Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by

${\displaystyle E_n=h\left(n+\frac{1}{2}\right)\mathrm{\nu <!--\; \nu \; -->}=h\left(n+\frac{1}{2}\right)\frac{1}{2\mathrm{\pi <!--\; \pi \; -->}}\sqrt{\frac{k}{m}}}$,

where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.

The difference in energy when n (or v) changes by 1 is therefore equal to ${\displaystyle h\mathrm{\nu <!--\; \nu \; -->}}$, the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency ${\displaystyle \mathrm{\nu <!--\; \nu \; -->}}$ (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,

${\displaystyle \mathrm{\Delta <!--\; \Delta \; -->}n=\pm <!--\; \pm \; -->1}$

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of an anharmonic oscillator, Dunham expansion is used.

Intensitiesedit

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate. Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.