# Ground State of a Diatomic Molecule

## Introduction

Two atoms that can form a bond will do so to create a diatomic molecule when they approach each other closely. We describe the energy minimum with a potential energy curve, called a potential well.

## Morse Potential

The potential energy, V(R), of a diatomic molecule can be described by the Morse potential:

where D_{e} is the well depth, R is internuclear distance, R_{e} is the equlibrium internuclear distance (bond length), and

_{e} is the vibrational constant and µ is the reduced mass.

The following figure shows the ground state potential well of the H_{2} molecule. The curve is calculated from the Morse potential and the energy levels are calculated using the harmonic oscillator model with the first anharmonic correction.

D_{o} is the dissociation energy, which is slightly different from the well depth, D_{e}. The vibrational energy levels in this plot are calculated using the harmonic oscillator model:

where v is the vibrational quantum number, v = 0,1,2,..., and *x*_{e} and *y*_{e} are the first and second anharmonicity constants, respectively. The v = 0 level is the vibrational ground state.

Copyright © 2000 by Brian M. Tissue, all rights reserved.