A **molecular vibration** occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 10^{13} to approximately 10^{14} Hz, corresponding to wavenumbers of approximately 300 to 3000 cm^{−1}.

## Contents

- Vibrational coordinates
- Internal coordinates
- Vibrations of a methylene group CH2 in a molecule for illustration
- Symmetry adapted coordinates
- Normal coordinates
- Newtonian mechanics
- Quantum mechanics
- Intensities
- References

In general, a molecule with *N* atoms has 3*N* – 6 normal modes of vibration, but a *linear* molecule has 3*N* – 5 such modes, because rotation about its molecular axis cannot be observed. A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds.

A molecular vibration is excited when the molecule absorbs a quantum of energy, *E*, corresponding to the vibration's frequency, *ν*, according to the relation *E* = *hν* (where *h* is Planck's constant). A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, because the potential energy of the molecule is more like a Morse potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra

## Vibrational coordinates

The coordinate of a normal vibration is a combination of *changes* in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

## Internal coordinates

*Internal coordinates* are of the following types, illustrated with reference to the planar molecule ethylene,

_{3}when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethene there are 12 internal coordinates: 4 C-H stretching, 1 C-C stretching, 2 H-C-H bending, 2 CH_{2} rocking, 2 CH_{2} wagging, 1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time.

## Vibrations of a methylene group (-CH2-) in a molecule for illustration

The atoms in a CH_{2} group, commonly found in organic compounds, can vibrate in six different ways: **symmetric and asymmetric stretching**, **scissoring**, **rocking**, **wagging** and **twisting** as shown here:

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

## Symmetry-adapted coordinates

Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four(un-normalised) C-H stretching coordinates of the molecule ethene are given by

where

Illustrations of symmetry-adapted coordinates for most small molecules can be found in Nakamoto.

## Normal coordinates

The normal coordinates, denoted as *Q*, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The advantage of working in normal modes is that they diagonalize the matrix governing the molecular vibrations, so each normal mode is an independent molecular vibration, associated with its own spectrum of quantum mechanical states. If the molecule possesses symmetries, it will belong to a point group, and the normal modes will "transform as" an irreducible representation under that group. The normal modes can then be qualitatively determined by applying group theory and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO_{2}, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch.

*Q = q*

_{1}+ q_{2}*Q = q*

_{1}- q_{2}When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined *a priori*. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

- principally C-H stretching with a little C-N stretching; Q
_{1}= q_{1}+ a q_{2}(a << 1) - principally C-N stretching with a little C-H stretching; Q
_{2}= b q_{1}+ q_{2}(b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.

## Newtonian mechanics

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a *force constant, k*. The anharmonic oscillator is considered elsewhere.

By Newton’s second law of motion this force is also equal to a reduced mass, *μ*, times acceleration.

Since this is one and the same force the ordinary differential equation follows.

The solution to this equation of simple harmonic motion is

*A* is the maximum amplitude of the vibration coordinate *Q*. It remains to define the reduced mass, *μ*. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, *m _{A}* and

*m*, as

_{B}The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy.

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies,*ν*_{i} are obtained from the eigenvalues,*λ*_{i}, of the matrix product * GF*.

*is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.*

**G***is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.*

**F**## 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

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
*n* to level *n+1* due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency

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,

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.

## Intensities

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. The intensity of Raman bands depends on polarizability.