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Molecule as a Harmonic Oscillator

The molecule as a harmonic Oscillator:

The near-infrared spectrum is obtained due to vibrations of the nuclei in the diatomic molecule along the internuclear axis.
This simplest possible assumption about the form of vibration is to treat the molecule as a harmonic oscillator.
Molecule as a harmonic Oscillator
Making this substitution, in eq. (i) , we get :
Molecule as a harmonic Oscillator
This gives the allowed energies for the harmonic oscillator. is called vibrational quantum number which can take integral values :
v = 0, 1, 2, 3, …………
 
A special feature of quantum -mechanical oscillator is the existence of zero-point energy.

Spectrum: 

Let us now investigate that expected spectrum of such an oscillator. The vibrational term ( energies in wave number unit m^-1 or cm^-1) are :
Molecule as a harmonic Oscillator
Thus, we have a series of equispaced discrete vibrational levels with common separation ω.
Molecule as a harmonic Oscillator
Fig. Discrete vibrational levels
When a transition takes place between an upper-level v’ and at a lower level v”, the wave number of the emitted or absorbed radiation is given by :
Molecule as a harmonic Oscillator

The selection rule (Δv = ±1)  gives :

                               v’ = v” + 1
 
Most of the electrons remains in ground state, i.e. the state corresponding to v = 0 
Therefore, the main vibrational absorption transition is from v = 0 to v = 1 ,i.e. lower level v” = 0 and upper level v’ = +1. Thus, the vibrational spectrum is expected to consists of a single band of ω (cm^-1).
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