Page 338 - Instant notes

Physical Chemistry 324

Vibrational energy levels

Vibrational energy, like all other molecular energy, is quantized (see Topic G4). The
permitted values of vibrational energy are obtained by solving the Schrödinger equation
for the motion of two atoms possessing the harmonic oscillator potential energy
2
V=½k(R−R e) . The allowed energy levels are:

where ω=√k/µ is the circular frequency of the oscillator (in units of radians per second),
and µ=m 1m 2/(m 1+m 2) is the reduced mass of the two atoms. The circular frequency is
−1
related to the frequency ν (in units of s , or hertz) by ω=2πν, so the permitted vibrational
energy levels can also be written as:

The integer υ is called the vibrational quantum number. The energy levels of a
harmonic oscillator are evenly spaced with separation hν (or ħ ), as shown in Fig. 2.
Note that the magnitude of the vibrational energy levels depend on the reduced mass
of the molecule, not the total mass. If one mass greatly exceeds the other (e.g.
) the reduced mass is approximately equal to the lighter mass, µ≈m 1. In effect, the center
of gravity is so close to the heavy mass that the light mass vibrates relative to a stationary
anchor, e.g. the H atom in HI.