum of all pairwise potential energies. 

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