Quantum Mechanics Tutorial: Vibration Modes of Polyatomic Molecules.

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Degrees of freedom:

In this context, "degree of freedom" means an unique way of a molecule to increase its kinetic energy. There are degrees of freedom of translation, of rotation, and of vibration. Each molecule has 3N degrees of freedom, where N is the number of atoms in this molecule. This number remains constant, even if the molecule is broken up into fragments, although the distribution in translational, rotational, and vibrational degrees of freedem may change. A single molecule, for example, has three degrees of translation, corresponding to the three dimensions in space. If the molecule dissociates, each of the fragments can move independently, so each of the fragments has three degrees of translation. If there are two fragments, the sum of translational degrees of freedom is six. Similarly, the total number of rotational degrees of freedom will change, and the total number of vibrational degrees of freedom has to decrease to keep the overall sum of 3N constant.
 
The sum of 3N degrees of freedom is easy to understand: if the molecule dissociates completely in N individual atoms, each of these atoms has three degrees of freedom for translation, but no chance to perform rotations or vibrations, since these movements require chemical bonds between the atoms.
 
The total number of 3N degrees of freedom decomposes as follows:

• non-linear molecules have
  • 3 degrees of freedom of translation,
  • 3 degrees of freedom of rotation, and
  • 3N-6 degrees of freedom of vibration;
• linear molecules have
  • 3 degrees of freedom of translation,
  • 2 degrees of freedom of rotation, and
  • 3N-5 degrees of freedom of vibration.

Diatomic molecules

are always linear, therefore they have 3 degrees of freedom of translation (corresponding to the three dimensions in space), 2 degrees of freedom of rotation (perpendicular to the molecular axis), and one degree of freedom of vibration (along the chemical bond).
 

Triatomic molecules

can be linear or non-linear, symmetric or asymmetric.

• Non-linear molecules have 3 degrees of freedom of translation (corresponding to the three dimensions in space), 3 degrees of freedom of rotation (also corresponding to the three dimensions in space), and 3N-6 = 3 degrees of freedom of vibration (in the asymmetric case ABC a stretch vibration for each bond and a deformation vibration of the angle A-B-C; the latter usually has lower frequency).
• Linear molecules have 3 degrees of freedom of translation (corresponding to the three dimensions in space), 2 degrees of freedom of rotation (perpendicular to the molecular axis), and 3N-5 = 4 degrees of freedom of vibration (in the asymmetric case ABC one stretch vibration for each bond and two deformation vibrations that change the angle A-B-C in two perpendicular planes; these two angle vibrations are degenerate in the absence of an external field, i.e., they respond to the same frequency).
• Symmetric molecules A₂B do not have individual stretch vibrations of the two bonds, but coupled movements: one symmetric stretch vibration, in which both bonds are elongated and compressed at the same time, and an asymmetric stretch vibration, in which one bond compresses while the other is elongated. The asymmetric combination always has a larger frequency than the symmetric one.

Molecules with more than three atoms

have an increasing number of vibrational degrees of freedom, which include torsions, movements of fragments relative to each other, and so on.
 

 

For symmetric molecules,

the classification of the vibrations according to irreducicle representations of the respective point group is of relevance, because only those vibrations can be observed in the IR spectrum, which transform like a vector or a component (x, y, or z) of a vector. The reason is that the essential property for the selection rule is the dipole moment, which is a vector. In contrast, for Raman spectra the polarizability is the essential property, which is a tensor with components (x², y², z², xy, xz, or yz).
 
Depending on symmetry, these limitations can make it easy or difficult to characterize molecular shape from spectroscopic results. Highly symmetric molecules may be extremely difficult; for example, C₆₀ with icosahedral symmetry, has a total of 174 vibration modes, but only four of them are IR-active! In contrast, the choice between point groups with different activities in IR and Raman often make molecular shape determination easy by quickly ruling out alternatives.
 
More on point group symmetry:
  Tutorial Group Theory,
  Web-Vorlesung Klassische Algebra, Universität Freiburg.

 Postulates.
 Harmonic oscillator.
 Anharmonic oscillator.
 Rovibrational spectra.
 
 Dr. Michael Ramek.