Consider an atom and a diatomic molecule colliding.

The questions that may be asked about this physical process are numerous, here are a few:

- How much energy did they collide with?
- Is their any energy released or absorbed during this collision?
- How many dimensions does this take place in?
- What is the mass and net charge of the colliding species?
- Is there electron transfer?
- Are there internal degrees of freedom for each species?
- Are these internal energies transferrable?
- In macroscopic situations what is the reaction rate?
- Are there fine spectral lines that we can predict?
- Does simulating this have a useful application?

To begin to answer any of these questions analytically we need to choose a dynamic formalism. The Hamiltonian for this system will include multiple expressons for potential energies. Here are some common choices.

The **Morse potential** is a convenient model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands.

The **Lennard-Jones potential** is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. A form of the potential was first proposed in 1924 by John Lennard-Jones.

The attractive long-range potential, however, is derived from dispersion interactions. The L-J potential is a relatively good approximation and due to its simplicity is often used to describe the properties of gases, and to model dispersion and overlap interactions in molecular models. It is particularly accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules.

The** Landau-Teller Potential **is a simple model for these inelastic collisions. The potential in the Hamilatonian is of the form

The dynamics are then given by the equations of motion. Classically, the solutions are trajectories along the potential energy surface with no saddle point. One can, however, parametrize the translation between the two colliding species and treat the vibrations in the diatomic molecule quantum dynamically. Semiclassical methods like the Born approximation may then be used to analyze modes of energy transfer during the inelastic collision.

When enough energy is involed, a bundle may lose or gain an atom from a collision. For example, consider an atom and a diatomic molecule colliding as seen in the reaction formula below.

The same questions from above may be asked about this physical process. However, we now have two systems, reactants and products. A saddle point appears on the potential energy surface thus parametrizing the translation is trickier. To see more go to Application.

Tim Wendler timoth500@yahoo.com

Manuel Berrondo Jean-Francois Van Huele J. Ward Moody Scott Bergesen Gus Hart

Manuel Berrondo Jean-Francois Van Huele J. Ward Moody Scott Bergesen Gus Hart