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 defined by the remaining atoms of the ethylene molecule. Another example is in BF3 when the boron atom moves in and out of the plane of the three fluorine atoms. In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane. In ethylene there are 12 internal coordinates: 4 C–H stretching, 1 C–C stretching, 2 H–C–H bending, 2 CH2 rocking, 2 CH2 wagging, 1 twisting. Note that the H–C–C angles cannot be used as internal coordinates as well as the H-C-H angle because the angles at each carbon atom cannot all increase at the same time. Note that these coordinates do not correspond to normal modes (see #Normal coordinates). In other words, they do not correspond to particular frequencies or vibrational transitions. Vibrations of a methylene group (–CH2–) in a molecule for illustration 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. These are shown here: (These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms). Symmetry–adapted coordinates Symmetry–adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four (un–normalized) C–H stretching coordinates of the molecule ethene are given by where are the internal coordinates for stretching of each of the four C–H bonds. Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto. Normal coordinates The normal coordinates, denoted as Q, refer to the positions of atoms away from their