Page 336 - Instant notes

I4
VIBRATIONAL SPECTROSCOPY

Key Notes
In a harmonic oscillator the magnitude of the restoring force is
directly proportional to the displacement from equilibrium and
the resulting potential energy is proportional to the square of the
displacement. At low vibrational energies, the vibration of a
molecular bond can be approximated to that of a harmonic
oscillator.
The energy of molecular vibration for the potential energy of a
harmonic oscillator is quantized according to E v =(υ+½)hν, where

υ is the vibrational quantum number and is the
frequency of the oscillator. The parameters k and µ are the force
constant and the reduced mass of the vibration.
A molecular vibration is active in absorption or emission
spectroscopy only if the electric dipole of the molecule changes
during the vibration. The specific selection rule requires ∆v=±1.
All allowed transitions of a harmonic oscillator of frequency, ν,
have energy hν.
Related topics The wave nature of matter (G4) Practical aspects of
spectroscopy (I2)
General features of Applied vibrational
spectroscopy (I1) spectroscopy (I5)

The harmonic oscillator

The general shape of the potential energy curve for the stretching or bending of a
molecular bond is shown in Fig. 1. The bottom of the potential energy well occurs at the
equilibrium bond length R e. The potential energy rises steeply for bond lengths R<R e
because of strong repulsion between the positively-charged nuclei of the atoms at each
end of the bond (Topic H6.). Potential energy rises as the atoms move further apart
because of molecular bond distortion. Complete dissociation of the bond occurs as R→∞.
Near the bottom of the well, for small perturbations from R e, the restoring force
experienced by the atoms can be assumed to be directly proportional to the bond length
displacement:
restoring force=−k(R−R e)