Lennard-Jones potentials energy models

Here Qi and Qj represent the two point charges, while Ry equals the distances between these two points. In some force fields, Coulombic interactions are modified by changing the dependence of the dielectric constant, e. In general, van der Waals interactions are modeled using a 6-12 Lennard-Jones potential energy term. This expression, shown in Eq. (28), consists of a repulsion and attraction term. 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

The Lennard-Jones potential is characterised by an attractive part that varies as r ° and a repulsive part that varies as These two components are drawn in Figure 4.35. The r ° variation is of course the same power-law relationship foimd for the leading term in theoretical treatments of the dispersion energy such as the Drude model. There are no... 

The values of C and D are evaluated at the critical point and normal boiling point. U. is the vertical molecule-cation interaction energy and U isJthe corresponding molecule-anion term. U and w are calculated as the sums of all the appropriate dielectric and Lennard-Jones potentials. The actual calculation of an x/m isotherm is the superposition of several solution models. The principal one corresponds to the partial filling by molecules on the cation sites. The value of x/m is a constant times Xg, summed over all sites, where the constant is the molecular weight ratio. 

There are presently two main difficulties which handicap attempts at exact calculation. The first concerns the intermolecular potential, and the hazards of extrapolation from models derived from viscosity measurements have been discussed. Furthermore, such a method is of dubious validity for polyatomic molecules, because the intermolecular repulsive potential will generally appear to become progressively shallower with increasing molecular dimensions if the viscosity data are cast, for example, in the Lennard-Jones form. Energy transfer depends... 

Interactions between neutral, nonpolar atoms or molecules are relatively weak, and can be accurately modeled by the Lennard-Jones potential discussed in the last section. Table 3.5 lists some specific examples. Notice that the well depth is less than 1% of typical bond energies in Table 3.2. In fact none of these atoms and molecules is condensed into a liquid at STP (standard temperature and pressure P = 1 atm, T = 273K). 

Figure 1.41. Potential energies for the bead-spring model LJ1—Lennard-Jones potential LJ2—van der Waals potential EXP1, EXP2—short-range polar potential FENE—finitely extensible nonlinear elastic potential.

In both equations, d is the separation between the atoms. The Lennard-Jones potential is simpler and computationally less demanding and is therefore favored for models of macromolecules such as proteins and DNA. The Buckingham function more closely resembles the energy relationship and is preferred when higher accuracy is required. The latter function is available in MOMEC and we will concentrate on this. 

Lennard-Jones potentials have been used widely in modeling rare gas and molecular crystals. Morse potentials become more appropriate when covalent systems are being studied D may then be interpreted as the covalent bond-dissociation energy and re the equihbrium bond length. Buckingham potentials have been very widely used in the study of ionic and semi-ionic sohds. ... 

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