Symmetry elements and operations are prerequisites for the discussion of the symmetry of molecules. This sections gives a short description of the basic terminology.

• Symmetry Elements and Operations
• Equivalence of Symmetry Operations
• Combination of Symmetry Operations

The symmetry of molecules is important for understanding the structures and properties of organic compounds. In chemistry, many phenomena may be easily explained by consideration of symmetry. The systematic discussion of symmetry is called group theory. the fundamental principles of symmetry analysis are explained in this section.

Atoms or groups in molecules are called symmetry equivalent if they have the same atomic number or chemical constitution, and if they can be transposed into each other by the application of symmetry operations (see below). This equivalency plays an important role in the chemical properties of these atoms or groups, and in the process of stereo differentiation of these positions.

Symmetry operations are geometrical operations applied to molecules (or better, molecular models). The application of a symmetry operation to a particular molecular geometry produces a spatial orientation which is indistinguishable from the initial orientation. For example, rotating a water (H₂O) molecule by 180° about an axis bisecting the H-O-H bond angle will produce a orientation of that molecule which looks the same as its original orientation. In fact, this rotation leads to an exchange of both protons H_a and H_b, but as these two protons may not be distinguished from each other, the resulting orientation of the water molecule is equivalent to the initial one. Another rotation of 180° about this axis leads to an orientation which is identical to the starting geometry.

The five basic symmetry operations that need to be considered are identity, rotation, reflection (mirroring), inversion, and rotary reflection. The symmetry properties of molecules are described as the set (combination) of valid symmetry operations for its molecular geometry. For each symmetry operation there is a corresponding symmetry element, with respect to which the symmetry operation is applied. For example, the rotation (a symmetry operation) is carried out around an axis which is the corresponding symmetry element. Other symmetry elements are planes (for reflections) and points (for inversions). All spatial points along these symmetry elements (i.e. the symmetry plane, axis, or center of inversion) do not change their positions during the corresponding symmetry operation. Common to all symmetry operations is that the geometrical center of a molecule does not change its position, all symmetry elements must intersect in this center. In fact, out of the five symmetry operations mentioned above only two (rotation and rotary reflection) are sufficient to describe all symmetry properties of molecules.

Different (valid) combinations of symmetry operations that leave at least a single common point (the molecular center) unchanged give rise to the so called point groups (not all arbitrary combinations of symmetry operations and elements are possible, and some combinations of symmetry operations implicitly imply others, see below). When considering crystals, additional symmetries arising from translations through three-dimensional space need to be considered, giving rise to the more extensive space groups.

The basic symmetry operations are:

• Identity Operation E

  The identity operation does nothing and leaves any molecule unchanged, the corresponding symmetry element is the entire object (molecule) itself. The reason for including this symmetry operation is that some molecules have only this symmetry element and no other symmetry properties. Another reason is the logical completeness of the mathematical description of group theory.

  All molecules which do not have any other symmetry element than the identity operation must belong to the C₁ point group (see below), and thus must be chiral. Many natural compounds like carbohydrates and α-amino acids belong to this group. However, we will see, that all molecules belonging to the C₁ point group must be chiral, but the reverse conclusion is not true. Chiral molecules may not necessarily be a member of the C₁ point group family.

• Inversion i

  The inversion (the symmetry operation) through a center of inversion (the symmetry element, which must be identical to the center of geometry of the molecule) takes any point in the molecule, moves its to the center, and then moves it out the same distance on the other side again (sometimes called point reflection). The benzene molecule, a cube, and spheres do have a center of inversion, whereas a tetrahedron does not.

  At this point, it should be noted, that molecules which contain a center of inversion as the only symmetry element (except for the identity operation, which occurs in all molecules), belong to the C_i point group (see below). However, if a center of inversion is present in a molecular geometry, the corresponding compound must be achiral, irrespective of any other type or number of symmetry elements (mirror planes, rotation or rotary-reflection axes) which may be present in addition.

