# Overtone

## Overtone and Fundamental Band

The infrared spectrum of a molecule results due to the transition between two different vibrational energy levels. The vibrational energy E_{vib.}= (v + 1/2)h𝜈.

where, v is the number of the vibrational level and can have the values 0,1,2,3,... etc. h is Plank constant and 𝜈 is the vibrational frequency of the band.

The energy difference between the two vibrational energy levels can be written as-

ΔE

_{vib.}= h𝜈.

The transition from ground state (v = 0) to the first excited state (v = 1) is called

**fundamental band**. The energy difference between any two adjacent energy levels either up or down is always same i.e. (h𝜈).

The transition from ground state (v = 0) to the second excited state (v = 2) with the absorption of IR radiation give rise to weak bands, called

**overtone**. Thus the energy difference between two levels can never be equal to h𝜈. If the transition occurs from v = 0 to v = 2 then-

ΔE

_{vib.}= E

_{vib.(v=2)}− E

_{vib.(v=0)}

or, ΔE

_{vib.}= (2 + 1/2)h𝜈 − (0 + 1/2)h𝜈

or, ΔE

_{vib.}= (5/2)h𝜈 − (1/2)h𝜈

or, ΔE

_{vib.}= 2h𝜈

This is the energy of first overtone.

In other words we can say the the transition which does not follow the selection rule (i.e. ΔV = ±1) is called overrtone.

The value of overtone is always greater than that of fundamental bands.

The transition from v = 0 to v = 2 is called first overtone. Similarly, transition from v = 0 to v = 3, 4, 5... are called second, third, fourth ... overtone respectively.

The fundamental transitions, v = ±1, are the most commonly occurring, and the probability of overtones rapid decreases as the number of quanta (Δv = ±n) increases. Based on the harmonic oscillator approximation, the energy of the overtone transition would be n times larger than the energy of the fundamental transition frequency, but the anharmonic oscillator calculations show that the overtones are less than a multiple of the fundamental frequency.

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