Lennard-Jones pair interaction energy

Rare-gas clusters can be produced easily using supersonic expansion. They are attractive to study theoretically because the interaction potentials are relatively simple and dominated by the van der Waals interactions. The Lennard-Jones pair potential describes the stmctures of the rare-gas clusters well and predicts magic clusters with icosahedral stmctures [139, 140]. The first five icosahedral clusters occur at 13, 55, 147, 309 and 561 atoms and are observed in experiments of Ar, Kr and Xe clusters [1411. Small helium clusters are difficult to produce because of the extremely weak interactions between helium atoms. Due to the large zero-point energy, bulk helium is a quantum fluid and does not solidify under standard pressure. Large helium clusters, which are liquid-like, have been produced and studied by Toennies and coworkers [142]. Recent experiments have provided evidence of... 

Extensive computer simulations have been caiTied out on the near-surface and surface behaviour of materials having a simple cubic lattice structure. The interaction potential between pairs of atoms which has frequently been used for inert gas solids, such as solid argon, takes die Lennard-Jones form where d is the inter-nuclear distance, is the potential interaction energy at the minimum conesponding to the point of... 

Finally, it was found necessary to add a Lennard-Jones (LJ) 12-6 intermolec-ular term between each pair of quantum-mechanical and MM atoms, in order to obtain good interaction energies as well as good geometries for intermolecular interactions. 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. 

The intermolecular interactions are usually assumed to be pair-additive functions such as the Lennard-Jones 12-6 or 9-6 potentials or the Buckingham expontential-6 type of potentials and are parameterized using methods similar to those described in the previous paragraph to reproduce the crystallographic structure and the lattice energy. For the case of liquid systems the parameterization of non-bonded interactions can be done to reproduce the liquid densities and the heats of vaporization. 

The last term in the formula (1-196) describes electrostatic and Van der Waals interactions between atoms. In the Amber force field the Van der Waals interactions are approximated by the Lennard-Jones potential with appropriate Atj and force field parameters parametrized for monoatomic systems, i.e. i = j. Mixing rules are applied to obtain parameters for pairs of different atom types. Cornell et al.300 determined the parameters of various Lenard-Jones potentials by extensive Monte Carlo simulations for a number of simple liquids containing all necessary atom types in order to reproduce densities and enthalpies of vaporization of these liquids. Finally, the energy of electrostatic interactions between non-bonded atoms is calculated using a simple classical Coulomb potential with the partial atomic charges qt and q, obtained, e.g. by fitting them to reproduce the electrostatic potential around the molecule. 

Exact calculations of AE have been carried out by Kelley and Wolfsberg [19] for colinear collisions between an atom and a diatomic molecule. The oscillator potential was considered to be both harmonic and Morse-type, and the interaction between the colliding pair was taken both as an exponential repulsion and as a Lennard-Jones 6 12 potential. Two important conclusions were reached First, when the initial energy of the oscillator increases, the total energy transferred from translation to vibration, AE, decreases. Second, the effect of using a Morse-oscillator potential in place of the harmonic oscillator was generally to decrease AE, often by more than a factor of 10. 

We have seen that the nearest-neighbor distance jn ionic solids is determined by the joint action of the Coulomb interaction and the predominantly repulsive overlap interaction that we approximate with the Lennard-Jones form. First, let us explore the properties of KCI, an ionic solid with ion charge Z = 1. We have seen that the Coulomb energy per ion pair in the face-centered cubic structure is... 

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