Chapter 3 - Evaluation of Lennard-Jones Potential Fields

Abstract

In this chapter a number of simple interaction energies for linear objects are determined, such as for a single atom with a plane, and for spherical objects such as for a sphere interacting with a plane graphene sheet. Our principal concern is the evaluation of the Lennard-Jones potential function interaction energy between atoms, molecules and nanostructures. When calculating the interaction energy between molecules containing a number of atoms, the pairwise interactions may be summed to derive a total interaction energy that is given by a double summation over all the atoms in each molecule. Here we assume that all the atoms are smeared over ideal lines or surfaces representing the molecules in order to replace the explicit summations with line or surface integrals, and the task of calculating the interaction energy is then made considerably easier. It turns out that many of the molecules and nanostructures can be modelled by the basic geometric objects of points, straight lines, flat planes, spheres, and right circular cylinders, and therefore integrals over these surfaces are needed and the purpose of this chapter is to address these integrals in a systematic way to facilitate the evaluation of ideal Lennard-Jones interactions.