Molecular vibration

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.

[1] A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond.

A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its ground state.

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.

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.

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

Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates.

[2][3] 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.

[2][3][4] The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule.

Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

Within the CH2 group, commonly found in organic compounds, the two low mass hydrogens can vibrate in six different ways which can be grouped as 3 pairs of modes: 1. symmetric and asymmetric stretching, 2. scissoring and rocking, 3. wagging and twisting.

[5] The projection operator is constructed with the aid of the character table of the molecular point group.

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, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.

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

In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate.

The vibration frequencies, νi, are obtained from the eigenvalues, λi, of the matrix product GF.

[9] 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

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.[10][11] The difference in energy when n (or v) changes by 1 is therefore equal to

, the product of the Planck constant and the vibration frequency derived using classical mechanics.

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,

When it comes to polyatomic molecules, it is common to solve the Schrödinger Equation using Watson's nuclear motion Hamiltonian.

For the anharmonic calculation of vibrational spectra of polyatomic molecules, more sophisticated approaches are used.

[13] 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.

[14] Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate.