The symmetry properties of molecules and orbitals are extremely useful in a consideration of their properties. In order to explore the symmetry of an object in a rigorous mathematical fashion, we need to introduce various terms to describe the concepts we shall encounter:

A symmetry operation is any action that may be carried out upon an object that causes it to overlap with itself exactly, or leaves it looking exactly the same,. i.e. any rotation of a sphere about an axis through its centre is a symmetry operation. Rotation about a different axis is not necessarily a symmetry operation (unless the rotation happens to be of 360º) , as even though the sphere looks exactly the same shape afterwards, it would be possible to see that the whole sphere had been moved in space.

Typical symmetry operations are rotations, reflections and inversions.

Every symmetry operation has an associated symmetry element, which is the point, line or plane with respect to which the symmetry operation is carried out. Thus the symmetry operation of inversion is carried out with respect to a symmetry element which is a point (called the centre of inversion). Rotations have symmetry elements which are lines – the axes of rotation. Reflections have symmetry elements which are planes – the mirror planes.

Molecules (and indeed objects in general) may be classified by identifying all their symmetry elements and then grouping together all the ones which have the same number of each type of symmetry element.

These groups are known as point groups if, as is common, the classification is carried out using only those symmetry elements corresponding to operations that leave at least one point completely unchanged. (Exceptions to this classification occur in crystals, whose regularly repeating structure means that they also possess translational symmetry operations. They belong to space groups.) There are five types of symmetry operation that leave at least one point unchanged:

The identity, represented E, which consists of doing nothing to the object. The corresponding symmetry element is the entire object. Every object possesses this symmetry operation; some objects only possess the identity.

An n-fold rotation, with symmetry element called an n-fold axis of symmetry, represented Cn. This operation consists of a rotation of 360º/n about the axis of symmetry. Note C1 is a rotation through 360º , and is equivalent to E, the identity, so is not listed separately.

e.g. A water molecule has one C2 axis. An ammonia molecule has one C3 axis, but note there are two symmetry operations associated with the axis – clockwise and anticlockwise rotations of 120º.

There are not two rotations associated with a C2 axis because clockwise and anticlockwise rotations of 180º give rise to identical objects. (If the two H atoms in water are assigned labels a and b, then the labels lie in the same place after a rotation of 180º regardless of the direction of rotation. If the three H atoms in ammonia are labeled a, b and c, then the labels lie in different places after clockwise or anticlockwise rotations of 120º. Note that since the hydrogens are actually indistinguishable, the rotations both count as symmetry operations.)

In molecules with multiple n-fold axes, the one with the highest number of n is called the principle axis. If there are multiple axes with the same highest value of n which are all equivalent (i.e. may be transformed into one another by symmetry operations of the molecule) then there is no principle axis. If there are multiple inequivalent axes (i.e. ones which are geometrically distinct.) with the same highest value of n, one is arbitrarily assigned to be the principle axis.

A reflection takes place in a mirror plane, denoted σ. If the mirror plane contains the principle axis, it is called a vertical mirror plane, σv. Note a special case: If a vertical mirror plane bisects two C2 axes (which must, of necessity, lie perpendicular to the principle axis), it is called a dihedral plane and denoted σd. A mirror plane which lies perpendicular to the principle axis is called a horizontal mirror plane, denoted σh.

An inversion occurs through a centre of symmetry, denoted i. (This operation involves taking each point in the object in a straight line to the inversion centre and then moving it an equal distance out the other side. i.e. if we consider the inversion centre to lie at (0,0,0) , then this operation turns a general point (x,y,z) into (-x,-y,-z). Spheres and cubes both have centres of inversion, as does a benzene molecule (a regular hexagon). Molecules such as water, ammonia and methane do not. An object which remains the same under inversion (eg an s orbital) is, by convention, labeled with the subscript letter g. An object which changes sign under inversion (eg a p orbital) is labeled with the subscript letter u.

An n-fold-improper rotation, about an n-fold axis of improper rotation denoted Sn, consists of a rotation of 360º/n about the axis followed by a reflection in a mirror plane perpendicular to the axis of rotation. Neither operation on its own need be a symmetry operation of the molecule, so long as their combined effect is to leave the molecule looking unchanged.
(eg a CH4 molecule has three S4 axes, each of which passes through the central carbon atom and bisects two H-C-H bond angles.)