Point Groups

Point groups are a method of classifying the shapes of molecules according to their symmetry elements.

Before we can talk about point groups, we need to describe the basic elements of symmetry. These are the proper axis of symmetry (or just axis of symmetry), improper axis of symmetry, plane of symmetry, and inversion center (or point of symmetry).

• The proper axis of symmetry is an imaginary line through a compound. Rotation of the compound by an integral fraction of a circle around this axis (1/2, 1/3, etc.) brings the compound to superposition on itself. An axis of symmetry is represented by C_n, where n is the integral fraction. All shapes have at least a C₁ axis (the identity axis).

• The improper axis of symmetry is also an imaginary line through a compound. Rotation of the compound by an integral fraction of a circle around this axis (1/2, 1/4, etc.) followed by reflection through a plane perpendicular to this axis brings the compound to superposition on itself. An improper axis of symmetry is represented by S_n, where n is the integral fraction (n must be even or 1).

• The plane of symmetry is a plane through a compound that relates its identical halves. A plane of symmetry is represented by σ and is the same as an S₁ axis.

• The inversion center (or point of symmetry) is an imaginary point in a compound. Reflection of the compound through this point brings the compound to superposition on itself. An inversion center is represented by i and is the same as an S₂ axis.

The main classes of point groups are C, D, S, T, O, and I. The first two classes are most common. Each of these classes is subdivided into different point groups.

• Compounds in the C class can be C_s, C_i, C_n, C_nv, or C_nh, where n is an integer.

  • Compounds in the C_s point group have a plane of symmetry and nothing more.

  • Compounds in the C_i point group have a center of inversion and nothing more.

  • Compounds in the C_n point group have exactly one C_n axis and nothing more. Compounds in this point group are always chiral. Examples.

    Compounds in the C₁ point group have only the identity axis; in other words, they have no symmetry. Most compounds have C₁ symmetry.

  • Compounds in the C_nv point group have a C_n axis and n planes of symmetry containing the C_n axis. They do not have a plane of symmetry perpendicular to the C_n axis. Examples.

    Linear molecules with conical symmetry (CO, ClC≡CH) are classified in the C_∞v point group.

  • Compounds in the C_nh point group have a C_n axis and a plane of symmetry perpendicular to the C_n axis. Examples.

• Compounds in the D class contain a C_n axis and n C₂ axes perpendicular to it. Compounds in the D class can be D_n, D_nd, or D_nh, where n is an integer.
  • Compounds in the D_n point group have no additional elements of symmetry. Compounds in this point group are always chiral. This point group is not common. Examples.

  • Compounds in the D_nd point group have n planes of symmetry containing the C_n axis. They do not have a plane of symmetry perpendicular to the C_n axis. Examples.

  • Compounds in the D_nh point group have a plane of symmetry perpendicular to the C_n axis. Examples.

    Linear molecules with cylindrical symmetry (N₂, HC≡CH) are classified in the D_∞h point group.

• Compounds in the S class are S_n, where n is an even integer. Compounds in the S class contain a S_n axis plus a C_n/2 axis coinciding with it.

• Compounds in the T (tetrahedral) class can be T, T_d, or T_h. Compounds in the T class contain four C₃ axes and three C₂ axes.

  • Compounds in the T point group have no additional elements of symmetry. Compounds in this point group are always chiral. This point group is not common. The compound C(X_S)₄, where X_S is a group with a stereocenter with the (S) configuration, is in this point group.

  • Compounds in the T_d point group have three planes of symmetry that contain the C₂ axes. This point group is not common. An octasubstituted cubane in which each vertex is alternately substituted with X_R and X_S is in this point group.

  • Compounds in the T_h point group have six planes of symmetry that contain the C₃ axes. Compounds such as CH₄ are in this point group.

• Compounds in the O (octahedral) and I (icosahedral) classes have even higher symmetry. These point groups are not common. PF₆^– is in the O_h point group, and C₆₀ is in the I_h point group.

You should also remember that most compounds are conformationally mobile (i.e., constantly changing their shape), and, as a result, the point group of a compound can depend on the time scale. For example, at a very short time scale, 1-propyne (HC≡CCH₃) is in the C_3v point group, but at longer time scales, rapid rotation about the C–CH₃ bond puts it in the C_∞v point group. Cyclohexane is in the D_3d point group at short time scales, when it is in a single chair form, but at longer time scales, at which it is in rapid equilibrium between its two chair forms, it is in the D_6h point group.

Additional vocabulary help

You might want to look at some definitions of stereoisomers.

Or you might want to look at the stereochemical glossary.

Or you might want to look at some examples of the different kinds of stereoisomers.

Or you might want to look at a flow chart showing how to determine the isomeric relationship between two structures.