Schoenflies notation

The Schoenflies notation is one of two conventions commonly used to describe crystallographic point groups. This notation is used in spectroscopy. The other convention is the Hermann-Mauguin notation, also known as the International notation. A point group in the Schoenflies convention is completely adequate to describe the symmetry of a molecule; this is sufficient for spectroscopy. The Hermann-Maunguin notation is able to describe the space group of a crystal lattice, while the Schoenflies notation isn't. Thus the Hermann-Maunguin notation is used in crystallography.

Symmetry elements

Symmetry elements are denoted by i for centers of inversion, C for proper rotation axes, and σ for mirror planes, and S for improper rotation axes (rotation-reflection axes). C and S are usually followed by a subscript n denoting the order of rotation possible.

By convention, the axis of proper rotation of greatest order is defined as the principle axis. All other symmetry elements are described in relation to it. Thus, mirror planes are denoted σ_v or σ_h for vertical mirror planes (parallel with the principal axis) and horizontal mirror planes (perpendicular with the principal axis).

Point groups

In three dimensions, there are 32 crystallographic point groups.

• The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (O_h) or without (O) improper operations (those that change handedness).
• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper operations, T excludes improper operations, and T_h is T with the addition of an inversion.
• C_n (for cyclic) indicates that the group has an n-fold rotation axis. C_nh is C_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C_nv is C_n with the addition of a mirror plane parallel to the axis of rotation.
• S_n (for Spiegel, German for mirror) denotes a group that contains only an n-fold rotation-reflection axis.
• D_n (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus a twofold axis perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nv has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.

Due to the crystallographic restriction theorem, n is restricted to the values of 1, 2, 3, 4, or 6.