The symmetry of the molecules can be described in terms of electrons of symmetry and the corresponding symmetry operations.
Clearly some molecules, like H_{2}O and CH_{4}, are symmetric. Now we shall develop a clear and precise way of describing such symmetries. Consider, for example, the plane containing the H2O molecule and the perpendicular plane shown in fig. 1.these planes are examples of planes of symmetry. You can test, what is here quite obvious, that the molecule is symmetric with respect to each plane by reflecting the molecule through the plane. This problem consists of treating the plane as a mirror. This reflection through the mirror, in both directions, produces a result that is indistinguishable from that existing originally. The reflection process is an example of a symmetry operation. It is the symmetry operation associated with a plane of symmetry. The plane of symmetry is an example of an element of symmetry.
As this example suggests, we can describe the symmetry of a molecule in terms of the elements of symmetry of the molecule. That a molecule has any particular element of symmetry can be checked by carrying out the symmetry operation associated with each symmetry element. If the operation does nothing more than leave the positions of atoms unchanged or carries one atom of a set of identical atoms into a position of another atom of the set, the result will be indistinguishable from that existing initially. When this is the result of the symmetry operation, the molecule does indeed have the corresponding symmetry element.
Planes of symmetry: the general symbol for a plane of symmetry is σ. If we draw the planes vertically, we refer t them as vertical planes of symmetry. Such planes are given the symbol σ_{v}. As if the plane of symmetry is drawn horizontally, we refer to it as a horizontal plane and label isσ_{h}, as if it is drawn in a way that can be described as diagonal, it is labeled σ_{d}.
Center of symmetry: now let us consider some other elements of symmetry and the symmetry operations associated with them. A molecule is said to have a center of symmetry if the operation of investigation through the center produces a result indistinguishable from that occurring originally.
Axis of symmetry: a molecule is said to have an axis of symmetry if rotation of the molecule about that axis gives us back our initial situation. A twofold axis is one for which a rotation by360/2 degrees, or ½ revolution, produces a result like axis of rotation denoted by C_{2}. A threefold axis is one for which a rotation, say clockwise, by a 1/3 revolution. The other is a clockwise rotation by 2/3 revolution or a counterwise rotation by 1/3 revolution.
The element of symmetry described as a fourfold axis consist of rotations by ¼, 2 (1/4) and 3(1/4) revolution. A C_{4} axis implies a C_{2 }axis since the operation consisting of a rotation by 2/4 revolution is equivalent to rotation by ½ revolution.
A sixfold axis of symmetry as the discussion of the fourfold axis suggests, a sixfold axis implies coincident two fold and three fold axes.
If a single molecule has a single axis of rotation, we agree to draw the molecule to that this axis is in the vertical direction. If a molecule has more than one axis of rotation, we draw the molecule so that the highest order axis is vertical. The ax of rotation of the benzene molecule, drawn in accordance with this rule is followed.
Rotation reflection axis: with only one additional type of symmetry and precisely indicated by stating the number and relative orientations of the four kinds of symmetry elements that have been described. Thus instead of saying, for example, that H_{2}O is a symmetric molecule, we now can show the elements of symmetry of the molecule.
Molecular symmetry: the symmetry of a molecule is completely and precisely indicated by stating the number and relative orientations of the four kinds of symmetry elements that have been described. Thus instead of saying, for example, H_{2}O is a symmetric molecule, we now can show the elements of the H_{2}O molecule.
One additional element of symmetry, which adds nothing to a description of the symmetry of a molecule but is helpful for an organised treatment of the consequences of this symmetry, should now be added. This symmetry element is called the identity and is given the symbol E. the symmetry operation associated with this symmetry element can be said to consist of no change. (Alternatively the operation can be described as a rotation about any axis by 360°.) Thus all molecules have the symmetry element E. (You will see that inclusion of this identity element is not as frivolous as it seems.)