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Molecule as a Harmonic Oscillator

The near infrared spectrum are obtained due to vibrations of the nuclie in the diatomic molecule along the inter nuclear axis.
This simplest possible assumption about the form of vibration is to treat the molecule as a harmonic oscilator. 
Making this substitution, in eq. (i) ,we get :
This gives the allowed energies for harmonic oscilator. 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 :
Thus, we have a series of equispaced discrete vibrational levels with common separation  ω.
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 :
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

Subsituting the values in above equation :
Thus, the vibrational spectrum is expected to consists of a single band of  ω(cm^-1).

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