A **molecular vibration** is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical **vibrational frequencies**, range from less than 10^{13} Hz to approximately 10^{14} Hz, corresponding to wavenumbers of approximately 300 to 3000 cm^{−1}.

In general, a non-linear molecule with *N* atoms has 3*N* – 6 normal modes of vibration, but a *linear* molecule has 3*N* – 5 modes, because rotation about the molecular axis cannot be observed.^{[1]} A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond. Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule.

A molecular vibration is excited when the molecule absorbs energy, *ΔE*, corresponding to the vibration's frequency, *ν*, according to the relation *ΔE* = *hν*, where *h* is Planck's constant. A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its ground state. When multiple quanta are absorbed, the first and possibly higher overtones are excited.

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 or more accurately, a Morse/Long-range 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.

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.^{[8]}

By Newton's second law of motion

Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto.^{}[6]

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 normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" an irreducible representation under its point group. The normal modes are 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:

- symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary.
*Q = q*_{1}+ q_{2} - asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the o
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.

^{[7]}## Newtonian mechanics

k Q {\displaystyle \mathrm {Force} =-kQ\!} By Newton's second law of motion this force is also equal to a reduced mass,

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

- principally C–H stretching with a little C–N stretching; Q