• Reflection σ

  The reflection (the symmetry operation) in a plane of symmetry or mirror plane σ (the corresponding symmetry element) produces a mirror image geometry of the molecule (this symmetry operation is not present in chiral molecules, see below). The mirror plane bisects the molecule and must include its center of geometry. If this plane is parallel to the principal axis (and includes it, see below), it is called a vertical mirror plane denoted σ_v, if it is perpendicular to the principal axis (and bisects it in the molecular center of geometry), it is denoted a horizontal mirror plane σ_h. Vertical mirror planes bisecting the angle between two C_n axes are called dihedral mirror planes σ_d (these mirror planes all include principal axis and intersect in it).

  The benzene molecule features all three different types of mirror planes (amongst other symmetry elements such as rotation axes rotary-reflection axes which will be discussed below; for benzene σ_v and σ_d planes coincide).

  Molecules which contain a single mirror plane as the only symmetry element belong to the C_s point group (see below), but any number of mirror planes will automatically result in achiral molecular geometries.

• n-Fold Rotation C_n

  The n-fold rotation (the symmetry operation) about a n-fold axis of symmetry (the corresponding symmetry element) produces molecular orientations indistinguishable from the initial for each rotation of 360°/n (clockwise and counter-clockwise). A water molecule has a single C₂ axis bisecting the H-O-H bond angle, and benzene has one C₆ axis (amongst one C₃ axis and seven C₂ axes of which the C₃ and one C₂ axis coincide with the C₆ axis). Linear molecules display a C_∞ axis (any infinitely small rotation about this axis produces unchanged orientations), and perfect spheres posses an infinite number of symmetry axes along any diameter with all possible integral values of n.

  If a molecule has one (or more) rotation axes C_n or S_n (see below), the axis with the greatest n is called the principal axis.

• n-Fold Rotary Reflection S_n

  The n-fold rotary reflection (or n-fold improper rotation, the symmetry operation) about an n-fold rotary reflection axis (or n-fold axis of improper rotation) is composed of two successive geometry transformations: first, a rotation through 360°/n about the axis of that rotation, and second, reflection through a plane perpendicular (and through the molecular center of geometry) to that axis. Neither of these two operations (rotation or reflection) alone is a valid symmetry operation, but only the outcome of the combination of both transformations.

  For example, a methane molecule has three S₄ axes (amongst other symmetry elements not shown in the image on the right). Any molecule featuring a S_n axes must be achiral. In fact, the most general description of chirality is the absence of any S_n rotary reflection in a molecular geometry, as we will see at the end of this page, this definition also includes inversion centers (= S₂ axis) and mirror planes (= S₁ axis).

Examples for the different basic symmetry operations and symmetry elements are given in the scheme below:

Out of the five symmetry operations mentioned above only two (rotation and rotary reflection) are sufficient to describe all symmetry properties of molecules. The identity operation is equivalent to a C₁ axis of rotation, as rotation of all objects around 360° about any axis produces the initial orientation only. On the other hand, a mirror plane is equivalent to a S₁ axis of rotary-reflection perpendicular to that plane. In addition, a center of inversion can be represented by a S₂ axis of improper rotation with any arbitrary orientation of this axis (all axes must intersect in the inversion center, which also must be the center of geometry for any molecule).

Symmetry operations may not be combined arbitrarily with each other, but only in a limited number of variations which are the so called point groups. Some symmetry operations implicitly imply others, as for example any S₄ or C₄ axis must be accompanied by a parallel C₂ axis. For example in benzene, the principal C₆ axis simultaneously must be a C₃ and a C₂ axis of symmetry, too. Similarly, any S_n axis with odd n is identical to a C_n axis in conjunction with a horizontal mirror plane σ_h.

However, certain combinations of some symmetry operations implicitly generate other symmetries. For example, the combination of any C_n axis with even n, together with a center of inversion i implicitly generates a horizontal mirror plane σ_h (which is perpendicular to the C_n axis and runs through the inversion center). Similarly, the combination of any C_n axis with a vertical mirror plane σ_v must imply (n - 1) additional vertical mirror planes σ_v.

In addition, combinations of two rotation axes or two rotary-reflection axes must imply a new rotation axis, and combining a rotation axis with a rotary-reflection generates another rotary-reflection axis.