Section 3.1: Symmetry Operations and Elements

Introduction

Figure \(\PageIndex{1}\): An example of a symmetry operation is a 180° rotation where the resulting position is indistinguishable from the original. A 180° rotation is called a C₂ operation; the axis of rotation is the symmetry element. (CC-BY-NC-SA; Kathryn Haas)

The symmetry of a molecule consists of symmetry operations and symmetry elements. A symmetry operation is an operation that is performed to a molecule which leaves it indistinguishable and superimposable on the original position. Symmetry operations are performed with respect to symmetry elements (points, lines, or planes).

Definition: Symmetry Operation

Manipulation of an object around a symmetry element. The object must be indistinguishable before and after the operation.

Definition: Symmetry Element

A geometric entity (point, axis, or plane) around which a symmetry operation is performed

An example of a symmetry operation is a 180° rotation of a water molecule in which the resulting position of the molecule is indistinguishable from the original position (see Figure \(\PageIndex{1}\)). In this example, the symmetry operation is the rotation and the symmetry element is the axis of rotation.

There are five types of symmetry operations including identity, reflection, inversion, proper rotation, and improper rotation. The improper rotation is the sum of a rotation followed by a reflection. The symmetry elements that correspond to the five types of symmetry operations are listed in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Table of elements and operations

Element	Operation	Symbol
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, followed by reflection in the plane perpendicular to the rotation axis	Sn

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Symmetry Operations and Elements

Identity (E)

All molecules have the identity element. The identity operation is a 360° rotation, essentially doing nothing to the molecule. It is included to fulfill the mathematical requirements of a group.

Proper Rotation and Proper Axis (C_n)

A "proper" rotation is just a simple rotation operation about an axis. The symbol for any proper rotation or proper axis is C_(360/n), where n is the degree of rotation. Thus, a 180° rotation is a C₂ rotation around a C₂ axis, and a 120° rotation is a C₃ rotation about a C₃ axis. Figure \(\PageIndex{2}\) shows both the C₃ principal rotation axis and one of the C₂ rotation axes in BF₃.

Definition: Principal Rotation Axis

The principal axis of a molecule is the highest order proper rotation axis. For example, if a molecule had C₂ and C₄ axes, the C₄ is the principal axis.

Figure \(\PageIndex{2}\):The C₃ and one of the C₂ rotation axes in BF₃. C₃ is the principal rotation axis. (Image source: https://symotter.org/gallery)

Exercise \(\PageIndex{1}\)

What is the principal rotation axis in benzene (C₆H₆)?

Answer

Benzene is a planar molecule. It has one C₆ axis and 6 C₂ axes. The principal rotation axis is the C₆ as it has the highest order.

Reflection and Mirror Planes (σ)

A reflection operation occurs with respect to a mirror plane. Mirror planes can be classified based on their relationship to the principal rotation axis:

1. σ_h (horizontal): horizontal reflection planes are perpendicular to principal rotation axis
2. σ_v (vertical): vertical reflection planes contain the principal rotation axis

• σ_d (dihedral): dihedral reflection planes are vertical reflection planes that bisect two C₂ rotation axes

Figure \(\PageIndex{3}\) shows the three types of mirror planes in the square planar PtCl₄. The horizontal reflection plane, shown in red, is in the plane of the molecule, and bisects the C₄ principal rotation axis. The vertical reflection planes, shown in orange, are along the Pt-Cl bonds and contain the C₄ principal rotation axis. The dihedral reflection planes, shown in yellow, bisect the Pt-Cl bonds (and the C₂ rotation axes along those bonds) and also contain the C₄ principal rotation axis.

Figure \(\PageIndex{3}\): The horizontal reflection plane (red), one of the vertical reflection planes (orange) and one of the dihedral reflection planes (yellow) in PtCl₄. (Image source: https://symotter.org/gallery)

Inversion and Inversion Center (i)

The inversion operation takes any point through the inversion center to an equidistant point on the opposite side of the inversion center. In other words, a point at the center of the molecule that can transform (x,y,z) into (-x,-y,-z) coordinates. The inversion center may be on the central atom or it may be at a point in space in the center of a molecule (Figure \(\PageIndex{4}\)).

Figure \(\PageIndex{4}\): The inversion center (dark green) in PtCl₄ and in N₂O₄. (Image source: https://symotter.org/gallery)

Improper Rotation and Improper Rotation Axis (S_n)

Improper rotation is by far the most difficult symmetry operation to visualize. It is a combination of a rotation with respect to an axis of rotation (C_n), followed by a reflection through a plane perpendicular to that C_n axis. The rotation and reflection may or may not be valid symmetry operations on their own. In short, an S_n operation is equivalent to C_n followed by \(\sigma_h\). In methane (Figure \(\PageIndex{5}\)), the S₄ improper rotation axis will rotate 90° and rotate the in plane H's out of plane and vice versa. The perpendicular reflection then exchanges the in and out of plane H's, returning them to the original confirmation. In this example, neither the C₄ rotation or the perpendicular reflection are symmetry operations on their own.

Figure \(\PageIndex{5}\): The improper rotation axis and reflection plane in methane. (Image source: https://symotter.org/gallery)

Successive Operations

Sometimes, new symmetry operations form by carrying out two or more simpler operations successively to result in an indistinguishable configuration. For example, the improper rotation axis above results from rotation about an axis of rotation (C_n) followed by reflection about a plane perpendicular to that axis (\(\sigma\)_h).

\[S_{n}=C_{n}\times\sigma_{h}\]

Note also that C_n in this case need not be a symmetry element for the molecule. For example, staggered ethane has an S₆ symmetry element although it does not have a C₆. Another example of a symmetry element that results from a combination of two different elements is the inversion center (i), which results from a C₂ rotation followed by a reflection about a plane perpendicular to that axis (\(\sigma\)_h).

\[i=C_{2}\times\sigma_{h}\]

Note that an inversion center is a special case of an improper rotation axis because it results when, in the first equation, \(n=2\). That is,

\[i=S_{2}\]

In fact, any symmetry operation can be carried out multiple time in a row. For example, when BH₃ is rotated twice by 120°, the two-step operation can be symboled by \(C_{3}^{2}\). When the C₃ operation is performed a third time, the molecule returns to its original configuration; i.e,

\[C_{3}^{3}=E\]

In general,

\[C_{n}^{n}=E\]

Exercise \(\PageIndex{1}\)

List all of the symmetry elements in ammonia (NH₃).

Answer

Ammonia has a trigonal pyramidal geometry. It has one C₃ rotation axis (the principal rotation axis) which will exchange the three H atoms. It also has three vertical reflection planes, one along each N-H bond.

Sources:

1. Introduction to Molecular Symmetry by J. S Ogden
2. Inorganic Chemistry by Catherine Housecroft And Alan G. Sharpe.
3. Symmetry@Otterbien: https://symotter.org/