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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 1013 Hz to approximately 1014 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. 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...

For a molecule with N atoms, the positions of all N nuclei depend on a total of 3 N coordinates, so that the molecule has 3 N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates. A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates. An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third r...

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 edit Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene, Stretching: a change in the length of a bond, such as C–H or C–C Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule. Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule, Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups. Out–of–plane: a change in the angle between any one of the C–H bonds and the plane defi...

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. F o r c e = − 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. F o r c e = μ d 2 Q d t 2 {\displaystyle \mathrm {Force} =\mu {\frac {d^{2}Q}{dt^{2}}}} Since this is one and the same force the ordinary differential equation follows. μ d 2 Q d t 2 + k Q = 0 {\displaystyle \mu {\frac {d^{2}Q}{dt^{2}}}+kQ=0} The solution to this e...

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 E n = h ( n + 1 2 ) ν = h ( n + 1 2 ) 1 2 π k m {\displaystyle E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over 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 h ν {\displaystyle h\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...