Normal Modes of Vibration

Vibrations in molecules can be excited by the application of radiation at the same frequency as the vibration itself. This gives rise to spectroscopic methods of studying molecular structure, which depend upon being able to predict or at least rationalise the vibrations that are observed. Symmetry considerations play an important role in this.

Note that there is only one possible mode of vibration for a diatomic molecule, stretching (or compression) of the bond. No bending modes exist, as neither end of the molecule has a fixed position that is defined by being attached to another atom.

For polyatomic molecules there are multiple vibrational modes, because all of the bonds may stretch or compress, and bond angles may also change, allowing the existence of bending modes.

For a non-linear molecule containing N atoms, there are 3N – 6 independent vibrational modes. If the molecule is linear, there are 3N – 5 modes of vibration:

Each atom may be displaced in any one of three directions (x, y, z), giving 3N possible displacements. Three coordinates are required to specify the location of the centre of mass of the molecule (fixing the location of the molecule in space). Further, three angles are needed to specify the orientation of a non-linear molecule in space. These six pieces of information restrict the possible vibrational modes to 3N – 6.

(Alternatively, the situation can be viewed as follows. There are 3N possible vibrational modes. However, if all the atoms are vibrating so that they move in the positive x direction, then this is actually a translation of the whole molecule through space in the x direction, with no change in the bond lengths or angles of the molecule. This does not count as a vibration. Similar remarks can be made about translations in the y and z directions. This reduces the number of vibrational modes by three. Similarly, it is possible for the atoms to vibrate together in such a way that the molecule is actually rotating rather than vibrating. Rotation can occur about any of the three Cartesian axes, so a further three of the possible vibrations are counted as rotations not vibrations. Thus of the 3N possible vibrations, 3 are translations, 3 are rotations and 3N – 6 are vibrations.)

For a linear molecule, only two angles are required to specify the orientation in space (the infinite rotational symmetry about the molecular axis means that all possible orientations about this axis are equivalent), hence there are 3N – 5 vibrations. (In terms of the alternative view of the situation, no vibrational motion of the atoms in a linear molecule can lead to rotation about the axis of the molecule, so one of the rotational modes is not counted.)

It is necessary for more advanced calculations to find a way of describing the modes (in terms of the different atomic motions that they represent). It proves simplest to do this in terms of the normal modes of a molecule. Normal modes are the independent, synchronous motions of atoms or groups of atoms that may be excited without exciting other normal modes. For example, the symmetric and antisymmetric stretches of CO2 , in which the oxygen atoms are displaced while the carbon atom remains stationary, are two of the normal modes of the molecule:

However, stretches of the two individual C=O bonds are not normal modes: When one C=O bond vibration is excited, the motion of the carbon atom causes the other C=O bond to stretch or compress, exciting the other vibrational mode.

Normal modes may be treated as independent harmonic oscillators (assuming that the anharmonicity of the vibrations is neglected). Note that the effective mass of a vibrational mode is a measure of the mass moved in the course of the vibration, and it is in general a highly complicated function of the masses of the atoms involved. It should further be noted that a case such as carbon dioxide, in which all the normal modes are purely stretching or bending is rare. More usually, a normal mode is a motion formed from simultaneous bending and stretching.