Potentials

      The Hamiltonian function, H, is the total energy of a system expressed in terms of coordinates and momenta of particles. When a particle of mass m is moving in one dimension, the classical Hamiltonian is defined as follows:
..........(1)

where px is the component of linear momentum in the x direction, and V(x) is the corresponding potential energy. The Hamiltonian can be expanded in various formulations depending on the components considered in the kinetic and potential energies.
A system is completely described by its Hamiltonian H, which consists of internal part H0 and external part H1. The internal part of Hamiltonian H0 is given as follows:
..........(2)

where pi is the momentum, mi is the mass of the particle i, and u and u(3) are the two-body and three-body interaction terms, respectively. The first term is the kinetic energy of the particles. One of the typical two-body pair-wise potentials, the second term in Eq. 2, is the Lennard-Jones (LJ) potential. Three-body or many-body potentials play an important role in the simulations of condensed matter, such as metals and semiconductors. The embedded atom method[1] and bond order potentials[2, 3] are some types of many-body potentials. The potentials are divided into two types of interactions, short-range and long-range, with respect to the distance between particles. Short-range interactions take only the nearest neighboring particles into account, and thus cover primarily chemical (covalent), metallic, and ionic (Coulomb) bonds. Long-range interactions such as van der Waals interactions and hydrogen bonding, are generally well-described by two-body potential functions. Many-body interactions are always as important as two-body interactions in the short range.

References
[1] Finnis, M.W. and J.E. Sinclair, A Simple Empirical N-Body Potential for Transition-Metals. Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, 1984. 50(1): p. 45-55.
[2] Tersoff, J., New Empirical-Model for the Structural-Properties of Silicon. Physical Review Letters, 1986. 56(6): p. 632-635.
[3] Brenner, D.W., et al., A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. Journal of Physics-Condensed Matter, 2002. 14(4): p. 783-802.