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. Number of vibrational modes For a molecule with atoms, the positions of all nuclei depend on a total of 3 coordinates, so that the molecule has 3 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 rotational coordinate. The number of vibrational modes is therefore 3 minus the number of translational and rotational degrees of freedom, or 3–5 for linear and 3–6 for nonlinear molecules. Vibrational coordinates 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 , the frequency of the vibration. Internal coordinates 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