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In vibrational spectroscopy, an overtone band is the spectral band that occurs in a vibrational spectrum of a molecule when the molecule makes a transition from the ground state (v=0) to the second excited state (v=2), where v is the vibrational quantum number that one gets after solving the Schrödinger equation for the molecule under consideration. It takes only non-negative integer values.
Generally, in order to study the vibrational spectra of molecules, the vibration of the chemical bonds is assumed to be simple harmonic.Thus the parabolic simple harmonic potential is used in the Schrödinger equation to solve for the vibrational energy eigenstates. When the Schrödinger equation is solved one gets the functional forms of the vibrational energy eigenstates. These energy states are found to be quantized, meaning they can assume only some "discrete" values of energy. When electromagnetic radiation is shined on a sample of a molecule, the molecules can absorb energy from the electromagnetic radiation and change their vibrational energy state. However, the molecules can absorb energy from radiation only under certain condition, namely- there should be a change in the electric dipole moment of the molecule when it is vibrating. This change in the electric dipole moment of the molecule leads to the transition dipole moment of the molecule, for transition from the lower to higher energy state, being non-zero which is an essential condition for any transition (process, change) to take place in the vibrational state of the molecule, as is the law of nature, explained thoroughly in Quantum mechanics.
It has been found that, when the bonds are considered vibrating simple-harmonically, the transition dipole moment is non-zero only for the transition where ∆v=±1. Hence; for the ideal, non-existent, simple-harmonically vibrating bond; there can never be any overtone observed in the vibrational spectrum. But, real molecules do not vibrate simple-harmonically. The potential function vibration is not parabolic, simple-harmonic potential, but is better approximated by the Morse potential. When Schrödinger equation is solved for the molecule under consideration with Morse potential as the potential function, one gets the functional forms of the vibrational energy eigenstates with the interesting property that when one calculates transition dipole moment corresponding to transitions from various energy levels to various other energy levels, the transition dipole moment does not become zero for the transitions where ∆v=±2,±3,±4, etc. Thus, for real molecules, the allowed transitions are those for which ∆v=±1,±2,±3,±4, etc. The overtone band observed in the IR spectrum is one such transition with ∆v=2, from v=0 to v=2 energy state.
1. C.N.Banwell and E.M.McCash:Fundamentals of Molecular Spectroscopy, Tata McGraw-Hill Fourth Edition
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