The infrared (IR) spectrum of a molecule can arise from transitions between different vibrational energy levels. For a harmonic oscillator, the vibrational energy of a molecule is given by:
where v is the vibrational quantum number and can have the values 0, 1, 2, 3, ...; h is Planck's constant; and ν is the fundamental vibrational frequency.
Fundamental Vibrational Transition
In the harmonic oscillator approximation, the energy difference between two adjacent vibrational energy levels is:
The transition from the ground vibrational state (v = 0) to the first excited vibrational state (v = 1) is called the fundamental transition or fundamental band.
The fundamental vibrational selection rule for an ideal harmonic oscillator is:
Thus, adjacent vibrational energy levels are separated by the same energy, hν, in the harmonic oscillator model.
What is an Overtone?
An overtone is a vibrational band that results from a transition in which the change in vibrational quantum number is greater than one, such as Δv = ±2, ±3, etc.
For example, the transition from the ground state v = 0 to the second excited state v = 2 gives rise to the first overtone.
Similarly, the transitions from v = 0 to v = 3, 4, 5, ... give the second, third, fourth, ... overtones, respectively.
Energy of the First Overtone
For a harmonic oscillator, the energy of the vibrational level with quantum number v is:
For the first overtone, the transition occurs from v = 0 to v = 2. Therefore:
E2 = (2 + 1/2)hν = 5/2 hν
E0 = (0 + 1/2)hν = 1/2 hν
ΔE = 5/2 hν - 1/2 hν
ΔE = 2hν
Thus, in the harmonic oscillator approximation, the energy required for the first overtone transition is 2hν.
Fundamental and Overtone Bands
| Transition | Band | Δv | Harmonic Approximation |
|---|---|---|---|
| v = 0 → 1 | Fundamental | ±1 | ν |
| v = 0 → 2 | First overtone | ±2 | 2ν |
| v = 0 → 3 | Second overtone | ±3 | 3ν |
| v = 0 → 4 | Third overtone | ±4 | 4ν |
Role of Anharmonicity
The harmonic oscillator is an idealized model. Real molecular vibrations are not perfectly harmonic. At higher vibrational energy levels, the vibrational energy levels become progressively closer together than predicted by the harmonic oscillator model.
Anharmonicity makes transitions such as Δv = ±2, ±3, ... weakly allowed. These transitions produce overtone bands in vibrational spectra.
In the harmonic oscillator approximation, the first overtone would occur at twice the fundamental frequency, the second overtone at three times the fundamental frequency, and so on. In real molecules, however, overtone bands occur at somewhat lower frequencies or wavenumbers than these exact integer multiples because of anharmonicity.
Intensity of Overtone Bands
Fundamental bands are generally much more intense than overtone bands. The intensity of overtone bands usually decreases rapidly as the magnitude of Δv increases.
Therefore, first overtones are generally stronger than second overtones, while higher overtones become progressively weaker and may be difficult to observe.
Summary
The transition from v = 0 to v = 1 is the fundamental vibrational transition. Transitions such as v = 0 to v = 2, v = 0 to v = 3, and higher transitions are called overtones. In an ideal harmonic oscillator, only Δv = ±1 transitions are allowed. Because real molecular vibrations are anharmonic, transitions with Δv = ±2, ±3, ... can acquire weak intensity and appear as overtone bands.