Anharmonic Oscillations of Chemical Bonds

Simple harmonic oscillator of a ball rolling in a dish Simple harmonic oscillator
Animated gif of a molecular bond oscillating between repulsive and attractive electrostatic forces. An anharmonic oscillator

Simple harmonic oscillators about a potential energy minimum can be thought of as a ball rolling frictionlessly in a dish (left) or a pendulum swinging frictionlessly back and forth. The restoring forces are precisely the same in either horizontal direction.

All bonds that hold matter together oscillate about a potential energy minimum (figure to the right) between the electrostatic forces that attract atoms and molecules together (on the right side of this figure) and the forces that repel them apart when they get too close (on the left side of this figure). Chemical bonds are often though of as equivalent to springs shown schematically here in the lower left. But the restoring forces for chemical bonds are not precisely symmetric, so the bond is an anharmonic oscillator that can be approximated very closely by a harmonic oscillator.

Chemical bonds have many degrees of freedom, which means they can oscillate in many ways. Atoms in a CH2 group, commonly found in organic compounds, can vibrate in six different ways shown below: symmetric and asymmetric stretching, scissoring, rocking, wagging and twisting:

Bonds of a carbon dioxide molecule stretching symetrically Symmetrical stretching
Bonds of a carbon dioxide molecule stretching asymetrically Asymmetrical stretching
Bonds of a carbon dioxide molecule scissoring. Scissoring
Bonds of a carbon dioxide molecule rocking. Rocking
Bonds of a carbon dioxide molecule wagging. Wagging
Bonds of a carbon dioxide molecule twisting. Twisting

Each degree of freedom has normal modes of oscillation determined by the masses of the atoms or molecules involved and the length and strength of the chemical bond. These can be compared to the fundamental and overtones of a vibrating string where each overtone has a higher frequency and therefore higher energy. In the figure top right, the x-axis is the distance across the bond oscillating around ro. The y-axis is the energy which we know from the Planck postulate is equal to the frequency times a constant. De is the energy (frequency) required to dissociate the bond. We know from experiment that it takes much more energy (De) to photoionize nitrogen (N2) into two nitrogen ions (N) than to photodissociate oxygen (O2) into two oxygen atoms (O).

The frequencies involved are very large. The frequency of oscillation of green light. in the middle of the visible spectrum is 5.6 x 1014 cycles per second, that is 560,000,000,000,000 cycles per second! When you heat a body of matter, most of the chemical bonds begin oscillating in the frequency range of 10 to 1000 terahertz (1012 cycles per second) depending on the temperature. Heat might be caused by the "friction" between molecules and atoms jossling against each other at these very high frequencies.

Top left image from Wikia: Astronomy. All other images from Wikipedia: Molecular vibration.

Last updated 05-Dec-2015    © 2015 Peter L. Ward. All Rights Reserved