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Symmetry Elements and Symmetry Operations

Symmetry elements and operations are two different things which are the keys of the molecular symmetry determination. A symmetry element is a geometrical structure which is a line, a plane or a point respected to a symmetry operation that we apply on molecule. Symmetry elements and symmetry operations are closely related. It is because of the symmetry operation is defined with respect to a symmetry element. Symmetry element existence proves the existence of corresponding symmetry operator and vice versa.¹

A symmetry operation is the movement of the molecule in one direction and get the equivalent molecular structure in each movement. This means that the molecule should be turn until it gives the same molecular arrangement again. Another way of defining the symmetry operation is to take the molecule into an equivalent configuration.¹

Identity Operation and Identity Element

Identity operation, E does not do anything on a molecule any way and keeps the molecule as it is. Every molecule has the identity element, even if it lacks the symmetry.¹

Image 1 : Three cubes with different planes of symmetry

Symmetry Planes and Reflection

A symmetry plane is an imaginary surface. It divides a molecule or an object into two equal parts. These two halves are exactly same. In molecular symmetry studies we learn multiple symmetry planes, as vertical, horizontal and diagonal planes. If a circle is considered, it has infinite number of symmetry planes through its center. In σ chemistry is used to denote a plane. Symmetry reflection is the mirror image of a molecule or and object through the plane of symmetry.¹

Image 2 : This image illustrates the 3D point reflection. All the points on one trigonal pyramid moved to the opposite point on the other pyramid through the black point marked on the center. It is the center of inversion

The Inversion Center

Let’s consider a molecule on 3D space. If the equivalent configuration of molecule can be obtained by changing the x, y, z coordinates into -x, -y, -z through a point, then that molecule has a center of inversion.¹

Image 3 : n- 2,3,4, and 6 fold symmetry axes

Proper Axes and Proper Rotations

C_n is the notation used to denote the proper rotations. Therefore, C_n axis is the proper rotational axis. C_n tells the rotation around the axis by (360/n)^o angle and results the same molecular configuration as original configuration. The principle axis is the highest-order proper rotation axis and corresponds to the most symmetrical rotation in a molecule.¹

Image 4 : Improper rotation axis -S₄

Improper Axes and Improper Rotations

There are two steps in improper rotations. The first step is the proper rotation and then a reflection through a plane perpendicular to the rotation axis. S_n is the notation used to denote the improper axis. N is the order. If an axis C_n and a plane perpendicular to that axis exit independently, then S_n exists. Even though there are not an axis or plane, S_n can exist.¹

Image 4: The molecule is rotated by 90^o around the axis go through the center atom(orange color) and in between C and D atoms (green color). Then the reflection has taken through the plane perpendicular to the axis.

Following table shows a summary of symmetry elements and symmetry operations:²

Element	Operation	Notation
Identity	identity	E
Proper axis	rotation by (360/n)o	Cn
Symmetry plane	reflection in the plane	σ
Inversion center	inversion of a point at (x,y,z) to (-x,-y,-z)	I
Improper axis	rotation by (360/n)o and reflect through the plane perpendicular to the rotation axis	Sn

Examples

Image 5 : Ball-stick model of water molecule

Water molecule

Image 6 : Water molecule a) z-axis b) y-axis c) x-axis

Image 6: The symmetry element of water molecule is the E. There is only one proper rotational (C₂) axis around the z-axis. There are two symmetrical planes. One of them is the plane through the zy plane (σ_zy) and the remaining plane lies on the zx plane (σ_zx).

Ammonia molecule

Image 7 : Ball-stick model of ammonia molecule

Image 7: As always ammonia also has the identity, E. There are three symmetry planes through N atom and each H atoms. There is only one C₃ axis which is going through the N atom.

References

1. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, 1990.
2. LibreTexts. Symmetry Elements and Operations Define the Point Groups. Physical Chemistry (LibreTexts).